# One way ANOVA for categorical data

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).

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

• that is an excellent explanation, gung, what is the between group and within group MSE in this example? – Heather Keturah Nov 25 '15 at 1:31
• @HeatherKeturah, there is no such thing. I tweaked my answer. – gung - Reinstate Monica Nov 25 '15 at 1:38
• @HeatherKeturah, you certainly can run an ANOVA, you just use $0$s & $1$s as your y values, but it isn't valid to do so. I don't have time to read the paper now; I'm not sure what's going on there. – gung - Reinstate Monica Nov 25 '15 at 2:29