Difference between Two Means (Correlated Pairs)

Learning Objectives

1. Determine whether you have correlated pairs or independent groups
2. Compute a t test for correlated pairs

Correlated Pairs

• Let's consider how to analyze the data from the ADHD treatment case study
• These data consist of the scores of 24 children with ADHD on a delay of gratification (DOG) task
• Each child was tested under four dosage levels
• In this section we will be concerned only with testing the difference between the mean in the placebo (D0) condition and mean in the highest dosage condition (D60)
• Why don't we use Differences between Two Means (Independent Groups)
• The answer lies in the fact that in this experiment we do not have independent groups
• The scores in the D0 condition are from the same subjects as the scores in the D60 condition
• There is only one group of subjects, each subject being tested in both the D0 and D60 conditions.

• The figure below shows a scatterplot of the 60 mg scores (D60) as a function of the 0 mg scores (D0)
• It is clear that children who get more correct in the D0 condition tend to get more correct in the D60 condition
• The correlation between the two conditions is high: r = 0.80
• Clearly these two variables are not independent

Computations

• The method to test the difference between these means was presented in the section on Testing a Single Mean
• The computational procedure is to compute the difference between the D60 and the D0 condition for each child and test whether the mean difference is significantly different from 0
• As shown in the section on testing a single mean, the mean difference score is 4.96 which is significantly different from 0: t = 3.22, df = 23, p = 0.0038
• This t test has various names including "correlated t test" and "related-pairs t test."

• In general, the correlated t test is computed by first computing the differences between the two scores for each subject. * Then, a test of a single mean is computed on the mean of these difference scores.

DOG scores as a function of dosage

D0	D60	D60-D0
57	62	5
27	49	22
32	30	-2
31	34	3
34	38	4
38	36	-2
71	77	6
33	51	18
34	45	11
53	42	-11
36	43	7
42	57	15
26	36	10
52	58	6
36	35	-1
55	60	5
36	33	-3
42	49	7
36	33	-3
54	59	5
34	35	1
29	37	8
33	45	12
33	29	-4

• If you had mistakenly used the method for an independent-groups t test with these data, you would have found that t = 1.42, df = 46, and p = 0.15
• That is, the difference between means would not have been found to be statistically significant
• This is a typical result: correlated t tests almost always have greater power than independent t-tests
• This is because in correlated t tests, each difference score is a comparison of performance in one condition with the performance of that same subject in another condition
• This makes each subject "their own control" and keeps differences between subjects from entering into the analysis
• The result is that the standard error of the difference between means is smaller in the correlated t test and, since this term is in the denominator of the formula for t, results in a larger t.

Details about the Standard Error of the Difference between Means (Optional)

• To see why the standard error of the difference between means is smaller in a correlated t test, consider the variance of difference scores
• As shown in the section on the Variance Sum Law, the variance of the sum or difference of the two variables X and Y is:

• Therefore, the variance of difference scores is the variance in the first condition (X) plus the variance in the second condition (Y) minus twice the product of (1) the correlation, (2) the standard deviation X, and (3) the standard deviation of Y. For the current example, r = .80 and the variances and standard deviations are shown in

Variances and Standard Deviations

	D0	D60	D60 - D0
Variance	128.02	151.78	56.82
Sd	11.31	12.32	7.53

The variance of the difference scores of 56.82 can be computed as:

128.02 + 151.78 - (2)(0.80)(11.31)(12.32)

which is equal to 56.82 except for rounding error. Notice that the higher the correlation, the lower the standard error of the mean.

Questions

1 Consider a within-subject experimental design with 2 conditions. The experimenter ran both a correlated pairs t-test and (incorrectly) an independent samples t-test. The experimenter found the correlated pair t-test was significant whereas the independent samples test was not. The experimenter conlcuded that this was because the mean difference score across subjects was greater than the difference between the means of the conditions. Is this the correct conclusion?

	Yes
	No

Answer >>

The two mean differences are the same. The correlated t test controlled for differences between subjects and thus reduced the standard error of the difference between the two means.

2 Correlated pairs t-tests generally have more power than independent sample t-tests because the former usually have smaller standard errors of the difference between means making the t-value bigger.

	True
	False

Answer >>

By using each subject as its own control, the standard error is lower.

3 Increasing the correlation between measures:

	increases the absolute value of t.
	decreases the absolute value of t.
	has no effect on the t.

Answer >>

All else being equal, the higher the correlation, the lower the standard error and therefore the higher the absolute value of t.

4 The null hypothesis in a correlated pairs t-test is that the population mean difference score is 0.

	True
	False

Answer >>

If the mean difference score is not 0, then the means of the two conditions will not be equal.

5 Using these data given, what is the t statistic using a correlated pairs t-test?

pre	post
 11	 11
  9	  5
 10	  8
 10	  7
 10	  6
 10	  8
 11	  9
 12	 12
  8	  4
 10	 11

Answer >>

Compute dependent t-test t value for these two variables. To use Analysis Lab, first compute difference scores and then do a t test on the mean difference score. t = 3.4641.