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Paired t-test

Definition: Used to compare means on the same or related subject over time or in differing circumstances.

Assumptions: The observed data are from the same subject or from a matched subject and are drawn from a population with a normal distribution.

Characteristics: Subjects are often tested in a before-after situation (across time, with some intervention occurring such as a diet), or subjects are paired such as with twins, or with subject as alike as possible. An extension of this test is the repeated measure ANOVA.

Test: The paired t-test is actually a test that the differences between the two observations is 0. So, if D represents the difference between observations, the hypotheses are:

H_o: D = 0 (the difference between the two observations is 0)

H_a: D 0 (the difference is not 0)

The test statistic is t with n-1 degrees of freedom. If the p-value associated with t is low (< 0.05), there is evidence to reject the null hypothesis. Thus, you would have evidence that there is a difference in means across the paired observations.

Location in WINKS: The paired t-test is located in the "Analysis --> t-test and Analysis of Variance" menu.

See also: Repeated Measures Analysis of Variance. Also, if the differences have already been calculated, a single sample test of u = 0 would be equivalent to the paired t-test. The non-parametric counterpart to the paired t-test is Friedman's test.

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Example in WINKS: Paired t-test

The DIET.DBF database contains information on 8 subjects who were placed on a diet, and observed for several weeks. Before and after weights were taken.

Step 1: In WINKS, oOpen the data set named DIET.SDA (Or create a data set with the data.)

Step 2: Select Analyze, t-tests and ANOVA/Paired/Rep. Measure (t-test/ANOVA).

Step 3: Select REP1 (BEFORE) as the first field and REP2(AFTER) as the second field and click Ok.

Step 4: The results are shown in the viewer. A portion of the output is shown below:

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Repeated Measures Analysis Summary              C:\WINKS\DIET.DBF
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Number of repeated measures is 2
Number of subjects read in 8

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 Means and standard deviations for 2 repeated measures:
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1)REP1: mean = 169.625 s.d. = 8.07001
2)REP2: mean = 150.25 s.d. = 11.04213

Mean Difference = 19.375 s.d.(difference) = 14.78356

95% C.I. about Mean Difference is (7.01367, 31.73633)

Paired t- test

Calculated t = 3.70687 with 7 D.F. p = 0.0076 (two- sided)
  

The means and standard deviations for each group are reported, but more importantly, the mean difference between BEFORE and AFTER measurements is given. The statistical procedures are performed on this average difference. These results are interpreted like those of a single sample t-test with null hypothesis: mean=0, and alternative hypothesis: mean <> 0.

The calculated t-statistic is t = 3.70687. The test is performed with 7 degrees of freedom, and p = 0.0076. A small p-value such as this is indicates rejection of the null hypothesis and leads to the conclusion that the average difference in BEFORE and AFTER weights is not zero, i.e., there is evidence of a significant (at the 0.05 level) change of weight in these eight subjects on average.

Step 5: Select Graph, Display Graph to display a graphical comparison.

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Exercise - Paired t-test

A company was wondering which style of pepperoni pizza was most popular. It set up an experiment where ten people were each given two types of pizza to eat, Type A and Type B. Each pizza was carefully weighed at exactly 16 oz. After fifteen minutes, the remainders of the pizza were weighed, and the amount of each type pizza remaining per person was calculated. It is assumed that the subject would eat more of the type of pizza he or she preferred. Here are the data:

Subject	Pizza A	Pizza B
1	12.9oz	16oz
2	5.7	7.5
3	16	16
4	14.3	15.7
5	2.4	13.2
6	1.6	5.4
7	14.6	15.5
8	10.2	11.3
9	4.3	15.4
10	6.6	10.6

1. Perform a paired t-test.

2. What value of the t-statistic is calculated, what are the degrees of freedom for the test, what p-value is reported for this data?

3. Do people seem to prefer one type of pepperoni pizza over another? If so, which seems to be most liked?

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