# How to perform pairwise comparisons

## Description

When analyzing data from a completely randomized single-factor design, suppose that you have performed an ANOVA and noticed that there’s a significant difference between at least one pair of treatment levels. How can pairwise comparisons help us explore which pairs of treatment levels are different?

## Using statsmodels, in Python

View this solution alone.

The solution below uses an example dataset that details the counts of insects in an agricultural experiment with six types of insecticides, labeled A through F. (See how to quickly load some sample data.)

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from rdatasets import data
df = data('InsectSprays')
df

count spray
0 10 A
1 7 A
2 20 A
3 14 A
4 14 A
... ... ...
67 10 F
68 26 F
69 26 F
70 24 F
71 13 F

72 rows × 2 columns

Before we perform any post hoc analysis, we need to see if the count of insects depends on the type of insecticide given by conducting a one way ANOVA. (See also how to do a one-way analysis of variance (ANOVA).)

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from statsmodels.formula.api import ols
model = ols('count ~ spray', data = df).fit()
import statsmodels.api as sm
sm.stats.anova_lm(model, typ=1)

df sum_sq mean_sq F PR(>F)
spray 5.0 2668.833333 533.766667 34.702282 3.182584e-17
Residual 66.0 1015.166667 15.381313 NaN NaN

At the 5% significance level, we see that the count differs according to the type of insecticide used. We assume that the model assumptions are met, but do not verify that here.

If we would like to compare the pairs without any corrections, we can use the ‘pairwise t test’ in the scikit_posthocs package.

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import scikit_posthocs as sp
sp.posthoc_ttest(df, val_col='count', group_col='spray', p_adjust=None, pool_sd=True )

A B C D E F
A 1.000000e+00 6.044761e-01 7.266893e-11 9.816910e-08 2.753922e-09 1.805998e-01
B 6.044761e-01 1.000000e+00 8.509776e-12 1.212803e-08 3.257986e-10 4.079858e-01
C 7.266893e-11 8.509776e-12 1.000000e+00 8.141205e-02 3.794750e-01 2.794343e-13
D 9.816910e-08 1.212803e-08 8.141205e-02 1.000000e+00 3.794750e-01 4.035610e-10
E 2.753922e-09 3.257986e-10 3.794750e-01 3.794750e-01 1.000000e+00 1.054387e-11
F 1.805998e-01 4.079858e-01 2.794343e-13 4.035610e-10 1.054387e-11 1.000000e+00

Techniques to adjust the above table for multiple comparisons include the Bonferroni correction, Fisher’s Least Significant Difference (LSD) method, Dunnett’s procedure, and Scheffe’s method. These can be used in place of ‘None’ for the p.adjust argument; see details here.

You can also determine the magnitude of these differences; see how to perform post-hoc analysis with Tukey’s HSD test.

See a problem? Tell us or edit the source.

## Solution, in R

View this solution alone.

The solution below uses an example dataset that details the counts of insects in an agricultural experiment with six types of insecticides, labeled A through F. (This is one of the datasets built into R for use in examples like this one.)

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df <- InsectSprays

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count spray
1  10    A
2   7    A
3  20    A
4  14    A
5  14    A
6  12    A
7  10    A
8  23    A
9  17    A
10 20    A


Before we perform any post hoc analysis, we need to see if the count of insects depends on the type of insecticide given by conducting a one way ANOVA. (See also how to do a one-way analysis of variance (ANOVA).)

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aov1 = aov(count ~ spray, data = df)
summary(aov1)

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Df Sum Sq Mean Sq F value Pr(>F)
spray        5   2669   533.8    34.7 <2e-16 ***
Residuals   66   1015    15.4
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


At the 5% significance level, we see that the count differs according to the type of insecticide used. We assume that the model assumptions are met, but do not verify that here.

If we would like to compare the pairs without any corrections, we can use the pairwise.t.test function built into R.

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pairwise.t.test(df$count, df$spray, p.adj="none")

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Pairwise comparisons using t tests with pooled SD

data:  df$count and df$spray

A       B       C       D       E
B 0.604   -       -       -       -
C 7.3e-11 8.5e-12 -       -       -
D 9.8e-08 1.2e-08 0.081   -       -
E 2.8e-09 3.3e-10 0.379   0.379   -
F 0.181   0.408   2.8e-13 4.0e-10 1.1e-11



Techniques to adjust the above table for multiple comparisons include the Bonferroni correction, Fisher’s Least Significant Difference (LSD) method, Dunnett’s procedure, and Scheffe’s method. These can be used in place of “none” for the p.adj argument; see details here.

You can also determine the magnitude of these differences; see how to perform post-hoc analysis with Tukey’s HSD test.