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
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.
What Is a P Value? Definition, Misinterpretations, and How to Read One
What a p value actually measures, the four most common misinterpretations (it is not the probability the null is true, not the probability of chance, not replication probability), the ASA’s 2016 principles, and correct-use guidance.
T-Test: One-Sample, Independent, and Paired — When to Use Each
A complete guide to the t-test: one-sample, independent-samples, and paired designs, Student’s vs. Welch’s, assumptions, effect size, APA reporting, and when to use ANOVA or a non-parametric test instead.
Regression Analysis: Assumptions, Interpretation, and How to Report It
A practical guide to choosing between linear, logistic, and other regression models, checking each model’s assumptions, interpreting coefficients and R² without overclaiming, and reporting results in a methods/results section reviewers will accept.
