# CA trend, monotone, Fisher exact
###### tags: `Categorical data analysis`

## Ix2 table
對於特殊的Ix2表格，即$y = \{0, 1\}$，可以檢測linear probability，即是$x$的值跟$y = 1$的機率有無線性關係。
定義:
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$\pi_{i} = P[Y = 1 | X = x_{i}] = p_{i}$
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想要檢驗有無線性關係，便可假設為

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$\pi_{i} = \alpha + \beta x_i$
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由least square method算出的結果便可得到


其中
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$\bar{x} = \frac{\sum_{i} x_{i}}{n}$
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### 假設檢定
要檢定是否有trend, that is, to test whether $\beta = 0$, we use <b>Cochran-Armitage trend test</b>
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$H_0: \beta = 0$
$H_a: \beta \neq 0$

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### SAS code

proc freq data = <data>;
weight <var>;
table <explanetory> * <response> / trend;
run;

### SAS implementation


第一個值是test statistic的z值，第二個值是單尾檢定，第三個值雙尾檢定

## Monotone relation
對於類別ordinal資料，有時候linear relation其實很難找，這時就退而求其次找monotone就好。要檢測monotone需要先定義一些名詞。

$i, j$為X跟Y的rank
<b>Concordant pair:</b> For $x_{ij}$, if $(i^{'} > i) \land (j^{'} > j)$, then $x_{i'j'}$ is called the concordant pair of $x_{ij}$

<b>Discordant pair:</b> if $(i^{'} > i) \land (j^{'} < j)$, then $x_{i'j'}$ is called the discordant pair of $x_{ij}$

此時numbers of concordant pair(NC): $\sum_{i}\sum_{j} x_{ij} * (\sum_{k} x_{k})$ where $x_k$ is the concordant pair of $x_{ij}$
number of discordant pair(ND): $\sum_{i}\sum_{j} x_{ij} * (\sum_{k} x_{k})$ where $x_k$ is the discordant pair of $x_{ij}$

### Test statistic
Here, we will use <b>Gamma measurement</b> to measure if the data has monotone relation
Define

Then the test statistic is

For the sample:
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$\hat{\gamma} = \frac{NC - ND}{NC + ND}$
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Note:
1. Gamma is symmertric, that is col跟row互換不會影響值
2. $|\gamma| \leq 1$

### SAS code

proc freq data = <data>;
weight <var>;
table <explanetory> * <response> / measures;
run;

### SAS implementation


## Fisher exact test
在cell的數量小時，卡方會不準確，在表格為Ix2的情況下，可以用fisher exact test
其假設檢定為

$H_0: X, Y$ independent
$H_a: X, Y$ dependent
而其p-value有很多種算法

SAS在計算two-sided pvalue是用第一種算法

### SAS code

proc freq data = <data>;
weight <var>;
table <explanetory> * <response> / exact;
run;

### SAS implementation