17.3.2.2 Algorithms (Two-Sample T-Test)tTest-TwoSample-Algorithm
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 /math-12f3be74f7266e038bdf61eb68a6208c.png?v=0 "\mu_d\,\!") (i.e. to test whether or not their means are equal, you can just test whether or not their difference is 0, /math-590eba6400c9e4df07e0c234f91e4460.png?v=0 "\mu_1-\mu_2=\mu_d=0\,\!") ). And the hypotheses take the form:
/math-f67a81ef315f0c59070ef53cb50b3932.png?v=0 "H_0:\mu_1-\mu_2=\mu_d\,\!") vs /math-5c09300d675833793616502094acaa0f.png?v=0 "H_1:\mu_1-\mu_2 \ne \mu_d") Two Tailed
/math-1fc800dcea3acfe50f9e90c310b1174a.png?v=0 "H_0:\mu_1-\mu_2 \le \mu_d") vs /math-0fbc9ae55928f128a94afcf65c2c2e39.png?v=0 "H_1:\mu_1-\mu_2 > \mu_d") Upper Tailed
/math-8d8812a65564ffde40d299d450d3de7f.png?v=0 "H_0:\mu_1-\mu_2 \ge \mu_d") vs /math-9a968e5033dd7110b7973529fa35c85d.png?v=0 "H_1:\mu_1-\mu_2 < \mu_d") Lower Tailed
Test Statistics
Consider two independent samples/math-d60f5062564e1ece65993038b62484fa.png?v=0 "x_1\,\!") and/math-9defdb27049cde0b5bdfd4e762a02d6b.png?v=0 "x_2\,\!") , of size /math-8d88a43872d6db6535d8672a15f09ce2.png?v=0 "n_1\,\!") and /math-2f6fac59fce80e76d698bcf5ea77bab6.png?v=0 "n_2\,\!") drawn from two normal population with means /math-4ed789d5f6cd3fea302e306da621cb83.png?v=0 "\mu_1\,\!") and /math-917bb8defee48f460766703f23fc0822.png?v=0 "\mu_2\,\!") , and variances /math-0c153418ac24896b413d1c3c8ad3e95c.png?v=0 "\sigma_1^2\,\!") and /math-f94e0c3605e0ebe3aa890455f27416e5.png?v=0 "\sigma_2^2\,\!") respectively, we have:
/math-6dc73b5670604b65621d1ad70209c678.png?v=0 "\bar{x}_1=\frac{1}{n_1}\sum_{j=1}^{n_1}x_{1j}") , /math-06a309ba82fba50c69561e78b4ffdf7f.png?v=0 "\bar{x}_2=\frac{1}{n_2}\sum_{j=1}^{n_2}x_{2j}") ,
/math-06931d63b90e0e6a6d7d81d64e149e48.png?v=0 "s_1^2=\frac{1}{n_1-1}\sum_{j=1}^{n_1}{(x_{1j}-\bar{x}_1)^2}") , /math-1ea85773f946cf6770cf4f76022e133d.png?v=0 "s_2^2=\frac{1}{n_2-1}\sum_{j=1}^{n_2}{(x_{2j}-\bar{x}_2)^2}")
where /math-875d0b783f9b167155a8f54a2c3f9e6a.png?v=0 "\bar{x}_1\,\!") and/math-68dc9b353bc272e9d32196f5e30bff4e.png?v=0 "\bar{x}_2\,\!") are sample means and /math-f963d3c66aa9a89e64a6db3d0dd63351.png?v=0 "s_1^2\,\!") and /math-8a65733320d52c6da4efeeb5888dba00.png?v=0 "s_2^2\,\!") are sample variances. Then we can compute the t test statistic by:
For equal variance is assumed, that is /math-3728dc0e9ed8d4ea6c77089a7c1bb2fb.png?v=0 "\sigma_1^2=\sigma_2^2\,\!") :
In this case the test statistic t:
/math-8992dd1682d9d05a300c39478ada35f5.png?v=0 "t=\frac{(\bar{x}_1-\bar{x}_2)-\mu_d}{s_p\sqrt{(1/n_1+1/n_2)}}")
has a t-distribution with /math-6350a1d09d381cb5b9c2420a49cec064.png?v=0 "(v = n_1+n_2-2)") degrees of freedom and
/math-49682ab10edeb0f5ed248d7195b85d78.png?v=0 "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:
/math-2f3931fdb28db4a39f9b7140cc965c81.png?v=0 "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
/math-fdd079fbba82fc1bb066171f062bab57.png?v=0 "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 /math-806277203dedea2ed8321f6cbd465a54.png?v=0 "H_0\,\!") if:
Two tailed test: /math-dc78eff8d710d926c9d712cebdb408d0.png?v=0 "|t| > t_{\sigma/2}\,\!") ;
Upper tailed test: /math-b3c71dc9421063ffaf6de1641540a5d5.png?v=0 "t > t_\sigma\,\!") ;
Lower tailed test: /math-b3c0361ff553afb248a469861134f399.png?v=0 "t < -t_\sigma\,\!") ;
The p-value will also be compared with a user-defined significance level,/math-3ac6004d77c0cc0055e95c99b9dfd7e0.png?v=0 "\sigma\,\!") , which commonly 0.05 is used. And the null hypothesis will be rejected if /math-350fad5bdd3c28b92d7b6a076336a463.png?v=0 "p < \mu\,\!") .
Confidence Intervals
The upper and lower /math-45828455265148a331fe94e5c5175f93.png?v=0 "(1-\sigma )\times 100\%") confident limits for mean difference /math-32a17f0051e09406b2f689fa5f0bb2c5.png?v=0 "(\mu_1 - \mu_2)\,\!") are calculated as:
For equal variance is assumed:
| Null Hypothesis
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Confidence Interval
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![\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]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(TwoSampletTest)/math-35e508ffa325356798bfeb78b55c5162.png?v=0 "\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]")
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![\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, \infty\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(TwoSampletTest)/math-438038a816d1733a3f4947f61e539c61.png?v=0 "\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, \infty\right]")
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![\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(TwoSampletTest)/math-043e91f03e23cf4cb47273b881ee6f02.png?v=0 "\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\right]")
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For equal variance is not assumed:
| Null Hypothesis
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Confidence Interval
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![\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]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(TwoSampletTest)/math-1dc3a8412c3e70c868d0def30834cdfd.png?v=0 "\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]")
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![\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \infty\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(TwoSampletTest)/math-6317c80637d203c8b436ce4d361f825c.png?v=0 "\left[(\bar{x}_1-\bar{x}_2)- t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \infty\right]")
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![\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(TwoSampletTest)/math-39eb7689ff8eaf4fc97f74a82869d201.png?v=0 "\left[-\infty, (\bar{x}_1-\bar{x}_2)+ t_{\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right]")
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where /math-f1004b741c7a8310ad2a3b7c0d726855.png?v=0 "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.
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