## Comparison of Means of Two Samples The t Test

The t test is used for measured variables in comparing two means. The unpaired t test compares the means of two independent samples. The paired t test compares two paired observations on the same individual or on matched individuals.

In our discussion of the z test, we used as a dependent variable (DV) an intelligence quotient (IQ) test, which has a known mean and standard deviation (SD), in order to compare our sample of interest —namely, readers of this magnificent opus—to the general population. We did not have to scrape up a control group of nonreaders for comparison because we knew in advance the true mean and SD of any control group we might draw from the general population.

This fortunate circumstance rarely occurs in the real world. As a result, many studies involve a comparison of two groups, treatment versus control, or treatment A versus treatment B. The statistical analysis of two samples is a bit more complicated than the previous comparison between a sample and a population. Previously, we used the population SD to estimate the random error we could expect in our calculated sample mean. In a two-sample comparison, this SD is not known and must be estimated from the two samples.

Let's consider a typical randomized trial of an antihypertensive drug. We locate 50 patients suffering from hypertension, randomize them into two groups, institute a course of drug or placebo, and measure the diastolic blood pressure. We then calculate a mean and SD for each group. Suppose the results are as shown in Table 4-1.

The statistical question is "What is the probability that the difference of 4 mm Hg between treatment and control groups could have arisen by chance?" If this probability is small enough, then we will assume that the

Table 4-1

Typical Randomized Trial

Treatment Group

Control Group

Sample size Mean

Standard deviation

102 mm Hg 8 mm Hg difference is not due to chance and that there is significant effect of the drug on blood pressure.

To approach this question, we start off with a null hypothesis that the population values of the two groups are not different. Then we try to show they are different. If we were to proceed as before, we would calculate the ratio of the difference between the two means, 4 mm Hg, to some estimate of the error of this difference. In Chapter 3, we determined the standard error (SE) as the population SD divided by the square root of the sample size; but this time, the population SD is unknown. We do, however, have two estimates of this population value, namely, the sample SDs calculated from the treatment and control groups. One approach to getting a best estimate of the population SD would be to average the two values. For reasons known only to statisticians, a better approach is to add the squares of the SDs to give 62 + 82 = 36 + 64 = 100. The next step is to divide this variance by the sample size and take the square root of the result to obtain the SE:

As usual, then, to determine the significance of this difference, you take the ratio of the calculated difference to its SE.

The probability we are looking for is the area to the right of 4 mm Hg on the curve (that is,2.0 X 2 mm Hg),which is close to 0.025 (one-tailed) (Figure 4-1).

The above formula works fine if the two samples are of equal size. However, if there are unequal samples, and you look in a real statistics book, you will find a complicated formula sprinkled liberally with n's and n - 1's. The basis for the formula, and the reason for its complexity, is twofold: (1) if one sample is bigger than the other, the estimated variance from it is more accurate, so it should weigh more in the calculation of the SE; and (2) because the SE involves a division by n, the formula has to account for the fact that there are now two n's kicking around. Don't worry about it; computers know the formula.

differencebetweenmeans SE of difference Difference (mm) Figure 4-1 Graphic interpretation of the t test.

This test statistic is called the Student's t test. It was developed by the statistician William Gossett, who was employed as a quality control supervisor at the Guinness Brewery in Dublin and who wrote under the pseudonym of Student, presumably because no one who knew his occupation would take him seriously. It turns out that this curve is not a normal curve when you get to small samples, so you have to look it up in a different appendix, under the "t" distribution. 