📖 How to Use This Guidebook: Content is now organized by Analysis Goal — Comparison, Correlation, Prediction, and Classification — matching the way researchers actually frame research questions. Each parametric test is paired with its non-parametric alternative. Technical assumptions have been corrected to specify that normality and homoscedasticity apply to residuals, not raw variables. New sections cover the n = 30 myth, VIF thresholds, and effect size interpretation. All code blocks now use a high-contrast light theme for full readability on RPubs and RStudio Viewer.
Before choosing any statistical test, pass through these four gates:
| Gate | Question | Action if Violated |
|---|---|---|
| Measurement Scale | What is the scale of the DV? (Nominal / Ordinal / Interval / Ratio) | Reclassify; choose appropriate test family |
| Independence | Are observations independent of each other? | Use multilevel or mixed models if clustered |
| Missing Data | Is missingness MCAR / MAR / MNAR? | Impute (MI or FIML); avoid listwise deletion |
| Sample Size | Is \(N\) adequate for the chosen test and expected effect size? | Run a priori power analysis; target \(1-\beta \geq .80\) |
Critical correction: Normality and homoscedasticity assumptions in parametric tests apply to residuals (errors) — not to the raw independent variables or raw dependent variable. Always diagnose residual distributions, never raw scores.
| Tool | Best For | Limitation |
|---|---|---|
| Shapiro–Wilk test | \(N < 50\); sensitive, formal test | At \(n > 100\), trivially rejects normality due to over-sensitivity — \(p\)-value loses diagnostic value |
| Kolmogorov–Smirnov | \(N \geq 50\) | Less powerful than Shapiro–Wilk; requires estimated parameters correction (Lilliefors variant) |
| Q-Q Plot (visual) | Preferred for \(n > 100\) | Requires judgment; not a formal test |
| Histogram + density | Any \(N\); useful for skew/kurtosis | Qualitative only |
Rule: For \(n \leq 100\), report Shapiro–Wilk \(W\) and \(p\)-value alongside a Q-Q plot. For \(n > 100\), rely on Q-Q plot inspection — a non-significant Shapiro–Wilk is no longer a reliable normality guarantee, and a significant one may flag trivial departures.
| Test | Use In | What It Tests |
|---|---|---|
| Levene’s Test | Group comparison designs (t-test, ANOVA) | Equality of variances across groups |
| Bartlett’s Test | Group designs with confirmed normality | Equality of variances — more powerful but less robust to non-normality |
| Breusch–Pagan Test | Regression models | Systematic relationship between residual variance and fitted values |
| White Test | Regression models | More general heteroscedasticity; no normality assumption required |
| Residual vs. Fitted Plot | Regression (visual) | Fan-shaped pattern signals heteroscedasticity |
Context rule: Use Levene’s Test for ANOVA-family designs. Use Breusch–Pagan (or White’s test) specifically for regression models. Using Levene’s for regression is a category error — it tests group variances, not the regression error structure.
The oft-cited rule “n ≥ 30 is sufficient for the Central Limit Theorem” is an oversimplification that causes systematic errors in practice.
| Distribution Type | Minimum Recommended \(N\) | Rationale |
|---|---|---|
| Approximately normal, symmetric | \(n \geq 20\) per group | CLT converges rapidly |
| Mild skew (skewness \(\lvert s \rvert < 1\)) | \(n \geq 30\) per group | Standard guidance applies |
| Moderate-to-heavy skew (\(\lvert s \rvert \geq 1\)) | \(n \geq 100\) per group | CLT convergence is substantially slower |
| Fat-tailed distributions (excess kurtosis \(> 3\)) | \(n \geq 100\)–\(200\) per group | Heavy tails create persistent sampling instability |
| Bimodal distributions | Use mixture models | CLT does not resolve structural bimodality |
The myth: \(n = 30\) is a minimum floor for mild departures from normality — it is not a universal pass for all distributions. For social science data (often skewed, bounded Likert composites), \(n \geq 100\) is the safer working rule before treating the sampling distribution of \(\bar{X}\) as approximately normal.
Variance Inflation Factor measures how much variance in a regression coefficient is inflated by collinearity with other predictors:
\[VIF_j = \frac{1}{1 - R^2_j}\]
where \(R^2_j\) is the coefficient of determination from regressing predictor \(X_j\) on all remaining predictors.
| VIF Value | Interpretation | Action |
|---|---|---|
| \(VIF < 3\) | No concern | Proceed normally |
| \(3 \leq VIF < 5\) | Mild | Monitor; report; no action required |
| \(5 \leq VIF < 10\) | Moderate concern | Investigate; consider combining or centering predictors |
| \(VIF \geq 10\) | Serious problem | Multicollinearity is distorting coefficients; act (ridge regression, remove predictors, PCA on correlated set) |
Also inspect the Condition Index (from eigenvalue decomposition of \(\mathbf{X}'\mathbf{X}\)): \(CI = \sqrt{\lambda_{\max}/\lambda_j}\); values \(> 30\) confirm serious collinearity.
\[n = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2 \cdot \sigma^2}{\delta^2}\]
where \(\delta\) is the minimum detectable effect, \(\sigma^2\) is the variance, and \(z\) values are critical values for Type I (\(\alpha\)) and Type II (\(\beta\)) error rates. Target \(1 - \beta \geq .80\); use \(\geq .95\) for confirmatory or high-stakes research. Use G*Power (Faul et al., 2007) for validated calculations across all major test families.
Use comparison methods when the research question is: “Do groups differ on a measured outcome?”
Use when: Comparing the means of two independent groups on a continuous DV.
\[t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}\]
| Assumption | Check | If Violated |
|---|---|---|
| Normality of residuals within each group | Shapiro–Wilk; Q-Q plot | Use Mann–Whitney U |
| Equal variances (homoscedasticity) | Levene’s Test | Apply Welch’s correction (do not pool variances) |
| Independence | Design review | Redesign or use paired test |
Rules of Thumb: \(n \geq 30\) per group (or \(\geq 100\) for skewed data); group size ratio \(\leq 3{:}1\) with unequal variances requires Welch’s correction.
Effect size — Cohen’s \(d\):
\[d = \frac{\bar{X}_1 - \bar{X}_2}{s_{\text{pooled}}}, \quad s_{\text{pooled}} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}\]
Benchmarks: small = .20, medium = .50, large = .80 (Cohen, 1988).
Red Flags 🚩
# Compare fuel efficiency (mpg) by transmission type
# am: 0 = automatic, 1 = manual
result_t <- t.test(mpg ~ am, data = mtcars, var.equal = FALSE)
print(result_t)#>
#> Welch Two Sample t-test
#>
#> data: mpg by am
#> t = -3.7671, df = 18.332, p-value = 0.001374
#> alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
#> 95 percent confidence interval:
#> -11.280194 -3.209684
#> sample estimates:
#> mean in group 0 mean in group 1
#> 17.14737 24.39231# Cohen's d
grp <- split(mtcars$mpg, mtcars$am)
n1 <- length(grp[[1]]); n2 <- length(grp[[2]])
sp <- sqrt(((n1-1)*var(grp[[1]]) + (n2-1)*var(grp[[2]])) / (n1+n2-2))
cohd <- abs(diff(sapply(grp, mean))) / sp
cat(sprintf("\nCohen's d = %.3f [large effect by Cohen (1988) benchmarks]\n", cohd))#>
#> Cohen's d = 1.478 [large effect by Cohen (1988) benchmarks]Use when: Normality is violated, the DV is ordinal, or \(N\) is small (\(< 30\)) and distribution is skewed.
\[U_1 = n_1 n_2 + \frac{n_1(n_1+1)}{2} - R_1, \quad U = \min(U_1, U_2)\]
Effect size: \(r = Z / \sqrt{N}\); benchmarks: small = .10, medium = .30, large = .50.
Red Flags 🚩 Applying Mann–Whitney to paired data (use Wilcoxon Signed-Rank); interpreting \(U\) as testing means when group distributions differ in shape.
Use when: Comparing two related or matched measurements (pre–post, twins, repeated measures on same unit).
\[t = \frac{\bar{D}}{s_D / \sqrt{n}}, \quad D_i = X_{1i} - X_{2i}\]
The normality assumption applies to the difference scores \(D_i\), not the raw scores.
Effect size — Cohen’s \(d_z\):
\[d_z = \frac{\bar{D}}{s_D}\]
Benchmarks: small = .20, medium = .50, large = .80.
Red Flags 🚩 Running two separate one-sample \(t\)-tests instead (inflates Type I error); ignoring the within-pair correlation (which drives the power advantage of this design).
Use when: Comparing means across three or more independent groups.
\[F = \frac{MS_{\text{Between}}}{MS_{\text{Within}}} = \frac{SS_B/(k-1)}{SS_W/(N-k)}\]
| Assumption | Check | If Violated |
|---|---|---|
| Normality of residuals | Shapiro–Wilk on residuals; Q-Q plot | Use Kruskal–Wallis |
| Homogeneity of variances | Levene’s Test | Use Welch’s ANOVA |
| Independence | Design review | Use multilevel model |
Post-hoc hierarchy: Tukey HSD (equal \(n\), equal variances) → Games–Howell (unequal variances) → Bonferroni (planned comparisons).
Effect size — Eta-squared \(\eta^2\):
\[\eta^2 = \frac{SS_{\text{Between}}}{SS_{\text{Total}}}\]
Also report \(\omega^2\) (less biased for small \(N\)):
\[\omega^2 = \frac{SS_B - (k-1)MS_W}{SS_T + MS_W}\]
| Metric | Small | Medium | Large |
|---|---|---|---|
| \(\eta^2\) / \(\omega^2\) | .01 | .06 | .14 |
| Cohen’s \(f\) | .10 | .25 | .40 |
Statistical significance ≠ practical importance. A one-way ANOVA with \(N = 800\) may yield \(F(2, 797) = 6.20\), \(p < .001\), yet \(\eta^2 = .015\) — accounting for less than 2% of variance. Always report and interpret effect sizes alongside \(p\)-values.
Red Flags 🚩 Running multiple \(t\)-tests instead of ANOVA (familywise error \(= 1-(1-\alpha)^k\)); reporting \(F\) without post-hoc tests when \(k > 2\); applying ANOVA to clustered data.
# Compare horsepower across cylinder groups
mtcars$cyl_f <- factor(mtcars$cyl, labels = c("4-cyl", "6-cyl", "8-cyl"))
fit_aov <- aov(hp ~ cyl_f, data = mtcars)
summary(fit_aov)#> Df Sum Sq Mean Sq F value Pr(>F)
#> cyl_f 2 104031 52015 36.18 1.32e-08 ***
#> Residuals 29 41696 1438
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Tukey HSD post-hoc
TukeyHSD(fit_aov)#> Tukey multiple comparisons of means
#> 95% family-wise confidence level
#>
#> Fit: aov(formula = hp ~ cyl_f, data = mtcars)
#>
#> $cyl_f
#> diff lwr upr p adj
#> 6-cyl-4-cyl 39.64935 -5.627454 84.92616 0.0949068
#> 8-cyl-4-cyl 126.57792 88.847251 164.30859 0.0000000
#> 8-cyl-6-cyl 86.92857 43.579331 130.27781 0.0000839# Eta-squared
ss <- summary(fit_aov)[[1]]$"Sum Sq"
eta_sq <- ss[1] / sum(ss)
omega_sq <- (ss[1] - (nlevels(mtcars$cyl_f)-1) * summary(fit_aov)[[1]]$"Mean Sq"[2]) /
(sum(ss) + summary(fit_aov)[[1]]$"Mean Sq"[2])
cat(sprintf("\neta-squared = %.3f\nomega-squared = %.3f\n", eta_sq, omega_sq))#>
#> eta-squared = 0.714
#> omega-squared = 0.687Use when: Normality is violated; DV is ordinal; cells have \(n < 20\).
\[H = \frac{12}{N(N+1)} \sum_{i=1}^{k} \frac{R_i^2}{n_i} - 3(N+1)\]
Requires \(n_i \geq 5\) per group for the \(\chi^2\) approximation to be valid. Post-hoc: Dunn’s test with Bonferroni or Holm correction. Effect size: \(\eta^2_H = (H - k + 1)/(N - k)\).
Use when: Examining the effects of two or more categorical IVs and their interaction on a continuous DV.
Additional rules: Always test the \(A \times B\) interaction before interpreting main effects. A significant interaction renders unconditional main effects misleading — interpret via simple effects analysis. Use Type III SS for unbalanced designs. \(n \geq 5\) per cell (ideally \(\geq 20\)).
Red Flags 🚩 Interpreting main effects as unconditional when a significant interaction is present; cells with \(n < 5\).
Use when: The same participants are measured across three or more conditions or time points.
Additional assumption — Sphericity:
\[\hat{\varepsilon} \geq \frac{1}{k-1}, \quad \text{tested by Mauchly's Test}\]
| Mauchly’s Result | Correction |
|---|---|
| \(p > .05\) — sphericity met | Standard \(F\); uncorrected \(df\) |
| \(\hat{\varepsilon} \geq .75\) | Huynh–Feldt correction |
| \(\hat{\varepsilon} < .75\) | Greenhouse–Geisser correction |
| Severe violation | Switch to MANOVA or mixed-effects model |
Red Flags 🚩 Ignoring Mauchly’s test; handling missing data by listwise deletion (use mixed-effects models instead).
Use when: Sphericity is severely violated; DV is ordinal; \(N\) is small. Effect size: Kendall’s \(W\) (concordance coefficient); \(W \geq .70\) indicates strong agreement across conditions.
Use when: Simultaneously comparing groups on two or more continuous DVs.
| Assumption | Diagnostic |
|---|---|
| Multivariate normality | Mardia’s test; Henze–Zirkler; Royston’s test |
| Homogeneity of covariance matrices | Box’s \(M\) (\(p > .001\) tolerance) |
| No multicollinearity among DVs | Bivariate \(r\) between DVs: .30–.90 |
| Independence | Design review |
| No multivariate outliers | Mahalanobis \(D^2\) at \(\chi^2_{p}\), \(p < .001\) |
Test statistic selection:
| Statistic | Best When |
|---|---|
| Wilks’ \(\Lambda\) | Most common; balanced power (Wilks, 1932) |
| Pillai’s Trace | Default choice — most robust to violations (Pillai, 1955; Olson, 1976) |
| Hotelling’s Trace | One dominant canonical dimension |
| Roy’s Largest Root | Maximum power with one variate; most sensitive to violations |
Effect size: Partial \(\eta^2_p\); benchmarks same as ANOVA (.01 / .06 / .14).
Red Flags 🚩 Running separate ANOVAs instead (inflates familywise error); DVs entirely uncorrelated (MANOVA loses advantage); Box’s \(M\) severely significant with unequal group sizes.
Use correlation methods when the research question is: “How strongly are variables associated?”
Use when: Both variables are continuous and interval/ratio scaled; relationship is expected to be linear.
\[r = \frac{\sum(X_i - \bar{X})(Y_i - \bar{Y})}{(n-1)\,s_X\,s_Y}\]
| Assumption | Check |
|---|---|
| Both variables continuous | Measurement review |
| Linear relationship | Scatterplot inspection |
| Bivariate normality (for inference) | Q-Q plots; Mardia’s test |
| No severe outliers | Scatterplot; leverage statistics |
| Independence of observations | Design review |
Benchmarks (Cohen, 1988): small = .10, medium = .30, large = .50.
The squared correlation \(r^2\) is the coefficient of determination — the proportion of variance in \(Y\) explained by \(X\).
Red Flags 🚩 Interpreting correlation as causation; failing to inspect scatterplot for non-linearity or heteroscedasticity; not checking for outliers driving spurious correlations.
# Pearson r: mpg vs weight
r_pearson <- cor.test(mtcars$mpg, mtcars$wt, method = "pearson")
cat("--- Pearson Correlation ---\n")#> --- Pearson Correlation ---print(r_pearson)#>
#> Pearson's product-moment correlation
#>
#> data: mtcars$mpg and mtcars$wt
#> t = -9.559, df = 30, p-value = 1.294e-10
#> alternative hypothesis: true correlation is not equal to 0
#> 95 percent confidence interval:
#> -0.9338264 -0.7440872
#> sample estimates:
#> cor
#> -0.8676594cat(sprintf("R-squared = %.3f (proportion of shared variance)\n",
r_pearson$estimate^2))#> R-squared = 0.753 (proportion of shared variance)# Spearman rho (non-parametric alternative)
r_spearman <- cor.test(mtcars$mpg, mtcars$wt, method = "spearman")
cat("\n--- Spearman Correlation (non-parametric) ---\n")#>
#> --- Spearman Correlation (non-parametric) ---print(r_spearman)#>
#> Spearman's rank correlation rho
#>
#> data: mtcars$mpg and mtcars$wt
#> S = 10292, p-value = 1.488e-11
#> alternative hypothesis: true rho is not equal to 0
#> sample estimates:
#> rho
#> -0.886422Use when: DV or IV is ordinal; or normality assumption for Pearson \(r\) is violated; or outliers are present.
\[\rho = 1 - \frac{6\sum d_i^2}{n(n^2-1)}, \quad \text{where } d_i = \text{rank}(X_i) - \text{rank}(Y_i)\]
Kendall’s \(\tau\) is preferred over \(\rho\) when there are many tied ranks or when \(N\) is small — it has better sampling properties in those conditions.
Red Flags 🚩 Using Spearman \(\rho\) when the research hypothesis is specifically about linear association (Pearson \(r\) is the appropriate test).
Use when: Testing association between two categorical (nominal) variables in a contingency table.
\[\chi^2 = \sum_{i,j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}, \quad E_{ij} = \frac{R_i \cdot C_j}{N}\]
| Assumption | Threshold |
|---|---|
| Independence of observations | No repeated measures on same case |
| Expected cell frequencies | \(E_{ij} \geq 5\) in \(\geq 80\%\) of cells; no cell with \(E < 1\) |
| Mutually exclusive categories | Categories do not overlap |
If expected cell frequencies are violated: Use Fisher’s Exact Test (\(2 \times 2\)) or collapse categories.
Effect size — Cramér’s \(V\):
\[V = \sqrt{\frac{\chi^2}{N \cdot \min(r-1,\, c-1)}}\]
Benchmarks: small = .10, medium = .30, large = .50.
Red Flags 🚩 Running \(\chi^2\) on dependent observations — use McNemar’s test (McNemar, 1947); interpreting \(\chi^2\) as a strength measure (it tests independence, not association magnitude).
Use when: One variable is truly dichotomous (binary); the other is continuous.
\[r_{pb} = \frac{\bar{X}_1 - \bar{X}_0}{s_X}\sqrt{\frac{n_1 n_0}{n^2}}\]
This is mathematically equivalent to Pearson \(r\) when one variable is coded 0/1. It relates directly to the independent \(t\)-test: \(r_{pb}^2 = t^2/(t^2 + df)\).
Use prediction methods when the research question is: “How well can we predict an outcome from a set of predictors?”
Use when: DV is continuous; predicting \(\hat{Y}\) from one or more IVs.
\[\hat{Y} = b_0 + b_1 X_1 + b_2 X_2 + \cdots + b_k X_k + \varepsilon\]
Correction from v2: The normality and homoscedasticity assumptions in OLS regression apply to the residuals \(\varepsilon_i = Y_i - \hat{Y}_i\), not to the raw IV or DV distributions. You can and should have non-normally distributed predictors. It is the error term that must satisfy distributional assumptions.
| Assumption | What It Tests | Diagnostic | Violation Remedy |
|---|---|---|---|
| Linearity | \(E[\varepsilon \mid X] = 0\) | Partial regression plots; RESET test | Polynomial terms; transformations |
| Independence of residuals | No autocorrelation | Durbin–Watson (\(DW \approx 2\)) | GLS; clustered SE; time-series models |
| Homoscedasticity of residuals | \(\text{Var}(\varepsilon_i) = \sigma^2\) (constant) | Breusch–Pagan test; Residual vs. Fitted plot | WLS; HC3 heteroscedasticity-robust SE |
| Normality of residuals | \(\varepsilon \sim \mathcal{N}(0,\sigma^2)\) | Shapiro–Wilk on residuals; Q-Q plot of residuals | Transform DV; robust regression |
| No multicollinearity | Predictors not collinear | \(VIF < 5\) (moderate); \(VIF < 10\) (serious) | Ridge regression; drop/combine predictors |
| No influential outliers | No single cases dominating fit | Cook’s \(D > 4/n\); leverage \(h_{ii} > 2(p+1)/n\) | Robust regression; investigate cases |
Standard Error of regression coefficient:
\[SE(b_j) = \sqrt{\frac{MS_{\text{Residual}}}{\sum(X_{ij} - \bar{X}_j)^2 \cdot (1 - R^2_j)}}\]
Note that \(1/(1-R^2_j) = VIF_j\) — multicollinearity directly inflates the SE of \(b_j\).
| Heuristic | Value |
|---|---|
| Events per variable (EPV) | \(N/k \geq 10\) (minimum); \(N/k \geq 20\) (recommended) |
| Minimum \(N\) for \(R^2\) test | \(50 + 8k\) |
| VIF — mild concern | \(3 \leq VIF < 5\) |
| VIF — moderate concern | \(5 \leq VIF < 10\) |
| VIF — serious problem | \(VIF \geq 10\) |
| Condition Index | \(> 30\) confirms serious collinearity |
| Cook’s \(D\) threshold | \(> 4/n\) — investigate case |
Effect size — Cohen’s \(f^2\):
\[f^2 = \frac{R^2}{1 - R^2}; \quad f^2 = 0.02 \text{ (small)},\; 0.15 \text{ (medium)},\; 0.35 \text{ (large)}\]
Red Flags 🚩
# Multiple regression: mpg ~ weight + horsepower + transmission
fit <- lm(mpg ~ wt + hp + am, data = mtcars)
summary(fit)#>
#> Call:
#> lm(formula = mpg ~ wt + hp + am, data = mtcars)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -3.4221 -1.7924 -0.3788 1.2249 5.5317
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 34.002875 2.642659 12.867 2.82e-13 ***
#> wt -2.878575 0.904971 -3.181 0.003574 **
#> hp -0.037479 0.009605 -3.902 0.000546 ***
#> am 2.083710 1.376420 1.514 0.141268
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 2.538 on 28 degrees of freedom
#> Multiple R-squared: 0.8399, Adjusted R-squared: 0.8227
#> F-statistic: 48.96 on 3 and 28 DF, p-value: 2.908e-11# --- Assumption checks ---
# 1. Normality of RESIDUALS (not raw variables)
shapiro.test(residuals(fit))#>
#> Shapiro-Wilk normality test
#>
#> data: residuals(fit)
#> W = 0.9453, p-value = 0.1059# 2. Homoscedasticity of RESIDUALS via Breusch-Pagan
if (requireNamespace("lmtest", quietly = TRUE)) {
cat("\nBreusch-Pagan test for heteroscedasticity:\n")
print(lmtest::bptest(fit))
} else {
cat("\nInstall 'lmtest' for Breusch-Pagan: install.packages('lmtest')\n")
}#>
#> Breusch-Pagan test for heteroscedasticity:
#>
#> studentized Breusch-Pagan test
#>
#> data: fit
#> BP = 5.534, df = 3, p-value = 0.1366# 3. VIF — multicollinearity
if (requireNamespace("car", quietly = TRUE)) {
cat("\nVariance Inflation Factors:\n")
vif_vals <- car::vif(fit)
print(vif_vals)
cat(sprintf("Max VIF = %.2f [threshold: 5 = moderate, 10 = serious]\n",
max(vif_vals)))
} else {
cat("\nInstall 'car' for VIF: install.packages('car')\n")
}#>
#> Variance Inflation Factors:
#> wt hp am
#> 3.774838 2.088124 2.271082
#> Max VIF = 3.77 [threshold: 5 = moderate, 10 = serious]# 4. Influential cases — Cook's Distance
thresh <- 4 / nrow(mtcars)
n_inf <- sum(cooks.distance(fit) > thresh)
cat(sprintf("\nCook's D > 4/n (%.4f): %d influential observation(s)\n",
thresh, n_inf))#>
#> Cook's D > 4/n (0.1250): 4 influential observation(s)# 5. Cohen's f-squared
r2 <- summary(fit)$r.squared
f2 <- r2 / (1 - r2)
cat(sprintf("\nR-squared = %.3f | Adjusted R-squared = %.3f | f2 = %.3f\n",
r2, summary(fit)$adj.r.squared, f2))#>
#> R-squared = 0.840 | Adjusted R-squared = 0.823 | f2 = 5.246Use when: DV is binary (0/1); model estimates the log-odds of the outcome.
\[\log\!\left(\frac{p}{1-p}\right) = b_0 + b_1 X_1 + \cdots + b_k X_k\]
\[\hat{p} = \frac{1}{1 + e^{-(b_0 + \mathbf{b}'\mathbf{X})}}\]
| Assumption | Diagnostic |
|---|---|
| Binary DV | Design review |
| No complete separation | Inspect max-rescaled \(R^2\); use Firth’s penalized likelihood if separated |
| No multicollinearity | \(VIF < 10\) |
| Independence of errors | Design review; GEE or mixed logistic if clustered |
| Adequate EPV | \(\geq 10\) events per predictor (Peduzzi et al., 1996) |
Reporting: \(OR = e^{b_k}\); \(OR > 1\) = increased odds; \(OR < 1\) = decreased odds. Report Nagelkerke \(R^2\) and AUC–ROC (\(\geq .70\) = acceptable, \(\geq .80\) = good, \(\geq .90\) = excellent).
Red Flags 🚩 EPV \(< 10\); applying to ordinal DV with \(> 2\) levels (use ordinal logistic); misinterpreting \(b_k\) as a probability change (it is a log-odds change).
Use when: Testing whether a theoretically motivated block of predictors accounts for incremental variance above a prior block.
\[\Delta R^2 = R^2_{\text{Model 2}} - R^2_{\text{Model 1}}\]
\[F_{\Delta R^2} = \frac{\Delta R^2 / \Delta k}{(1 - R^2_{\text{Model 2}}) / (N - k_2 - 1)}\]
Rules: Block entry order must be theory-driven; report \(\Delta R^2\), \(\Delta F\), and exact \(p\)-value per block; recheck all OLS assumptions at each step.
Use these methods when the research question is: “What underlying structure exists in the data?” or “How can cases or variables be grouped?”
| Feature | PCA | EFA |
|---|---|---|
| Goal | Data reduction; explain total variance | Identify latent constructs; explain common variance |
| Variance modeled | 100% of observed variance | Shared variance only (communalities) |
| Output | Components = linear composites of observed variables | Factors = hypothetical latent variables |
| Use when | Reducing variables with minimal information loss | Hypothesizing underlying theoretical constructs |
| Validation step | Not applicable | CFA required for confirmatory testing |
Core assumptions for EFA:
| Rule | Notes |
|---|---|
| Kaiser’s Criterion (\(\lambda > 1.0\)) | Known to overextract (Kaiser, 1960) — use as lower bound only |
| Scree Plot elbow | Subjective; combine with quantitative criteria (Cattell, 1966) |
| Parallel Analysis | Gold standard — compare observed eigenvalues to random-data eigenvalues (Horn, 1965) |
| Velicer’s MAP | Minimizes average squared partial correlation (Velicer, 1976) |
Rotation: Oblique (Promax/Oblimin) for correlated factors (default for most social science constructs); Orthogonal (Varimax) only when factors are theoretically and empirically independent.
Reliability:
\[\alpha = \frac{k\bar{r}}{1+(k-1)\bar{r}} \quad \text{(Cronbach, 1951)} \qquad \omega_h = \frac{(\sum\lambda_i)^2}{(\sum\lambda_i)^2+\sum\delta_i} \quad \text{(McDonald, 1999 — preferred)}\]
| Value | Interpretation |
|---|---|
| \(\geq .90\) | Excellent |
| \(\geq .80\) | Good |
| \(\geq .70\) | Acceptable |
| \(\geq .60\) | Questionable |
| \(< .60\) | Poor |
Red Flags 🚩 Using PCA for construct validation; reporting Kaiser’s criterion alone; Cronbach’s \(\alpha\) as evidence of unidimensionality (it is not — use \(\omega_h\); McNeish, 2018).
Use when: Testing a pre-specified factor structure grounded in theory or prior EFA.
Additional assumptions: \(N \geq 200\); \(N \geq 5\) per free parameter; use MLR or WLSMV for ordinal indicators.
| Index | Acceptable | Good | Note |
|---|---|---|---|
| \(\chi^2/df\) | \(< 5.0\) | \(< 2.0\) | \(\chi^2\) alone is \(N\)-sensitive |
| CFI | \(\geq .90\) | \(\geq .95\) | — |
| TLI | \(\geq .90\) | \(\geq .95\) | Penalizes complexity |
| RMSEA | \(< .08\) | \(< .06\) | Report 90% CI; test \(H_0\): RMSEA \(\leq .05\) |
| SRMR | \(< .10\) | \(< .08\) | — |
Hu & Bentler (1999)
| Type | Indicator | Threshold |
|---|---|---|
| Convergent | AVE | \(\geq .50\) |
| Convergent | Factor loadings | \(\lambda \geq .50\) (ideally \(\geq .70\)) |
| Discriminant | Fornell–Larcker: \(AVE > \phi^2_{ij}\) | For each construct pair |
| Discriminant | HTMT | \(< .85\) (strict) or \(< .90\) (liberal) |
| Composite reliability | CR | \(\geq .70\) |
\[AVE = \frac{\sum\lambda_i^2}{\sum\lambda_i^2+\sum\delta_i}, \qquad CR = \frac{(\sum\lambda_i)^2}{(\sum\lambda_i)^2+\sum\delta_i}\]
Red Flags 🚩 Relying on \(\chi^2\) alone; applying modification indices \(> 10\) atheoretically; Heywood cases (negative residual variances); not testing discriminant validity.
Use when: Testing complex theoretical models with latent variables, mediation, and/or moderation.
\[\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\Lambda}\,\boldsymbol{\Phi}\,\boldsymbol{\Lambda}' + \boldsymbol{\Theta}\]
Rules: \(N \geq 200\); \(N \geq 10\) per free parameter; bootstrap \(B \geq 5000\) for indirect effects; report bias-corrected 95% CI for mediation (Preacher & Hayes, 2008).
Mediation indirect effect:
\[a \cdot b = a \times b, \quad a: X \to M,\quad b: M \to Y\]
Avoid Sobel test — use bias-corrected bootstrapped CIs instead (MacKinnon et al., 2004).
Red Flags 🚩 Not checking temporal precedence for mediation; treating non-significant direct paths as full mediation without bootstrapped CIs; ignoring equivalent models.
Use when: Discovering natural groupings in data with no predefined criterion variable.
| Method | Best When |
|---|---|
| Hierarchical (Ward’s linkage) | Small \(N\); unknown number of clusters (Ward, 1963) |
| \(K\)-Means | Large \(N\); hypothesized or fixed \(k\) |
| GMM (Gaussian Mixture Models) | Overlapping clusters; probabilistic assignment |
| DBSCAN | Non-spherical clusters; noise/outlier handling needed |
Rules: Always standardize variables before clustering; determine \(k\) using Elbow + Silhouette (\(\bar{s} \geq .50\); Rousseeuw, 1987) + Gap statistic; validate on split-half samples.
Red Flags 🚩 Unstandardized variables; researcher-chosen \(k\) without empirical support; treating cluster membership as a measured IV in subsequent regression (circular).
Logic flow: If [Data Type] + If [Analysis Goal] → Use [Test]
IF goal = DESCRIBE distribution → Descriptive statistics, histograms, box plots, density plots
IF goal = COMPARE groups IF groups = 2 IF independent + continuous DV + normal residuals → Independent t-test IF independent + non-normal / ordinal DV → Mann-Whitney U IF related/paired + continuous DV + normal diffs → Paired t-test IF related/paired + non-normal / ordinal DV → Wilcoxon Signed-Rank IF groups >= 3 IF independent + continuous + normal + equal var → One-Way ANOVA IF independent + continuous + violated assump. → Kruskal-Wallis H IF repeated measures + continuous → RM-ANOVA IF repeated measures + non-normal / ordinal → Friedman Test IF 2+ IVs (factorial) → Factorial ANOVA IF 2+ continuous DVs simultaneously → MANOVA
IF goal = CORRELATE variables IF both continuous + linear + normal → Pearson r IF ordinal / non-normal / outliers present → Spearman rho IF many ties or small N → Kendall tau IF 1 continuous + 1 binary → Point-Biserial r_pb IF both categorical (nominal) → Chi-Square (or Fisher Exact) IF both ordinal in contingency table → Gamma or Somer’s d
IF goal = PREDICT an outcome IF DV continuous + predictors any scale → OLS Multiple Regression IF DV continuous + blocks of predictors → Hierarchical Regression IF DV binary (0/1) → Logistic Regression IF DV ordinal (3+ ordered levels) → Ordinal Logistic Regression IF DV is a count → Poisson / Negative Binomial IF DV continuous + predictors correlated (VIF>10) → Ridge / LASSO Regression IF complex model with latent variables → SEM / CFA
IF goal = FIND STRUCTURE in variables IF goal = data reduction, no theory → PCA IF goal = latent constructs, exploratory → EFA IF goal = test pre-specified factor structure → CFA IF goal = test complex latent path model → SEM
IF goal = FIND GROUPS in cases IF N small, k unknown → Hierarchical Cluster (Ward) IF N large, k hypothesized → K-Means Cluster IF overlapping groups, probabilistic → Gaussian Mixture Model IF non-spherical, with noise → DBSCAN
Core principle: Statistical significance (\(p < .05\)) tells you that an effect is unlikely to be zero. Effect size tells you how large the effect is in practice. With large samples, even trivially small effects achieve \(p < .001\). With small samples, important effects may not reach \(p < .05\). Always report and interpret effect sizes alongside \(p\)-values (Cohen, 1994; Cumming, 2014).
Worked example of the distinction:
A study of \(N = 1000\) finds: \(t(998) = 3.16\), \(p = .002\), Cohen’s \(d = 0.20\) (small effect). The difference is real but trivial — it explains only \(\approx 1\%\) of variance.
Another study with \(N = 40\) finds: \(t(38) = 1.90\), \(p = .065\), Cohen’s \(d = 0.62\) (medium effect). The effect is practically meaningful but underpowered — it would be significant with \(N \approx 70\).
\[d = \frac{\bar{X}_1 - \bar{X}_2}{s_{\text{pooled}}}\]
| Value | Interpretation | Variance explained (\(r^2\)) |
|---|---|---|
| \(.20\) | Small | \(\approx 1\%\) |
| \(.50\) | Medium | \(\approx 6\%\) |
| \(.80\) | Large | \(\approx 14\%\) |
\[\eta^2 = \frac{SS_{\text{Between}}}{SS_{\text{Total}}} \qquad \omega^2 = \frac{SS_B - (k-1)MS_W}{SS_T + MS_W}\]
\(\omega^2\) is preferred over \(\eta^2\) for small samples — \(\eta^2\) is a biased overestimate.
Partial \(\eta^2_p\) is used in factorial designs and MANOVA:
\[\eta^2_p = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_{\text{error}}}\]
| Metric | Small | Medium | Large |
|---|---|---|---|
| \(\eta^2\) / \(\omega^2\) | .01 | .06 | .14 |
| Cohen’s \(f = \sqrt{\eta^2/(1-\eta^2)}\) | .10 | .25 | .40 |
| Test Family | Metric | Small | Medium | Large | Formula |
|---|---|---|---|---|---|
| \(t\)-test | Cohen’s \(d\) | .20 | .50 | .80 | \((\bar{X}_1-\bar{X}_2)/s_p\) |
| ANOVA | \(\eta^2\) | .01 | .06 | .14 | \(SS_B/SS_T\) |
| ANOVA | \(\omega^2\) | .01 | .06 | .14 | Less biased than \(\eta^2\) |
| ANOVA | Cohen’s \(f\) | .10 | .25 | .40 | \(\sqrt{\eta^2/(1-\eta^2)}\) |
| Correlation | Pearson \(r\) | .10 | .30 | .50 | \(\text{cov}(X,Y)/s_Xs_Y\) |
| Regression | Cohen’s \(f^2\) | .02 | .15 | .35 | \(R^2/(1-R^2)\) |
| \(\chi^2\) | Cramér’s \(V\) | .10 | .30 | .50 | \(\sqrt{\chi^2/[N\min(r{-}1,c{-}1)]}\) |
| Non-parametric | \(r\) | .10 | .30 | .50 | \(Z/\sqrt{N}\) |
| MANOVA | \(\eta^2_p\) | .01 | .06 | .14 | \(SS_{\text{eff}}/(SS_{\text{eff}}+SS_{\text{err}})\) |
| Logistic Reg. | Nagelkerke \(R^2\) | .02 | .13 | .26 | Pseudo-\(R^2\) |
Benchmarks: Cohen (1988, 1992); Lakens (2013)
Apply when parametric assumptions are violated and non-parametric alternatives are not preferred (Osborne, 2002).
| Skew Pattern | Transformation | Formula | Notes |
|---|---|---|---|
| Moderate positive | Square Root | \(X' = \sqrt{X}\) | Requires \(X \geq 0\) |
| Substantial positive | Logarithmic | \(X' = \ln(X)\) or \(\log_{10}(X)\) | Requires \(X > 0\); add constant if zeros present |
| Severe positive | Inverse (Reciprocal) | \(X' = 1/X\) | Reverses rank order |
| Moderate negative | Reflect + Square Root | \(X' = \sqrt{k-X}\), \(k=\max(X)+1\) | Restore sign after analysis |
| Substantial negative | Reflect + Log | \(X' = \ln(k-X)\) | Same reflection strategy |
| Proportions \([0,1]\) | Arcsine Square Root | \(X' = \arcsin\!\sqrt{p}\) | Stabilizes binomial variance |
| Count data | Square Root or Log | \(X' = \sqrt{X}\) or \(\ln(X+1)\) | Consider Poisson regression |
| Bimodal | — | Do not transform | Investigate subpopulations |
Back-transformation: Log-transformed means become geometric means on the original scale. Always report results in original units.
\[t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{s_1^2/n_1 + s_2^2/n_2}} \qquad\qquad F = \frac{MS_{\text{Between}}}{MS_{\text{Within}}}\]
\[r = \frac{\sum(X_i-\bar{X})(Y_i-\bar{Y})}{(n-1)s_Xs_Y} \qquad\qquad \rho = 1 - \frac{6\sum d_i^2}{n(n^2-1)}\]
\[\chi^2 = \sum\frac{(O-E)^2}{E} \qquad OR = e^{b} \qquad \text{logit}(p) = \ln\!\frac{p}{1-p}\]
\[d = \frac{\bar{X}_1-\bar{X}_2}{s_p} \qquad \eta^2 = \frac{SS_B}{SS_T} \qquad VIF_j = \frac{1}{1-R^2_j}\]
\[SE(b_j) = \sqrt{\frac{MS_{\text{Res}}}{\sum(X_{ij}-\bar{X}_j)^2}\cdot VIF_j} \qquad CI_{95\%}: \hat{\theta} \pm 1.96\cdot SE_{\hat{\theta}}\]
\[1-\beta = \text{Power} \qquad \alpha = \text{Type I Error} \qquad \beta = \text{Type II Error}\]
Foundational & General Statistics
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Parametric Tests
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Non-Parametric Tests
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Chi-Square & Association
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Regression Analysis
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MANOVA
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Factor Analysis, Reliability & SEM
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Cluster Analysis, Effect Sizes & Transformations
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