Kinds

Independence Test for Two Discrete Variables
Definition

Let category variables $A_{1}, \dots, A_{r}$ and $B_{1}, \dots, B_{c}$ and consider a null hypothesis $H_{0} : A, B$ are independent. Then, the test Statistic, which follows Chi-squared Distribution, is defined as $Q = j = 1 \sum b i = 1 \sum a \frac{( X _{ij} - n p ^ _{ij} ) ^{2}}{n p ^ _{ij}} \to D χ^{2} ((r - 1) (c - 1))$ where $n = j \sum i \sum X_{ij}$ , $\overset{p}{^}_{ij} = \overset{p}{^}_{i .} \overset{p}{^}_{. j} = \frac{X _{i .}}{n} \frac{X _{. j}}{n}$ , $X_{i} = j \sum X_{ij}$ , $X_{j} = i \sum X_{ij}$
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Fisher’s Exact Test

The test statistic is defined as $T = j = x_{1} \sum x_{2} \frac{( j n _{1} ) ( m _{1} - j n _{2} )}{( m _{1} n )}$ where $x_{1} = max (0, m_{1} - n_{2})$ , $x_{2} = min (n_{1}, m_{1})$

We reject $H_{0}$ if $T \leq α$
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McNemar's Test
Definition

McNemar’s test is used to analyze paired nominal data (same samples are used for both conditions), particularly in before-and-after studies or matched-pair designs.

The test statistic is defined as $Q = \frac{( n _{12} - n _{21} ) ^{2}}{n _{12} + n _{21}} \sim χ_{2} (1)$

Examples

After treatment / No insomnia After treatment / Insomnia
Before treatment / No insomnia 45 15
Before treatment / Insomnia 25 15

Consider a null hypothesis $H_{0} :$ The treatment is not effective. $Q = \frac{( 15 - 25 ) ^{2}}{15 + 25} = 2.5 < χ_{0.05}^{2} (1)$ We can not reject the null hypothesis.
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