Other Statistical Assumptions
Beyond normality, statistical tests have several other important assumptions to check.
🔍 Key Assumptions by Test
Common to All Tests
1. Independence of Observations
What it means: Each data point should be independent—one observation shouldn't influence another.
Violations: - Measuring same person multiple times (without accounting for it) - Siblings, classmates, or other clusters - Time series data (consecutive measurements) - Multiple measurements from same source
How to check: - Study design review (not a statistical test!) - Think about how data were collected - Ask: "Could one person's score affect another's?"
Solutions: - Use repeated measures designs - Use mixed models/hierarchical models - Account for clustering in analysis
For t-tests and ANOVA
2. Homogeneity of Variance (Homoscedasticity)
What it means: Groups should have similar spread (variance).
Check with Levene's Test:
# Load required package
library(car)
# Test for equal variances
leveneTest(outcome ~ group, data = data)
# If p > .05: variances are equal (assumption met)
# If p < .05: variances differ (assumption violated)
Visual check:
# Boxplot to compare spreads
boxplot(outcome ~ group, data = data,
main = "Check for Equal Spread")
# Groups should have similar box heights
Solutions if violated:
- Use Welch's t-test (R's default for t.test)
- Use Welch's ANOVA: oneway.test(outcome ~ group, data = data)
- Transform data (log, sqrt)
Good News for t-tests
R's t.test() defaults to Welch's t-test, which doesn't assume equal variances!
For Regression
3. Linearity
What it means: The relationship between X and Y should be linear (straight line).
Check with scatterplot:
# Scatterplot
plot(data$predictor, data$outcome,
xlab = "Predictor", ylab = "Outcome",
pch = 19, col = "steelblue")
# Add line of best fit
abline(lm(outcome ~ predictor, data = data),
col = "red", lwd = 2)
# Look for: points scattered around line
# Red flags: clear curve, fan shape
Check residual plot:
model <- lm(outcome ~ predictor, data = data)
# Residuals vs Fitted plot
plot(fitted(model), residuals(model),
xlab = "Fitted Values", ylab = "Residuals",
main = "Residuals vs Fitted")
abline(h = 0, col = "red", lty = 2)
# Should see: random scatter around zero
# Red flags: systematic curve, pattern
Solutions if violated:
- Transform variables (log, sqrt, polynomial)
- Add polynomial terms: lm(y ~ x + I(x^2))
- Use non-linear regression
4. No Multicollinearity (Multiple Regression)
What it means: Predictor variables shouldn't be too highly correlated with each other.
Check with VIF (Variance Inflation Factor):
library(car)
model <- lm(outcome ~ pred1 + pred2 + pred3, data = data)
# Calculate VIF
vif(model)
# Interpretation:
# VIF = 1: No correlation
# VIF < 5: Usually okay
# VIF 5-10: Moderate concern
# VIF > 10: Serious problem
Check correlation matrix:
# Correlations between predictors
cor(data[, c("pred1", "pred2", "pred3")])
# Look for: correlations > .80 or .90
Solutions if violated: - Remove one of the correlated predictors - Combine correlated predictors (e.g., average them) - Use principal component analysis - Use ridge regression
5. Homoscedasticity of Residuals
What it means: Residual variance should be constant across fitted values.
Check with residual plot:
model <- lm(outcome ~ predictor, data = data)
plot(fitted(model), residuals(model),
xlab = "Fitted Values", ylab = "Residuals",
main = "Check for Equal Spread")
abline(h = 0, col = "red", lty = 2)
# Should see: constant spread across x-axis
# Red flag: "funnel shape" (spread increases)
Formal test (Breusch-Pagan):
library(lmtest)
bptest(model)
# p > .05: equal variance (good)
# p < .05: heteroscedasticity (concern)
Solutions if violated: - Transform outcome variable (log, sqrt) - Use weighted least squares - Use robust standard errors
6. No Influential Outliers
What it means: No single point should overly influence the results.
Check Cook's Distance:
model <- lm(outcome ~ predictor, data = data)
# Cook's distance plot
plot(model, which = 4)
# Rule of thumb: Cook's D > 1 is concerning
# Also check: Cook's D > 4/(n-k-1) where n = sample size, k = predictors
Identify influential points:
# Points with high Cook's distance
cooks_d <- cooks.distance(model)
influential <- which(cooks_d > 4/(nrow(data) - 2))
# View influential points
data[influential, ]
Solutions: - Investigate: Is it a data entry error? - Consider removing (with justification!) - Use robust regression - Report results with and without outlier
For Chi-Square Tests
7. Expected Frequencies ≥ 5
What it means: Each cell should have expected count ≥ 5.
Check expected frequencies:
result <- chisq.test(table(data$var1, data$var2))
# View expected counts
result$expected
# All values should be ≥ 5
Solutions if violated: - Combine categories (if logical) - Use Fisher's Exact Test (2×2 tables only) - Collect more data
📋 Quick Assumption Checklist
Before Running t-test or ANOVA
# ☐ Independence (design check)
# ☐ Normality
shapiro.test(residuals(model)) # or by group for t-test
# ☐ Equal variances (for standard t-test/ANOVA)
library(car)
leveneTest(outcome ~ group, data = data)
Before Running Regression
# ☐ Independence (design check)
# ☐ Linearity
plot(data$x, data$y)
# ☐ Normality of residuals
shapiro.test(residuals(model))
# ☐ Homoscedasticity
plot(fitted(model), residuals(model))
# ☐ No multicollinearity (if multiple predictors)
library(car)
vif(model)
# ☐ No influential outliers
plot(model, which = 4)
Before Running Chi-Square
# ☐ Independence (design check)
# ☐ Expected frequencies ≥ 5
result <- chisq.test(table)
result$expected