07_chi-Square_tests.qmd

# Chi-Square tests

Pearson's chi-square test is used to make two types of comparisons for
categorical variables. A *goodness of fit test* determines whether a
particular hypothesized distribution is reasonable for a categorical
variable. A *test of independence* assesses whether two categorical
variables are independent of each other.

## Goodness of Fit

Suppose a company wants to determine if employees are equally likely to
call in sick on any day of the week. That is, test the hypothesis
$$H_0: p_1 = p_2 = p_3 = p_4 = p_5$$ where $p_1$ represents the
probability that a random call-in sick occurs on Monday, etc. The
employer collects data and obtains the following table

::: center
  ----- ----- ----- ----- -----
    Mon   Tue   Wed   Thu   Fri
     34    18    19    12    26
  ----- ----- ----- ----- -----
:::

If $H_0$ is true, the expected value in cell $i$ is $E_i = np_i$, where
$n = 34 + 18 + 19 + 12 + 26 = 109$ and $p_i = 0.2$. The differences
between the observed and expected values are combined into the $\chi^2$
statistic. $$\chi^2 = \sum_i\frac{(O_i - E_i)^2}{E_i}$$ If the null
hypothesis is true, the $\chi^2$ statistic approximately follows a
$\chi^2(k-1)$ distribution, where $k$ is the number of groups (5). This
approximation is poor if the expected value in any cell is small.

We first need to get the data into **[R]{.sans-serif}**.

`> days = c(34, 18, 19, 12, 26)`

`> names(days) = c("Mon", "Tue", "Wed", "Thu", "Fri")`

`> days`

Many chi-square tests are performed using the `chisq.test` function.

`> chisq.test(days)`

**[R]{.sans-serif}** returns the $\chi^2$ test statistic, the degrees of
freedom, and the p-value. Notice that we didn't specify that we were
testing the hypothesis that were testing the hypothesis of *equal*
probabilities. By default, `chisq.test` assumes a test of equal
probabilities unless the user specifies otherwise. If, for some reason,
we wanted to test the hypothesis that $p_1 = 0.35$, $p_2 = 0.1$,
$p_3 = 0.1$, $p_4 = 0.1$, $p_5 = 0.35$, we can do so by passing the
probabilities to `chisq.test` in this way.

`> chisq.test(days, p=c(0.35, 0.1, 0.1, 0.1, 0.35))`

So far, the output for the chi-square tests have been limited. What if
we wanted to obtain expected values or cell residuals? We unleash more
of **[R]{.sans-serif}**'s capabilities by storing the value of the test
as an object.

`> cs.test = chisq.test(days)`

`> names(cs.test)`

`> cs.test$expected`

`> cs.test$statistic`

`> cs.test$residuals`

## Chi-Square Test of Independence

A Chi-Square Test of Independence tests for a relationship between two
categorical variables. We will use the "UCBAdmissions data set" to
illustrate.

`> data()`

`> UCBAdmissions`

`> class(UCBAdmissions)`

`> dim(UCBAdmissions)`

Let's select the second 2 by 2 table to perform a chi-square test of
independence.

`> dat = UCBAdmissions[,,2]`

`> dat`

When we pass a vector of values to `chisq.test`, a test of homogeneity
is performed by default. When we pass a table or matrix, a test of
independence is performed.

`> chisq.test(dat)`

The default in **[R]{.sans-serif}** is to use Yates' continuity
correction for 2x2 tables. You can opt not to use it.

`> chisq.test(dat, correct=FALSE)`

## Fisher's Exact Test

Pearson Chi-Square tests rely on an approximation that becomes poor as
cell sizes get small. With smaller samples, Fisher's Exact Test might be
a good alternative. However, there are many flavors of Fisher's test.
**[R]{.sans-serif}** has implemented many of them, including one in the
base package.

`> fisher.test(dat)`