September 27, 2005
Documentation for Two-sample Independent t Test
Kevin M. Sullivan, PhD, MPH, MHA: cdckms@sph.emory.edu
This analysis is conducted to observe whether there is a significant difference in means between two independent samples, given respective standard deviation or standard error. The data input screen is as follows:
The
input values requested are:
·
Two-sided confidence intervals (%) that can be
chosen are between 0 and 100.
·
Enter individual sample means.
·
Enter standard deviation (or) standard error of individual sample mean.
The result of the
calculation is shown next:
The interpretation of the test is that there is no significant difference between the means of these two groups. It is noteworthy that before interpreting as above, F test for the equality of variances from these two independent, normally distributed samples should be first checked. If the two variances are not significantly different, ie. p-value of test for equality of variance is >0.05, the result of difference in means should be interpreted from t statistics and p-value based on equal variance. In the example above, the two variances are significantly different, ie. p-value of F test is 0.004, and therefore, the p-value of difference in means is 0.2424.
In addition, the confidence interval of difference in means is also displayed.
The formulae for two-sample t
test are as follows:
All statistics are derived from formulae of the text 'Fundamentals of Biostatistics' (5th edition) by Bernard Rosner; For two-sample t test with equal variance, statistics were based on equation 8.11 to 8.13; If assuming unequal variance, statistics were based on equation 8.21 to 8.23.
·
Two-sample
t test with equal variance:
S = pooled estimate of the variance.
= degree of freedom
· Two-sample t test with unequal variance:
Satterthwaite’s method (see also Welch's modified t test)
= approximate degree of
freedom.
Standard error=Standard deviation/√n
·
Hartley's
f test for equal variance:
S2L = the larger
of two variances;
S2S = the
smaller of two variances
Note: test for equality of variance is based on equation 8.15-8.16.
Reference:
· Bernard Rosner. Fundamentals of Biostatistics (5th edition).
· Welch, B. L. (1938). The significance of the difference between two means when the population variances are unequal. Biometrika 29, 350-362.
· Statterthwaite, F. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin 2, 110-114.
Acknowledgement:
Default values were obtained from example 8.18 (pg. 297-8) described in 'Fundamentals of Biostatistics' (5th edition) by Bernard Rosner.