Statistical Hypothesis Testing with SAS and R

by Dirk Taeger and Sonja Kuhnt

(c) John Wiley & Sons, Ltd

Test 2.2.3: Welch test

Description:

Tests if two population means \(\mu_1\) and \(\mu_2\) differ less than, more than or by a value \(d_0\).

Assumptions:

• Data are measured on an interval or ratio scale.
• Data are randomly sampled from two independent Gaussian distributions.
• The standard deviations \(\sigma_1\) and \(\sigma_2\) of the underlying Gaussian distributions are unknown and not equal and estimated through the pooled population standard deviation.

Hypotheses:

1. \(H_0\) : \(\mu_1 - \mu_2 = d_0\) vs \(H_1\) : \(\mu_1 - \mu_2 \neq d_0\).
2. \(H_0\) : \(\mu_1 - \mu_2 \le d_0\) vs \(H_1\) : \(\mu_1 - \mu_2 \gt d_0\).
3. \(H_0\) : \(\mu_1 - \mu2 \ge d_0\) vs \(H_1\) : \(\mu_1 - \mu_2 \lt d_0\).

Test statistic:

\[T = \frac{(\bar{X_1} - \bar{X_2}) - d_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\] with \[S_j = \sqrt{\frac{1}{n_j-1}\sum_{i=1}^n (X_j - \bar{X_j})^2}\]

for \(j= 1,2\)

Test decision:

Rejection \(H_0\) if for the observed value t of T:

1. \(t \lt t_{\alpha/2, v}\) or \(t \gt t_{1-\alpha/2, v}\)

2. \(t \gt t_{1-\alpha, v}\)

3. \(t \lt t_{\alpha,v}\)

P-value:

1. \(\rho = 2P(T \le (-|t|))\)

2. \(\rho = 1 - P(T \le t)\)

3. \(\rho = P(T \le t)\)

Annotation:

• This test is also known as a two-sample t-test or Welch-Satterthwaite test.
• The test statistic \(T\) approximately follows a t-distribution with

\[v = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\left(\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}\right)} \] degree of freedom of [Bernard Welch (1947) and Franklin Satterthwaite (1946) approximation].

• \(t_{\alpha,v}\) is the \(\alpha\) quantitle of t-distribution with \(v\) degree of freedom.

• William Cochran and Gertrude Cox (1950) proposed an alternative way to calculate critical values for the test statistic.

• The assumtion of underlying Gaussian distribution can be relaxed if the sample sizes of both samples are large. Usually sample sizes \(n_1, n_2 \ge 25\) or \(30\) for both distributions are considered to be large enough.

Example

To test the hypothesis that the mean systolic boold pressures of healthy subjects (status=0) and subjects with hypertention (status=1) are equal, hence \(d_0 = 0\). The dataset contains \(n_1 = 25\) subjects with status 0 and \(n_2 = 30\) with status 1.

#Blood_pressure dataset
no <- seq(1:55)
status <- c(rep(0, 25), rep(1, 30))
mmhg <- c(120,115,94,118,111,102,102,131,104,107,115,139,115,113,114,105,
          115,134,109,109,93,118,109,106,125,150,142,119,127,141,149,144,
          142,149,161,143,140,148,149,141,146,159,152,135,134,161,130,125,
          141,148,153,145,137,147,169)
blood_pressure <-data.frame(no,status,mmhg)

status0<-blood_pressure$mmhg[blood_pressure$status==0]
status1<-blood_pressure$mmhg[blood_pressure$status==1]

t.test(status0,status1,mu=0,alternative="two.sided",var.equal=FALSE)

## 
##  Welch Two Sample t-test
## 
## data:  status0 and status1
## t = -10.451, df = 50.886, p-value = 2.887e-14
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -37.32904 -25.29763
## sample estimates:
## mean of x mean of y 
##  112.9200  144.2333

Remarks:

• \(mu=value\) is optional and defines the value \(mu_0\) to test against. Default is 0.
• \(alternative="value"\) is optional and defines the type of alternative hypothesis: “two.sided”= two sided (A); “greater”=true mean difference is greater (B); “less”=true mean difference is lower (C). Default is “two.sided”.

Noted: Sang Nguyen

Nashville,TN - NOV 2016