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date-created: 2023-05-20T21:26:03++0800
date-modified: 2023-05-20T21:26:03++0800
Fisher’s exact test is a statistical test used to determine the significance of the association between two categorical variables in a 2x2 [[Contingency table|contingency table]]. It was developed by Sir [[Ronald Fisher]] in the early 20th century.
To perform Fisher’s exact test, you start with
$$
\begin{array}{|r|c|c|}
\hline & \text { B1 } & \text { B2 } \
\hline \text { A1 } & a & b \
\hline \text { A2 } & c & d \
\hline
\end{array}
$$
They hypothesis is
$$
H_0: \text{no association} \text{ v. s. } H_a: \text{there’s association}
$$
一般来说, 我们进一步扩展这个联表
$$
\begin{array}{|r|c|c|c|}
\hline & \text { B1 } & \text { B2 } & \text { Total } \
\hline \text { A1 } & a & b & a+b \
\hline \text { A2 } & c & d & c+d \
\hline \text { Total } & a+c & b+d & n=a+b+c+d \
\hline
\end{array}
$$
- 扩展出的 $a+b$ 等项称为 [[Marginal total|marginal totals]], 而
- $n$ 称为 [[Grand total|grand total]].
在 Fisher’s exact test 中, 在零假设成立的前提下, a 符合[[超几何分布]]
$$p=\frac{\left(\begin{array}{c} a+b \ a \end{array}\right)\left(\begin{array}{c} c+d \ c \end{array}\right)}{\left(\begin{array}{c} n \ a+c \end{array}\right)}=\frac{\left(\begin{array}{c} a+b \ b \end{array}\right)\left(\begin{array}{c} c+d \ d \end{array}\right)}{\left(\begin{array}{c} n \ b+d \end{array}\right)}=\frac{(a+b)!~(c+d)!~(a+c)!~(b+d)!}{a!~b!~c!~d!~n!}$$
这就直接给出了 p-value.
CALCULATIONS WITH R
Pearson's Chi-squared test with Yates' continuity correction
data: counts
X-squared = 5.5646, df = 1, p-value = 0.01833
Fisher's Exact Test for Count Data
data: counts
p-value = 0.02552
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
0.9918169 15.9604612
sample estimates:
odds ratio
5.039206
Warning message:
In chisq.test(counts) : Chi-squared近似算法有可能不准
FEATURES
- Fisher’s Exact Test的一个主要优点是，与其他一些用于列联表的统计方法（如卡方检验）相比，它不需要对数据的分布做出任何假设。这使得它在样本量较小的情况下特别有用。
- 然而，对于较大的样本或者类别较多的分类变量，计算Fisher’s Exact Test的p值可能会非常复杂和耗时。
REFERENCES
RELATED NOTES
- [[Chi-squared test]]
- [[P-value]]
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