Applied Nonparametric Statistics-lec9

Ref：https://onlinecourses.science.psu.edu/stat464/print/book/export/html/12

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前面我们考虑的情况是：response是连续的，variable是离散的。举例：如果打算检查GPA的中位数是否与学生坐在教室的位置有关，

那么GPA的中位数是连续的，是响应变量；学生坐的位置（前中后）是离散的，是解释变量。

现在考虑解释变量也是连续的情况，即检查两个连续变量之间的因果关系。其中，我们最关心的是关系的强弱和方向。

首先，我们考虑线性相关的情况，计算Pearson's correlation coefficient

计算Pearson's Correlation Coefficient

cor.test(x, y)

结果将给出系数cor，置信区间，p-value

计算斜率（最小二乘法拟合时）

> h=c(67, 62, 64, 65)
> w=c(120, 172, 167, 145)
> lm(w~h)

注意：这里使用的是y~x来拟合，应该是可以选用不同的公式的。结果里，斜率是-10.85

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Spearman's Rank Correlation

使用两个变量的rank值，置换计算Pearson's，就是Spearman's

Kendall's Tau Rank Correlation

measuring association by counting the number of concordant and disconcordant pairs

concordant pairs

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Bootstrap

The sample we get from sampling from the data with replacement is called the bootstrap sample

sample=sample(data, 10, replace=T)

Steps for Creating a Bootstrap Estimate of Correlation

1. Gather a bootstrap sample of size n (Think carefully how to do this).

2. Calculate the sample correlation, ri , from the bootstrap sample.

3. Repeat steps (1)-(2) B times. Typically want B to be larger than 100. I would say B = 1000 is a good number.

4. To find the (1-α)100\%CI for ρ, you would order the data and find the α/2 and 1-α/2 percentiles as the lower and upper bounds.