Goodness of Fit Example

Mars candies manufactures M&Ms. According to Mars, the proportions of M&Ms with each color are given in the table below.

colors=c("Brown","Yellow","Red","Orange","Blue","Green")
proportions = c(0.13,0.14,0.13,0.20,0.24,0.16)
A=matrix(proportions,nrow=1)
colnames(A)=colors
rownames(A)=c("Proportions")
A

	Brown	Yellow	Red	Orange	Blue	Green
Proportions	0.13	0.14	0.13	0.20	0.24	0.16

Suppose we observe the following counts in a large bag of M&Ms.

observations = c(61,59,49,77,141,88)
B = matrix(observations,nrow=1)
colnames(B)=colors
rownames(B)=c("Counts")
B

	Brown	Yellow	Red	Orange	Blue	Green
Counts	61	59	49	77	141	88

First, let's carry out a Goodness of Fit test. The command is below. Notice, this command doesn't need any fancyness with matrices or column names, etc. I just added that stuff above to make things easier to read.

chisq.test(observations, p = proportions)

	Chi-squared test for given probabilities

data:  observations
X-squared = 15.188, df = 5, p-value = 0.00959

Since the p-value is so small (less than 1%), it seems highly likely that there is a non-random reason why the distribution of colors in the sample does not match the advertised distribution of colors. It could be that our sample was not well mixed, or someone tampered with the sample, or maybe the advertised proportions are not correct.

Approximating Goodness of Fit Using Two-Way Tables

N=100000
A=rbind(observations,N*proportions)
colnames(A)=colors
rownames(A)=c("observations","enlarged proportions")
A

	Brown	Yellow	Red	Orange	Blue	Green
observations	61	59	49	77	141	88
enlarged proportions	13000	14000	13000	20000	24000	16000

chisq.test(as.table(A))

	Pearson's Chi-squared test

data:  as.table(A)
X-squared = 15.113, df = 5, p-value = 0.009889

As long as we scale the proportions by a large factor N, the chi-squared test for two-way tables will be close to the test for goodness of fit.