17.3.2.2 Algorithms (Two-Sample T-Test)

The two sample t-test calculates a Student's t statistic and the associated probability to test whether or not the difference of the two sample means equals to \(\mu_d\,\!\) (i.e. to test whether or not their means are equal, you can just test whether or not their difference is 0, \(\mu_1-\mu_2=\mu_d=0\,\!\) ). And the hypotheses take the form:

\(H_0:\mu_1-\mu_2=\mu_d\,\!\) vs \(H_1:\mu_1-\mu_2 \ne \mu_d\) Two Tailed

\(H_0:\mu_1-\mu_2 \le \mu_d\) vs \(H_1:\mu_1-\mu_2 > \mu_d\) Upper Tailed

\(H_0:\mu_1-\mu_2 \ge \mu_d\) vs \(H_1:\mu_1-\mu_2 < \mu_d\) Lower Tailed

Test Statistics

Consider two independent samples\(x_1\,\!\)and\(x_2\,\!\), of size \(n_1\,\!\) and \(n_2\,\!\) drawn from two normal population with means \(\mu_1\,\!\) and \(\mu_2\,\!\), and variances \(\sigma_1^2\,\!\) and \(\sigma_2^2\,\!\) respectively, we have:

\(\bar{x}_1=\frac{1}{n_1}\sum_{j=1}^{n_1}x_{1j}\), \(\bar{x}_2=\frac{1}{n_2}\sum_{j=1}^{n_2}x_{2j}\), \(s_1^2=\frac{1}{n_1-1}\sum_{j=1}^{n_1}{(x_{1j}-\bar{x}_1)^2}\), \(s_2^2=\frac{1}{n_2-1}\sum_{j=1}^{n_2}{(x_{2j}-\bar{x}_2)^2}\)

where \(\bar{x}_1\,\!\)and\(\bar{x}_2\,\!\) are sample means and \(s_1^2\,\!\) and \(s_2^2\,\!\) are sample variances. Then we can compute the t test statistic by:

For equal variance is assumed, that is \(\sigma_1^2=\sigma_2^2\,\!\):

In this case the test statistic t:

\[t=\frac{(\bar{x}_1-\bar{x}_2)-\mu_d}{s_p\sqrt{(1/n_1+1/n_2)}}\]

has a t-distribution with \((v = n_1+n_2-2)\) degrees of freedom and

\[s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}\]

is the pooled variance of the two samples.

For equal variance is not assumed:

In this case the usual two sample t-statistic no longer has a t-distribution and an approximate test statistic, t'is used:

\[t'=\frac{(\bar{x}_1-\bar{x}_2)-\mu_d}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]

And a t-distribution with v degrees of freedom is used to approximate the distribution of t'where

\[v=\frac{(s_1^2/n_1+s_2^2/n_2)^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}}\]

Then compare the t value with the critical value and we will reject \(H_0\,\!\) if:

Two tailed test: \(|t| > t_{\sigma/2}\,\!\);

Upper tailed test: \(t > t_\sigma\,\!\);

Lower tailed test: \(t < -t_\sigma\,\!\);

The p-value will also be compared with a user-defined significance level,\(\sigma\,\!\), which commonly 0.05 is used. And the null hypothesis \(H_0\,\!\) will be rejected if \(p < \mu\,\!\).

Confidence Intervals

The upper and lower \((1-\sigma )\times 100\%\) confident limits for mean difference \((\mu_1 - \mu_2)\,\!\) are calculated as:

For equal variance is assumed:

Null Hypothesis	Confidence Interval
\[H_0:\mu_1-\mu_2=\mu_d\,\!\]	\[\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, (\bar{x}_1-\bar{x}_2)+ t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]\]
\[H_0:\mu_1-\mu_2 \le \mu_d\]	\[\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, \infty\right]\]
\[H_0:\mu_1-\mu_2 \ge \mu_d\]	\[\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]\]

For equal variance is not assumed:

Null Hypothesis	Confidence Interval
\[H_0:\mu_1-\mu_2=\mu_d\,\!\]	\[\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, (\bar{x}_1-\bar{x}_2)+ t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]\]
\[H_0:\mu_1-\mu_2 \le \mu_d\]	\[\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \infty\right]\]
\[H_0:\mu_1-\mu_2 \ge \mu_d\]	\[\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]\]

where \(t_{\sigma/2}\,\!\) is the critical value of the t-distribution with v degrees of freedom.

Power Analysis

The power of a two sample t-test is a measurement of its sensitivity. Detail algorithm about calculating power please read the help of Power and Sample Size.

Reference

The two-sample t-test is implemented with a Nag function, nag_2_sample_t_test (g07cac). Please refer to the corresponding Nag document for more details on the algorithm.