SMHS ANOVA
Contents
Scientific Methods for Health Sciences - Analysis of Variance (ANOVA)
Overview
Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more group means. While the t-test is limited to comparing two groups, ANOVA generalizes this to multiple groups, partitioning the total observed variance into components attributable to different sources (e.g., between‑group and within‑group variation). This chapter introduces one‑way and two‑way ANOVA, discusses underlying assumptions, provides step‑by‑step calculations, and illustrates applications with real data and R examples.
Motivation
Suppose a plant biologist wants to compare the yield of five different varieties of wheat. Each variety is planted in four randomly assigned plots, yielding 20 plots total. The yields (in bushels/acre) are:
| Variety A | Variety B | Variety C | Variety D | Variety E |
|---|---|---|---|---|
| 26.2 | 29.2 | 29.1 | 21.3 | 20.1 |
| 24.3 | 28.1 | 30.8 | 22.4 | 19.3 |
| 21.8 | 27.3 | 33.9 | 24.3 | 19.9 |
| 28.1 | 31.2 | 32.8 | 21.8 | 22.1 |
If we denote the population means of the five varieties by \(\mu_1, \mu_2, \mu_3, \mu_4, \mu_5\), we could perform \(\binom{5}{2}=10\) pairwise t‑tests. However, this approach inflates the Type I error rate and lacks a single overall test. ANOVA provides an integrated framework that simultaneously tests \(H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu_5\) vs. \(H_a:\) at least one mean differs, while controlling the family‑wise error rate.
Theory
One‑Way ANOVA
One‑way ANOVA compares the means of \(k\) independent groups under the assumption that the populations are normally distributed with equal variances.
- Notation:
- \(y_{ij}\) = \(j\)‑th observation in group \(i\) (\(i=1,\dots,k\); \(j=1,\dots,n_i\)).
- \(n_i\) = number of observations in group \(i.\)
- \(n = \sum_{i=1}^k n_i\) = total sample size.
- Group mean\[\bar{y}_{i.} = \frac{1}{n_i}\sum_{j=1}^{n_i} y_{ij}.\]
- Grand mean\[\bar{y}_{..} = \frac{1}{n}\sum_{i=1}^k \sum_{j=1}^{n_i} y_{ij}.\]
- Variance decomposition:
- Total sum of squares\[SS_{total} = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_{..})^2\] with \(df_{total}=n-1.\)
- Between‑groups sum of squares\[SS_{between} = \sum_{i=1}^k n_i (\bar{y}_{i.} - \bar{y}_{..})^2\] with \(df_{between}=k-1.\)
- Within‑groups (error) sum of squares\[SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar{y}_{i.})^2\] with \(df_{within}=n-k.\)
The fundamental identity is\[SS_{total} = SS_{between} + SS_{within}\], and likewise \(df_{total} = df_{between} + df_{within}.\)
- Mean squares and F‑statistic:
- \(MS_{between} = \frac{SS_{between}}{df_{between}}\)
- \(MS_{within} = \frac{SS_{within}}{df_{within}}\)
- Test statistic\[F = \frac{MS_{between}}{MS_{within}}.\]
Under \(H_0\) (all population means equal), \(F\) follows an F‑distribution with \((k-1,\; n-k)\) degrees of freedom. A large \(F\) value suggests that the between‑group variation is substantial relative to within‑group variation, providing evidence against \(H_0\).
- ANOVA table (general form):
| Source of Variation | Degrees of Freedom (df) | Sum of Squares (SS) | Mean Square (MS) | F‑statistic | P‑value |
|---|---|---|---|---|---|
| Between Groups | \(k-1\) | \(SS_{between}\) | \(MS_{between}\) | \(F = \frac{MS_{between}}{MS_{within}}\) | \(P(F_{k-1,\; n-k} > F_{\text{obs}})\) |
| Within Groups (Error) | \(n-k\) | \(SS_{within}\) | \(MS_{within}\) | ||
| Total | \(n-1\) | \(SS_{total}\) |
- Assumptions and diagnostics:
- Independence: Observations are independent within and across groups.
- Normality: The residuals ( \(e_{ij} = y_{ij} - \bar{y}_{i.}\) ) should be approximately normally distributed for each group. This can be checked with Q‑Q plots, Shapiro‑Wilk, or Kolmogorov‑Smirnov tests.
- Homoscedasticity (equal variances): Group populations should have the same variance \(\sigma^2\). Formal tests (Levene’s, Bartlett’s) or visual inspection of residual‑vs‑fitted plots can be used.
If assumptions are violated, consider transformations (log, square‑root) or non‑parametric alternatives (Kruskal‑Wallis test).
- Effect size – Eta‑squared and partial Eta‑squared:
- \(\eta^2 = \frac{SS_{between}}{SS_{total}}\) (proportion of total variance explained by group differences).
- Partial \(\eta^2 = \frac{SS_{between}}{SS_{between} + SS_{within}}\).
Interpretation: 0.01 = small, 0.06 = medium, 0.14 = large effect (Cohen, 1988).
- Post‑hoc comparisons: If the overall \(F\)-test is significant, we conduct pairwise comparisons to identify which groups differ. Common methods (with correction for multiple testing) include:
- Tukey’s HSD (Honest Significant Difference)
- Bonferroni correction
- Scheffé’s method
Two‑Way ANOVA
Two‑way ANOVA extends the idea to two categorical factors (A and B), allowing us to test:
- Main effect of factor A.
- Main effect of factor B.
- Interaction effect between A and B (whether the effect of one factor depends on the level of the other).
- Model\[y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \varepsilon_{ijk}\]
where:
- \(\mu\) = grand mean.
- \(\alpha_i\) = effect of level \(i\) of factor A (\(i=1,\dots,a\)).
- \(\beta_j\) = effect of level \(j\) of factor B (\(j=1,\dots,b\)).
- \((\alpha\beta)_{ij}\) = interaction effect.
- \(\varepsilon_{ijk} \sim N(0,\sigma^2)\) (error), with \(k=1,\dots,r\) replicates per cell.
- Sum of squares decomposition\[SS_{total} = SS_A + SS_B + SS_{AB} + SS_{error}\]
with degrees of freedom\[df_{total} = N-1,\; df_A = a-1,\; df_B = b-1,\; df_{AB} = (a-1)(b-1),\; df_{error} = N - ab\], where \(N = a \times b \times r\) is the total number of observations.
- Hypotheses:
- \(H_0^{(A)}: \alpha_1 = \alpha_2 = \dots = \alpha_a = 0\) (no main effect of A).
- \(H_0^{(B)}: \beta_1 = \beta_2 = \dots = \beta_b = 0\) (no main effect of B).
- \(H_0^{(AB)}: (\alpha\beta)_{ij} = 0\) for all \(i,j\) (no interaction).
- ANOVA table for two‑way design (balanced, with replication):
| Source | df | SS | MS | F | P‑value |
|---|---|---|---|---|---|
| Factor A | \(a-1\) | \(SS_A\) | \(MS_A = \frac{SS_A}{df_A}\) | \(F_A = \frac{MS_A}{MS_{error}}\) | … |
| Factor B | \(b-1\) | \(SS_B\) | \(MS_B = \frac{SS_B}{df_B}\) | \(F_B = \frac{MS_B}{MS_{error}}\) | … |
| Interaction A×B | \((a-1)(b-1)\) | \(SS_{AB}\) | \(MS_{AB} = \frac{SS_{AB}}{df_{AB}}\) | \(F_{AB} = \frac{MS_{AB}}{MS_{error}}\) | … |
| Error | \(N-ab\) | \(SS_{error}\) | \(MS_{error} = \frac{SS_{error}}{df_{error}}\) | ||
| Total | \(N-1\) | \(SS_{total}\) |
- Assumptions: same as one‑way ANOVA (independence, normality, homoscedasticity). Additionally, the interaction model assumes that the effects are additive unless the interaction term is included.
- Interactions: If the interaction is significant, the main effects cannot be interpreted separately. Simple‑effects analysis (comparing levels of one factor at fixed levels of the other) is then appropriate.
Applications
Example 1: One‑Way ANOVA with R
A clinical trial tests three new analgesics (Drug A, B, C) against a placebo for pain relief (score 0‑10, lower is better). Data:
Pain = c(5,6,4,5,7, 3,4,5,4,3, 2,3,4,3,2, 6,7,6,5,7)
Group = factor(rep(c("Placebo","DrugA","DrugB","DrugC"), each=5))
Step 1: Exploratory data analysis
boxplot(Pain ~ Group, col="lightblue", main="Pain Score by Treatment",
xlab="Treatment", ylab="Pain Score")
Step 2: Fit ANOVA model
fit <- aov(Pain ~ Group) summary(fit)
Output:
Df Sum Sq Mean Sq F value Pr(>F) Group 3 40.00 13.333 20.00 1.25e-05 *** Residuals 16 10.67 0.667 --- Significant at <0.001
Step 3: Check assumptions
# Normality of residuals shapiro.test(residuals(fit)) # Homogeneity of variances (Levene’s test) library(car) leveneTest(Pain ~ Group)
Step 4: Post‑hoc comparisons (Tukey)
TukeyHSD(fit, conf.level=0.95) plot(TukeyHSD(fit))
Step 5: Effect size
eta_squared <- summary(fit)[[1]][1,2] / sum(summary(fit)[[1]][,2]) eta_squared
Example 2: Two‑Way ANOVA with Interaction
A study examines how gender (M/F) and exercise regimen (None, Light, Heavy) affect cholesterol level (mg/dL). Data are balanced with 5 subjects per cell.
R analysis:
chol <- c(220,210,230,215,225, 200,195,205,210,200,
190,185,195,190,180, 210,205,215,200,210,
180,175,185,180,170, 170,165,175,170,160)
gender <- factor(rep(rep(c("M","F"), each=5), 3))
exercise <- factor(rep(c("None","Light","Heavy"), each=10))
# Two‑way ANOVA with interaction
fit2 <- aov(chol ~ gender * exercise)
summary(fit2)
# Check interaction visually
interaction.plot(exercise, gender, chol, type="b", col=c("blue","red"),
pch=c(16,18), main="Interaction Plot: Cholesterol")
If the interaction is not significant, we may refit the model without it:
fit2_additive <- aov(chol ~ gender + exercise) summary(fit2_additive)
Software
R Commands for ANOVA
- One‑way ANOVA: aov(y ~ group, data)
- Two‑way ANOVA (with interaction): aov(y ~ factor1 * factor2, data)
- Check assumptions:
- Normality: shapiro.test(residuals(model)) - Homogeneity of variances: leveneTest(y ~ group, data)
- Post‑hoc tests: TukeyHSD(model), pairwise.t.test(y, group, p.adjust="bonferroni")
- Effect size: library(effectsize); eta_squared(model)
Complete R Example (One‑Way)
# Simulated data: three diets (A, B, C) and weight loss (lbs)
set.seed(123)
n <- 20
diet <- factor(sample(rep(c("A","B","C"), c(7,7,6))))
weight_loss <- c(rnorm(7,5,1.2), rnorm(7,7,1.2), rnorm(6,4,1.2))
data <- data.frame(diet, weight_loss)
# ANOVA
model <- aov(weight_loss ~ diet, data=data)
summary(model)
# Diagnostic plots
par(mfrow=c(2,2))
plot(model)
par(mfrow=c(1,1))
# Post‑hoc (if significant)
if(summary(model)[[1]][1,5] < 0.05) {
TukeyHSD(model)
}
# Effect size
library(effectsize)
eta_squared(model)
Problems
Problem 1 (One‑Way ANOVA)
A manufacturer compares the assembly times (seconds) of three brands of ping‑pong tables.
| Assembly Time (sec) | Brand |
|---|---|
| 93, 67, 77, 92, 97, 62 | 1 |
| 136, 120, 115, 104, 115, 121, 102, 130 | 2 |
| 198, 217, 209, 221, 190 | 3 |
At \(\alpha=0.05\), test whether the average assembly times differ across brands.
- Solution outline*:
1. State hypotheses\[H_0: \mu_1 = \mu_2 = \mu_3\] vs. \(H_a\): at least one mean differs. 2. Check assumptions (normality, equal variance). 3. Compute ANOVA table. 4. Make decision based on F‑test. 5. If significant, perform post‑hoc comparisons.
Problem 2 (Two‑Way ANOVA)
A researcher investigates how gender (M, F) and education level (High School, Bachelor, Graduate) affect starting salary. Data (in thousands):
| Gender | Education | Salaries |
|---|---|---|
| M | High School | 45, 48, 50 |
| M | Bachelor | 65, 68, 70 |
| M | Graduate | 85, 88, 90 |
| F | High School | 42, 44, 46 |
| F | Bachelor | 60, 62, 64 |
| F | Graduate | 80, 82, 84 |
Test for main effects and interaction. If the interaction is significant, conduct simple‑effects analysis.
References
- SOCR ANOVA Chapter
- ANOVA – Wikipedia
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
- Kutner, M.H., Nachtsheim, C.J., Neter, J., & Li, W. (2005). Applied Linear Statistical Models (5th ed.). McGraw‑Hill.
See Also
- AP Statistics: One‑Way ANOVA
- AP Statistics: Two‑Way ANOVA
- SOCR One‑Way ANOVA Activity
- SOCR Two‑Way ANOVA Activity
- SOCR Home page: http://www.socr.umich.edu
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