Research Methods & Statistics

Quantitative Analysis Methods

This sub-cluster covers the statistical techniques used to analyze numeric data once it has been collected: regression analysis (linear, logistic, and their extensions), analysis of variance (ANOVA) and its variants, t-tests, confidence intervals, p-values and the ongoing debate over their interpretation and misuse, and multilevel/hierarchical models for nested data. Pages here explain what each technique assumes, what it can and cannot show, and how results should be reported — the practical, applied layer that sits between raw data and a results section a peer reviewer will accept.

Guides

ANOVA in Python with statsmodels: anova_lm, Type III Contrasts, and the AnovaRM Limits

statsmodels will return a clean-looking F table for a model that answers a different question than you asked. The three failure points are a missing C(), anova_lm’s Type I default, and AnovaRM’s silent lack of a sphericity correction.

Kruskal-Wallis Test in R: kruskal.test(), Dunn’s Post-Hoc, and the Assumption You Probably Didn’t Violate

A practical guide to running the Kruskal-Wallis test in R: checking which ANOVA assumption actually failed, the defaults kruskal.test() applies without telling you, and why three R packages give three different Dunn’s test p-values from the same data.

Shapiro-Wilk Test in R: shapiro.test(), Its Sample-Size Limits, and When Not to Use It

How to run shapiro.test() in R, why it stops at n = 5000, why W and n must be reported together, and why using the p-value to pick between a t-test and a rank test inflates conditional Type I error.

Wilcoxon Test in R: wilcox.test(), the W Statistic, and the Silent Exact/Approximate Switch

One R function runs both Wilcoxon tests, and three of its defaults routinely trip people up: the statistic it names W is not the one SPSS reports, it drops to the normal approximation without being asked, and its confidence interval does not estimate a difference in medians.

Mixed-Effects Models: Multilevel, Hierarchical, and What the Random Part Does

Mixed-effects, multilevel and hierarchical models are one model under three names. Clustered data break OLS standard errors, and the random structure you choose feeds back into fixed-effect inference.

ROC Curves and AUC: What They Plot, What AUC Means, and Where It Misleads

What an ROC curve plots (sensitivity vs. 1-specificity across every threshold), what AUC actually means, common benchmark bands, and the real limits of AUC — class imbalance, calibration, and clinical utility.

Log and Other Data Transformations: When to Use Them and What They Cost

When to use log, square root, reciprocal, and Box-Cox transformations for skewed research data, how to interpret a log-transformed coefficient correctly, the zero-value problem, and when a non-parametric test or GLM is the more honest choice instead.

Winsorizing vs. Trimming: Choosing How to Handle Outliers

Winsorizing replaces extreme values with a percentile cutoff instead of deleting them, preserving sample size while capping their influence. This guide covers how it differs from trimming and outright deletion, when each is defensible, the researcher-degrees-of-freedom risk of picking a method after seeing the result, and what to disclose when reporting it.

Variance Inflation Factor and Multicollinearity: What It Actually Breaks

VIF is 1/(1-R2) from regressing each predictor on the others. Multicollinearity inflates standard errors and destabilises individual coefficients, but it does not bias them or harm prediction.

Events per Variable and the Minimum Sample Size for Regression

The EPV 10 rule for regression sample size, where it came from, and the Riley et al. (2019) performance-based criteria that replaced it for prediction models.

MCAR, MAR and MNAR: Diagnosing Your Missing Data

Little’s MCAR test can rule out MCAR, but distinguishing MAR from MNAR is structurally untestable from the data alone. A reproducible simulation shows exactly where the test’s blind spot is, and how to route each diagnosis to a valid method.

Bayesian Sample Size Determination

Three genuinely different Bayesian answers to how much data is enough: precision-based estimation, assurance (power averaged over the prior), and sequential Bayes factor stopping rules that remove the fixed-N requirement entirely.

Prior Sensitivity Analysis

A prior sensitivity analysis re-runs a Bayesian model under a small set of alternative, defensible priors to check whether the conclusion that matters actually changes — the step a reviewer looks for when a paper reports a weakly informative prior without saying what would happen under a different one.

RERI: Relative Excess Risk Due to Interaction

How to compute RERI, AP, and the synergy index for additive interaction between two exposures, from a regression model, with a worked example and reporting guidance.

Competing Risks: Cumulative Incidence and the Fine-Gray Model

Competing risks bias the standard Kaplan-Meier estimate upward. This guide covers the cumulative incidence function, the cause-specific vs. subdistribution hazard distinction, and how to fit a Fine-Gray model in R, Stata, and SAS.

Recurrent Event Analysis: Andersen-Gill, PWP, and WLW Models

Recurrent events — readmissions, exacerbations, infections, device failures — need a different Cox extension than a single-event survival model. This guide covers Andersen-Gill, Prentice-Williams-Peterson, and Wei-Lin-Weissfeld models, when to pick each, robust variance, and count-based and non-parametric alternatives.

ROPE: The Region of Practical Equivalence

The region of practical equivalence (ROPE) is a Bayesian decision rule: compare a posterior’s highest density interval to a pre-specified band of negligible effect sizes to decide reject, accept, or undecided.

Genome-Wide Significance Thresholds in GWAS: Why p < 5×10⁻⁸?

The conventional p < 5×10⁻⁸ threshold GWAS uses for genome-wide significance, where the number comes from, the suggestive-significance threshold, and when a looser bar is defensible.

Simple Effects Analysis After a Significant Interaction

How to decompose a significant ANOVA interaction into simple effects: which factor to slice by, pooled vs. separate error terms, SPSS/R/Stata/SAS syntax, multiple-comparisons correction, effect size, and reporting.

Reading SPSS Output: The Tables That Matter

A general guide to the SPSS output every procedure shares: the Case Processing Summary, footnote letters, rounding to .000, and how to read a Sig. column correctly.

Pattern-Mixture Models for MNAR Sensitivity Analysis

How pattern-mixture models stratify subjects by missingness pattern and apply delta-adjustment to test whether a conclusion holds under missing-not-at-random (MNAR) assumptions, complementing tipping-point analysis with pattern-specific and arm-differential sensitivity checks, plus a fully reproducible worked simulation.

Diagnostic Odds Ratio: What It Combines, and What It Hides

A guide to the diagnostic odds ratio (DOR): the formula, its relationship to likelihood ratios, why it is prevalence-independent but not threshold-independent, and a reproducible simulation showing what a single summary number can hide about a test’s sensitivity/specificity balance.

Love Plots: Reading Covariate Balance After Matching or Weighting

Love plots are the standard graphic for propensity-score balance: SMD before/after adjustment, the <0.1 threshold convention, the variance-ratio check it misses, and a reproducible matching-vs-weighting simulation showing what an unbalanced covariate actually means.

The Breslow-Day Test: Checking Homogeneity Before You Pool Odds Ratios

The Breslow-Day test checks whether stratum-specific odds ratios are homogeneous enough to justify a single Mantel-Haenszel pooled estimate. Covers the test statistic, the Tarone correction, and two reproducible worked examples — one where the pooled odds ratio should not be trusted, one illustrating the low-power caveat with few strata.

How to Choose a Statistical Test: A Decision Tree

A navigable decision tree for picking the right statistical test from four questions — outcome type, group count, paired or independent, and whether parametric assumptions hold — routing to a full CASRAI guide for every test.

Quantitative Bias Analysis: Modeling Bias Instead of Naming It

How quantitative bias analysis (QBA) turns a suspected bias into a modeled, bias-adjusted estimate — simple, probabilistic and multidimensional bias analysis, plus a worked, script-verified misclassification correction.

Q-Values and the Positive False Discovery Rate: The FDR Analog of a P-Value

The q-value is the FDR analog of a p-value — the minimum false discovery rate at which a test is called significant. Defines it against the Benjamini-Hochberg adjusted p-value it’s routinely confused with, and runs a reproducible seeded simulation comparing Bonferroni, BH-FDR, and Storey’s q-value across four scenarios, including an honest low-power case where FDR’s advantage disappears.

Suppressor Variables in Regression: How They Work and How to Spot One

A worked, reproducible simulation shows a variable with almost no correlation to the outcome still raising another predictor’s coefficient and R-squared, plus a checklist for telling real suppression from a coding error.

Last Observation Carried Forward (LOCF): Why It’s Discouraged

LOCF carries a dropout’s last score forward as if their trajectory went flat. A seeded simulation shows how much bias that assumption introduces, even under pure random dropout, and why MMRM and multiple imputation replaced it as the regulatory default.

Targeted Maximum Likelihood Estimation (TMLE): The ML-Compatible Doubly Robust Estimator

TMLE is a doubly robust causal-effect estimator that updates the outcome model itself through a targeting step, so flexible machine-learning nuisance models still support valid inference. Includes a reproducible simulation comparing TMLE and AIPW under near-positivity stress.

Multiple Imputation: A Practical Protocol

A step-by-step protocol for running multiple imputation correctly: building the imputation model with auxiliary variables, matching each variable’s imputation method to its data type, choosing how many imputations is enough, and pooling with Rubin’s rules.

KMO and Bartlett’s Test: Is Your Data Factorable?

Bartlett’s test of sphericity and the KMO measure of sampling adequacy are the two checks before running EFA. Covers Kaiser’s interpretation bands, the per-variable MSA diagnostic most guides skip, and — via a reproducible simulation — what it means when Bartlett’s is significant only because n is large.

Frailty Models for Clustered Survival Data

Frailty models add a random-effect term to Cox regression to handle clustered or recurrent event times whose independence assumption would otherwise be violated. Covers shared vs. individual frailty, distribution choice, a reproducible Kendall’s-tau simulation, and how frailty compares to robust-variance (GEE-style) alternatives.

Interpreting Bayes Factors: Evidence Scales and What They Mean

The Jeffreys (1961) and Kass-Raftery (1995) scales for reading Bayes factor magnitude, a reproducible worked example, and why a Bayes factor — unlike a p value — can express evidence for the null hypothesis.

Directed Acyclic Graphs for Confounder Selection: Finding the Minimal Adjustment Set

How to turn a directed acyclic graph into an operational adjustment-set procedure (the backdoor criterion) instead of a diagramming convention — including a worked collider that must not be adjusted for, and a reproducible simulation showing why.

Fuzzy Regression Discontinuity: The Instrumental-Variables Interpretation

Fuzzy RDD explained: why imperfect compliance at the cutoff turns regression discontinuity into an instrumental-variables problem, what LATE it identifies, and how to check the first stage.

Games-Howell Test: The Post Hoc Test for Unequal Variances

Games-Howell replaces Tukey HSD as the all-pairwise post hoc test once Levene’s test flags unequal variances, computing a separate Welch-Satterthwaite degrees of freedom for every pairwise comparison instead of pooling variance across the whole design.

Mann-Whitney U Test Assumptions: What It Actually Requires

The Mann-Whitney U test isn’t assumption-free: it requires independent groups and ordinal/continuous data, just not normality. Covers what’s actually required, the equal-shape caveat for reporting a median difference vs. stochastic dominance, a reproducible worked example, and APA reporting.

Hayes’ PROCESS Macro: Model Numbers, Syntax, and Reporting

A lookup guide to Hayes’ PROCESS macro model numbers (1, 4, 6, 7, 8, 14, 15) mapped to their path diagrams, plus SPSS/SAS/R syntax and how to report PROCESS output in a manuscript.

The Sobel Test and Its Alternatives

How the Sobel test’s normal-theory formula for the mediated effect (ab) works, why its normality assumption fails for a skewed product term, the named Aroian/Goodman variants, and what a seeded simulation shows about the power lost versus bootstrapping.

Tipping-Point Analysis for Missing Outcome Data

How tipping-point sensitivity analysis stress-tests a study’s conclusion against its missing outcome data: systematically varying the assumed value for the missing subjects until the finding flips, and reporting how extreme that assumption would need to be — with a fully reproducible worked example.

Full Information Maximum Likelihood (FIML): Missing Data Without an Imputation Step

How FIML estimates parameters directly from all available raw data under the MAR assumption, with no separate imputation step, and when it beats (or loses to) multiple imputation.

Rubin’s Rules for Pooling Multiply Imputed Estimates

How to combine point estimates and standard errors across multiply imputed datasets using Rubin’s rules: the within-imputation and between-imputation variance components, the small-sample degrees-of-freedom adjustment, and a fully worked, reproducible pooling example.

ANCOVA Assumptions: Homogeneity of Regression Slopes

ANCOVA adds two assumptions to ANOVA’s standard set: linearity and homogeneity of regression slopes. This guide tests the slopes assumption with a reproducible worked example and covers what to do — moderated regression, simple slopes, Johnson-Neyman regions — when it fails.

Cohen’s f² Effect Size for Regression: Formula, Benchmarks, and How It Differs From Cohen’s d

Cohen’s f² is the regression-specific effect size, derived from R² (or R² change), with its own small/medium/large conventions distinct from Cohen’s d.

Autocorrelation in Time-Series Data: ACF, PACF, and What It Breaks

What autocorrelation is, why it violates the independence assumption, and how to read the ACF and PACF correlograms used to diagnose it — the concept behind the Durbin-Watson test and ARIMA modeling.

Cramér’s V: Effect Size for Categorical Association

Cramér’s V rescales the chi-square statistic so association strength can be compared across contingency tables of different sizes — something phi, which only works on 2×2 tables, can’t do. This guide covers the formula, a full worked calculation, and how to interpret the result using Cohen’s effect-size bands, which change with degrees of freedom.

Doubly Robust Estimation: The Guarantee, and Where It Breaks Down

Doubly robust estimation (AIPW) is consistent if either the propensity model or the outcome model is correctly specified, not both. A reproducible simulation shows it succeeding where IPW alone and regression alone fail from the same misspecification — and shows where the guarantee itself breaks down.

The Scheffé Test: Testing Complex Contrasts After ANOVA

How the Scheffé test differs from Tukey’s HSD: it tests any linear contrast among group means, not only pairwise ones, at the cost of being conservative for simple pairwise comparisons.

Instrumental Variables: The Three Assumptions and the Weak-Instrument F-Test

The three IV assumptions (relevance, exclusion restriction, exogeneity), what violates each, the first-stage F-statistic and the F>10 weak-instrument rule of thumb, 2SLS mechanics, and a reproducible Monte Carlo simulation showing weak-instrument bias toward OLS.

Verification Bias in Diagnostic Accuracy Studies

Verification (work-up) bias happens when the reference standard isn’t applied to all index-test-negative patients, inflating apparent sensitivity and deflating specificity. A seeded numeric simulation shows the size of the distortion and how the Begg-Greenes correction recovers it.

How to Interpret the F-Statistic in ANOVA and Regression

The F-statistic compares explained variance to unexplained variance the same way a t-statistic compares a signal to its noise. This guide works through the ratio in both its common forms, one-way ANOVA and simple regression, with fully computed worked examples showing where each number in the output comes from.

Restricted Mean Survival Time (RMST): The Hazard-Ratio Alternative When Proportional Hazards Fails

Restricted mean survival time (RMST) is the area under the Kaplan-Meier curve to a fixed time point — a model-free alternative to the hazard ratio when proportional hazards fails. Includes a worked, reproducible simulation where the pooled hazard ratio and RMST disagree.

The Mantel-Haenszel Method: Pooling Stratum-Specific Odds Ratios into One Confounder-Adjusted Estimate

The Mantel-Haenszel method pools stratum-specific odds ratios into a single confounder-adjusted estimate. This guide covers the formula, a worked stratified 2×2-table example contrasting the crude and pooled OR, the variance/CI, and the Breslow-Day homogeneity check that decides whether pooling is legitimate.

Accelerated Failure Time Models: The Time-Ratio Alternative to Cox

How accelerated failure time models parameterize covariates as time ratios rather than hazard ratios, when a violated proportional-hazards assumption makes AFT the better fit, and how to choose between Weibull, log-normal, and log-logistic.

G-Computation: Simulating Counterfactual Outcomes for Time-Varying Confounding

G-computation (the parametric g-formula) simulates counterfactual outcomes forward from a fitted outcome model to handle time-varying confounding, an alternative to IPTW-weighted marginal structural models — with a worked, reproducible comparison of both mechanisms on the same simulated data.

Repeated-Measures ANOVA Assumptions: Normality and Outliers in the Difference Scores

RM-ANOVA has two assumptions beyond sphericity that get little attention: normality and outlier-freedom of the pairwise difference scores between conditions. Here is how to check each and what to do when one fails.

How to Interpret the t-Statistic

A t-statistic is an estimate expressed relative to its own standard error, not a size. This guide covers the one formula behind every t-statistic, why a bigger t is not a bigger effect, and how to read it in a t-test and in a regression coefficient table.

Clustered Standard Errors: Naive vs. Cluster-Robust SEs

Clustering within classrooms, clinics, or repeated subjects violates the independence assumption behind classical standard errors. A seeded, reproducible simulation compares naive vs. cluster-robust SEs on the same data, then shows why cluster-robust corrections themselves under-perform with too few clusters.

Landmark Analysis: The Practical Fix for Immortal Time Bias

Landmark analysis fixes immortal time bias by fixing a landmark time, restricting to survivors, and classifying exposure at that point. Includes a worked before/after simulation and guidance on choosing the landmark.

Decision Curve Analysis: Net Benefit, Threshold Probability, and a Worked Example

Decision curve analysis evaluates whether acting on a prediction model beats treating everyone or no one, across a range of clinical decision thresholds — a question AUC and calibration cannot answer on their own.

Marginal Structural Models: Solving Time-Varying Confounding with IPTW

A time-varying confounder that is also affected by prior treatment breaks standard regression adjustment. Marginal structural models, fit via inverse probability of treatment weighting (IPTW), solve it — with a worked, reproducible numeric example.

Healthy Worker Effect: Hire vs. Survivor Bias and How to Control It

The healthy worker effect makes employed cohorts look healthier than the general population. This guide splits it into the healthy hire effect and the healthy worker survivor effect, and covers the study-design and analytic strategies (internal comparisons, active-worker restriction, g-methods) that address each.

The Phi Coefficient: Correlation for Two Binary Variables

How the phi coefficient measures correlation between two binary variables, its exact identity with chi-square and Pearson’s r, and why unequal marginal distributions cap its maximum value below 1.

How to Read a Correlation Matrix

A practical guide to reading a correlation matrix in a results table: what the diagonal and symmetry mean, how to read significance flags, how to spot multicollinearity risk before it reaches a regression, and how to tell a genuinely strong relationship from one that is only “significant” because the sample is large.

Standardized Mortality Ratio (SMR): Indirect Standardization, Worked Calculation, and the Comparability Trap

How the standardized mortality ratio (SMR) works: indirect standardization against a reference population’s age-specific rates, a fully worked calculation with a 95% confidence interval, and why two SMRs can’t be validly compared to each other directly.

Causal Mediation Analysis: Potential Outcomes, Natural Effects, and Sequential Ignorability

The formal causal-inference layer beneath applied mediation analysis: potential-outcomes definitions of the natural direct and indirect effects, the sequential-ignorability assumption, nonparametric identification, and sensitivity analysis for the untestable part of that assumption.

Robust (Heteroscedasticity-Consistent) Standard Errors: HC0 Through HC3

How HC0, HC1, HC2, and HC3 robust standard errors differ, why HC3 is usually preferred in small samples, and when they fix genuine heteroscedasticity versus mask a misspecified functional form.

Hosmer-Lemeshow Test and Logistic Model Calibration: Mechanics, Limits, and Alternatives

How the Hosmer-Lemeshow deciles-of-risk goodness-of-fit test works, a reproducible worked example, why it becomes oversensitive at large sample sizes, and what to report instead.

The Hausman Test: Fixed vs. Random Effects

How the Hausman test compares fixed- and random-effects panel estimators, what a significant result actually implies, and the limitations (heteroskedasticity, negative statistics, pretesting) that make “reject means use FE” too simple on its own.

Incidence Rate Ratio: Why It Isn’t a Risk Ratio (Worked Example)

An incidence rate ratio compares person-time-adjusted rates rather than a fixed-denominator risk ratio. This guide walks through a fully worked calculation showing why that distinction changes the result when follow-up duration varies, plus the Poisson-based confidence interval.

Calibration Plots for Prediction Models: Reading Slope, Intercept, and the Calibration Hierarchy

Discrimination (AUC) measures whether a model ranks cases correctly; calibration measures whether its predicted probabilities can be trusted at face value. A well-discriminating model can still be badly miscalibrated — here’s how to read a calibration plot, the calibration slope and intercept, and the four-level calibration hierarchy TRIPOD+AI reporting now expects.

Blocking in Experimental Design: RCBD Mechanics and When It Beats Simple Randomization

Blocking groups experimental units on a known nuisance source before treatments are assigned, then randomizes only within blocks — a design-stage control, not a post-hoc covariate adjustment. Covers RCBD mechanics, block sizing, the ANOVA change it obliges, and when it beats simple randomization for power.

How to Interpret T-Test Results

A t-test’s output is five numbers read together, not one. This guide walks the t-statistic, df, p-value, confidence interval, and effect size as a set, and flags the two most common over-interpretation traps.

Moderation Analysis: Building and Interpreting an Interaction Term

A step-by-step walkthrough of moderated regression: centering predictors, building and interpreting the interaction coefficient, and probing it with simple slopes and a simple-slopes plot.

Cochran’s Q Test: The Repeated-Measures Extension of McNemar’s Test

A guide to Cochran’s Q test for three or more related dichotomous outcomes: how it extends McNemar’s test, the formula and degrees of freedom, a worked contingency-table example, and the pairwise McNemar-with-Bonferroni post-hoc procedure a significant result requires.

The Tobit Model for Censored Outcomes: Censoring vs. Truncation, and Why OLS Gets It Wrong

A real floor or ceiling in the outcome itself — not missing data — is what makes Tobit the right tool. This guide separates censoring from truncation and works a numeric example showing exactly how much bias plain OLS produces on censored data.

Generalized Estimating Equations (GEE): Working Correlation and When to Use Them Instead of a Mixed Model

GEE as a population-averaged alternative to mixed models for correlated and clustered data: choosing a working correlation structure (exchangeable, AR(1), unstructured), why the robust sandwich estimator survives misspecifying it, and when a subject-specific mixed model is the better tool instead.

Hierarchical (Block-Entry) Regression Analysis

How to order predictor blocks by theory, run the ΔR² significance test, and report it — with a full worked example, vs. stepwise entry.

Confidence Interval for an Odds Ratio: Why the Bounds Aren’t Symmetric

An odds ratio’s 95% CI is computed on the log-odds scale, then exponentiated back — which is exactly why the reported bounds aren’t symmetric around the point estimate.

Dunnett’s Test: Comparing Treatment Groups to a Control

Dunnett’s test compares several treatment groups against a single shared control rather than every possible pair, which is what gives it more statistical power than Tukey HSD for many-to-one designs.

Levene’s Test for Equality of Variances

Levene’s test checks whether groups have equal variance, a precondition behind the t-test and ANOVA. Covers the absolute-deviation logic, why the Brown-Forsythe median-based version resists skew better than the original mean-based version, and why a significant result should route you to Welch’s correction rather than a data transformation.

Breusch-Pagan Test for Heteroscedasticity: Logic, Worked Example, and the Robust-SE Fix

The Breusch-Pagan test’s logic — regressing squared residuals on the predictors — a worked numeric example, and why heteroscedasticity-consistent standard errors, not a transformation, are the routine fix once it’s confirmed.

Youden’s J Index: ROC Threshold Selection, the Equal-Cost Assumption, and What to Use Instead

Youden’s J (sensitivity + specificity − 1) is the default ROC cutpoint rule. It assumes false positives and false negatives cost the same — here’s when that’s wrong, and the cost-weighted, closest-to-(0,1), and constrained-sensitivity alternatives to use instead.

How to Interpret an Odds Ratio: Odds, Not Probability

An odds ratio describes odds, not probability. The exact language that stays correct, the “X times more likely” trap that misstates it as a risk ratio, and a worked example where OR (3.5) and the true risk ratio (2.0) diverge.

Elastic Net Regression: The Alpha Parameter, Correlated Predictors, and Cross-Validation

How elastic net’s alpha parameter blends the LASSO and ridge penalties, why the resulting grouping effect handles correlated predictors better than LASSO alone, and how to tune alpha and lambda honestly with nested cross-validation.

Negative Binomial Regression: The Overdispersion Fix, NB1 vs. NB2, and a Worked Model Comparison

How negative binomial regression fixes Poisson overdispersion via its dispersion parameter, the NB1 vs. NB2 parameterizations, the formal likelihood-ratio test for whether you need it, and a full worked comparison against a Poisson fit on the same data.

Mauchly’s Test of Sphericity and the Greenhouse-Geisser Correction

The statistical theory behind sphericity, Mauchly’s test, and the Greenhouse-Geisser vs. Huynh-Feldt correction decision for repeated-measures ANOVA — software-agnostic and separate from the SPSS how-to.

Berkson’s Bias: How Hospital and Clinic Sampling Manufactures a Spurious Association

Berkson’s bias distorts associations in hospital- and clinic-based samples whenever the exposure and disease under study each independently affect the odds of being admitted. Covers Berkson’s 1946 hospital-data example, a worked numeric table, and the modern registry/EHR equivalents of the same sampling trap.

Path Analysis: Direct, Indirect, and Total Effects

Path analysis is structural equation modeling with a structural model but no measurement model. This guide works a full path diagram end to end: standardized path coefficients, the tracing rule for direct/indirect/total effects, and how to tell whether a recursive model is even testable.

Bayesian Hierarchical Models: Partial Pooling Explained (With a Worked Example)

Hierarchical (multilevel) Bayesian models use partial pooling to shrink each group’s estimate toward the overall mean by an amount set by its own sample size and the between-group variance — beating both no pooling and complete pooling.

Confidence Interval for a Proportion: Wald vs. Wilson vs. Clopper-Pearson

Why the Wald interval for a proportion fails near 0/1 or with small n, and when to use Wilson score, Clopper-Pearson, or Agresti-Coull instead.

The Friedman Test and Its Post-Hoc Procedures: Nemenyi vs. Wilcoxon-Bonferroni

The Friedman test is the nonparametric alternative to repeated-measures ANOVA for 3+ related conditions. A significant result only says conditions differ somewhere — the Nemenyi test and Wilcoxon signed-rank tests with a Bonferroni correction are how you find out which pairs.

How to Interpret R-Squared: What It Measures and What It Doesn’t

R-squared is the proportion of variance explained, not proof a model is correct, causal, or that any coefficient matters. The specific misinterpretation traps, and adjusted R-squared for multiple predictors.

Immortal Time Bias: How It Inflates Drug Effects and How to Correct It

How a period where the outcome cannot yet occur gets misclassified into the exposed group, illustrated with a worked pharmacoepidemiology cohort, and the time-varying-exposure, landmark-analysis, and prescription-time-distribution-matching fixes.

Permutation Tests: When They Beat Parametric Tests

Permutation tests build their own null distribution by reshuffling group labels instead of assuming a normal population — a worked example shows when that beats a t-test, and when the compute cost is worth paying.

Pooled Standard Deviation: Formula, Worked Example, and Why It Matters for Cohen’s d

How to calculate the pooled standard deviation correctly, why it’s weighted by degrees of freedom rather than averaged, and a worked example showing how using the wrong SD changes a Cohen’s d result.

Lead-Time Bias and Length-Time Bias in Screening

Lead-time bias moves the diagnosis date earlier without changing the death date; length-time bias skews screen-detected cases toward slower disease. Both inflate apparent screening survival — this guide separates them with worked timelines and covers the mortality-endpoint design that resists both.

Ordinal Logistic Regression and the Proportional-Odds Assumption

Ordinal outcomes need a model that uses their order without assuming equal spacing between categories. This guide covers when ordinal logistic regression beats multinomial logit, what the proportional-odds (parallel-lines) assumption actually claims, how to test it with the Brant test, and the partial-proportional-odds model to use when it fails.

The Log-Rank Test and Its Variants: When Equal Weighting Loses Power

The log-rank test compares Kaplan-Meier survival curves with equal weight at every event time — which is exactly why it loses power against crossing curves and non-proportional hazards, and when the Wilcoxon, Tarone-Ware, and Peto weighted variants take over.

Synthetic Control Method: Constructing a Weighted Counterfactual for a Single Treated Unit

A synthetic control builds a weighted composite of untreated donor units to serve as a counterfactual for a single treated unit. This guide covers donor-pool selection, weight construction, pre-treatment fit diagnostics, and placebo-test validation in space and time.

Collider Bias: How Controlling for a Shared Effect Fabricates an Association

Collider bias is the reverse of confounding: adjusting for a variable that is a common effect of your exposure and outcome creates a spurious association where none exists. Covers the DAG mechanics, Berkson’s paradox, the published birth weight paradox case, and how to tell a collider from a confounder before you add it to a model.

Granger Causality: What It Does and Does Not Show

Granger causality tests whether one time series improves prediction of another — a narrower claim than true causation. This guide covers the two main ways it misleads (shared driving factors, lagged reverse causality), the stationarity and lag-selection prerequisites, and a worked time-series example.

Attributable Risk and Attributable Fraction: Absolute vs. Proportional Measures of Exposure Impact

Attributable risk, attributable fraction, and population attributable fraction are four related but distinct measures, easily conflated. This guide separates the absolute from the proportional, the exposed-group measure from the population measure, and works one example through all four so the difference from relative risk is concrete.

ARIMA Models for Research Time Series: AR, I, MA, and Box-Jenkins Identification

A researcher-facing guide to ARIMA: what the AR, I, and MA components each correct for, how to test stationarity with ADF and KPSS before fitting, and how to read ACF/PACF correlograms through the Box-Jenkins identification-estimation-diagnostic loop.

Summarizing a Posterior Distribution: Point Estimates, Credible Intervals, and HDI vs. Equal-Tailed Intervals

A guide to summarizing a Bayesian posterior distribution: point estimates (mean, median, mode), credible intervals, and why HDI vs. equal-tailed intervals give different answers for a skewed posterior.

Likelihood Ratio Test for Nested Models

A complete guide to the likelihood ratio test for comparing nested maximum-likelihood models: why the nesting requirement invalidates non-nested comparisons, how the chi-square test statistic is built from the -2 log-likelihood difference, degrees of freedom, the boundary problem for variance components in mixed models, and when to use an LRT versus AIC or BIC.

Mediation Analysis: Methods and Reporting

A guide to testing mediation with bootstrapped indirect effects rather than the outdated Baron-and-Kenny causal-steps method, including Hayes’ PROCESS macro, sample-size planning, and how to report results.

Stepwise Regression: Why Reviewers Reject It (and What to Use Instead)

The specific statistical problems with stepwise variable selection – inflated Type I error from repeated testing, unstable model selection, and overstated R² – and the alternatives reviewers actually accept: theory-driven entry, hierarchical blocks, and LASSO.

Durbin-Watson Test for Autocorrelated Residuals: Reading the Statistic and the Inconclusive Zone

Autocorrelated residuals don’t bias OLS coefficients — they invalidate the standard errors. This guide covers reading the Durbin-Watson statistic’s 0–4 scale, the inconclusive dL/dU zone, the lagged-dependent-variable trap, and when the test applies (time-series and panel data) versus when it doesn’t (cross-sectional data).

McNemar’s Test: The Discordant-Cell Test for Paired Binary Data

A guide to McNemar’s test for paired binary outcomes: why a chi-square test of independence is the wrong tool for before/after and matched-pairs designs, the discordant-cell (b, c) logic, a worked 2×2 example, the continuity-corrected and exact-binomial versions, when to use Cochran’s Q or the Stuart-Maxwell test instead, and how to report the result.

Survival Analysis: Choosing Between Kaplan-Meier, Cox, and Parametric Models

A decision framework for choosing Kaplan-Meier, Cox regression, or a parametric survival model based on the research question you are actually asking, not a single-method tutorial.

Tukey HSD Test: When to Use It and How to Report It

Tukey’s HSD is the post hoc test for all pairwise ANOVA comparisons under equal variance — how it differs from Games-Howell and Bonferroni, a worked post-hoc table, and the correct APA reporting line.

Inverse Probability Weighting: When It Beats Propensity Score Matching

IPW reweights the full sample instead of discarding unmatched units the way propensity score matching does — a real advantage that comes with its own failure mode: a few units with propensity scores near 0 or 1 can receive enormous weights and dominate the estimate. This guide covers weight construction, stabilization, the extreme-weight diagnostics that catch the problem, and when IPTW is the better choice over matching for a given causal question.

Multinomial Logistic Regression: Setup and Interpretation

Multinomial logistic regression produces one full set of relative-risk-ratio coefficients per non-reference outcome category, not a single odds ratio. This guide covers reference-category selection, how RRR differs from the odds ratio binary logistic regression trains you to expect, the IIA assumption, and a worked three-category example.

Cook’s Distance and Influential Observations: Competing Thresholds, Leverage vs DFBETAS, and What to Do Next

The three Cook’s distance thresholds in circulation disagree with each other and none is what Cook proposed. A worked R demonstration separating outliers, high leverage and genuine influence, and the honest options for an influential point that is not a data error.

The Kolmogorov-Smirnov Test: One-Sample vs Two-Sample, and the Estimated-Parameter Trap

The one-sample Kolmogorov-Smirnov test and the two-sample test answer different questions and have different validity conditions. The one-sample version is only valid when the reference distribution is fully specified in advance; estimating its parameters from the same data makes the test extremely conservative, which is what the Lilliefors correction fixes.

Difference-in-Differences: Evidencing Parallel Trends and the Staggered-Adoption Problem

Parallel trends is a counterfactual assumption, not something a pre-trend plot can confirm. This guide sets out what each diagnostic actually establishes, and shows — in a seeded R simulation where parallel trends holds exactly and every true effect is positive — a two-way fixed effects regression returning a wrong-signed estimate that a modern group-time estimator recovers correctly.

Endogeneity: The Three Sources, and the Remedy That Matches Each

Endogeneity is three different problems – omitted-variable bias, simultaneity and measurement error – and the remedy that fixes one does nothing for the others. A source-by-source map of fixed effects, IV, control functions and selection models, with what each assumes and where propensity scores do not help.

The Wilcoxon Signed-Rank Test: Assumptions, Exact vs. Normal Approximation, and How to Report It

A decision guide to the Wilcoxon signed-rank test: what its null actually asserts (symmetry of the differences, not equal medians), why it is the paired t-test’s counterpart rather than Mann-Whitney’s, Wilcoxon vs Pratt zero handling, the exact vs normal-approximation rule, matched-pairs rank-biserial effect size, and the APA reporting line.

Net Reclassification Improvement (NRI): The Calculation, and the Case Against It

The NRI compares two risk prediction models by counting who moved in the right direction. This guide gives the arithmetic for both the category-based and category-free versions with a worked reclassification table, then sets out the published methodological case against the statistic — including simulations in which a marker with no predictive information at all produced a positive, statistically significant NRI.

Exploratory Factor Analysis: Extraction, Rotation, and How Many Factors to Retain

Exploratory factor analysis turns on three decisions usually made by accepting a default: extraction method, number of factors retained, and rotation. This guide makes each defensible — why parallel analysis should replace the eigenvalue-greater-than-one rule, why oblique rotation is the right default in social science, and why rotation changes interpretability but never model fit.

Structural Equation Modeling (SEM): Fit Indices, Cut-Offs and Model Evaluation

A judgment-layer guide to evaluating structural equation models: which fit indices to report, why the widely cited Hu and Bentler cut-offs differ from what that paper actually recommended, modification-index discipline, and how to justify sample size.

Cohen’s d Effect Size: Calculation & Interpretation Guide

How to calculate and report Cohen’s d: the pooled standard deviation formula, when the Hedges’ g correction is needed for small samples, what the small/medium/large benchmarks do and do not mean, and how APA style expects effect sizes to appear alongside p-values.

Poisson Distribution: Formula, Assumptions, and When to Use It for Count Data

How to recognize count data that fits a Poisson distribution, the formula and its single parameter (lambda), the four assumptions that must hold, worked examples, and when to switch to negative binomial or Poisson regression instead.

Intention-to-Treat Analysis: What It Protects Against and When Per-Protocol Misleads

Intention-to-treat analysis preserves the comparability randomization created; per-protocol analysis discards it by conditioning on adherence. A worked trial example, plus why non-inferiority trials invert the usual advice.

How to Read a Phylogenetic Tree: Nodes, Branch Lengths and Bootstrap Values

Nodes, branch lengths, tip order and the numbers on the branches (bootstrap values vs Bayesian posterior probabilities) explained with a worked interpretation example.

The Cox Proportional Hazards Model: Assumptions and How to Check Proportionality

What the proportional hazards assumption in a Cox model actually asserts, three standard ways to check it (log-log plots, Schoenfeld residuals, time-varying covariate tests), a worked interpretation of a Schoenfeld test output, and what to do when the assumption fails.

Mixed-Effects Models: Choosing Random vs. Fixed Effects for Nested Research Data

Choosing between fixed and random effects is determined by data structure, not preference. A decision table for nested, clustered, and repeated-measures data, plus a worked interpretation of real-looking model output.

How to Read a Kaplan-Meier Curve: Censoring Marks, Median Survival, and Risk Tables

How to correctly read a Kaplan-Meier survival curve: what the censoring tick marks mean, why the line is a step function, how to read off median survival, and why the numbers-at-risk table decides whether the tail of the curve is trustworthy.

How to Interpret a Hazard Ratio (and Why It Is Not a Risk Ratio)

A hazard ratio from a Cox proportional-hazards model is an instantaneous rate ratio, not a risk ratio. This guide walks through a worked example, a plain-language lookup table, and the proportional-hazards assumption check that determines whether a single HR is a valid summary at all.

Bonferroni Correction and Multiple Comparisons: When to Correct and When FDR Is Better

A lookup table for Bonferroni-adjusted alpha, a worked interpretation of a real-looking multiple-testing output, and the decision rule for when family-wise error control (Bonferroni/Holm) is right versus when false discovery rate control (Benjamini-Hochberg) is the better tool.

Propensity Score Matching: How It Works and What It Cannot Fix

Propensity score matching balances measured covariates between treated and untreated groups — it cannot address confounding from variables that were never measured. How the method works, how to read a covariate balance table, and what it cannot fix.

The Kruskal-Wallis Test: When to Use It Instead of ANOVA and How to Report It

A decision guide to the Kruskal-Wallis test: when it replaces a one-way ANOVA, what its assumptions actually require, a fully worked H-statistic and Dunn’s post-hoc example, and how to report both in a results section.

Multicollinearity and VIF in Regression: Detection and What to Do About It

A high VIF means a predictor’s coefficient is imprecise, not that the model is wrong. This guide covers what multicollinearity does and does not do, how VIF is calculated, why VIF > 5 or > 10 are conventions not rules, and a worked coefficient table showing the fix.

How to Read a Residual Plot: Checking Homoscedasticity and Regression Assumptions

Four annotated residual plot readings side by side — well-behaved, funnel, curved, and outlier-driven — and exactly what each pattern tells you about homoscedasticity and your regression assumptions.

How to Interpret Cohen’s d: Benchmarks, Hedges’ g, and Why 0.2/0.5/0.8 Is Field-Dependent

Cohen’s d of 0.2/0.5/0.8 are conventions, not rules. Field-specific benchmarks, a worked interpretation example, and when to report Hedges’ g instead.

Z-Score (Standard Score): Formula, Worked Example, and How to Read the Z-Table

How to calculate a z-score, what standardization does (and doesn’t) do to a distribution, and how to correctly read a standard normal (z) table, including a full table.

Statistical Significance: What the Verdict Means and Doesn’t Mean

A decision-framework guide to statistical significance: what the verdict asserts, how the alpha threshold is chosen, why sample size distorts it, the multiple-comparisons problem, and the research-integrity risks (p-hacking, HARKing) around the 0.05 line.

ANOVA (Analysis of Variance): One-Way, Two-Way, Assumptions, Kruskal-Wallis and Effect Size

ANOVA compares means across three or more groups in a single test. This guide covers the F-ratio and variance partitioning, a fully worked ANOVA table, one-way/two-way/repeated-measures/factorial variants, MANOVA and ANCOVA, assumptions and Welch’s ANOVA, the Kruskal-Wallis nonparametric alternative and Dunn’s test, post-hoc tests (Tukey, Bonferroni, Scheffe, Dunnett), effect size (eta-squared, partial eta-squared, omega-squared), and correct APA-style reporting.

Standard Deviation: Formula, Worked Example, and How to Interpret It

Standard deviation measures how far observations typically fall from the mean. This guide builds the formula step by step, explains population vs. sample calculation and Bessel’s correction, works a labelled example, and clarifies the difference between standard deviation and standard error.

Sampling Distribution: Definition, Standard Error & CLT

What a sampling distribution is, how it differs from a population or sample distribution, and why standard error and the Central Limit Theorem make statistical inference possible.

Absolute Risk Reduction, Relative Risk Reduction, and Number Needed to Treat (NNT)

A worked example walking through absolute risk reduction, relative risk reduction, NNT, NNH, odds ratio vs. risk ratio, confidence intervals, and CONSORT reporting requirements.

Monotonic Relationships in Statistics: What They Are and Why They Matter

A monotonic relationship is one where two variables consistently move in the same direction (or consistently in opposite directions) without necessarily doing so at a constant rate. Every linear relationship is monotonic, but most monotonic relationships are not linear — a distinction that determines whether Pearson’s r or a rank-based statistic like Spearman’s rho is the right tool.

Markov Chain Monte Carlo (MCMC): What It Is and How to Read the Diagnostics

What MCMC does, how Metropolis-Hastings, Gibbs sampling, and Hamiltonian Monte Carlo/NUTS work, and the diagnostics (R-hat, ESS, trace plots, divergent transitions) that show whether a Bayesian model actually converged.

Geometric Mean: Formula, Uses, and When to Use It Instead of the Average

The geometric mean is the correct average for growth rates, ratios, and log-normally distributed data — this guide covers the formula, when it beats the arithmetic mean, the geometric standard deviation, a common back-transformation error, and the related harmonic mean.

Relative Frequency and Frequency Distributions: Definitions, Formulas, and Reporting

How to calculate and report relative frequency, frequency distributions, cumulative frequency, and relative cumulative frequency — including the empirical-probability link via the law of large numbers, class-interval trade-offs for continuous data, and row/column/total percentages in contingency tables.

Coefficient of Variation (CV): Formula, Examples, and Limitations

The coefficient of variation (CV = SD / mean) explained: worked examples, why it fails for interval-scale or near-zero data, its role in assay precision, geometric CV, and how it compares to SD, variance, and IQR.

Logistic Regression (the Logit Model): Interpretation, Diagnostics, and Reporting

A guide to logistic regression (the logit model): the logit link, interpreting log-odds and odds ratios correctly, maximum-likelihood fitting, diagnostics including separation, ROC/calibration, variants, and reporting.

Cumulative Distribution Function (CDF): Definition, Properties, and How to Read One

A plain guide to the cumulative distribution function (CDF): what F(x) = P(X ≤ x) means, its four defining properties, how it relates to the PDF/PMF, and how researchers use the empirical CDF, percentiles, K-S tests, Q-Q plots, and survival analysis.

Prevalence vs. Incidence: Definitions, Formulas, and How They Relate

Prevalence counts existing cases; incidence counts new ones. This guide defines both, works through P ≈ I × duration, and covers risk ratio, rate ratio, odds ratio, and attributable risk.

Spurious Correlation: A Guide to Spurious Relationships

What a spurious relationship actually is, and the mechanisms that produce false statistical associations: confounding, selection bias and collider stratification, reverse causation, chance, Simpson’s paradox, time-series artefacts, and measurement artefacts — plus how to investigate one and the research-integrity risks of undisclosed data dredging.

Degrees of Freedom in Statistics: What df Means and How to Calculate It

What degrees of freedom (df) means in statistics, why sample variance divides by n-1 (Bessel’s correction), the df formula for every common test, and why df is a useful check for reporting errors.

Sensitivity vs. Specificity: The 2×2 Table, PPV/NPV, Likelihood Ratios, and ROC Curves

A complete guide to sensitivity and specificity in diagnostic and screening research: how they are calculated from the 2×2 table, why they trade off against each other, why predictive values depend on prevalence while sensitivity/specificity do not, likelihood ratios, ROC/AUC, diagnostic study design biases, and the STARD reporting guideline.

Skewness: Left-Skewed vs. Right-Skewed Distributions Explained

Skewness measures how asymmetric a distribution is. This guide explains left-skewed vs. right-skewed (the skew is named for the tail, not the bulk), the mean/median/mode diagnostic, how to measure skewness, and how to handle skewed data in analysis.

Interquartile Range & Five-Number Summary: How to Calculate and Use Them

How to calculate the five-number summary and interquartile range step by step, use the 1.5×IQR rule to flag outliers, build a box plot, and report IQR correctly.

Mean Absolute Deviation: Formula, Worked Example, and How It Differs from Standard Deviation

Mean absolute deviation (average deviation from the mean) explained with a worked example, plus how it differs from standard deviation and the commonly confused median absolute deviation.

Causal Analysis: A Guide to Causal Inference

Causal analysis (causal inference) asks whether an outcome would have differed had the exposure differed, not just whether two variables move together. This guide covers the potential outcomes framework, the Bradford Hill viewpoints, DAGs (confounders, colliders, mediators), study designs ordered by causal strength (RCTs, difference-in-differences, regression discontinuity, instrumental variables, matching, propensity scores), the Table 2 fallacy, E-value sensitivity analysis, and Mendelian randomization.

Confirmatory Factor Analysis (CFA): A Complete Guide

Confirmatory factor analysis tests a pre-specified measurement model against new data — unlike exploratory factor analysis, which discovers structure. This guide covers specification, identification, estimation, fit indices, modification-index risk, and measurement invariance.

Normality of Distribution: How to Check the Normal Distribution Assumption in Research Data

What the normal distribution is, why the Central Limit Theorem makes most explanations of it misleading, and how to actually assess normality using Q-Q plots, Shapiro-Wilk, and skewness/kurtosis before choosing a parametric or non-parametric analysis.

Descriptive Statistics: Central Tendency, Dispersion, Shape, and How to Report Them

A complete guide to descriptive statistics: mean, median, and mode; range, IQR, variance, SD, and coefficient of variation; skewness and kurtosis; percentiles and z-scores; which statistic fits which measurement scale; outlier detection; and APA reporting conventions.

Histograms for Research Data: What They Show and How to Build One

A histogram plots how a continuous variable is distributed by binning values into ranges. This guide covers how histograms differ from bar charts, how bin-width choice changes apparent shape, how to read distribution shape, and how histograms compare to density plots, box plots, violin plots, and Q-Q plots — plus how to build one in R, Python, SPSS, and Excel.

Correlation Coefficient: What It Measures, Pearson vs. Spearman, and How to Report It

A correlation coefficient is a single number, always between -1 and +1, that summarizes the direction and strength of a linear (Pearson) or monotonic (Spearman, Kendall) relationship between two variables. This guide covers which coefficient to use, how to interpret magnitude honestly, and why you must plot your data before trusting any r value.

Chi-Square Test: Independence, Goodness-of-Fit, Assumptions, and How to Report It

A complete guide to the chi-square test: the test of independence vs. goodness-of-fit, observed vs. expected frequencies, assumptions (and Fisher’s exact test / Yates’ correction when they fail), effect size (phi, Cramér’s V, odds ratio), McNemar’s test for paired data, post-hoc residual analysis, APA reporting, and syntax for R, Python, and SPSS — including the SPSS Crosstabs dialog’s row/column/total percentage options and adjusted standardized residuals.