Chapter 15: Testing for a difference between two dependent (Correlated) Groups.

Example: Suppose you wanted to test the drug that may affect IQ, but this time you look for changes within each subject. This is done by measuring the IQ for each subject before and after taking the drug. For 9 subjects, you might get a result like this:

The most common application for this test is called a ‘repeated-measures design’ where two measures are obtained from the same group of subjects, and you are looking for a difference in the means of these measures.

Subject IQ before (X)

IQ after (Y)

1 117 132

2 109 105

3 101 105

4 105 125

5 90 96

6 125 136

7 101 107

8 127 130

9 104 101

The book discusses two ways of conducting t-tests for a repeated-measures design. We will only cover the second (easy) way. So we’ll be skipping sections 15.1 and 15.2The easy (‘alternative’) way is to treat each subject as a single measure by calculating the difference (D) for the scores within each subject, and conducting a t-test on these values of D.

Subject IQ before (X)

IQ after (Y)

D=Y-X

1 117 132 15

2 109 105 -4

3 101 105 4

4 105 125 20

5 90 96 6

6 125 136 11

7 101 107 6

8 127 130 3

9 104 101 -3

Just perform a simple single sample t-test (like in Chapter 13) on the values of D using n-1 degrees of freedom, where n is the number of pairs.

Ds

Dt

Subject IQ before (X)

IQ after (Y)

D=Y-X

1 117 132 15

2 109 105 -4

3 101 105 4

4 105 125 20

5 90 96 6

6 125 136 11

7 101 107 6

8 127 130 3

9 104 101 -3

All we are doing here is conducting a single-sample t-test on the differences.

Ds

Dt

Subject IQ before (X)

IQ after (Y)

D=Y-X

1 117 132 15

2 109 105 -4

3 101 105 4

4 105 125 20

5 90 96 6

6 125 136 11

7 101 107 6

8 127 130 39 104 101 -3

46.262.2

044.6

t

44.6D

Looking in Table D, df = 8, area in one tail: 0.025

tcrit = ±2.306

Decision: reject H0 and conclude that the drug had a significant effect on IQ using a two-tailed test with a = .05.

This is the same equation for a single sample t-test, but with D instead of X:

86.7Ds

62.29

86.7

n

ss DD

Using APA Format:

“There was a statistically significant change in IQ after taking the drug (M = 6.44, SD = 7.86), t(8)= 2.46, p<.05

The effect size is computed just as a single sample t-test. Again, just use ‘D’ for ‘X’:

819.086.7

44.6

Ds

Dg

Remember, the general convention is that g=.2 is small, g=.5 is medium and g=.8 is large, so this is a large effect size

What is the power of this test using this effect size?

We will use the power curve for one mean: (not two means, because we’re really just conducting a hypothesis test on the mean of the differences)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n=8

10

12

15

20

25

30

40

50

75

100

150

250

500

1000

Effect size (d)

Po

we

r = 0.05, 2-tails, 1 mean

Here’s the power curve for a two-tailed test for one mean with a =.05 from Chapter 13:

For an effect size of 0.819, sample size of 9, what is the power?

Answer: about .525 or so.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n=8

10

12

15

20

25

30

40

50

75

100

150

250

500

1000

Effect size (d)

Po

we

r = 0.05, 2-tails, 1 mean

Power with correlated groups is calculated the same way as for a single sample design. Here’s the power curve for a two-tailed test with a =.05 from Chapter 13:

For an effect size of 0.8, how many pairs (n) do we need for a power of 0.8?

Answer: about 15 pairs (yellow curve)

Other examples (besides repeated measures) of testing difference of means between dependent (correlated) groups. The methods are the same as for repeated measures – calculate a t-test on the difference between each pair.

Matched pairs experiments: Where pairs of subjects are naturally grouped together. For example, in identical twin studies a measure is made from each twin in a pair. D is the difference in scores for each pair of twins.

Matched-subjects design: Where the experimenter matches pairs of subjects together based on some other variable such as age or visual acuity.

Example from the tutorial:

25) We measure the visual acuity of 31 republicans under two conditions: 'itchy' and 'tiny'. You then subtract the visual acuity of the 'tiny' from the 'itchy' conditions for each republicans and obtain a mean pair-wise difference of -2.15 with a standard deviation is 7.37.

Using an alpha value of 0.01, is the visual acuity from the 'itchy' condition significantly different than from the 'tiny' condition?What is the effect size?

Answer:

= -2.15D̄�sD̄� = 1.32tobs = -1.63tcrit = ± 2.75 (df = 30)

We fail to reject H0.The visual acuity of itchy republicans is not significantly different than the visual acuity of tiny republicans (M=-2.15, SD = 7.37), t(30) = -1.6288, p = 0.1138.Effect size: g = 0.29 (small)