Political association

Description

From example 4.2

A 2×3 contingency table of political party association (Democrat / Independent / Republican) by gender (female / male). Used to illustrate bar charts, mosaic plots, and the chi-squared test for $r\times c$ tables.

	Dem	Ind	Rep
female	762	327	468
male	484	239	477

Bar chart

From Bar charts

A row-proportion bar chart shows the political-party mix within each gender. The key step is dividing each cell by its row total so we compare distributions (not raw counts).

R code

library(lattice)
 
x <- matrix(c(762, 327, 468,
              484, 239, 477), ncol=3, byrow=TRUE)
dimnames(x) <- list(c("female", "male"),
                    c("Dem", "Ind", "Rep"))
political_tab <- as.table(x)
 
barchart(political_tab / rowSums(political_tab),
         main="Political association by gender",
         horizontal = FALSE, auto.key=TRUE)

Mosaic plot

From Mosaic Plots

Unlike a bar chart, a mosaic plot encodes both the count (tile area) and the conditional proportions. With shade=TRUE, cells are coloured by their standardised residual under the independence model:

• Blue — more observations than expected under independence
• Red — fewer observations than expected

This makes it a visual partner to the chi-squared test: coloured cells are the cells driving the test’s significance.

R code

mosaicplot(political_tab, shade=TRUE,
           main="Political association by gender")

Python code

from statsmodels.graphics.mosaicplot import mosaic
import numpy as np
 
political_tab = np.asarray([[762, 327, 468],
                            [484, 239, 477]])
mosaic(political_tab, statistic=True, gap=0.05);

Chi-squared test (r × c)

From Chi-squared Test for r x c Tables

For an $r\times c$ table the test is identical in form to the 2×2 case; under $H_0$ the statistic follows ${\chi }_{(r-1)(c-1)}^2$. Here $df=(2-1)(3-1)=2$.

R code

chisq.test(political_tab)
 
# Standardised residuals — tell us which cells drive the (non-)fit
chisq.test(political_tab)$stdres

Standardised residuals

$$r_{ij}=\frac{n_{ij}-{\mu }_{ij}}{\sqrt{{\mu }_{ij}(1-p_{i+})(1-p_{+j})}}$$

should be roughly $N(0,1)$ under $H_0$. Large $|r_{ij}|$ flags where the data deviates from independence — the colours in the mosaic plot above correspond to the sign of these residuals.

See Chi-Square & Fisher for the general procedure and interpretation tips.

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See also: L4 Exploring Categorical Data · Chi-Square & Fisher