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Chapter 4. The Special Case of Two Groups: The *t* Test

As we have just seen in Chapter 3, many investigations require comparing only two groups. In addition, as the last example in Chapter 3 illustrated, when there are more than two groups, analysis of variance only allows you to conclude that the data are not consistent with the hypothesis that all the samples were drawn from a single population. It does not help you decide *which* one or ones are most likely to differ from the others. To answer these questions, we now develop a procedure that is specifically designed to test for differences in two groups: the *t test* or *Student's t test.* While we will develop the *t* test from scratch, we will eventually show that it is just a different way of doing an analysis of variance. In particular, we will see that *F* = *t*2 when there are two groups.

The *t* test is the most common statistical procedure in the medical literature; you can expect it to appear in more than half the papers you read in the general medical literature. In addition to being used to compare two group means, it is widely applied incorrectly to compare multiple groups, by doing all the pairwise comparisons, for example, by comparing more than one intervention with a control condition or the state of a patient at different times following an intervention. As we will see, this incorrect use increases the chances of rejecting the null hypothesis of no effect above the nominal level, say 5%, used to select the cutoff value for a “big” value of the test statistic *t.* In practical terms, this boils down to increasing the chances of reporting that some therapy had an effect when the evidence does not support this conclusion.

Suppose we wish to test a new drug that may be an effective diuretic. We assemble a group of 10 people and divide them at random into two groups, a control group that receives a placebo and a treatment group that receives the drug; then we measure their urine production for 24 hours. Figure 4-1A shows the resulting data. The average urine production of the group receiving the diuretic is 240 mL higher than that of the group receiving the placebo. Simply looking at Figure 4-1A, however, does not provide very convincing evidence that this difference is due to anything more than random sampling.

###### Figure 4-1.

**(A)** Results of a study in which five people were treated with a placebo and five people were treated with a drug thought to increase daily urine production. On the average, the five people who received the drug produced more urine than the placebo group. Are these data convincing evidence that the drug is an effective diuretic? **(B)** Results of a similar study with 20 people in each treatment group. The means and standard deviations associated ...

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