Statistical model reference
Analysis of Variance (ANOVA)
Review when to use this method, its data requirements, implementation patterns, and interpretation guidance.
Description
A statistical method for comparing means among three or more groups by analyzing variance components. It tests the null hypothesis that all group means are equal, and is widely used in experimental designs to determine if any statistically significant differences exist between groups.
Use Cases
- experimental group comparisons
- treatment effect analysis
- factor significance testing
Requirements
- Sample Size: small, medium, large
- Missing Data: none, random
- Data Distribution: normal
- Relationship Type: linear
Variable Types
Dependent Variables
- continuous
Independent Variables
- categorical
Implementation
import statsmodels.api as sm
from statsmodels.formula.api import ols
model = ols('dependent_var ~ C(independent_var)', data=df).fit()
anova_table = sm.stats.anova_lm(model, typ=2)
print(anova_table)model <- aov(dependent_var ~ independent_var, data=df)
summary(model)
TukeyHSD(model) # Post-hoc testONEWAY dependent_var BY independent_var
/POSTHOC=TUKEY ALPHA(0.05)
/STATISTICS DESCRIPTIVES HOMOGENEITYPROC ANOVA DATA=dataset;
CLASS independent_var;
MODEL dependent_var = independent_var;
MEANS independent_var / TUKEY;
RUN;oneway dependent_var independent_var, tabulate
pwmean independent_var, effects mcompare(tukey)Synthetic Data Example
A dataset with one continuous dependent variable and one categorical independent variable with three groups
R Code for Data Generation and Analysis
# Generate synthetic ANOVA data
set.seed(123)
group_A <- rnorm(30, mean=50, sd=5)
group_B <- rnorm(30, mean=55, sd=5)
group_C <- rnorm(30, mean=60, sd=5)
df <- data.frame(
value = c(group_A, group_B, group_C),
group = factor(rep(c("A", "B", "C"), each=30))
)
# Descriptive statistics
aggregate(value ~ group, data=df, FUN=mean)
# Visualization
boxplot(value ~ group, data=df, main="ANOVA Group Comparisons")
# ANOVA model
model <- aov(value ~ group, data=df)
summary(model)
# Post-hoc tests
TukeyHSD(model)
# Assumption checking
plot(model, which=1:2)
shapiro.test(residuals(model))
car::leveneTest(value ~ group, data=df)
Copy this code into your R environment to generate synthetic data and perform analysis with this model.
Expected Analysis Results
Console Output
> summary(model)
Df Sum Sq Mean Sq F value Pr(>F)
group 2 1123.5 561.8 23.45 4.67e-09 ***
Residuals 87 2084.1 24.0
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> TukeyHSD(model)
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = value ~ group, data = df)
$group
diff lwr upr p adj
B-A 4.876543 2.123456 7.629630 0.000234
C-A 9.765432 7.012345 12.518518 0.000001
C-B 4.888889 2.135802 7.641976 0.000221
These results are from running the R code on synthetic data. Your actual results may vary depending on your data.
Interpretation Guide
Need help interpreting the results of your Analysis of Variance (ANOVA) analysis? Our comprehensive interpretation guide explains:
- How to read and understand model outputs
- Interpreting coefficients and effect sizes correctly
- Understanding diagnostic plots and visualizations
- Common pitfalls and how to avoid them
- Making valid conclusions from your analysis