One-Way ANOVA

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One-way ANOVA

Definition

One-way ANOVA (Analysis of Variance) tests whether there are significant differences between the means of three or more independent groups. It extends the t-test to multiple groups while controlling the Type I error rate.

Formula

Key Components:

SS_{between} = \sum_{i=1}^{k} n_i(\bar{x}_i - \bar{x}_g)^2

Between-groups sum of squares, where:

$\bar{x}_i$ = mean of group $i$
$\bar{x}_g$ = grand mean
$n_i$ = sample size of group $i$

SS_{within} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2

Within-groups sum of squares, where:

$x_{ij}$ = $j$ th observation in group $i$
$\bar{x}_i$ = mean of group $i$
$n_i$ = sample size of group $i$

Final Test Statistic:

F = \frac{MS_{between}}{MS_{within}} = \frac{SS_{between}/(k-1)}{SS_{within}/(N-k)}

Where:

$SS_{within}$ = within-groups sum of squares
$k$ = number of groups
$N$ = total sample size

Key Assumptions

Independence: Observations must be independent

Normality: Data within each group should be normally distributed

Homogeneity of Variance: Groups should have equal variances

Practical Example

Step 1: State the Data

Group A	Group B	Group C
8	6	9
9	5	10
7	8	10
10	7	8

Step 2: State Hypotheses

$H_0: \mu_1 = \mu_2 = \mu_3$ (all means equal)
$H_a:$ at least one mean is different
$\alpha = 0.05$

Step 3: Calculate Summary Statistics

Group A: $\bar{x}_A = 8.50, s_A = 1.29$
Group B: $\bar{x}_B = 6.50, s_B = 1.29$
Group C: $\bar{x}_C = 9.25, s_C = 0.96$
Grand mean: $\bar{x} = 8.08$

Step 4: Calculate Sums of Squares

SS_{between} = 16.17

SS_{within} = 12.75

SS_{total} = 28.92

Step 5: Calculate Mean Squares

MS_{between} = \frac{SS_{between}}{k-1} = 8.08

MS_{within} = \frac{SS_{within}}{N-k} = 1.42

Step 6: Calculate F-statistic

$F = \frac{MS_{between}}{MS_{within}} = 5.71$

Step 7: Draw Conclusion

The critical value $F_{(2,9)}$ at $\alpha = 0.05$ is $4.26$ .

The calculated F-statistic ( $F = 5.71$ ) is greater than the critical value ( $4.26$ ), and the p-value ( $p = 0.025$ ) is less than our significance level ( $\\alpha = 0.05$ ). We reject the null hypothesis in favor of the alternative. There is statistically significant evidence to conclude that not all group means are equal. Specifically, at least one group mean differs significantly from the others.

Effect Size

Eta-squared ( $\eta^2$ ) measures the proportion of variance explained:

\eta^2 = \frac{SS_{between}}{SS_{total}}

Guidelines:

Small effect: $\eta^2 \approx 0.01$
Medium effect: $\eta^2 \approx 0.06$
Large effect: $\eta^2 \approx 0.14$

For the example above, the effect size is: $\eta^2 = \frac{16.17}{28.92} = 0.56$ which indicates a large effect.

Code Examples

1library(tidyverse)
2group <- factor(c(rep("A", 4), rep("B", 4), rep("C", 4)))
3values <- c(8, 9, 7, 10, 6, 5, 8, 7, 9, 10, 10, 8)
4
5
6# calculate SS between manually
7mean_values <- tapply(values, group, mean)
8grand_mean <- mean(values)
9ss_between <- sum(table(group) * (mean_values - grand_mean)^2)
10print(str_glue("SS between: {ss_between}")))
11
12# calculate everything manually
13ss_within <- sum((values - ave(values, group, FUN = mean))^2)
14print(str_glue("SS within: {ss_within}"))
15
16ss_total = sum((values - grand_mean)^2)
17print(str_glue("SS total: {ss_total}"))
18
19ms_between = ss_between / (length(unique(group)) - 1)
20print(str_glue("MS between: {ms_between}"))
21
22ms_within = ss_within / (length(values) - length(unique(group)))
23print(str_glue("MS within: {ms_within}"))
24
25f = ms_between / ms_within
26print(str_glue("F: {f}"))
27
28p = 1 - pf(f, length(unique(group)) - 1, length(values) - length(unique(group)))
29print(str_glue("p value: {p}"))
30
31
32# Perform One-Way ANOVA using aov()
33data <- tibble(group, values)
34anova_result <- aov(values ~ group, data = data)
35summary(anova_result)

Python

1import numpy as np
2from scipy import stats
3
4# Data
5group_A = [8, 9, 7, 10]
6group_B = [6, 5, 8, 7]
7group_C = [9, 10, 10, 8]
8
9
10# calculate ss for each group
11ss_A = np.sum((group_A - np.mean(group_A)) ** 2)
12ss_B = np.sum((group_B - np.mean(group_B)) ** 2)
13ss_C = np.sum((group_C - np.mean(group_C)) ** 2)
14
15ss_whithin = ss_A + ss_B + ss_C
16print(f'SS within: {ss_whithin:.4f}')
17
18# calculate ss between
19grand_mean = np.mean([np.mean(group_A), np.mean(group_B), np.mean(group_C)])
20ss_between = len(group_A) * (np.mean(group_A) - grand_mean) ** 2 + len(group_B) * (np.mean(group_B) - grand_mean) ** 2 + len(group_C) * (np.mean(group_C) - grand_mean) ** 2
21print(f'SS between: {ss_between:.4f}')
22
23print(f'SS total: {ss_whithin + ss_between:.4f}')
24ms_within = ss_whithin / (len(group_A) + len(group_B) + len(group_C) - 3)
25print(f'MS within: {ss_whithin:.4f}')
26ms_between = ss_between / 2
27print(f'MS between: {ss_between:.4f}')
28
29f = ms_between / ms_within
30print(f'F-statistic: {f:.4f}')
31
32
33
34# Perform one-way ANOVA
35print('Perform one-way ANOVA using scipy.stats')
36f_stat, p_value = stats.f_oneway(group_A, group_B, group_C)
37
38# Print results
39print(f'F-statistic: {f_stat:.4f}')
40print(f'p-value: {p_value:.4f}')

Alternative Tests

Consider these alternatives when assumptions are violated:

Kruskal-Wallis Test: Non-parametric alternative when normality is violated
Welch's ANOVA: When variances are unequal
Brown-Forsythe Test: Robust to violations of normality

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