Test for Independent

  • Two categorical variables
  • Are they related?

Test for Homogeniety

  • One categorical variable
  • Does it act the same for different populations?

Test for Independent

Took a random sample of 300 students from some university.

For each student, recorded their status as an undergraduate student or a graduate student.

Also recorded whether the majority of their classes are given in-person, remotely, or in a hybrid manner

Question: Is student status independent of class delivery type?

  In-person Remote Hybrid Total
Ungrads 121 47 31 199
Grads 33 57 11 101
Total 154 104 42 300

  • Estimated probability person in sample is an undergraduate is $\frac{199}{300}$.

  • Estimated probability persion in sample is taking a majority of in-person courses is $\frac{154}{300}$.

  • If delivery type and student type are independent, the probability that fall into “Ungrads - In-person” category should be $\frac{199}{300}.\frac{154}{300}$

  • The expected number of people, out of the total of 300 sampled, who fall into “Ungrads - In-person” category should be

    \[\frac{199}{300}.\frac{154}{300}.300=\frac{(199)(154)}{300}\approx102.5\]

  • Under the assumption of independence, the expected counts are

  In-person Remote Hybrid Total
Ungrads $\frac{(199)(154)}{300}$ $\frac{(199)(104)}{300}$ $\frac{(199)(42)}{300}$ 199
Grads $\frac{(101)(154)}{300}$ $\frac{(101)(104)}{300}$ $\frac{(101)(42)}{300}$ 101
Total 154 104 42 300
\[\begin{align} &H_0:\small{\text{Student status and class delivery type are independent}}\\ &H_1:\small{\text{Student status and class delivery type are not independent}}\\ \end{align}\]

Test statistic

\[W:=\sum_{i}\frac{(O_i-E_i)^2}{E_i}\sim\chi^2(?)\]

Degrees of freedom parameter is

\[\color{red}{ (\text{number of rows}-1)(\text{number of cols}-1) }\]

Reject $H_0$, in favor of $H_1$ if $W$ is “large”.

obs <- matrix(c(121,47,31,33,57,11), nrow = 2, ncol = 3, byrow = T)

total <- sum(obs)

exp <- matrix(0, 2, 3)

n_rows <- dim(exp)[1]
n_cols <- dim(exp)[2]

for(i in 1:n_rows){
   for(j in 1:n_cols){
        exp[i,j] = sum(obs[i,])*sum(obs[,j])/total
   }
}

W <- sum((obs - exp)^2/exp)
W

32.1930892630927

For our example, the test statistic is

\[W\approx32.193\]

The critical value $(\alpha=0.10)$ is

\[\chi^2_{0.10,2}=4.60517\] 
dof = (n_rows - 1)*(n_cols - 1)
dof

qchisq(1-0.10, dof)

2

4.60517018598809

Reject $H_0$ because $W\gt\chi^2_{0.10,2}$.

There is sufficient evidence in the data to conclude that student status and class delivery type are dependent at level $0.10$.

Test for Homogeneity

  • Took a radom sample of 199 undergraduate students.
  • Took an independent random sample of 101 graduate students.
  • For each sample, recorded whether the majority of their classes are given in-person, remotely, or in a hybrid manner.

Question: Is the distribution of in-person, to remote, to hybrid the same for both groups?

  In-person Remote Hybrid Total
Ungrads 121 47 31 199
Grads 33 57 11 101
Total 154 104 42 300
\[\begin{align} &H_0:\small{\text{The distribution of class type is the same for undergraduate and graduate students.}}\\ &H_1:\small{\text{Not }H_0.}\\ \end{align}\]
  • Overall “in-person” probability is estimated to be $\frac{154}{300}$.
  • Since there are 199 undergrads, the expected number in the in-person group under $H_0$ is $\frac{154}{300}.199$.

Note:

  • Expected number under the assumption of independence:
\[\frac{199}{300}.\frac{154}{300}.300\quad\text{(1)}\]
  • Expected number under the assumption of homogeneity:
\[\frac{154}{300}.199\quad\text{(Same as (1))}\]

Because we use the same data, so we will have the same result as the assumption of independence.