One way ANOVA for categorical data

Question

I know how to do one way ANOVA when the response variable is continuous. I need help learning how to do one way ANOVA when the response variable is binary (i.e., how to calculate MSE within group and between groups when the response is binary).

Answer 1

When your response is binary, you don't do an ANOVA. (For example, there is no within or between MSE.) Instead, you can do a chi-squared test, or you can fit a logistic regression model. Here is a simple example (with made up data), coded in R:

tab = as.table(matrix(c(12, 33, 19,
                        38, 17, 31 ), nrow=2, byrow=TRUE))
rownames(tab)        = c("Yes", "No")
names(dimnames(tab)) = c("y", "x")
tab
#      x
# y      A  B  C
#   Yes 12 33 19
#   No  38 17 31
chisq.test(tab)
#         Pearson's Chi-squared test
# X-squared = 18.695, df = 2, p-value = 8.716e-05
d = as.data.frame(tab)
d
#     y x Freq
# 1 Yes A   12
# 2  No A   38
# 3 Yes B   33
# 4  No B   17
# 5 Yes C   19
# 6  No C   31
lr.mod = glm(y~x, d, family=binomial, weights=Freq)
summary(lr.mod)
# ...
# Coefficients:
#             Estimate Std. Error z value Pr(>|z|)    
# (Intercept)   1.1527     0.3311   3.481 0.000499 ***
# xB           -1.8160     0.4458  -4.073 4.64e-05 ***
# xC           -0.6631     0.4410  -1.504 0.132701    
# ...
#     Null deviance: 204.71  on 5  degrees of freedom
# Residual deviance: 185.62  on 3  degrees of freedom
# ...
1-pchisq(q=(204.71 - 185.62), df=(5 - 3))
# [1] 7.155816e-05