Brown-Mood Median Test
对于两独立样本尺度中的位置参数（中位数）检验问题：
\(H_0: med_x = med_y\) \(H_1=med_x > med_y\)
在\(H_0\)假设下，两组数据具有相同的中位数
核心是比较\(med_xy\) 和\(med_x\) \(med-y\)的关系。
| X | Y | sum | |
|---|---|---|---|
| >\(M_{xy}\) | A | B | t |
| <\(M_{xy}\) | C | D | (m+n)-(A+B) |
| \(\sum\) | m | n | m=n=A+B+C+D |
when m,n,t are fixed.
则A的密度函数如下：
\(P(A=k)=\frac{\binom{m}{k}\binom{n}{n-k}}{\binom{m+n}{t}}\), k\(\leqslant\) min{m,t}
以下是Brown-Mood R代码
#Brown-Mood median test
BM.test <- function(x,y,alt)
{
xy <- c(x,y)
md.xy <-median(xy)
t <- sum(xy > md.xy)
lx <- length(x)
ly <- length(y)
lxy <- lx + ly
A <- sum(x > md.xy)
if(alt == "greater")
{w <- 1 - phyper(A,lx,ly,t)}
else if (alt == "less")
{w <- phyper(A,lx,ly,t)}
conting.table = matrix(c(A,lx-A,lx,t-A,ly-(t-A),ly,t,lxy-t,lxy),3,3)
col.name <- c("X","Y","X+Y")
row.name <- c(">MXY","<MXY","TOTAL")
dimnames(conting.table)<-list(row.name,col.name)
list(contingencu.table=conting.table,p.value = w)
}这里要特别注意phyper()函数的参数中关于 lower.tail = TRUE 和 FALSE的定义，防止出现多余计算。
\[phyper(q,m,n,k) = \frac{\binom{m}{k}\binom{n}{n-k}}{\binom{m+n}{t}}, k\leqslant min(m,t)\]
大样本计算时，趋于正态分布:
binom 分布的 \[E[X] = μ = t\cdot\frac{m}{m+n}\]
\[D[x]\]
example:
x <- c(698,688,675,656,655,648,640,639,620)
y <- c(780,754,740,712,693,680,621)
BM.test(x,y,"less")
$contingencu.table
X Y X+Y
>MXY 2 6 8
<MXY 7 1 8
TOTAL 9 7 16