September 27, 2005
Documentation for Two-sample Independent t Test
Minn M. Soe, MD, MPH, MCTM : m[email protected].edu
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' (5
th
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.
df
= degree of freedom
Two-sample t test with unequal variance:
Satterthwaite’s method
(see also
Welch's modified t test)
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df
df
= approximate degree of freedom.
Standard error=Standard deviation/√n
Hartley's f test for equal variance:
S
2
L
= the larger of two variances;
S
2
S
= 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.
f statistics = S
2
L
/ S
2
S
