Research reference
Glossary
Understand the method. Check the assumptions. Interpret with care.
86 methods and measures · 29 research concepts
115 of 115 entries
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Analysis of covariance (ANCOVA)Group comparisonsCompares group means while modelling the association of a numeric outcome with selected covariates.
What it means
ANCOVA is a regression model containing grouping factors and covariates. A common-slope specification assumes the covariate association is the same across groups; interaction terms allow that association to vary.
When it fits
Use to estimate a planned adjusted comparison, such as comparing randomized groups on follow-up scores while including baseline scores.
When to reconsider
Do not claim adjustment automatically removes confounding. Adjusting for post-treatment variables can change the estimand or introduce bias, and comparisons outside shared covariate ranges require extrapolation.
Assumptions and checks
The conditional mean form and variance model must be appropriate, with independent errors for ordinary ANCOVA. Common slopes are a model restriction, not an inherent requirement of every covariate-adjusted analysis.
Research example
Illustration: two programs differ by 6 raw follow-up points but by 3 adjusted points at a baseline score of 50. The adjusted comparison describes that model and covariate value, not an automatic causal correction.
What to report
Identify covariates and why they were selected, interactions, the values or population used for adjusted means, adjusted contrasts with intervals, and any extrapolation or sensitivity concerns.
Read the source
APA reportingResearch practiceA framework for communicating how a study was designed, analyzed and interpreted so readers can assess its evidence.
What it means
APA Journal Article Reporting Standards specify information for the relevant quantitative design. Good reporting extends beyond punctuation and rounding to sampling, exclusions, measurement, missingness, analytic decisions and uncertainty.
When it fits
Use the checklist appropriate to the study design, together with the destination journal requirements.
When to reconsider
Do not treat one formatted test sentence as complete reporting or conceal unplanned analyses.
Assumptions and checks
Reported numbers and methods must match the actual analysis and distinguish prespecified from exploratory work.
Research example
Illustration: report a 4-point mean difference with its interval, analyzed group sizes and the missing-data rule, then explain its substantive meaning. A bare p = .02 is insufficient.
What to report
Describe design, participants, measures, analysis, effects and uncertainty; disclose deviations, limitations and data/code availability where appropriate.
ARIMA time-series modelTime series and growthARIMA predicts a time series from its own past values and past prediction errors, after any needed differencing.
What it means
ARIMA(p, d, q) combines autoregressive order p, differencing order d and moving-average order q. Seasonal extensions add structure at a defined seasonal period; differencing changes the series being modeled.
When it fits
Use for regularly spaced numeric series with temporal dependence and enough history to estimate and check the chosen structure.
When to reconsider
Do not randomly shuffle observations for forecast validation or assume stable forecasts through major structural breaks.
Assumptions and checks
After modeled transformations and differencing, the remaining process should satisfy the specified stability and error structure; inspect residual autocorrelation and changing variance.
Research example
Use five years of monthly library visits to forecast the next six months, retaining the last year for rolling forecast evaluation and checking annual seasonality.
What to report
Report orders, seasonal period, transformations, estimation, residual diagnostics, forecast horizon and prediction intervals.
Read the source
Arithmetic meanDescriptive statisticsThe mean is the total of the observed values divided by their number.
What it means
For n observations, x-bar = sum(x)/n. A weighted mean uses specified weights and answers a different question from an unweighted mean.
When it fits
Use for meaningful numeric measurements, including mean expenditure when total resource use is the target.
When to reconsider
Do not use alone to represent the typical individual when a few extreme values dominate, or average arbitrary category codes.
Assumptions and checks
Calculation needs no normality assumption; population inference still depends on sampling, dependence and missingness.
Research example
Three laboratories process 4, 7 and 10 samples per hour: their unweighted mean is 7 samples per hour.
What to report
Give the mean, units, SD, n and any weighting or trimming.
Read the source
Association and causationStudy designVariables moving together does not by itself show that changing one would change the other.
What it means
An association describes observed distributions. A causal contrast concerns outcomes under different interventions. Randomization supports causal identification, while observational identification requires justified additional assumptions.
When it fits
Use association language for descriptive relationships; reserve causal wording for a design and analysis that support the intervention question.
When to reconsider
Do not turn a large correlation, small p-value or adjusted regression coefficient into automatic proof of causality.
Assumptions and checks
Consider timing, confounding, selection, measurement and whether the treatment contrast is well defined. Random assignment does not guarantee perfect balance in every realized sample.
Research example
Illustration: a positive association between library visits and grades may reflect prior motivation; assigning additional visits is a different question.
What to report
State the design, target contrast, identification rationale and plausible alternative explanations.
Average variance extractedReliability and validitySummarizes how much indicator variance a reflective factor explains relative to total indicator variance.
What it means
In a simple standardized one-factor measurement model, AVE is the mean squared loading. More general formulas account for factor and indicator variances; AVE concerns the indicators and is distinct from reliability of their summed score.
When it fits
Evaluating convergent measurement evidence alongside loadings, content and broader validity evidence.
When to reconsider
Do not interpret AVE above 0.50 as proof of validity, or apply the simple formula unchanged when indicators cross-load or form an index.
Assumptions and checks
A defensible reflective model and consistent standardization; ordinal models refer to an underlying response scale unless otherwise specified.
Research example
Standardized loadings of 0.80, 0.70 and 0.60 yield AVE = (0.64 + 0.49 + 0.36)/3 ≈ 0.50. Content coverage and measurement uncertainty still need evaluation.
What to report
Give the formula, loadings, scale and model, and interpret AVE alongside other convergent and discriminant evidence.
Bartlett test of equal variancesModel diagnosticsTests whether independent normal populations have the same variance.
What it means
The statistic compares a pooled variance with the group variances using a logarithmic combination and an approximate chi-square reference. It is particularly sensitive to departures from normality, which can resemble evidence of unequal variances.
When it fits
Use when normal group distributions are well justified and comparing population variances is the intended question.
When to reconsider
Do not confuse this procedure with Bartlett sphericity testing in factor analysis. With skewness or heavy tails, consider a less normality-sensitive variance procedure and graphical assessment.
Assumptions and checks
Independent observations, normal distributions within groups, and estimable positive group variances. A large p-value does not establish variance equality.
Research example
Illustration: compare measurement precision across three instruments using normally distributed calibration errors. If the errors contain long-tailed contamination, Bartlett rejection may reflect that contamination as well as variance differences.
What to report
State that the equal-variance version was used, show group sample sizes and variances, and report the statistic, degrees of freedom, p-value, and evidence concerning normality.
Read the source
Bayesian ANOVABayesian methodsBayesian ANOVA estimates group and interaction effects, or compares models containing different combinations of those effects.
What it means
Bayes-factor ANOVA integrates each model’s likelihood over its parameter priors. An inclusion Bayes factor additionally depends on the specified set of models and their prior probabilities, so it is not a standalone property of a factor.
When it fits
Use for group or factorial comparisons with explicit prior assumptions and clearly defined model comparisons.
When to reconsider
Do not interpret support for a model containing an interaction as showing that every pair of groups differs.
Assumptions and checks
Specify residual and variance assumptions, contrast coding and dependence; repeated observations require participant or other appropriate structure.
Research example
For three teaching methods measured twice per participant, compare a model with method-by-time interaction against a stated additive model while retaining participant effects.
What to report
Report model space, priors, comparison direction, group contrasts and posterior intervals, numerical uncertainty and prior sensitivity.
Bayesian regressionBayesian methodsBayesian regression combines a model for the outcome with prior information to estimate relationships and predictive uncertainty.
What it means
The posterior is proportional to the likelihood multiplied by the prior. Gaussian, binary, count and multilevel models use different likelihoods and links; posterior uncertainty remains conditional on those choices.
When it fits
Use when prior knowledge, partial pooling or direct probabilistic statements about model parameters are useful.
When to reconsider
Do not interpret a narrow credible interval as protection against omitted confounding or accept unstable sampling simply because estimates appear plausible.
Assumptions and checks
Justify priors and likelihood, check prior and posterior predictions, and assess convergence, effective sample size and sampler pathologies when using MCMC.
Research example
Estimate examination score from study hours across schools, allowing school intercepts to vary; a slope interval then reflects the stated model and priors, not proof that additional study causes improvement.
What to report
Give model, scales, priors, posterior summaries, interval definition, computation diagnostics and predictive checks.
Bayesian structural equation modelStructural equation modelsEstimates a relationship between two unobserved constructs while accounting for their measured indicators and explicitly stated prior information.
What it means
The qualified release fits two Gaussian reflective factors with 3 to 8 indicators each and one directed path. Complete-case indicators are z standardized, the first loading fixes each factor scale, and indicator intercepts are fixed to zero. Four Stan NUTS chains estimate the posterior.
When it fits
A prespecified two-factor measurement and structural model with continuous indicators, independent observations and enough complete cases is substantively defensible.
When to reconsider
Do not use this release for ordinal indicators, feedback, residual correlations, multilevel models or arbitrary model constraints. Credible intervals do not establish causality or validate the priors.
Assumptions and checks
Continuous reflective measurement, Gaussian residual likelihood, independent cases, defensible complete-case selection and explicit priors. Free loadings and the path use Normal(0,1); indicator and latent SDs use Gamma(shape=2,rate=2). Inspect measurement orientation and prior sensitivity.
Research example
Estimate a latent trust-to-satisfaction path using three continuous measurements per construct and report its posterior interval alongside measurement and sampler diagnostics.
What to report
Report the model and markers, standardization, complete/excluded rows, priors, chains and seed, posterior means and credible intervals, R-hat, effective sample sizes, divergences, BFMI and the covariance posterior predictive check. The latter is not a frequentist p-value or a test of Gaussianity.
Bayesian t-testBayesian methodsA Bayesian t-test compares mean hypotheses using explicit prior assumptions, or estimates the mean difference with a posterior distribution.
What it means
A Bayes factor compares the data’s marginal likelihood under two specified models; posterior model odds equal prior odds multiplied by the Bayes factor. Posterior estimation and testing a point null are related but distinct tasks.
When it fits
Use for one-sample, paired or independent-group mean questions when the likelihood and effect-size prior can be defended.
When to reconsider
Do not call a Bayes factor the probability that a hypothesis is true or treat a default prior as scientifically neutral.
Assumptions and checks
The selected model must represent pairing, residual distribution and variance structure; common default two-group models assume a shared variance.
Research example
BF10 = 4 means the observed data are four times as probable under the specified alternative model as under the null; with equal prior odds, posterior odds are 4:1.
What to report
State hypotheses, prior family and scale, Bayes-factor direction, numerical accuracy and sensitivity to reasonable priors.
Binary logistic regressionPredictive modelsModels the probability of an outcome with two categories, such as completion versus noncompletion.
What it means
The model relates the log odds of the designated event to a linear predictor. Exponentiating a coefficient yields a conditional odds ratio for the specified unit or contrast, not a risk ratio.
When it fits
A binary event is the outcome and adjusted associations or event probabilities are required.
When to reconsider
Do not use the binary model for unordered outcomes with three categories or ignore separation, where predictors perfectly distinguish events from nonevents.
Assumptions and checks
Suitable logit specification, adequate information in both outcome groups, and independence or an explicit dependence model. Continuous predictors need not be normally distributed.
Research example
If a fitted odds ratio is 2 and the reference probability is 0.20, doubling its odds of 0.25 gives a probability of 0.50/1.50 = 0.33, not 0.40.
What to report
State event coding, reference categories, predictor units, odds ratios and intervals, event counts, separation checks, and calibration and discrimination for prediction.
Read the source
Central tendencyDescriptive statisticsCentral tendency describes a typical value, but different definitions of typical answer different questions.
What it means
The mean, median and mode summarize arithmetic balance, the middle rank and the most frequent value, respectively. These can differ substantially in skewed or multimodal distributions.
When it fits
Choose a summary that matches the research question and measurement scale, alongside a distribution plot.
When to reconsider
Do not describe a mixture of distinct groups using one average alone.
Assumptions and checks
Numeric codes for unordered categories do not justify arithmetic averaging.
Research example
For waiting times of 2, 3, 3, 4 and 28 minutes, the mean is 8, the median is 3 and the mode is 3.
What to report
Name the summary, units, sample size, missing-value rule and a measure of spread.
Read the source
Chi-square test for categorical dataCategorical dataCompares observed category counts with counts expected under a specified null model.
What it means
For a contingency table the usual null is independence between the row and column variables; a one-way goodness-of-fit test instead checks prespecified category probabilities. Pearson X-squared sums squared observed-minus-expected deviations divided by expected counts.
When it fits
Use for counts in mutually exclusive categories, such as comparing completion status across independent program groups.
When to reconsider
Do not enter percentages without their counts, count one participant repeatedly as independent, or judge approximation quality from observed counts alone. Sparse expected cells may require exact or appropriately simulated inference.
Assumptions and checks
Sampling units are independent under the chosen model, categories are well defined, and the expected-count pattern supports the approximation. Complex survey weights and clustering need design-aware alternatives.
Research example
Illustration: 70 of 100 program A participants and 55 of 100 program B participants complete training. A 2-by-2 table tests association; the 15-percentage-point difference adds practical context.
What to report
Report the table with denominators, X-squared, degrees of freedom, p-value method and any correction, expected-count concerns, and an interpretable association estimate with uncertainty.
Read the source
Cohen's dModel interpretationExpresses a mean difference in units of a specified standard deviation.
What it means
Independent-group d commonly divides the difference by a pooled within-group SD. Paired-data variants use different denominators, including the SD of differences. Hedges g applies a small-sample bias correction to a standardized difference.
When it fits
Use to communicate a contrast on a standardized scale when its denominator is scientifically defensible.
When to reconsider
Do not interchange paired and independent variants, or treat .20, .50 and .80 as universal practical thresholds.
Assumptions and checks
Comparisons across studies require compatible constructs, designs and standardizers; interval methods have their own distributional assumptions.
Research example
Illustration: means 76 and 70 with pooled SD = 12 give d = .50. Reversing the group order gives -.50.
What to report
State the precise variant, contrast direction, denominator, sample sizes and interval; retain the raw mean difference.
Composite reliabilityReliability and validityEstimates how much variation in a specified composite score is attributable to the modeled common factors.
What it means
For a simple one-factor unit-weighted score, reliability uses squared summed loadings and factor variance relative to total score variance. Formulas change with correlated errors, multiple factors, weighting and ordinal response scales; some versions coincide with omega.
When it fits
A defensible factor model allows unequal item loadings and the reliability target is a clearly defined score.
When to reconsider
Do not call a value “CR” without specifying its formula, or apply reflective reliability formulas to a formative index.
Assumptions and checks
A credible fitted measurement model, correct residual structure and agreement between the modeled score and the score actually used.
Research example
Three standardized indicators each load 0.80 with independent residual variances of 0.36. Reliability is 2.40² / (2.40² + 1.08) ≈ 0.84 under this model.
What to report
State score weights, exact coefficient or formula, model fit, indicator scale, sample and uncertainty; justify adequacy for the score’s intended use.
Confidence interval (CI)Statistical foundationsAn interval showing estimation uncertainty under a stated statistical procedure.
What it means
A 95% confidence procedure is designed to cover the fixed target parameter in 95% of repeated samples under its assumptions. Actual coverage may differ for approximate methods or misspecified models.
When it fits
Use with a point estimate to assess both precision and which effect sizes remain compatible with the analysis.
When to reconsider
Do not say a realized frequentist interval gives a 95% probability for its fixed parameter, or that it contains 95% of individual observations.
Assumptions and checks
Coverage depends on the sampling design, model and interval method, including any selection or multiplicity adjustment.
Research example
Illustration: a mean difference of 4 points with 95% CI [1, 7] excludes zero but remains compatible with both modest and larger differences. Whether these matter requires a substantive benchmark.
What to report
Give the estimate, limits, confidence level, units and interval method.
Confirmatory factor analysisFactor analysis and SEMEvaluates how well a prespecified pattern of latent factors and measured indicators accounts for their relationships.
What it means
CFA estimates loadings, factor variances or covariances and residual parameters under explicit constraints. Identification requires fixing factor scales and having sufficient independent information; acceptable global fit does not establish unique truth.
When it fits
Theory or prior evidence specifies which items measure which reflective constructs.
When to reconsider
Do not add correlated residuals solely to improve fit and then call the revised model independently confirmed.
Assumptions and checks
Identified measurement structure, appropriate estimator for continuous or ordinal indicators, defensible residual restrictions and suitable missing-data handling. Group comparisons require relevant measurement invariance.
Research example
A 12-item survey proposes three correlated four-item factors. Evaluate this structure, inspect localized misfit and compare a substantively plausible alternative before interpreting factor correlations.
What to report
Provide model specification, estimator, identification, sample size, loadings and intervals, residuals, fit statistics and all post hoc changes.
ConfoundingStudy designA comparison mixes the association of interest with differences in other causes of the outcome.
What it means
Confounding prevents an observed exposure-outcome contrast from identifying its intended causal contrast without adequate design or adjustment. Covariates should be selected using causal knowledge, not p-values alone.
When it fits
Assess it before interpreting an observational association causally.
When to reconsider
Do not assume including every available variable removes bias; post-exposure variables can change the target or introduce bias.
Assumptions and checks
Causal interpretation requires a defensible adjustment strategy and other identification conditions.
Research example
Illustration: students choosing tutoring may already have lower baseline scores; comparing final scores alone can misrepresent the tutoring effect.
What to report
Explain covariate selection, timing, adjusted contrasts and remaining unmeasured-confounding concerns.
Cook’s distanceModel diagnosticsCook’s distance indicates how much a fitted regression changes when one observation is omitted.
What it means
For ordinary least squares it combines residual size and leverage into a scaled change in fitted values. A large residual alone or unusual predictor values alone need not make a case influential.
When it fits
Use after fitting a regression to identify cases needing inspection and sensitivity analysis.
When to reconsider
Do not automatically delete cases crossing a rule such as 4/n; valid observations may reveal important heterogeneity or model misspecification.
Assumptions and checks
Interpret it relative to the fitted model; generalized-model diagnostics may use approximations and require additional care.
Research example
A school with unusually high spending changes the spending coefficient from 0.2 to 0.8 when included: inspect its data and compare substantively justified models.
What to report
Describe influential cases, verification, coefficient sensitivity and the reason for any exclusion.
Covariance-based structural equation modelingFactor analysis and SEMCombines measurement models and structural relationships to evaluate a theory about observed and latent variables.
What it means
The model implies covariances and often means that are compared with observed data using an estimator such as maximum likelihood. Different structural explanations can reproduce the same covariance structure, so good fit alone does not identify causation.
When it fits
An explicit, identifiable theory links measured indicators, latent constructs and structural relationships.
When to reconsider
Do not use fit cutoffs as an automatic theory-validity certificate or search numerous paths without disclosing exploration.
Assumptions and checks
Identification, adequate information, credible measurement specification and an estimator suitable for the distributions and sampling structure. Causal paths need additional design and confounding assumptions.
Research example
A longitudinal model connects a baseline support factor to later engagement, adjusting prior engagement. Evaluate measurement quality and alternative structural explanations as well as path estimates.
What to report
Provide a diagram or equations, constraints, estimator, missing-data treatment, fit and residual diagnostics, parameter intervals and model revisions.
Cox proportional hazards regressionSurvival analysisCox regression relates predictors to the instantaneous event rate among individuals still at risk.
What it means
The model expresses hazard as h0(t) × exp(xβ), leaving the baseline hazard unspecified. Exp(β) is a conditional hazard ratio, not a risk ratio or a ratio of median survival times.
When it fits
Use for time-to-event associations with censoring when proportional effects are reasonable, or model departures explicitly.
When to reconsider
Do not summarize strongly time-varying effects with one constant hazard ratio without qualification, or interpret adjusted associations as automatically causal.
Assumptions and checks
Check proportional hazards, continuous-predictor form, influential observations and conditional non-informative censoring; account for clustering and delayed entry.
Research example
An adjusted completion hazard ratio of 1.4 compares instantaneous completion rates among candidates still enrolled at the same follow-up time; it does not mean 40% more finish overall.
What to report
Give hazard ratios with intervals, reference groups, covariates, event count, tie handling and proportional-hazards assessment.
Read the source
Cronbach’s alphaReliability and validitySummarizes how item covariances contribute to the consistency of a summed or averaged scale score.
What it means
Alpha equals k/(k−1) times one minus the ratio of summed item variances to total-score variance. Interpreting it as reliability requires additional measurement assumptions; high alpha does not establish unidimensionality or validity.
When it fits
Assessing an intended reflective multi-item score together with its dimensional structure and scoring rules.
When to reconsider
Do not calculate alpha to validate a formative index or remove important content simply to raise the coefficient.
Assumptions and checks
Correct item orientation and a meaningful composite. Equality with score reliability requires essentially tau-equivalent items and uncorrelated errors; violations can distort interpretation.
Research example
For four items with average standardized inter-item correlation 0.40, standardized alpha is 4 × 0.40 / (1 + 3 × 0.40) = 0.73. This is not a universal pass mark.
What to report
Give item count, sample, raw or standardized alpha with uncertainty, reverse scoring, missing-item rules and evidence about dimensionality.
Decision trees and tree ensemblesPredictive modelsPredicts outcomes through branching decision rules, sometimes combining many trees to improve performance.
What it means
A single tree recursively partitions predictors into regions with fitted class probabilities or numeric predictions. Ensembles combine trees through procedures such as bagging or boosting, so their behavior differs from one interpretable tree.
When it fits
Nonlinear patterns and interactions are plausible and prediction can be evaluated on representative data outside model training.
When to reconsider
Do not treat an attractive training-set tree as stable evidence or interpret feature importance as a causal effect.
Assumptions and checks
Useful measured predictors and appropriate validation splits, including grouping or time ordering when needed. Tree complexity, class imbalance and leakage require explicit control.
Research example
A retention model splits on missed sessions and prior attainment. Check whether the rules and prediction error remain useful for students in a later intake.
What to report
Specify tree or ensemble type, tuning, depth or regularization, validation design, baseline comparison, prediction metrics and uncertainty or stability checks.
Read the source
Difference-in-differencesCausal inferenceDifference-in-differences compares an outcome’s change in a treated group with its change in a comparison group.
What it means
In the basic two-group, two-period design, the estimator subtracts the comparison group’s change from the treated group’s change. Causal identification uses parallel trends in untreated potential outcomes, not equal starting levels.
When it fits
Use for interventions with credible comparison units and before-and-after outcomes when untreated trends can be defended.
When to reconsider
Do not treat a nonsignificant pretrend test as proof of parallel trends, or apply a conventional two-way fixed-effects coefficient uncritically under staggered adoption and heterogeneous effects.
Assumptions and checks
Require parallel untreated trends, no anticipation, consistent composition and no relevant spillovers; account for dependence at the treatment-assignment level.
Research example
Retention rises from 70% to 80% in intervention colleges and from 65% to 69% in comparison colleges: the basic DiD estimate is 6 percentage points.
What to report
Report treatment timing, comparison definition, event-time patterns, estimator, clustered uncertainty and sensitivity to identification assumptions.
Durbin-Watson testModel diagnosticsAssesses first-order serial correlation in regression errors using ordered residuals.
What it means
The statistic is the sum of squared successive residual differences divided by the residual sum of squares. Values near 2 suggest little first-order correlation, but significance depends on the design and reference calculation, not a universal numeric cutoff.
When it fits
Use for a suitable regression with a genuine observation order, such as equally spaced monthly measurements.
When to reconsider
Do not run it on arbitrarily sorted cross-sectional rows or interpret a value near 2 as proof of independent errors. Models containing lagged outcomes can require other serial-correlation procedures.
Assumptions and checks
The ordering and regression assumptions of the test must fit the data; classical calibration makes specific assumptions about errors and regressors. It addresses a particular serial pattern rather than every possible dependence structure.
Research example
Illustration: monthly demand-model residuals remain positive for several months at a time. A low DW statistic can support investigating positive serial dependence and revising the error model.
What to report
Report ordering and spacing, model specification, DW, alternative hypothesis, p-value method, and residual autocorrelation plots or follow-up diagnostics relevant to the design.
Read the source
Effect sizeStatistical foundationsA quantity describing the magnitude and, where relevant, direction of a difference or association.
What it means
Effect sizes include raw mean differences, standardized differences, correlations and ratio measures. Their estimators have sampling uncertainty and can be biased; standardization changes the interpretation and comparison scale.
When it fits
Choose a measure aligned with the research question and report original units whenever useful.
When to reconsider
Do not infer importance from p alone or apply universal small, medium and large labels without context.
Assumptions and checks
Interpretation depends on the outcome scale, reference group, denominator, model and target population.
Research example
Illustration: a 3-point improvement on a 100-point assessment and d = .30 describe the same contrast in different units if the chosen SD is 10.
What to report
Identify the effect-size definition, sign convention, units or denominator, and confidence or credible interval.
Effect sizes with confidence intervalsPower and effect sizesAn effect estimate describes magnitude, and its confidence interval shows the uncertainty supported by the analysis.
What it means
Effects may use original units, standardized differences or ratios; each requires an interval appropriate to its estimator and design. Bootstrap percentile, studentized and bias-corrected accelerated intervals are different procedures, not interchangeable labels.
When it fits
Use to compare the estimated effect and its precision with a prespecified threshold of practical importance.
When to reconsider
Do not conclude equivalence just because the interval includes zero, or treat the interval as covering all possible bias.
Assumptions and checks
Coverage depends on the model and sampling or resampling scheme; ratio intervals commonly require transformation or specialized methods.
Research example
An estimated improvement of 3 points with a 95% CI from -1 to 7 leaves both negligible change and a worthwhile 5-point benefit compatible with the procedure.
What to report
Name the estimand, units, interval level and method, covariate adjustment and practical threshold.
Elastic net regressionPredictive modelsCombines lasso-style variable selection with ridge-style stabilization of correlated predictors.
What it means
Elastic net combines absolute-value and squared-coefficient penalties. Overall penalty strength and the mixture of the two penalties control sparsity and shrinkage.
When it fits
Prediction involves many predictors, some may be unnecessary, and meaningful predictors occur in correlated groups.
When to reconsider
Do not pick the mixing parameter by inspecting final test performance or equate a retained group with an established underlying mechanism.
Assumptions and checks
Suitable regression specification, consistent scaling and validation that includes tuning both penalty dimensions within training data.
Research example
A study predicts wellbeing from 100 correlated survey items. Tune penalty strength and mixture together, then compare predictions on an untouched participant sample.
What to report
Report penalty parameterization, both tuning settings, resampling design, selected coefficients, stability and performance against simpler models.
EstimandStudy designThe exact quantity your study aims to learn, defined before choosing how to calculate it.
What it means
An estimand is a target parameter; an estimator is a rule for estimating it, and an estimate is the resulting value. In intervention studies, specify population, treatments, outcome, summary measure and handling of intercurrent events.
When it fits
Define it during design and revisit it when missingness, nonadherence or competing events alter interpretation.
When to reconsider
Do not assume changing the analysis population or excluding treatment switchers leaves the target question unchanged.
Assumptions and checks
The analysis and identification assumptions must connect observed data to the stated target.
Research example
Illustration: the mean 12-week score difference under assignment to tutoring is distinct from the difference if every student attended all sessions.
What to report
State the target population, contrast, outcome, time point and handling of events that affect outcome interpretation.
Exploratory factor analysisFactor analysis and SEMExplores whether shared relationships among measured items can be represented by fewer underlying dimensions.
What it means
A common-factor model separates shared covariance from item-specific variance. Extraction, factor count and rotation affect interpretation; EFA differs from principal components analysis, which decomposes total variance.
When it fits
Developing a measurement instrument or investigating its dimensional structure before specifying a confirmatory model.
When to reconsider
Do not label exploratory discovery as confirmation or retain factors solely because eigenvalues exceed one. Formative indicators need not share a common factor.
Assumptions and checks
A plausible common-factor structure, informative item correlations and enough information for stable estimation. Estimator and correlation type should suit continuous or ordinal responses; ML inference has distributional assumptions.
Research example
Twenty engagement items may organize into behavioral and emotional dimensions. Compare theory, parallel analysis and interpretable rotated solutions, then assess the proposed structure in new data.
What to report
State correlation type, extraction, factor-retention rationale, rotation, full loadings, communalities, cross-loadings and any item-removal decisions.
Factor loadingsFactor analysis and SEMQuantifies how an indicator relates to a latent factor under a particular measurement model.
What it means
A loading is a coefficient in the factor measurement equation; its magnitude depends on factor and indicator scaling. In a standardized single-factor model without cross-loadings, a loading of 0.80 corresponds to 0.64 of item variance explained by that factor.
When it fits
Interpreting and evaluating a fitted reflective measurement model alongside item content and residual structure.
When to reconsider
Do not equate a correlation with a computed composite score to a common-factor loading, or delete items by a universal loading threshold alone.
Assumptions and checks
An identified factor scale and defensible measurement model. Squaring one loading does not generally isolate explained variance when correlated factors and cross-loadings are present.
Research example
An item loading of 0.80 and another of 0.45 suggest different relationships with the factor; wording, content coverage and estimation uncertainty also matter.
What to report
State standardization, factor scaling, all relevant loadings, uncertainty, cross-loadings and any fixed reference loadings.
Fisher exact testCategorical dataTests association in a contingency table using a conditional probability calculation rather than a large-sample chi-square approximation.
What it means
For a 2-by-2 table it conditions on the margins and tests an odds-ratio null, usually 1. Exact describes the reference calculation under its assumptions, not perfect precision or freedom from sampling bias.
When it fits
Use for independent categorical observations, particularly small or sparse 2-by-2 tables where the chi-square approximation is questionable.
When to reconsider
Do not use an independent-table test for paired binary outcomes; those require a matched analysis such as McNemar. Different two-sided conventions can produce different exact p-values.
Assumptions and checks
The conditional sampling model and independence of observational units must fit the design. Computation for larger tables can be demanding; simulated p-values must be labelled as simulated.
Research example
Illustration: an event occurs in 1 of 12 participants in one group and 6 of 12 in another. Fisher inference can assess association despite sparse cells, while the effect interval reveals substantial uncertainty.
What to report
Report all cell counts, the alternative and two-sided convention when relevant, exact or simulated method, and an odds ratio with a clearly identified confidence interval.
Read the source
Fornell–Larcker criterionReliability and validityCompares variance captured by each reflective construct’s indicators with variance shared between constructs.
What it means
The usual criterion requires each construct’s AVE to exceed its squared correlation with another construct, equivalently comparing square-root AVE with the absolute correlation. Passing is limited evidence because the criterion can miss problematic lack of distinction.
When it fits
Using it as one descriptive check within a broader evaluation of reflective constructs’ discriminant validity.
When to reconsider
Do not declare constructs distinct solely because every diagonal exceeds nearby correlations, or transfer the criterion directly to formative measurement.
Assumptions and checks
Credible reflective measurement models, meaningful AVE estimates and compatible construct correlations. Measurement misspecification can undermine the comparison.
Research example
Two constructs each have AVE 0.64 and correlate 0.70. Their square-root AVE of 0.80 passes this comparison, but cross-loadings and HTMT may still reveal overlap.
What to report
Present AVE and construct correlations, measurement specification and complementary evidence, including conceptual distinction and uncertainty.
Friedman testRank-based testsCompares conditions using ranks assigned within each participant or complete block.
What it means
It is a rank-based procedure for an unreplicated complete block design. The null is no systematic condition effect after blocking; the usual statistic has an approximate chi-square reference distribution.
When it fits
Use when each independent participant rates all conditions, or each complete block contains every treatment, and within-block ordering is meaningful.
When to reconsider
Do not apply it to unrelated groups or treat it as a universal repeated-measures model. Missing conditions and treatment-by-block patterns require special care or another analysis.
Assumptions and checks
Blocks are independent and comparable measurements are available for every condition within each analyzed block. Ties require adjustment; small samples may need exact or permutation methods.
Research example
Illustration: 25 teachers rate three lesson formats. Ranking formats within each teacher controls for teachers who generally rate everything more highly; the test assesses whether condition ranks differ systematically.
What to report
State the number of complete blocks and conditions, rank summaries, statistic, degrees of freedom, p-value method, missing-block handling, and multiplicity-adjusted paired follow-ups when appropriate.
Read the source
Full-information maximum likelihood regressionMissing dataEstimates a model using the information each participant actually provides, without filling in a single completed dataset.
What it means
FIML maximizes an observed-data likelihood, integrating over missing components under the specified joint model. It is an estimation strategy for a regression or structural model, rather than a separate regression family.
When it fits
A suitable likelihood model can be specified and incomplete cases retain informative measurements.
When to reconsider
Do not assume FIML automatically handles every missing predictor or makes nonrandom dropout harmless; treatment of exogenous covariates depends on model specification.
Assumptions and checks
Ignorable missingness, commonly MAR with distinct model and missingness parameters; a correctly specified distributional model and identifiable parameters.
Research example
A longitudinal study records baseline stress for 400 people but follow-up stress for 340. A joint model can use baseline information from all 400, subject to its missingness assumptions.
What to report
Report the modeled variables, missingness patterns, estimator, distributional assumptions, covariate handling, sample contribution and sensitivity analyses.
Generalized additive model (Gaussian)RegressionModels a continuous outcome as the sum of potentially curved relationships with numeric predictors.
What it means
The qualified mgcv 1.9-4 implementation fits Gaussian identity models with cubic regression splines and REML smoothing. Smooth-term F tests and pointwise mean intervals are approximate and conditional on smoothing parameters.
When it fits
Independent observations have a continuous outcome and plausible additive nonlinear relationships with one to five numeric predictors.
When to reconsider
Do not use this release for non-Gaussian outcomes, clustered errors, interactions, categorical predictors or extrapolation.
Assumptions and checks
Independent Gaussian errors, adequate basis flexibility and additive mean structure. Concurvity can make individual smooth terms difficult to identify.
Research example
Estimate how a continuous response varies smoothly with age while holding another numeric predictor at its median.
What to report
Report outcome, predictors, complete observations, family and link, basis dimensions, REML selection, effective degrees of freedom, approximate tests, pointwise mean intervals and limitations.
Generalized estimating equationsRegressionEstimates population-averaged associations for correlated observations within clusters.
What it means
RSearch uses gee 4.13-29 with Gaussian identity, binomial logit or Poisson log means, independence or exchangeable working correlation, and robust cluster sandwich uncertainty.
When it fits
The question concerns a marginal mean across a population and observations are grouped in independent clusters.
When to reconsider
Do not interpret coefficients as conditional random effects or report likelihood-based AIC. Small-cluster corrections, offsets and time-based correlation are outside scope.
Assumptions and checks
Independent clusters, a correctly specified marginal mean and a justified missingness mechanism. The release requires at least 30 clusters; this threshold does not guarantee reliable asymptotic inference.
Research example
Estimate the population-average association between workload and repeated employee outcomes, clustering by employee.
What to report
Report observations and clusters, family and link, working correlation, coefficients and robust intervals, missing-data policy and lack of a small-sample correction.
Read the source
Generalized linear mixed modelRegressionModels binary or count outcomes while allowing clusters to have different baseline tendencies.
What it means
The qualified glmmML 1.1.7 implementation uses a Gaussian random intercept and Laplace integration with a binomial logit or Poisson log link. Coefficients are conditional on the random effect.
When it fits
Repeated or grouped outcomes require a cluster-specific model with a plausible random-intercept structure.
When to reconsider
Random slopes, crossed effects, offsets, categorical predictors and zero-inflated models are outside the current release.
Assumptions and checks
Independent clusters, a suitable outcome distribution and link, Gaussian random intercepts and a justified complete-case analysis.
Research example
Model binary responses from participants nested in sites, allowing each site its own baseline log odds.
What to report
Report clusters and observations, family and link, estimation method, fixed-effect coefficients and intervals, random-intercept SD and model boundaries.
Read the source
Generalized linear modelPredictive modelsExtends regression to outcomes such as counts or binary events by choosing an appropriate distribution and link.
What it means
A GLM combines a response distribution from an exponential family with a link connecting the conditional mean to a linear predictor. The distribution determines a mean-variance relationship; the link determines coefficient interpretation.
When it fits
The outcome’s support and variation suggest a model such as binomial-logit or Poisson-log rather than ordinary normal-error regression.
When to reconsider
Do not choose a family solely because software accepts the data, or ignore excess variation, excess zeros, exposure time or clustering.
Assumptions and checks
Correct mean and link specification, suitable variance structure, and independent observations unless extended. Overdispersion can invalidate conventional Poisson standard errors.
Research example
Model annual publication counts using a log link and log years observed as an offset. An exponentiated coefficient of 1.20 indicates a modeled rate ratio of 1.20.
What to report
Name the family and link, offsets, dispersion assessment, coefficients or meaningful response-scale contrasts, intervals and diagnostic checks.
Read the source
Goodness-of-fit measuresModel diagnosticsSummarize how a model reproduces observed data or balances fit against complexity.
What it means
Different measures answer different questions: SEM chi-square assesses exact fit, CFI and TLI compare with a baseline model, and RMSEA and SRMR summarize particular discrepancies. Information criteria such as AIC and BIC support relative comparisons under compatible likelihoods, rather than a universal pass mark.
When it fits
Use a justified set of measures alongside residuals, parameter plausibility, substantive theory, and, when prediction is the goal, validation on new data.
When to reconsider
Do not accept or reject a theory because one index crosses a conventional threshold. Good global fit neither proves causality nor guarantees that a particular path or local residual is well modelled.
Assumptions and checks
Indices must match the model class, estimator, sample structure, and treatment of ordinal or nonnormal data. Likelihood-based comparisons require compatible outcomes, observations, and likelihood definitions.
Research example
Illustration: a questionnaire model has CFI = 0.96 but several large item-pair residuals. The favorable global index does not remove the need to investigate those local discrepancies.
What to report
Identify the estimator and exact index variants, model degrees of freedom, baseline model where relevant, RMSEA interval, residual findings, and whether comparisons or modifications were prespecified.
Gradient-boosted trees (binary classification)RegressionBuilds an additive sequence of small trees to predict whether a binary outcome is class 0 or class 1.
What it means
The qualified gbm 2.2.3 implementation uses Bernoulli loss, 150 depth-one trees, learning rate 0.05 and no subsampling. Fixed settings precede stratified five-fold held-out validation.
When it fits
Independent observations have a binary 0/1 target and numeric predictors available before the outcome occurs.
When to reconsider
Do not use this release for grouped or temporal validation, multiclass outcomes, interactions or causal inference.
Assumptions and checks
Validation observations are exchangeable with training observations. Prior preprocessing and selection must not leak target information. Predicted probabilities need not be calibrated.
Research example
Estimate held-out discrimination and probability error for a binary outcome using numeric baseline measurements.
What to report
Report target coding, sample and class counts, seed, tree count, depth, learning rate, fold design, held-out metrics and external-validation limitations.
Read the source
Heterotrait–monotrait ratioReliability and validityAssesses whether indicators intended to measure different reflective constructs are too similar relative to indicators of the same construct.
What it means
HTMT relates between-construct item correlations to within-construct item correlations. Variants differ, including arithmetic-mean HTMT and geometric-mean HTMT2; the latter accommodates unequal loadings under its model conditions.
When it fits
Evaluating empirical separation of reflective constructs alongside their theoretical distinction and measurement model.
When to reconsider
Do not treat 0.85 or 0.90 as universal laws, apply the ratio blindly to formative indices, or ignore unstable denominators from weak within-construct correlations.
Assumptions and checks
Appropriate indicator grouping, correct reverse scoring and suitable correlation estimates. Variant, sign handling and measurement assumptions affect interpretation.
Research example
A trust-versus-satisfaction HTMT of 0.93 suggests substantial overlap. Examine its interval, item content and competing models before deciding whether the constructs can be distinguished.
What to report
Specify HTMT variant, correlation and missing-data handling, prespecified interpretive criterion, uncertainty procedure and actions taken when constructs overlap.
Hierarchical clusteringClusteringHierarchical clustering builds nested groups that can be displayed as a branching dendrogram.
What it means
Agglomerative algorithms begin with individual observations and repeatedly merge clusters according to a dissimilarity measure and linkage rule. Single, complete, average and Ward linkage optimize different notions of proximity.
When it fits
Use to explore grouping at several resolutions or when a nested representation is useful.
When to reconsider
Do not interpret dendrogram branches as a biological or causal hierarchy without external evidence; single linkage can chain observations through bridges.
Assumptions and checks
Distance and linkage must match the variable types and objective. Ward’s variance criterion requires the appropriate Euclidean formulation.
Research example
Group 40 laboratories by standardized assay profiles, then compare a three-group cut with a five-group cut and inspect membership changes.
What to report
Report distance, scaling, linkage, cut rule, cluster sizes and sensitivity to these choices.
Read the source
HomoscedasticityModel assumptionsThe outcome variation left after modeling is similar across predictor values or comparison groups.
What it means
Constant conditional error variance is written Var(error | X) = sigma squared. Unequal variance can invalidate conventional standard errors even when the conditional mean model is correct.
When it fits
Inspect residual spread for linear models and compare group variability when using equal-variance procedures.
When to reconsider
Do not assume equal variances because a test is nonsignificant, or use a robust standard error to excuse a wrong mean model.
Assumptions and checks
Assess variance conditional on an appropriate model; clustering and serial dependence are separate issues.
Research example
Illustration: prediction errors spread roughly from -2 to 2 at low income and -10 to 10 at high income, suggesting changing conditional variance.
What to report
Describe diagnostics and the chosen response, such as Welch inference, justified weighting or appropriate robust covariance estimation.
Read the source
Independent and paired observationsStudy designPaired observations belong together, such as two measurements from one participant; independent groups contain unrelated observational units.
What it means
Within-pair covariance contributes to the variance of a difference. A paired analysis preserves the matching, while repeated or clustered designs may require a more general dependence model.
When it fits
Use paired procedures for genuine one-to-one repeated or matched measurements.
When to reconsider
Do not create pairs by sorting independent groups or treat multiple visits from one participant as independent people.
Assumptions and checks
Pair identities must be correct, and pairs must be independent for standard paired tests; assumptions then concern within-pair differences.
Research example
Illustration: 40 students assessed before and after a course provide 40 pairs, not 80 independent participants.
What to report
Describe the matching unit, complete-pair count, difference direction and handling of incomplete pairs.
Read the source
Independent-samples t-testGroup comparisonsCompares the population means of two groups containing different, independent observational units.
What it means
The statistic divides the estimated mean difference by its standard error. Welch inference estimates the two group variances separately, whereas the pooled Student procedure assumes a common population variance.
When it fits
Use when the mean of a numeric outcome is the intended comparison, such as scores from independently randomized groups.
When to reconsider
Do not use for before-and-after observations on the same people, repeated records, or clustered sampling without accounting for dependence. Severe outliers or very small, skewed samples require further assessment.
Assumptions and checks
Observations are independent within and between groups. The pooled test has exact t inference for normal groups with a common variance. Welch uses approximate degrees of freedom and does not require equal variances; larger-sample mean inference still depends on tails and balance.
Research example
Illustration: two independent groups average 74 and 69 points. The estimated difference is 5 points; its confidence interval, rather than the difference alone, shows the precision of the comparison.
What to report
Report group sizes, means and standard deviations, the signed mean difference with confidence interval, t, degrees of freedom, p-value, and whether Welch or pooled inference was used.
Read the source
Inverse probability of treatment weightingCausal inferenceIPTW reweights observations to make measured pretreatment covariates more comparable across treatment groups.
What it means
For a binary-treatment average treatment effect, basic weights are 1/e(x) for treated and 1/(1 - e(x)) for untreated observations. Other estimands use different weights; stabilization and truncation must be distinguished and documented.
When it fits
Use for a prespecified observational treatment effect when confounders are measured and overlap supports reliable weighting.
When to reconsider
Do not ignore extreme weights or interpret a large weighted total as a larger independent sample; trimming can change the target population.
Assumptions and checks
Require consistency, conditional exchangeability and positivity, plus an adequate weight model; check weighted covariate balance and use appropriate uncertainty estimation.
Research example
An untreated student with estimated treatment probability .95 receives an unstabilized ATE weight of 20, highlighting weak overlap and potentially unstable inference.
What to report
Give target effect, propensity model, weight formula and distribution, effective sample size, balance, truncation and variance estimator.
K-means clusteringClusteringK-means groups observations around k numeric centers so members are relatively close to their own center.
What it means
It minimizes within-cluster squared Euclidean distances through an iterative algorithm that can converge to a local optimum. The selected k, scaling and initialization can change the partition.
When it fits
Use for exploratory segmentation of numeric measurements when compact groups are plausible.
When to reconsider
Avoid unmodified k-means for arbitrary categorical codes, strongly irregular clusters or data dominated by outliers. Cluster labels are not proof of natural types.
Assumptions and checks
Euclidean distance must express meaningful similarity; normalize scales deliberately and check stability across starts and samples.
Research example
Cluster 200 schools using standardized attendance and attainment measures, comparing k = 2, 3 and 4 for stability and interpretability.
What to report
Report variables, scaling, k rationale, algorithm, random starts, seed, cluster sizes and validation.
Read the source
Kaplan-Meier survival estimateSurvival analysisA Kaplan-Meier curve estimates the probability of remaining event-free beyond each time while accounting for incomplete follow-up.
What it means
The estimate multiplies conditional survival proportions across event times, using the risk set immediately before each event. Right-censored individuals contribute follow-up until censoring, rather than being treated as having the event or surviving forever.
When it fits
Use to describe a defined time-to-event outcome with right censoring, overall or in groups.
When to reconsider
Do not read precise long-term survival from a tail with very few at risk; for competing event probabilities, use an appropriate cumulative-incidence method.
Assumptions and checks
Censoring must be non-informative for survival within the analyzed groups; define time origin consistently and handle delayed entry when present.
Research example
Follow 100 doctoral candidates from enrollment to completion; a candidate still studying when follow-up ends is censored at that date.
What to report
Provide event and censor definitions, follow-up, numbers at risk, confidence bands and whether median survival was reached.
Kolmogorov-Smirnov testModel diagnosticsMeasures the largest gap between cumulative distributions and tests whether that gap is larger than expected under a null model.
What it means
The one-sample version compares an empirical distribution with a fully specified continuous reference; the two-sample version compares independent distributions. The ordinary one-sample null distribution is not valid when reference parameters are estimated from the same observations.
When it fits
Use to assess a prespecified continuous distribution or compare two independent continuous distributions when any distributional difference matters.
When to reconsider
Do not fit a normal mean and standard deviation to a sample and then report the ordinary KS p-value as a valid normality test. Discrete data, rounding, and ties need suitable calibration.
Assumptions and checks
Observations follow the independence and continuity conditions of the selected variant. Estimated parameters call for an appropriate adjusted test or calibrated bootstrap that repeats the fitting step.
Research example
Illustration: compare observed instrument errors with a calibration distribution fixed before collecting the data. If its parameters are instead fitted from those errors, the reference calculation must change.
What to report
Report D, sample sizes, one- or two-sample form, the reference distribution and parameter origin, alternative, and exact, asymptotic, or simulated p-value method.
Read the source
Kruskal-Wallis testRank-based testsTests for differences in rank patterns across independent groups.
What it means
The statistic compares group rank sums under a null of common distributions. A location or median interpretation needs additional assumptions about comparable distribution shapes; a significant omnibus result does not specify the differing groups.
When it fits
Use for an ordinal or numeric outcome across several independent groups when a rank comparison matches the estimand.
When to reconsider
Do not substitute it for repeated-measures ANOVA on the same participants. Do not label a result a median difference if group distribution shapes differ materially.
Assumptions and checks
Independent units and a common ordering of the outcome are required. Ties need correction, and small groups can make the usual chi-square approximation unreliable.
Research example
Illustration: compare satisfaction ratings from three independently sampled service centers. A significant test warrants planned or adjusted rank comparisons, not a claim that all centers differ.
What to report
Give group sample sizes, medians and spread alongside distribution plots, H, degrees of freedom, p-value method, tie treatment, and any adjusted follow-up comparisons.
Read the source
KurtosisDescriptive statisticsKurtosis describes the prominence of extreme deviations, not simply the height of a distribution’s peak.
What it means
Pearson kurtosis is the fourth central moment divided by variance squared; a normal distribution has 3. Excess kurtosis subtracts 3, giving a normal reference of 0.
When it fits
Use alongside plots when tail behavior matters.
When to reconsider
Do not compare coefficients without checking conventions or diagnose normality from one value.
Assumptions and checks
The population fourth moment must exist and variance be positive.
Research example
Pearson kurtosis 5 and excess kurtosis 2 describe the same coefficient convention-adjusted.
What to report
Specify Pearson or excess, estimator correction and n.
Read the source
Lasso regressionPredictive modelsShrinks regression coefficients and can set some to zero, producing a smaller predictive model.
What it means
The lasso adds a penalty proportional to the sum of absolute coefficient values to the fitting loss. Its coefficient estimates are deliberately biased; selected variables can change substantially among correlated alternatives.
When it fits
A sparse prediction model is useful and there are enough data to tune the penalty and assess stability.
When to reconsider
Do not interpret selection as proving that a variable matters scientifically or apply ordinary regression p-values as if selection had been prespecified.
Assumptions and checks
A suitable regression form, comparable predictor scaling, and representative validation. Reliable recovery of true predictors requires stronger conditions than useful prediction.
Research example
From 80 questionnaire predictors, cross-validation selects a model with 9 nonzero coefficients. Repeat resampling to see whether the same 9 consistently appear.
What to report
Report scaling, penalty selection, validation error, retained coefficients, selection stability and whether the goal was prediction or inference.
Latent growth curve modelTime series and growthA latent growth curve model estimates average change over time and differences in people’s trajectories.
What it means
Repeated outcomes load on latent growth factors such as an intercept and slope, with time scores defining the trajectory. Growth-factor means describe average change, while their variances and covariance describe individual differences.
When it fits
Use for repeated measurements when development or change is central and the measurement occasions support an identifiable trajectory model.
When to reconsider
Do not impose linear growth despite clear curvature or interpret between-person slope associations as causal effects.
Assumptions and checks
Time coding, residual structure and missing-data assumptions must be justified; latent constructs require comparable measurement across occasions.
Research example
Measure research confidence at months 0, 3, 6 and 12, using those time scores to distinguish initial confidence from monthly change.
What to report
State time loadings, measurement model, growth means and variances, residual correlations, estimator, missingness handling and model fit.
Read the source
Levene testModel diagnosticsAssesses whether outcome variability differs across groups.
What it means
Levene-type procedures compare absolute deviations from a group center. The original uses means; the median-centered Brown-Forsythe variant is often less sensitive to skewness or extreme observations.
When it fits
Use as part of examining group variance differences, with group standard deviations and distribution plots.
When to reconsider
Do not treat a nonsignificant result as proof of equal variances or use a preliminary test as the sole rule for selecting pooled versus Welch inference. Dependence between observations remains a separate issue.
Assumptions and checks
Independent observations and an appropriate grouping structure are required. The chosen center and reference approximation affect sensitivity, particularly with small or highly irregular samples.
Research example
Illustration: two groups have standard deviations of 4 and 11 points. A Levene-type test quantifies evidence about unequal variability, while the observed spread and sample sizes inform the practical consequence.
What to report
Specify mean- or median-centering, group sizes and dispersion, test statistic, degrees of freedom, p-value, and the variance strategy used in the substantive analysis.
Listwise deletionMissing dataAnalyzes only records with no missing values in the variables required for that analysis.
What it means
Complete-case validity depends jointly on the missingness process, estimand and model. MCAR is sufficient in many standard settings, but some regression estimands remain valid under weaker conditions.
When it fits
Use when the relevant validity conditions are defensible and the resulting information loss is acceptable.
When to reconsider
Do not justify deletion using only a small missing-data percentage, or silently compare models fitted to different participants.
Assumptions and checks
Explain why selecting complete records should preserve the target parameter; MAR alone does not guarantee validity for every complete-case analysis.
Research example
Illustration: 10% missingness in each of five independently incomplete variables leaves about .9^5 = 59% of records complete.
What to report
Give the original and analyzed sample sizes, variable-specific missingness, rationale and a suitable sensitivity comparison.
Mahalanobis distanceModel diagnosticsMeasures how unusual a combination of numeric values is after accounting for scales and correlations.
What it means
Squared Mahalanobis distance is (x - center)-transpose times the inverse covariance matrix times (x - center). Its distribution depends on the population and how the center and covariance were obtained; estimated-sample distances do not automatically have an exact chi-square calibration.
When it fits
Use to explore unusual multivariate profiles or covariance-adjusted distances when the variables and covariance estimate are appropriate.
When to reconsider
Do not delete every observation above a cutoff. Outliers can distort ordinary covariance estimates and hide one another, and singular covariance prevents the ordinary inverse calculation.
Assumptions and checks
A meaningful, nonsingular covariance estimate is needed. Robust estimation or regularization may be necessary with contamination or many variables; probability cutoffs require additional distributional and estimation assumptions.
Research example
Illustration: a participant has average reading and unusually high writing scores. Neither value alone may look extreme, but their combination can be unusual when these scores are normally strongly related.
What to report
Specify the variables, center and covariance estimator, whether distance or squared distance is reported, the justification for any flagging threshold, and the investigation or sensitivity analysis that follows.
Read the source
Mann-Whitney U testRank-based testsCompares the ranked outcomes of two independent groups.
What it means
The usual null calibration assumes equal outcome distributions, and U summarizes pairwise ordering between groups. Interpreting rejection as a median or location difference requires additional common-shape assumptions; different spreads can also affect the result.
When it fits
Use when an ordinal or numeric outcome and the scientific question support a rank-based comparison between independent groups.
When to reconsider
Do not describe it automatically as a test of medians or as assumption-free. It is not the paired Wilcoxon test, and it does not directly estimate a difference in arithmetic means.
Assumptions and checks
Observations are independent within and across groups, with a meaningful common ordering. Account for ties, and choose exact, permutation, or approximate inference appropriate to the data and implementation.
Research example
Illustration: two independent clinics report waiting times with long upper tails. U asks about their rank ordering; if one clinic has much wider dispersion, a simple median-shift explanation may be misleading.
What to report
Report group sizes and distributions, U or the software-specific W convention, p-value method, tie handling, and a clearly defined rank effect or location-shift interval when its assumptions fit.
Read the source
McDonald’s omegaReliability and validityEstimates the contribution of modeled common factors to the reliability of a specified scale score.
What it means
Omega total includes variance attributed to all modeled common factors; omega hierarchical isolates a general factor in an appropriate hierarchical or bifactor model. The two answer different questions and cannot be substituted without explanation.
When it fits
Item loadings differ or the score’s general versus multidimensional common variance needs explicit evaluation.
When to reconsider
Do not fit a poor one-factor model merely to obtain omega or assume omega is automatically correct whenever it exceeds alpha.
Assumptions and checks
An adequate factor model, well-defined score and appropriate handling of item scales and residual correlations. A general-factor interpretation requires evidence for that factor structure.
Research example
An instrument has omega total 0.90 but omega hierarchical 0.55. Much reliable score variance may reflect narrower domains, so a single general-score interpretation needs scrutiny.
What to report
Name the omega variant, factor model, estimator, scoring weights, interval and fit evidence; distinguish observed-score reliability from latent-response reliability for ordinal items.
Measurement levels and numeric codesStudy designA column containing numbers is not necessarily a quantitative measurement.
What it means
Nominal categories identify classes; ordinal categories add ordering; interval scales support interpretable equal differences; ratio scales additionally have a meaningful zero for ratios. Numeric labels do not create equal category spacing.
When it fits
Check variable meaning and coding before selecting summaries, transformations or model families.
When to reconsider
Do not average identification numbers or assume a 1-to-5 response item has equal psychological distances. Composite-scale treatment requires a defensible measurement model.
Assumptions and checks
Measurement interpretation depends on the instrument and construct, not the spreadsheet storage type.
Research example
Illustration: school codes 1, 2 and 3 identify categories. A mean school code of 2.1 has no substantive interpretation. Ordered satisfaction responses convey more information but still need appropriate modeling choices.
What to report
Provide labels, units, response options, coding direction and the rationale for any scale-score construction.
Read the source
MedianDescriptive statisticsThe median is the middle of the ordered observations.
What it means
For an odd numeric sample it is the central observation; the usual even-sample convention averages the two central values. It depends on order, so extreme magnitudes have less influence than on the mean.
When it fits
Use for skewed numeric outcomes such as completion times, or report a median category for ordinal responses.
When to reconsider
Do not substitute it for the mean when average total cost is the research target.
Assumptions and checks
Ordering must be meaningful; interpolating between ordinal categories needs care.
Research example
Completion times of 6, 8, 9, 12 and 40 minutes have a median of 9 minutes.
What to report
Give the median, interquartile range, n and handling of missing values.
Read the source
Mediation analysisPredictive modelsExamines whether an exposure-outcome relationship includes a pathway through an intermediate variable.
What it means
In a linear additive model without exposure-mediator interaction, the indirect path is often represented by the product a × b. Causal direct and indirect effects require explicit identification assumptions and may differ from that product in nonlinear models.
When it fits
Theory and measurement timing support an intermediate process, and the design can address confounding of the relevant pathways.
When to reconsider
A significant indirect path from a cross-sectional survey does not establish a mechanism. Randomizing treatment alone does not randomize the mediator.
Assumptions and checks
Temporal ordering, suitable models and, for common causal estimands, strong no-unmeasured-confounding conditions; exposure-induced mediator-outcome confounding requires particular care.
Research example
Training increases confidence by 0.5 points; each confidence point predicts 4 performance points conditional on training. The linear indirect association is 2 points, pending causal assumptions.
What to report
Define the estimand, diagram, timing, adjustment set, direct and indirect estimates with intervals, resampling method and sensitivity to confounding.
Meta-analysisRegressionCombines comparable independent study effects with uncertainty based on their sampling standard errors.
What it means
The qualified metafor 5.0-1 implementation fits common-effect inverse-variance models or REML random-effects models with Knapp-Hartung coefficient inference and up to five numeric moderators.
When it fits
Independent study effects and valid standard errors are available on one suitable common analysis scale.
When to reconsider
Do not pool dependent effects as independent studies, mix incompatible effect scales or confuse sampling standard errors with standard deviations.
Assumptions and checks
Independent comparable effects, appropriate sampling approximations and a defensible model for remaining heterogeneity. Complete cases do not resolve selective reporting.
Research example
Combine independent studies reporting a log risk ratio and its sampling standard error.
What to report
Report the number of effects, analysis scale, model, uncertainty method, coefficients and intervals, heterogeneity, missing-effect handling and study-quality limitations.
Read the source
Missing at random (MAR)Missing dataAfter accounting for observed information, missingness does not additionally depend on the missing values themselves.
What it means
For a given missingness pattern, its probability depends on the observed data but not the unobserved data. Ignoring the missingness mechanism in likelihood inference also requires appropriate distinctness conditions and a correctly specified data model.
When it fits
Use as an explicit working assumption for suitable multiple-imputation or likelihood analyses with relevant observed predictors.
When to reconsider
Do not translate MAR as missingness being unrelated to everything, or claim observed data alone prove it.
Assumptions and checks
Include adequate information about missingness and the outcome in a compatible model; MAR is relative to the variables observed.
Research example
Illustration: follow-up is less common among younger participants. MAR assumes no further dependence on the missing follow-up score after conditioning on the relevant observed information, including age.
What to report
State why MAR is plausible, variables used and sensitivity to departures.
Missing completely at random (MCAR)Missing dataWhether a value is missing is unrelated to the values that would have been observed.
What it means
The missingness pattern is independent of the complete data, both observed and unobserved components.
When it fits
Use as a justified simplifying assumption when the collection process makes it plausible.
When to reconsider
A nonsignificant MCAR diagnostic does not establish MCAR; a low missingness rate does not establish it either.
Assumptions and checks
The mechanism must not depend on relevant observed or unobserved measurements.
Research example
Illustration: a random-number generator selects 20 of 200 completed forms for accidental removal independently of their contents.
What to report
Explain the collection mechanism, missingness patterns and uncertainty about the assumption.
Read the source
Missing not at random (MNAR)Missing dataMissingness still depends on unobserved values after accounting for observed information.
What it means
The observed-data distribution generally cannot identify the full-data target under unrestricted MNAR. Identification requires additional assumptions, design information or suitable external information.
When it fits
Examine MNAR scenarios when the response process plausibly depends on unrecorded outcomes.
When to reconsider
Do not expect ordinary MAR imputation to automatically remove MNAR bias or label a dataset MNAR from a single diagnostic.
Assumptions and checks
Sensitivity models must state their untestable restrictions and justify a scientifically plausible range.
Research example
Illustration: at the same observed baseline score, participants with worse unrecorded follow-up wellbeing are less likely to respond. Explore how lower missing scores would change the conclusion.
What to report
Describe the mechanism, sensitivity parameters or external information, and where conclusions change.
Mixed-effects modelsGroup comparisonsModel an outcome while allowing observations from the same person, site, or other cluster to share variation.
What it means
Fixed effects describe specified population-level relationships, while random effects represent variation among grouping units. A linear mixed model uses a continuous-response error model; generalized mixed models extend the framework to outcomes such as counts or binary responses.
When it fits
Use for repeated measurements, students within schools, or crossed sources of dependence, with a random-effects structure justified by the design.
When to reconsider
Do not assume a random intercept captures every dependence pattern. Few clusters, singular fits, or overly complex random slopes can undermine variance estimation and inference.
Assumptions and checks
Specify the conditional mean, random-effects distribution, and residual covariance appropriately. Likelihood analyses with incomplete outcomes need an ignorable missingness mechanism under the fitted model, not simply permission to ignore missingness.
Research example
Illustration: 300 pupils attend 20 schools. A school intercept allows shared school variation, while a time slope per pupil can represent different learning trajectories when pupils have repeated scores.
What to report
Report the full fixed and random structure, cluster counts, estimation method, variance components, effect intervals, convergence or singularity findings, and how inferential degrees of freedom were obtained.
Read the source
ModeDescriptive statisticsThe mode is the value or category occurring most often.
What it means
A sample can have several tied modes. For continuous measurements, an estimated density peak or modal histogram interval depends on smoothing or bin choices.
When it fits
Use to identify the most common response or category.
When to reconsider
Do not treat a modal category as a majority unless its proportion exceeds one half.
Assumptions and checks
Categories and counting rules must be consistent.
Research example
Among 20 respondents, 8 prefer online, 7 hybrid and 5 classroom teaching: online is the mode, but only 40% chose it.
What to report
Give modal values, counts, percentages and ties; explain bins for continuous data.
Read the source
Moderation and interactionPredictive modelsExamines whether an association differs depending on another variable’s value.
What it means
In Y = b0 + b1X + b2Z + b3XZ, the slope for X is b1 + b3Z. Interaction depends on the model’s scale; interaction in log odds need not imply the same pattern in probability differences.
When it fits
A prespecified question asks for whom or under which conditions an association differs.
When to reconsider
One significant subgroup result and one nonsignificant result do not establish an interaction. Avoid splitting continuous moderators at arbitrary cutoffs.
Assumptions and checks
Adequate observations across combinations of predictors, suitable functional form, and the parent model’s assumptions. Causal effect modification requires causal identification.
Research example
If the study-hours slope is 2 and the hours-by-support coefficient is 0.5, the fitted slope is 3 points per hour at support = 2.
What to report
Give interaction estimates with intervals, variable centering and coding, and predicted outcomes or simple slopes over an observed, meaningful moderator range.
Monte Carlo simulation and bootstrapPower and effect sizesSimulation generates repeated datasets or draws to study uncertainty; bootstrap methods use the observed sample or a fitted model as their starting point.
What it means
The ordinary nonparametric bootstrap samples observations with replacement to approximate a statistic’s sampling distribution. Monte Carlo is a broader computational approach, including simulation under prespecified data-generating models; its numerical error differs from sampling uncertainty.
When it fits
Use bootstrap intervals for suitable statistics or simulations to assess power, bias and interval coverage under plausible designs.
When to reconsider
Do not resample individual rows independently when clusters or temporal dependence define the sampling units, or expect resampling to correct selection bias.
Assumptions and checks
The resampling scheme must represent the design and the estimator’s regularity conditions; sparse tails and very small samples can undermine approximation.
Research example
Bootstrap 60 classrooms by resampling whole classrooms, retaining their students; separately simulate new classroom studies to evaluate a planned estimator.
What to report
Give the generating or resampling scheme, repetitions, seed, interval method, failed fits and Monte Carlo precision.
MulticollinearityModel assumptionsPredictors contain overlapping information, making their separate regression contributions difficult to estimate precisely.
What it means
Exact linear dependence makes coefficients non-identifiable; near dependence can inflate coefficient uncertainty and sensitivity. Pairwise correlations alone can miss dependence involving several predictors.
When it fits
Investigate unstable coefficients, wide intervals or unexpected sign changes in multivariable models.
When to reconsider
Do not delete a required confounder solely to reduce VIF or treat a fixed VIF threshold as a universal failure rule.
Assumptions and checks
Interpret diagnostics relative to model parameterization, sample information and whether the goal is prediction or coefficient estimation.
Research example
Illustration: entering total spending together with its two components, food spending and non-food spending, creates exact linear dependence.
What to report
Describe dependencies, relevant VIF or condition diagnostics, and the scientific reason for any reparameterization.
Multiple imputationMissing dataCreates several plausible versions of incomplete data so the final analysis reflects uncertainty about missing values.
What it means
Each completed dataset is analyzed separately; compatible estimates and their variances are then pooled, commonly using Rubin’s rules. Chained equations model incomplete variables conditionally and can accommodate different variable types.
When it fits
Missingness can reasonably be explained by observed information, and useful predictors of missingness and missing values are available.
When to reconsider
Do not average imputed datasets into one supposedly complete dataset or treat imputed values as observed facts. Standard imputation does not resolve missing-not-at-random bias.
Assumptions and checks
A defensible MAR assumption for standard procedures, suitable imputation models, and compatibility with the analysis, including interactions, nonlinear terms and clustering.
Research example
In a survey with 15% missing income, impute using employment, education and the study outcome. Fit the planned regression in each imputation and pool coefficients and uncertainty.
What to report
Describe missingness, predictors, model choices, number of imputations, convergence checks, pooling rules and sensitivity to departures from MAR.
Multiple linear regressionPredictive modelsModels a numeric outcome using several predictors and estimates each association conditional on the others.
What it means
Ordinary least squares minimizes squared residuals in a model linear in its coefficients; predictors may include transformations and interactions. A coefficient describes an adjusted association under the chosen specification, not automatically a causal effect.
When it fits
The question concerns a conditional mean or prediction of a numeric outcome with substantively justified covariates.
When to reconsider
Do not add every available variable indiscriminately, extrapolate far outside the data, or interpret adjustment as eliminating all confounding.
Assumptions and checks
Correct conditional-mean form, no perfect collinearity and appropriate handling of dependence. Conventional small-sample inference additionally assumes normal, constant-variance errors; robust procedures address particular violations.
Research example
A coefficient of 2.4 for weekly study hours means 2.4 higher predicted exam points per additional hour at the same modeled attendance and prior score.
What to report
Give the formula, coding and units, coefficients with intervals, sample size, missing-data handling, residual and influence checks, and validation for prediction.
Multiple testing and multiplicityStatistical foundationsTesting many claims increases opportunities for false discoveries unless the analysis addresses that search.
What it means
Familywise error rate is the probability of at least one false rejection in a family. False discovery rate is the expected proportion of false rejections among all rejections, with zero when none are rejected.
When it fits
Define the claim family and error target before confirmatory testing; use a suitable adjustment or planned testing hierarchy.
When to reconsider
Do not report only the smallest p-value or interpret an FDR target as a guaranteed false-positive fraction in one study.
Assumptions and checks
Control depends on valid individual p-values and method-specific dependence conditions. Holm tolerates arbitrary dependence; ordinary BH requires suitable conditions.
Research example
Illustration: with 20 independent true-null tests at .05, the chance of at least one false rejection is 1 - .95^20, about 64%.
What to report
Give the family, number of tests, method, error target and clearly labeled adjusted results.
Multivariate analysis of covariance (MANCOVA)Group comparisonsCompares a joint profile of numeric outcomes across groups while including covariates.
What it means
MANCOVA extends the multivariate linear model by estimating outcome-specific covariate effects and testing group contrasts in the adjusted mean vector. The covariate values or target population used for adjustment are part of the question being answered.
When it fits
Use when several related outcomes and prespecified baseline covariates jointly address a group comparison.
When to reconsider
Do not regard statistical adjustment as equivalent to randomization. Avoid unexplained post-treatment adjustment, poor covariate overlap, or interpreting a multivariate finding as significance on every outcome.
Assumptions and checks
A correctly specified multivariate conditional mean and covariance model is needed. Common covariate slopes apply only if interactions are omitted; ordinary inference also assumes independent units and appropriate multivariate error behavior.
Research example
Illustration: compare two education programs on reading and writing while adjusting for baseline achievement. Report the adjusted reading and writing contrasts as well as the joint test, because the pattern may differ between outcomes.
What to report
Report the adjustment set, target covariate values, interactions, multivariate test, adjusted outcome contrasts and confidence intervals, missing-data handling, and follow-up multiplicity strategy.
Multivariate analysis of variance (MANOVA)Group comparisonsTests group differences in a set of numeric outcomes considered jointly.
What it means
The null concerns equality of population mean vectors. Statistics such as Pillai trace and Wilks lambda use the relationship between hypothesis and residual covariance matrices; an omnibus result does not identify which outcome or group contrast differs.
When it fits
Use when a joint outcome profile is substantively meaningful, such as reading and numeracy scores from the same students.
When to reconsider
Do not combine unrelated outcomes merely to obtain one p-value. Very many outcomes relative to available observations, redundant measures, and covariance instability can make the analysis unreliable.
Assumptions and checks
Independent observational units, an estimable residual covariance matrix, and for classical inference multivariate normal errors with common covariance across groups. Robustness depends on design, sample size, and the statistic.
Research example
Illustration: three curricula are compared jointly on reading and numeracy. A significant multivariate result supports a difference in the mean profile, but does not establish improvement in both outcomes.
What to report
State the outcomes, group sizes, multivariate statistic, approximate F and degrees of freedom when used, p-value, and planned outcome-specific or contrast follow-ups with uncertainty and multiplicity control.
Negative binomial regressionRegressionModels expected counts when their variability exceeds what a Poisson model allows.
What it means
RSearch fits an NB2 log-link model using MASS 7.3-65. Its variance is the mean plus the squared mean divided by the estimated dispersion parameter.
When it fits
A nonnegative integer outcome has overdispersion compatible with the NB2 model.
When to reconsider
Do not use this release for exposure-adjusted rates, zero-inflated mixtures or clustered data. Those extensions are not implemented.
Assumptions and checks
Independent observations, a suitable conditional mean and NB2 distribution, numeric predictors and a defensible complete-case analysis.
Research example
Model the number of service requests as a function of customer tenure when observation periods are equal.
What to report
Report analytic N, dispersion, coefficients or incidence-rate ratios, confidence intervals, model diagnostics and the absence of causal identification.
Read the source
NormalityModel assumptionsA normal distribution is a symmetric bell-shaped model; the relevant assumption concerns a specific part of an analysis.
What it means
In a Gaussian linear model, errors conditional on predictors are normal. A paired t-test concerns the distribution of within-pair differences. Marginal predictors or pooled outcomes need not all be normal.
When it fits
Examine relevant residuals or differences using plots, study design and sensitivity analysis.
When to reconsider
Do not automatically switch tests because one raw column fails Shapiro-Wilk, or take a nonsignificant normality test as proof.
Assumptions and checks
Finite-sample inference may require normal errors; asymptotic robustness also depends on independence, tails, sample size and balance.
Research example
Illustration: two groups centered at 40 and 70 can produce a bimodal pooled histogram while each group has approximately Gaussian errors.
What to report
State what distribution was assessed, diagnostics used, influential departures and any robust alternative.
Odds ratio (OR)Model interpretationCompares event odds, where odds equal the event probability divided by the probability of no event.
What it means
For two groups, OR = [p1/(1-p1)]/[p0/(1-p0)]. In logistic regression, exponentiating an appropriate coefficient gives a conditional odds ratio for the stated predictor contrast.
When it fits
Use for suitable binary-outcome analyses, explaining the reference category and adjustment set.
When to reconsider
Do not describe OR = 2 as twice the probability. Adjusted and unadjusted ORs can differ even without confounding.
Assumptions and checks
The outcome coding, predictor contrast and model must be specified; interactions can make an OR depend on other predictors.
Research example
Illustration: probabilities .50 and .25 give odds 1 and 1/3, so OR = 3 while the risk ratio is 2.
What to report
Give OR with its interval, reference group, predictor units and covariates; include absolute probabilities when identifiable.
One-sided and two-sided testsStatistical foundationsA two-sided test detects departures in either direction; a one-sided test targets one prespecified direction.
What it means
The alternative determines which outcomes count against the null and how rejection probability is allocated. The one-sided null generally includes the boundary and effects in the opposite direction.
When it fits
Use a directional test only when the scientific question and decision rule justify it before outcomes are examined.
When to reconsider
Do not select the direction after seeing the estimate or halve a two-sided p-value automatically without checking the test and direction.
Assumptions and checks
The specified model and prespecified alternative must match the reported p-value and interval.
Research example
Illustration: testing whether tutoring changes scores in either direction is two-sided; a prespecified test of an increase does not establish evidence for a decrease if scores fall.
What to report
State the alternative, direction, alpha and scientific rationale; label one-sided confidence bounds explicitly.
One-way ANOVAGroup comparisonsTests whether the population means differ across levels of one grouping factor.
What it means
The omnibus null states that all group means are equal; rejection indicates that at least one differs, without identifying the pair. Classical ANOVA pools within-group variance, while Welch ANOVA allows unequal variances.
When it fits
Use for a numeric outcome across independent groups, especially three or more, when group mean differences address the question.
When to reconsider
Do not interpret the omnibus p-value as showing that every group differs. Repeated observations need a model for dependence, and unadjusted testing of every pair inflates multiplicity.
Assumptions and checks
Independent observational units and a suitable within-group error model. Classical F inference assumes normal errors and equal group variances; unequal variances and unequal sample sizes can be particularly problematic together.
Research example
Illustration: three teaching groups average 61, 65, and 74 points. An omnibus test asks whether all population means could be equal; planned contrasts or adjusted follow-up comparisons locate relevant differences.
What to report
Report group summaries, the ANOVA variant, F and both degrees of freedom, p-value, an appropriately defined effect size, and confidence intervals for substantive contrasts.
Read the source
Ordinal logistic regressionRegressionModels an ordered categorical outcome using predictors and common odds ratios across category cut points.
What it means
The qualified ordinal 2025.12-29 implementation fits cumulative logit models with flexible thresholds and numeric linear predictors. It reports asymptotic Wald uncertainty and Holm-adjusted nominal-effect screens for proportional odds.
When it fits
Independent observations have three to seven ordered integer outcome categories with adequate support and numeric predictors.
When to reconsider
Do not treat unordered labels as ordered outcomes, ignore proportional-odds violations or interpret odds ratios as risk ratios.
Assumptions and checks
Independent observations, a defensible ascending category order, proportional odds, adequate category support and linear effects on the cumulative logit scale.
Research example
Model an ordered satisfaction score from numeric baseline measurements while estimating conditional higher-category odds ratios.
What to report
Report category order and counts, complete observations, predictors, link, threshold policy, coefficients, odds ratios and Wald intervals, proportional-odds diagnostics and missingness limitations.
Read the source
p-valueStatistical foundationsDescribes how incompatible the observed result is with a specified null model, using the test statistic you chose.
What it means
The tail probability, under the null hypothesis and the other model assumptions, of a test statistic at least as extreme as observed. Extremeness depends on the test and its stated alternative.
When it fits
Use alongside the estimated effect, its uncertainty, the design and the full analysis plan.
When to reconsider
Do not read p as the probability the null is true, the probability the result happened by chance, or the chance a finding will replicate.
Assumptions and checks
The test model, sampling structure and handling of repeated testing must justify the reference distribution.
Research example
Illustration: p = .03 for a prespecified difference test means that equally or more extreme test statistics occur with probability .03 under the null model. It does not mean a 97% probability of a real effect.
What to report
Name the test, report p and the effect with an interval, and disclose alternative analyses and selection decisions.
Paired-samples t-testGroup comparisonsTests the average within-pair change or difference when observations are meaningfully matched.
What it means
A paired t-test is a one-sample t-test on the pairwise differences. The standard error is based on the variability of those differences, not on treating the two measurements as independent groups.
When it fits
Use for two measurements on each participant or prespecified matched pairs when the mean difference is the target.
When to reconsider
Do not pair unrelated rows because their positions match. A before-and-after difference without a suitable comparison design does not isolate a treatment effect from time trends or regression to the mean.
Assumptions and checks
Pairs are independent of other pairs, with correct identifiers and consistent subtraction order. Normality concerns the differences for exact small-sample inference, not the two raw score distributions separately.
Research example
Illustration: five people change by 2, 4, 3, 5, and 1 points. Their mean change is 3 points; a paired test evaluates this change using the spread of these five differences.
What to report
Report the number of complete pairs, mean and standard deviation of differences, confidence interval, t, degrees of freedom, p-value, and how incomplete pairs were handled.
Read the source
Parametric and nonparametric methodsModel assumptionsThese labels describe how a method specifies the population or data distribution, not whether it has assumptions.
What it means
Parametric models use a distributional family described by finitely many parameters. Nonparametric methods leave aspects of that distribution unrestricted; some use ranks, but many do not.
When it fits
Choose the estimand and design first, then a method with credible assumptions and useful precision.
When to reconsider
Do not replace a mean test with a rank test and assume the research question remains identical. Rank tests are not generally tests of medians without additional conditions.
Assumptions and checks
Independence, exchangeability, continuity, symmetry or common distributional shapes may be needed, depending on the method.
Research example
Illustration: comparing average waiting time asks a different question from whether observations in one clinic tend to rank above those in another.
What to report
Name the method and target quantity, explaining assumptions and any exact or approximate inference.
Partial least squares regressionPredictive modelsCompresses correlated predictors into outcome-informed components and uses those components for prediction.
What it means
PLS constructs components using relationships between predictor and response matrices, unlike principal components regression, which constructs components without the outcome. Component count controls model complexity.
When it fits
Many correlated measurements, such as spectral wavelengths, are available for predicting one or more numeric outcomes.
When to reconsider
Do not interpret predictive components as validated psychological constructs or equate PLS regression with a fully specified PLS structural equation model.
Assumptions and checks
Relevant predictor-response structure, suitable scaling, and representative validation data. Component selection and preprocessing must occur inside training folds to prevent information leakage.
Research example
Predict protein concentration from 200 correlated wavelengths measured in 120 specimens, selecting the number of components by cross-validation and evaluating untouched specimens.
What to report
State scaling, algorithm, selected component count, tuning procedure, held-out prediction errors and the interpretation limits of loadings and coefficients.
Read the source
Partial least squares structural equation modelingFactor analysis and SEMModels relationships among constructs represented through weighted indicator composites, with the measurement approach specified explicitly.
What it means
Traditional PLS path modeling estimates component scores and structural relationships. When the scientific target is a reflective common-factor model, uncorrected composite estimates and consistent PLS estimates target different quantities.
When it fits
The research aim and measurement theory justify composite-based modeling, with prediction or structural relationships evaluated using suitable validation.
When to reconsider
Do not choose PLS solely because the sample is small or assume it removes identification, power or confounding problems. PLS regression alone is not a complete SEM analysis.
Assumptions and checks
Explicit distinction between reflective factors and formative or composite constructs, credible structural specification, sufficient information and an appropriate uncertainty procedure.
Research example
Model how a composite of access resources relates to a reflective satisfaction construct. Evaluate the two measurement specifications differently and assess predictive performance outside estimation data.
What to report
State construct definitions, indicator modes, algorithm and corrections, weights or loadings, uncertainty, structural estimates and predictive validation.
Path analysisFactor analysis and SEMRepresents a system of relationships among observed variables using connected regression equations.
What it means
Path analysis is SEM without a latent measurement model; specified paths and residual covariances imply restrictions on the data. Separate regressions reproduce some recursive models, but do not generally replace estimation of every path model.
When it fits
A theory specifies several connected relationships among adequately measured observed variables.
When to reconsider
Do not assume measured scores are error-free in substance, or use a saturated model’s perfect fit as evidence that the proposed direction is correct.
Assumptions and checks
An identifiable system, appropriate treatment of residual dependence and exogeneity, and suitable functional and distributional assumptions. Directed arrows require design-based justification for causal interpretation.
Research example
Model whether supervisor support predicts workload and whether both predict later exhaustion. Report direct and indirect associations while acknowledging possible unmeasured workplace differences.
What to report
Show all paths, residual covariances and exclusions; give unstandardized estimates and intervals, degrees of freedom and fit where the model imposes testable restrictions.
Pearson correlationCorrelationMeasures the direction and strength of a linear relationship between two numeric variables.
What it means
Pearson r is the covariance divided by the product of the standard deviations, ranging from -1 to 1. A sample r of zero indicates no linear association in that sample, but can coexist with a strong curved relationship and does not establish zero population correlation.
When it fits
Use when a scatterplot supports a roughly linear relationship and the numeric scales have meaningful distances.
When to reconsider
Do not treat correlation as a causal effect, agreement between instruments, or an adequate summary of a U-shaped pattern. Investigate influential observations before interpreting r.
Assumptions and checks
Pairs must be independent across observational units for the usual test. Its classical small-sample inference uses a bivariate normal model; computing the descriptive coefficient does not itself require normality.
Research example
Illustration: among 80 students, r = 0.45 between study hours and exam scores indicates positive linear association. It does not show that adding an hour would cause a particular score increase.
What to report
Report r, the number of complete pairs, a confidence interval, the inferential method and p-value when relevant, and a scatterplot. Explain missing-value and outlier handling.
Read the source
Point-biserial correlationCorrelationSummarizes the association between a binary group indicator and a numeric outcome.
What it means
It is Pearson correlation when one variable is coded 0 and 1. Its sign depends on which group is coded 1, and its magnitude depends on both the mean difference and the group proportions.
When it fits
Use to describe association between a genuine binary characteristic and a continuous score, such as program participation and test performance.
When to reconsider
Do not split a continuous variable into two groups merely to calculate this coefficient. Do not mistake it for the biserial correlation, which invokes an underlying continuous latent variable.
Assumptions and checks
Independent observations and nonconstant variables are needed. Classical significance inference corresponds to the equal-variance two-group t framework; unequal variances may warrant a Welch mean comparison instead.
Research example
Illustration: code scholarship receipt as 1 and nonreceipt as 0. A positive coefficient means recipients have higher average scores in this sample, without establishing that the scholarship caused the difference.
What to report
State the binary coding, group sizes, group means and standard deviations, coefficient, uncertainty, and inference method.
Read the source
Power and sample-size analysisPower and effect sizesPower analysis estimates how often a planned test would detect a specified effect under stated conditions.
What it means
Power is 1 - beta for a particular alternative, sample size, variance and decision rule. Solving for sample size requires a scientifically justified effect, not merely an observed pilot effect assumed certain.
When it fits
Use before collecting data to compare feasible designs or determine the smallest effect a fixed design can detect.
When to reconsider
Do not use observed-effect post hoc power to explain a completed nonsignificant result; inspect its estimate and interval instead.
Assumptions and checks
Match allocation, pairing, clustering, attrition, multiplicity and analysis model to the actual study; uncertain inputs require sensitivity scenarios.
Research example
Plan to detect a 5-point difference with SD 12, two-sided alpha .05 and 90% power; recalculate for SD 15 and anticipated attrition.
What to report
Report target effect, its rationale, all design inputs, calculation method and sensitivity range.
Read the source
Propensity score matchingCausal inferencePropensity score matching compares treated and untreated observations with similar measured probabilities of receiving treatment.
What it means
The propensity score is P(treatment = 1 | pretreatment covariates). Matching constructs a comparison sample; the target effect and its population depend on the matching design and any units discarded.
When it fits
Use in observational studies with defensible confounder measurement and sufficient overlap, after defining the causal effect of interest.
When to reconsider
Do not match on post-treatment variables or assume matching removes unmeasured confounding; matching alone does not guarantee balance.
Assumptions and checks
Identification requires conditional exchangeability, positivity and consistency, including a defensible no-interference assumption; assess covariate balance after matching.
Research example
Match scholarship recipients to nonrecipients on prior attainment and household resources, then assess the effect on retention in the matched target population.
What to report
State estimand, covariates, score model, distance or caliper, replacement, exclusions, balance, matched-design uncertainty and sensitivity to hidden bias.
Read the source
Quantile regressionRegressionModels a chosen conditional percentile of an outcome, such as the median.
What it means
Linear quantile regression minimizes asymmetrically weighted absolute residuals. RSearch supports five selected quantiles with quantreg 6.1 and nid coefficient uncertainty.
When it fits
The question concerns a conditional percentile rather than the conditional mean.
When to reconsider
Do not interpret a conditional quantile as a fixed subgroup of participants or read pseudo-R1 as ordinary explained variance.
Assumptions and checks
A suitable linear conditional quantile model and independent observations. The qualified scope uses complete cases and numeric predictors.
Research example
Estimate how the conditional median of waiting time changes with workload.
What to report
Report the selected quantile, complete-case count, coefficients, uncertainty method, confidence intervals and model limitations.
Read the source
R-squared (R²)Model interpretationSummarizes how much better a regression fits observed outcomes than a stated baseline.
What it means
For ordinary least squares with an intercept, R² = 1 - SSE/SST, comparing squared residuals with deviations from the sample mean. Adjusted R² penalizes model size. No-intercept and pseudo-R² definitions differ.
When it fits
Use as one descriptive fit measure alongside residual checks and prediction error.
When to reconsider
Do not interpret R² as the percentage caused by predictors, a guarantee of correct specification, or proof of performance on new data.
Assumptions and checks
Identify the model, baseline and evaluation sample. Training R² usually increases when predictors are added.
Research example
Illustration: SSE = 400 and SST = 1,000 give R² = .60. This describes the fitted sample, not 60% causal explanation.
What to report
State R² definition, adjusted R² if relevant, and separate training fit from external validation.
Random forest (binary classification)RegressionCombines many randomized trees to predict whether a binary target is class 0 or class 1.
What it means
The qualified randomForest 4.7-1.2 implementation uses 300 trees and fixed split settings. Stratified five-fold held-out predictions provide internal validation, while final-forest out-of-bag permutation importance is descriptive.
When it fits
Independent observations have a binary 0/1 target and numeric predictors available before the outcome occurs.
When to reconsider
Do not use this release for clustered or temporal validation, multiclass targets, categorical predictors or causal inference.
Assumptions and checks
Validation rows are exchangeable with training rows. Prior preprocessing and variable selection must not leak target information. Vote fractions need not be calibrated probabilities.
Research example
Estimate held-out discrimination and classification accuracy for a binary outcome using numeric baseline measurements.
What to report
Report target coding, sample and class counts, seed, fixed settings, fold design, held-out metrics, importance limitations and the need for external validation.
Read the source
Repeated-measures ANOVAGroup comparisonsCompares mean responses across conditions or occasions measured on the same observational units.
What it means
The analysis accounts for the shared participant rather than treating every record as independent. The usual univariate test for a within-subject factor with more than two levels assumes sphericity: equal variances of the relevant pairwise differences.
When it fits
Use for a planned set of repeated conditions or visits with a numeric response and a design compatible with the chosen ANOVA model.
When to reconsider
Do not use an ordinary independent-groups ANOVA on repeated rows. Irregular timing, substantial missing visits, or complex clustering can make a mixed-effects model more suitable.
Assumptions and checks
Participants are independent of one another; within-participant dependence is modelled. Normal error assumptions support classical inference; assess sphericity and use an appropriate correction or alternative when needed.
Research example
Illustration: the same 40 students complete a scale at baseline, week 4, and week 8. The test asks whether the population mean is constant across those occasions, not whether all individual trajectories are identical.
What to report
Report complete participant counts, occasion summaries, within-subject contrasts, F and corrected degrees of freedom, any Greenhouse-Geisser or Huynh-Feldt correction, uncertainty, and missing-data handling.
Reproducibility and replicabilityResearch practiceReproducibility checks whether the same data and procedure yield consistent results; replication asks whether new evidence supports a consistent finding.
What it means
The National Academies distinguishes computational reproducibility using the original inputs from replicability across studies collecting their own data. Disciplines sometimes use different terminology, so define the intended meaning.
When it fits
Preserve analysis provenance and evaluate how conclusions behave across independent studies or settings.
When to reconsider
Do not treat a reproducible calculation as proof that the design, assumptions or substantive claim are correct.
Assumptions and checks
Re-execution requires sufficiently specified data versions, preprocessing, software, settings and randomness; replication also requires comparable target questions.
Research example
Illustration: rerunning the same 200-person analysis from documented inputs checks reproducibility. Recruiting another 200 people to examine the same target effect investigates replicability.
What to report
Record data and code versions, transformations, software, seeds where relevant, deviations and access conditions.
Ridge regressionPredictive modelsStabilizes regression predictions by shrinking coefficients, particularly when predictors overlap strongly.
What it means
Ridge adds a penalty proportional to the sum of squared coefficient values. It trades bias for reduced variability and generally retains predictors rather than selecting an exact subset.
When it fits
Many correlated predictors may each carry some predictive information and unstable ordinary least-squares coefficients are a concern.
When to reconsider
Do not describe shrunken coefficients as unbiased causal estimates or expect ridge to remove irrelevant variables automatically.
Assumptions and checks
Appropriate functional form, thoughtfully scaled predictors and an independently evaluated tuning procedure. Shrinkage does not repair omitted confounding or data leakage.
Research example
Predict reading scores from several strongly correlated language measures. Tune the penalty on training folds and compare held-out error with ordinary regression.
What to report
State preprocessing, whether the intercept was penalized, selected penalty, validation error and the practical interpretation of coefficient shrinkage.
Shapiro-Wilk normality testModel diagnosticsTests whether a sample shows evidence against a normal-distribution model.
What it means
The W statistic compares the ordered observations with the pattern expected under normality. A large p-value is insufficient evidence to reject that model, not proof that the data or a later analysis meet all assumptions.
When it fits
Use as one diagnostic alongside a Q-Q plot and an assessment of the departures relevant to the planned analysis.
When to reconsider
Do not automatically switch methods at p = 0.05. Small samples may hide important departures, while large samples may detect harmless deviations; fitted regression residuals also do not behave exactly like independent raw observations.
Assumptions and checks
The ordinary reference test concerns independent observations from a continuous distribution. For a paired t-test inspect differences; for regression assess the conditional error pattern rather than requiring every predictor to be normal.
Research example
Illustration: a sample of 2,000 observations yields p < 0.001 with only mild tail curvature. Whether this matters depends on the estimand and robustness of the intended method, not the normality p-value alone.
What to report
Give W, sample size, p-value, what was tested, and graphical evidence. Explain the analytical consequence of any departure rather than simply stating that normality passed or failed.
Read the source
Significance level (alpha)Statistical foundationsA prespecified limit on the false-positive rate of a testing procedure when its null hypothesis holds.
What it means
A level-alpha test has Type I error probability no greater than alpha across the null parameter space under its assumptions. Approximate procedures may only achieve nominal control asymptotically.
When it fits
Set alpha before examining outcomes, considering the consequences of errors and the family of hypotheses.
When to reconsider
Do not change .05 to .10 after seeing p = .08. Alpha is not the probability that a particular rejected hypothesis is true.
Assumptions and checks
Error control requires the stated test, sampling and stopping procedure. Repeated looks may require a sequential design.
Research example
Illustration: a valid alpha = .01 procedure has a long-run false rejection probability of at most 1% under its null, not a guarantee of exactly one error in every 100 studies.
What to report
State alpha, whether it applies to one test or a family, and any multiplicity or interim-analysis adjustment.
SkewnessDescriptive statisticsSkewness summarizes asymmetry in a distribution.
What it means
Moment skewness is the third central moment divided by SD cubed. Positive values generally indicate a longer right tail; sample bias corrections differ between implementations.
When it fits
Use with plots to investigate asymmetric outcomes or residuals.
When to reconsider
Do not infer normality from skewness near zero.
Assumptions and checks
Population moment skewness requires a finite third moment and nonzero variance.
Research example
Most of 100 waiting times may lie below 10 minutes, with several near 60, creating right skew.
What to report
Give the coefficient convention, n and a histogram or quantile plot.
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Spearman rank correlationCorrelationMeasures whether higher values of one variable generally accompany higher or lower values of another.
What it means
Spearman rho is Pearson correlation applied to ranks, with tied observations assigned appropriate ranks. It targets monotonic association, which can be curved, rather than a linear change in the original measurement units.
When it fits
Use for ordinal measures or a monotonic numeric relationship where ranks answer the research question.
When to reconsider
Do not use it as a universal remedy for nonnormal data: a U-shaped relationship can still yield rho near zero. It does not remove clustering, measurement error, or confounding.
Assumptions and checks
Observational pairs should be independent across units. Ties and sample size affect inference; exact, permutation, and asymptotic procedures need to be distinguished.
Research example
Illustration: 60 participants rank service quality from 1 to 5 and report waiting time. A negative rho means longer waits generally accompany lower ratings, without assuming equal spacing between rating categories.
What to report
Report rho, complete-pair count, tie handling, p-value method, and an interval when supported. Describe the direction using the actual variable coding.
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Standard deviationDescriptive statisticsStandard deviation describes how widely observations vary around their mean, in the original units.
What it means
The usual sample SD is the square root of sum((x - x-bar)^2)/(n - 1). It measures variation among observations, whereas a standard error measures uncertainty in an estimate.
When it fits
Use with means to describe numeric outcomes when their spread is meaningful.
When to reconsider
Do not replace SD with standard error to make observations appear less variable, or presume that every distribution follows the 68% rule.
Assumptions and checks
At least two observed values are needed for sample SD; extreme values can dominate it.
Research example
Scores of 8, 10 and 12 have mean 10 and sample SD 2 points.
What to report
State SD, units, n and whether sample or population scaling was used.
Read the source
Standard deviation versus standard errorStatistical foundationsSD describes variation among observations; SE describes sampling uncertainty in an estimated statistic.
What it means
An SE estimates the standard deviation of an estimator across repeated samples. For an independent, identically distributed sample with finite variance, the usual estimated SE of the mean is s divided by square root of n.
When it fits
Use SD to describe participants or measurements, and SE or an interval to describe precision of an estimate.
When to reconsider
Do not substitute SE for SD to make a sample look less variable, or apply s/sqrt(n) to clustered observations without adjustment.
Assumptions and checks
SE formulas must respect the sampling and dependence structure. SD itself does not require normally distributed data.
Research example
Illustration: 100 independent scores with SD = 15 have estimated mean SE = 1.5. Participants still vary by SD = 15.
What to report
Label error bars and table quantities explicitly; provide sample size and units.
Statistical and practical significanceModel interpretationA statistical signal and a useful real-world improvement answer different questions.
What it means
Statistical significance concerns a testing rule relative to sampling variation. Practical importance concerns the effect magnitude, uncertainty, costs, benefits and a context-specific decision threshold.
When it fits
Define meaningful differences before analysis and compare the entire interval with those benchmarks.
When to reconsider
Do not label an effect important solely because p < .05 or unimportant solely because p > .05.
Assumptions and checks
The benchmark must be justified for the target population and outcome, rather than selected to favor the result.
Research example
Illustration: an estimated 0.2-point improvement with CI [0.1, 0.3] may exclude zero while falling well below a justified 3-point educational benchmark.
What to report
Report magnitude and uncertainty in interpretable units and explain the benchmark and trade-offs.
Statistical powerStatistical foundationsThe chance a planned test will reject its null at a particular true effect.
What it means
Power is a function of the alternative parameter values, design, sample size, variability and decision rule. It is not a single universal property of a dataset.
When it fits
Plan a study around a scientifically meaningful effect and examine plausible alternatives for variability and attrition.
When to reconsider
Do not treat .80 as universally adequate or calculate observed-effect power as new evidence after an inconclusive test.
Assumptions and checks
The calculation must match the eventual analysis, including pairing, clustering, sidedness and missingness allowances.
Research example
Illustration: a design with 90% power for a 5-point difference may have much less power for a 2-point difference. Both belong on a planning sensitivity curve.
What to report
State target effect, alpha, target power, variance assumptions, allocation, design and resulting sample size.
Support vector machinesPredictive modelsLearns a predictive boundary or regression function while controlling how much complexity and error are tolerated.
What it means
Classification SVMs optimize a margin-based objective; kernels allow nonlinear boundaries through implicit feature representations. Support vector regression uses an error-insensitive region with penalties outside it.
When it fits
Prediction from numerous features is the aim and kernel and regularization settings can be tuned with adequate validation.
When to reconsider
Do not interpret an SVM decision score as a calibrated probability or expect a nonlinear kernel to explain causal mechanisms.
Assumptions and checks
Training examples represent the intended application, feature scales are handled appropriately, and tuning occurs without test-set leakage. Performance can be sensitive to kernel parameters.
Research example
Classify research abstracts into two topics using text features. Select regularization on training folds, then evaluate a separate set of independently labeled abstracts.
What to report
Give kernel, scaling, regularization and other tuning settings, validation design, class-specific errors, and any separate probability-calibration procedure.
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Synthetic controlCausal inferenceSynthetic control builds a weighted comparison from untreated units to approximate a treated unit before an intervention.
What it means
The conventional method selects nonnegative donor weights summing to one to fit pretreatment predictors and outcome history. The post-intervention gap is interpreted through assumptions supporting that synthetic unit as the untreated counterfactual.
When it fits
Use for one or a few aggregate interventions with a credible donor pool and substantial pretreatment history.
When to reconsider
Do not use contaminated donors or claim a credible effect when pretreatment fit is poor; a good fit alone cannot exclude an unrelated concurrent shock.
Assumptions and checks
Require no relevant anticipation or spillovers, suitable untreated donors and stability of the relationship generating the counterfactual.
Research example
Construct a weighted comparison for one university introducing a fee policy from unaffected universities, then compare enrollment gaps before and after introduction.
What to report
Give donor eligibility, weights, predictor choices, pretreatment fit, intervention timing, placebo comparisons and sensitivity to donors; explain limits of inferential claims.
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Two-way ANOVAGroup comparisonsExamines how a numeric outcome varies with two categorical factors and their interaction.
What it means
A factorial model represents the two main effects and, when included, an interaction describing how one factor contrast changes across levels of the other. In unbalanced designs, coding, weighting, and the tested sums of squares determine the precise hypotheses.
When it fits
Use for a crossed design such as two teaching approaches at three class sizes, with replication in the relevant combinations.
When to reconsider
Do not interpret an average main effect as constant across all settings when an important interaction is present. Empty cells can make intended contrasts nonestimable.
Assumptions and checks
Independent errors, an appropriate mean structure, and common error variance for the classical model; normal errors support exact F inference. Repeated or clustered designs require corresponding extensions.
Research example
Illustration: approach B exceeds A by 8 points in small classes but by 1 point in large classes. The 7-point difference between these contrasts describes an interaction.
What to report
Report cell sizes and means, model terms, contrast coding and sums-of-squares convention, interaction estimates and intervals, and justified follow-up contrasts with multiplicity handling.
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Type I errorStatistical foundationsRejecting a null hypothesis that is actually true.
What it means
This is an error of the testing decision. Its probability is evaluated under the null and controlled by a valid significance-level procedure.
When it fits
Consider its consequences when planning confirmatory claims.
When to reconsider
Do not equate it with the fraction of published positive findings that are false.
Assumptions and checks
The null, decision rule and data-generating assumptions must be specified.
Research example
Illustration: declaring two teaching approaches different when their population mean outcomes are equal is a Type I error for that mean-difference hypothesis.
What to report
Describe the decision rule and error rate being controlled.
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Type II errorStatistical foundationsFailing to reject a null hypothesis when a specified alternative is true.
What it means
Its probability, beta, varies with the actual alternative; power is 1 minus beta.
When it fits
Evaluate missed-effect costs during study planning.
When to reconsider
A nonsignificant result does not establish equivalence or absence of a meaningful effect.
Assumptions and checks
Specify an alternative effect, variability, sample size and test.
Research example
Illustration: with power .80 for a 5-point difference, beta is .20 at that difference.
What to report
Report the planned alternative and power; interpret inconclusive results through their intervals.
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VarianceDescriptive statisticsVariance summarizes squared deviations from the mean.
What it means
Sample variance uses sum((x - x-bar)^2)/(n - 1), with squared measurement units. Under independent, identically distributed sampling with finite variance, this estimates population variance without bias.
When it fits
Use in variance decomposition, model estimation and comparing variability on the same scale.
When to reconsider
Do not compare raw variances across differently scaled variables as if the larger value showed greater relative variability.
Assumptions and checks
The unbiasedness statement requires the stated sampling conditions; a finite sample calculation does not establish finite population variance.
Research example
Scores of 8, 10 and 12 have sample variance 4 points squared and SD 2 points.
What to report
Specify denominator, scale, missingness handling and any weighting.
Variance inflation factor (VIF)Model diagnosticsIndicates how strongly overlap among predictors increases uncertainty in an individual regression coefficient.
What it means
For an ordinary single-degree-of-freedom predictor, VIF = 1 / (1 - R-j-squared), where R-j-squared comes from regressing that predictor on the others. Generalized VIFs handle multi-parameter terms and require attention to their degrees-of-freedom scaling.
When it fits
Use to investigate unstable coefficient estimates or wide intervals when predictors convey overlapping information.
When to reconsider
Do not delete a confounder or a required main effect solely because VIF exceeds 5 or 10. Collinearity affects precision, and its importance depends on the estimand, design, sample size, and model terms.
Assumptions and checks
VIF describes a specified design matrix, so coding, interactions, and included terms matter. A finite VIF does not establish causal identification, good prediction, or an adequate outcome model.
Research example
Illustration: if one predictor has R-j-squared = 0.80 against the others, its VIF is 5. Under the corresponding linear-model comparison, its coefficient standard error is inflated by approximately the square root of 5.
What to report
Report the model terms, VIF or adjusted GVIF convention, affected coefficients and intervals, and the substantive reason for retaining, combining, or removing predictors.
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Wilcoxon signed-rank testRank-based testsUses the directions and ranked magnitudes of paired differences to assess a central shift.
What it means
The signed-rank procedure tests a symmetry-and-location null for the distribution of differences. Under symmetry the location coincides with the median; without that condition it is not a general test solely of a zero median.
When it fits
Use for two related measurements when differences have meaningful magnitudes and a symmetric location model is plausible.
When to reconsider
Do not choose it solely because raw scores fail a normality test. Strongly asymmetric differences may call for a sign test or another method aligned with the intended estimand.
Assumptions and checks
Pairs are independent of other pairs, correctly matched, and measured so that absolute differences can be ranked meaningfully. Zeros, ties, and numerical rounding affect the statistic and its reference distribution.
Research example
Illustration: participants complete a fatigue scale before and after a shift. Rank the absolute within-person changes, retain their signs, and interpret the result in the chosen subtraction direction.
What to report
Report complete pairs, zero and tie handling, statistic convention, p-value method, and a location estimate with interval when supported. A Hodges-Lehmann estimate is a pseudomedian, not automatically the sample median change.
Read the source
Examples are illustrative. Choose a method from your research question, study design, and data structure; a diagnostic test alone cannot establish that a method is appropriate.