Fisher Exact Test
Fisher exact model.
We consider following 2x2 contingency table:
- $N$,
- the total number of people
- $K$,
- the number of people under YES condition
- $n$,
- the number of people in treatment group
| Condition | treatment group | control group | total |
|---|---|---|---|
| YES | $k$ | $K − k$ | $K$ |
| NO | $n − k$ | $N + k − n − K$ | $N − K$ |
| total | $n$ | $N − n$ | $N$ |
In Fisher exact model, we interpret $k$ is a observed value of r.v. $X^{t}$ which follows hypergeometric distribution with parameters $N$, $K$, $n$.
The p.d.f. and c.d.f. of $X^{t}$ is given by
\[\begin{eqnarray} \mathrm{Hyper}(k; N, K, n) & = & P \left( X^{t} = k \right) \nonumber \\ F_{\mathrm{Hyper}}(k; N, K, n) & = & \sum_{i=0}^{k} P \left( X^{t} = k \right) . \nonumber \end{eqnarray}\]Let $\alpha \in [0, 1]$ be significance level. Then p-value $p_{\alpha}$ is calculated by
\[\begin{eqnarray} k_{\alpha} & := & \inf \{ k \mid F_{\mathrm{Hyper}}(x^{t}; N, K, n) \le \alpha \} \\ p_{\alpha} & := & F_{\mathrm{Hyper}}(k_{\alpha}; N, K, n) \end{eqnarray}\]Example
- $N = 30$,
- $K = 19$,
- $n = 15$,
- \(x_{i}^{c} \in \{0, 1\}\),
- 1 means that $i$-th person became infected with influenza
- people innoculated with a recombinant DNA influenza vaccine
- control group
- \(x_{i}^{t} \in \{0, 1\}\),
- 1 means that $i$-th person became infected with influenza
- people innoculated with a placebo
- treatment group
- $x^{c} := \sum_{i=1}^{n} x_{i}^{c}$,
- the number of infected people in the control group
- $x^{t} := \sum_{i=1}^{n} x_{i}^{t}$,
- the number of infected people in the treatment group
| Infection status | Vacctine | Placebo | Total |
|---|---|---|---|
| Yes = 1 | 7 = $x^{t}$ (47%) | 12 = $x^{c}$ (80%) | 19 |
| No = 0 | 8 (53%) | 3 (20%) | 11 |
| Totals | 15 | 15 | 30 |
Fisher exact model
- $X^{t}$,
- hypergemetric distribution with $N := 30$, $K := 19$, $n := 15$.
- \(X^{t}(\omega) = x^{t}\),
- $\Theta := [0, 1]$,
One-side test
\[\begin{eqnarray} F_{\mathrm{Hyper}}(x^{t}; N, K, n) & = & \sum_{k = 0}^{x^{t}} \mathrm{Hyper}(k; N, K, n) \nonumber \\ & \approx & 0.0640679660169915 \end{eqnarray}\]