Dunnett's Test

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Dunnett's Test

Definition

Dunnett's Test is a multiple comparison procedure used to compare several treatments against a single control group. It maintains the family-wise error rate while providing more statistical power than methods that compare all pairs.

Formula

Test Statistic:

t = \frac{\bar{X}_i - \bar{X}_c}{SE}

Where:

$\bar{X}_i$ = mean of treatment group $i$
$\bar{X}_c$ = mean of the control group
$n_i$ = sample size of treatment group $i$
$n_c$ = sample size of the control group
$SE$ = standard error of the mean

The standard error of the mean is calculated as:

SE = s \times \sqrt{\frac{1}{n_i} + \frac{1}{n_c}}

Where $s^2$ is the pooled variance, combining the variances of all groups:

s^2 = \frac{\sum_{i=1}^k(n_i-1)s_i^2}{\sum_{i=1}^k(n_i-1)}

In ANOVA, the pooled variance corresponds to the within-group variance:

s^2 = MS_{\text{Within}} = \frac{SS_{\text{Within}}}{df_{\text{Within}}}

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	Data	N	Mean	SD
Control	8.5, 7.8, 8, 8.2, 7.9	5	8.08	0.27
Treatment A	9.1, 9.3, 8.9, 9, 9.2	5	9.10	0.16
Treatment B	7.2, 7.5, 7, 7.4, 7.3	5	7.28	0.19
Treatment C	10.1, 10.3, 10, 10.2, 9.9	5	10.10	0.16

Step 2: State Hypotheses

For each treatment group i vs control:

$H_0: \mu_i = \mu_c$
$H_a: \mu_i \neq \mu_c$
$\alpha = 0.05$

Step 3: Calculate SS and MSE

$SS_{Between} = 22.532$ with $df = 19$
$SS_{Within} = 0.656$ with $df = 16$
$SS_{Total} = SS_{Between} + SS_{Within} = 23.188$
$MS_{E} = \frac{0.656}{16} = 0.041$

Step 4: Calculate Test Statistics

For each treatment vs control:

Treatment A: $t = \frac{9.10 - 8.08}{\sqrt{\frac{2(0.041)}{5}}} = 7.965$
Treatment B: $t = \frac{7.28 - 8.08}{\sqrt{\frac{2(0.041)}{5}}} = -6.247$
Treatment C: $t = \frac{10.10 - 8.08}{\sqrt{\frac{2(0.041)}{5}}} = 15.774$

Step 5: Compare with Critical Value

Critical value for $\alpha = 0.05$ , $k = 3$ treatments, $df = 16$ : $t_{crit} = 2.74$

Compare $|t|$ with critical value:

Treatment A: $|7.965| > 2.74$ (Significant)
Treatment B: $|-6.247| > 2.74$ (Significant)
Treatment C: $|15.774| > 2.74$ (Significant)

Step 6: Draw Conclusions

All treatments show significant differences from the control group ( $p \lt 0.05$ ):

Treatment A significantly increases response
Treatment B significantly decreases response
Treatment C shows largest significant increase

Code Examples

1library(multcomp)
2library(tidyverse)
3# Data preparation
4df = tibble(
5  Group = rep(c("Control", "Treatment A", "Treatment B", "Treatment C"), each = 5),
6  Response = c(8.5, 7.8, 8, 8.2, 7.9, 
7               9.1, 9.3, 8.9, 9, 9.2, 
8               7.2, 7.5, 7, 7.4, 7.3, 
9               10.1, 10.3, 10, 10.2, 9.9)
10)
11
12# multcom package requires the Group variable to be a factor
13df$Group <- as.factor(df$Group)
14
15# Perform one-way ANOVA
16model <- aov(Response ~ Group, data = df)
17
18# Perform Dunnett's test
19dunnett_test <- glht(model, linfct = mcp(Group = "Dunnett"))
20summary(dunnett_test)

Python

1import numpy as np
2from statsmodels.stats.multicomp import pairwise_tukeyhsd
3import pandas as pd
4
5# Create example data
6data = pd.DataFrame({
7    'Response': [8.5, 7.8, 8.0, 8.2, 7.9,  # Control
8                9.1, 9.3, 8.9, 9.0, 9.2,   # Treatment A
9                7.2, 7.5, 7.0, 7.4, 7.3,   # Treatment B
10                10.1, 10.3, 10.0, 10.2, 9.9], # Treatment C
11    'Group': np.repeat(['Control', 'Treatment A', 
12                       'Treatment B', 'Treatment C'], 5)
13})
14
15# Perform multiple pairwise comparisons using Tukey's test
16# (Note: Dunnett's test focuses only on comparisons against a control group, 
17# but statsmodels does not provide a direct implementation of Dunnett's test.)
18results = pairwise_tukeyhsd(data['Response'], 
19                           data['Group'])
20
21print(results)

Alternative Tests

Tukey's HSD: When comparing all groups to each other
Bonferroni: When a simpler but more conservative approach is needed
Games-Howell: When variances are unequal

Related Calculators

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Dunnett's Test

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Dunnett's Test

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Calculate SS and MSE

Step 4: Calculate Test Statistics

Step 5: Compare with Critical Value

Step 6: Draw Conclusions

Code Examples

Alternative Tests

Related Calculators

One-Way ANOVA Calculator

Tukey's HSD Test Calculator

Two-Way ANOVA Calculator

Effect Size Calculator

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