Worked statistical analyses for researchers, each with the R script and output shown.https://celeus.ai/blog/https://celeus.ai/blog/confounding-variables/https://celeus.ai/blog/confounding-variables/What a confounding variable is, how adjusting for one changes an odds ratio, and which variables to leave out of the model, worked on the birthwt data.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/correlation-matrix-in-r/https://celeus.ai/blog/correlation-matrix-in-r/Build a correlation matrix in R with a p-value and confidence interval for every pair, adjust for testing many pairs at once, and read the table honestly.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/correlation-vs-causation/https://celeus.ai/blog/correlation-vs-causation/Why a strong correlation does not show cause: a seeded simulation where a third variable drives two measures, and a checklist of what a design needs.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/cox-regression-in-r/https://celeus.ai/blog/cox-regression-in-r/Fit a Cox model in R with coxph() on the survival lung data, read the output line by line, run cox.zph for proportional hazards, and see the script.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/hazard-ratio/https://celeus.ai/blog/hazard-ratio/What a hazard ratio of 2 does and does not mean, how the reference group and the unit change it, and how to report it with a confidence interval.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/inter-rater-reliability/https://celeus.ai/blog/inter-rater-reliability/Cohen's kappa computed by hand and in R on a two-rater table, weighted kappa for ordered grades, and why high agreement can still give a low kappa.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/multicollinearity-vif/https://celeus.ai/blog/multicollinearity-vif/When correlated predictors matter: how to compute and read the variance inflation factor, what it does to confidence intervals, and when to leave it alone.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/regression-to-the-mean/https://celeus.ai/blog/regression-to-the-mean/Why extreme baseline scores drift back toward the average with no treatment at all, shown in a seeded simulation, and how a control group and ANCOVA remove it.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/residual-standard-error/https://celeus.ai/blog/residual-standard-error/Where the residual standard error sits in R's lm() output, what it means in your outcome's units, and how to read it alongside the residual plot.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/sensitivity-vs-specificity/https://celeus.ai/blog/sensitivity-vs-specificity/How sensitivity, specificity and predictive values are estimated from a 2x2 table, why prevalence moves the predictive values, and how to report intervals.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/standard-error-vs-margin-of-error/https://celeus.ai/blog/standard-error-vs-margin-of-error/Standard deviation, variance, standard error, margin of error and IQR: what each measures, how they connect, and which to report, with a worked example.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/type-i-and-type-ii-errors/https://celeus.ai/blog/type-i-and-type-ii-errors/Null and alternative hypotheses, alpha, type I and type II errors and power, shown by counting false positives and misses across 2,000 seeded studies.Fri, 23 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/anova-explained/https://celeus.ai/blog/anova-explained/What an ANOVA test does, how to choose one-way, Welch, two-way or repeated-measures ANOVA, and a worked Welch example in R with eta squared and its interval.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/breusch-pagan-test/https://celeus.ai/blog/breusch-pagan-test/How the Breusch-Pagan test checks a regression for non-constant spread, how to read it in R, and why large samples flag changes that may not matter.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/chi-square-test-explained/https://celeus.ai/blog/chi-square-test-explained/What the chi-square test of independence computes, how to read expected counts, Cramer's V and residuals, and a worked R example on real survey data.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/cohens-d-vs-hedges-g/https://celeus.ai/blog/cohens-d-vs-hedges-g/Which standardised mean difference to report, Cohen's d, Hedges' g or Glass's delta, by sample size, unequal spread and purpose, with a worked R example.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/confidence-interval-conclusion/https://celeus.ai/blog/confidence-interval-conclusion/What a correct confidence interval conclusion says, built from one worked example in R, with common incorrect wordings side by side and how to fix each.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/cramers-v/https://celeus.ai/blog/cramers-v/What Cramer's V and phi measure, why their small, medium and large cut-offs depend on table size, the bias-corrected V, and a worked example in R.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/data-cleaning-checklist/https://celeus.ai/blog/data-cleaning-checklist/An ordered data cleaning checklist for statistical analysis: structure, codes, duplicates, ranges, units, missing data and outliers, run on R's airquality data.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/dissertation-statistics/https://celeus.ai/blog/dissertation-statistics/Choose the statistics for a thesis from the study design: five common designs, the test, the assumption checks and the effect size to report for each.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/dunns-test/https://celeus.ai/blog/dunns-test/What Dunn's test compares after Kruskal-Wallis, how Holm and Bonferroni adjust its p-values, and a worked R example on six insecticides with effect sizes.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/effect-size-mann-whitney/https://celeus.ai/blog/effect-size-mann-whitney/The rank-biserial correlation for Mann-Whitney and Wilcoxon tests: what it measures, how its interval works, why it is not Z over root N, how to report it.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/effect-sizes-explained/https://celeus.ai/blog/effect-sizes-explained/What an effect size is, which one goes with which test, the SD each version of Cohen's d uses, and a worked R example with d and its 95% interval.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/eta-squared/https://celeus.ai/blog/eta-squared/Eta squared, partial eta squared and omega squared differ only in the denominator. A two-way ANOVA in R shows which to report and the interval to give.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/f-test-vs-t-test/https://celeus.ai/blog/f-test-vs-t-test/Which F-test you mean decides the answer: the ANOVA F on two groups equals Student's t squared, while the variance-ratio F compares spreads, not means.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/heteroscedasticity/https://celeus.ai/blog/heteroscedasticity/What non-constant spread does to a regression, how to see it in a residual plot, and when a log transform or robust standard errors is the right fix.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/homogeneity-of-variance/https://celeus.ai/blog/homogeneity-of-variance/What unequal group variances do to Student's t and classic ANOVA, why Welch beats testing first, and a worked Welch ANOVA in R with effect sizes.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/how-to-report-a-chi-square-test/https://celeus.ai/blog/how-to-report-a-chi-square-test/A complete results paragraph for a chi-square test of independence: counts, expected counts, the statistic, Cramer's V and the odds ratio with intervals.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/how-to-report-a-t-test/https://celeus.ai/blog/how-to-report-a-t-test/Fill-in APA 7 sentences for Welch, Student, paired and one-sample t-tests, the formatting rule behind each symbol, and a worked example from a real R run.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/independence-assumption/https://celeus.ai/blog/independence-assumption/What repeated or clustered observations do to standard errors, shown on real tree growth data in R with and without a random intercept, and how to spot it.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/iqr-rule-upper-lower-fence/https://celeus.ai/blog/iqr-rule-upper-lower-fence/How to compute Tukey's upper and lower fences by hand and with R's boxplot.stats(), why the two can disagree, and what to do with a value beyond a fence.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/likert-scale-data-analysis/https://celeus.ai/blog/likert-scale-data-analysis/How to analyse Likert data in a thesis: when a single item needs an ordinal method, when a summed scale can take a t-test, and a worked ordinal regression in R.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/linear-regression-assumptions/https://celeus.ai/blog/linear-regression-assumptions/The four assumptions of linear regression, the R check and the fix for each, and which failures bias the estimates and which only distort the standard errors.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/logistic-regression-assumptions/https://celeus.ai/blog/logistic-regression-assumptions/Which assumptions logistic regression really makes, which ones it never made, and how to check each one in R, worked through on the birthwt data.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/one-sample-t-test/https://celeus.ai/blog/one-sample-t-test/When a one-sample t-test fits, where the reference value has to come from, and a worked R example on Newcomb's 1882 light measurements with Cohen's d.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/outliers-before-statistical-test/https://celeus.ai/blog/outliers-before-statistical-test/A decision rule for outliers before a test: correct errors, exclude by a written criterion, keep genuine extremes, with seeded data run with and without them.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/p-value-from-test-statistic/https://celeus.ai/blog/p-value-from-test-statistic/Turn a reported t, z, F or chi-square statistic into its p-value by hand and with R's pt(), pnorm(), pf() and pchisq(), compared with full test output.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/p-values-and-confidence-intervals/https://celeus.ai/blog/p-values-and-confidence-intervals/What a p value and a confidence interval each tell you, how they relate, how to read p = 0.06, and how to report both, with a worked example in R.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/paired-vs-unpaired-t-test/https://celeus.ai/blog/paired-vs-unpaired-t-test/How to tell whether your two sets of measurements are paired or independent, the designs people misclassify, and what ignoring the pairing costs, shown in R.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/parametric-test-assumptions/https://celeus.ai/blog/parametric-test-assumptions/The assumptions behind t-tests, ANOVA and regression, how to check each without the pre-test traps, and what to do when one fails, with a worked Welch ANOVA.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/post-hoc-tests-after-anova/https://celeus.ai/blog/post-hoc-tests-after-anova/Which post hoc test follows which ANOVA: Tukey after the classic F-test, Games-Howell after Welch, Dunn after Kruskal-Wallis, all three run on one dataset.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/pretest-posttest-design/https://celeus.ai/blog/pretest-posttest-design/How to analyse a two-group pretest posttest design: change scores, ANCOVA or a mixed model, when each one fits, and a worked ANCOVA example in R.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/regression-analysis-guide/https://celeus.ai/blog/regression-analysis-guide/Choosing a regression model from your outcome type (linear, logistic, Poisson, negative binomial or ordinal), with a multiple linear regression worked in R.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/reporting-statistics-apa-style/https://celeus.ai/blog/reporting-statistics-apa-style/The APA 7 rules for reporting statistics (italics, leading zeros, p values, degrees of freedom, intervals) and a fill-in template for each common test.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/survival-analysis-guide/https://celeus.ai/blog/survival-analysis-guide/Survival analysis in R on public lung cancer data: censoring, Kaplan-Meier, log-rank, Cox regression with its proportional hazards check, and AFT.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/t-tests-explained/https://celeus.ai/blog/t-tests-explained/What a t-test does, how to choose the one-sample, paired, Student or Welch version, and a worked Welch example in R with Cohen's d and its interval.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/test-for-normality-before-t-test/https://celeus.ai/blog/test-for-normality-before-t-test/Should you test normality before a t-test? Usually not. Why a Shapiro-Wilk pre-test misleads at small and large n, what to check instead, and an R example.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/two-sample-t-test/https://celeus.ai/blog/two-sample-t-test/The two-sample t-test formula worked by hand and confirmed in R, its assumptions, and Student's versus Welch's version on the same unequal-sized groups.Thu, 22 Oct 2026 00:00:00 GMThttps://celeus.ai/blog/welchs-t-test/https://celeus.ai/blog/welchs-t-test/Why Welch's t-test is the safer default for two groups, how its degrees of freedom work, and a worked R example where group sizes and spreads both differ.Thu, 22 Oct 2026 00:00:00 GMT