# Answer to Question #3145 in Statistics and Probability for cynthia Hicks

Question #3145

What formula would be used for a within-participants design in statistics?

Expert's answer

The basic formula for hypothesis testing statistics:

& differences between groups

& variability within the groups

The bottom part of the equation (variability within groups) is composed of two parts:

1.& due to measurement error

2.& due to individual differences

& differences between groups

& measurement error + individual differences

If individual differences are large, then the overall ration will be low.& This means even for important effects you may not be able to pick them up in your stats because of large individual differences.

&

the goal of within-subject designs

The goal of within-subjects variables is to remove individual differences from the bottom part of the equation.& Consider the example.

within-subject t-test

For each item you compute a difference score.

di = Xi1 - Xi2

This difference score ignores the range of scores for that item -- it equates items which started high and items which started low by looking to see if they changed.

It is then possible to compute a mean of these difference scores:

Md = SUMi (di)

What we then ask with the within-subjects t-test is:& Is the average difference score large compared to the variability in the difference scores?

&

t = Md / SEMd& = Md / (sd / sqrt N)

Example:

Text Complexity

Student simple complex d

1 10.4& & & 14.3 -3.9

2 16.9 & 20.4 -3.5

3 & 6.7 & & 7.5 -0.8

4 25.9 26.8 -0.9

5 12.4 14.6 -2.2

6 18.5 20.3 -1.8

7 13.5 12.3 1.2

8 & & & & & & & 8.2 10.2 -2.0

. . . .

M 14.06 15.80 -1.74

SD & & & 6.24 & 6.31 & 1.62

&

Within-subjects t-test

t = Md / SEMd & = Md / (sd / sqrt N)

& t& =& - 1.74 / (1.62 / sqrt 8)

t (7)& =& - 3.038, p < .02

& differences between groups

& variability within the groups

The bottom part of the equation (variability within groups) is composed of two parts:

1.& due to measurement error

2.& due to individual differences

& differences between groups

& measurement error + individual differences

If individual differences are large, then the overall ration will be low.& This means even for important effects you may not be able to pick them up in your stats because of large individual differences.

&

the goal of within-subject designs

The goal of within-subjects variables is to remove individual differences from the bottom part of the equation.& Consider the example.

within-subject t-test

For each item you compute a difference score.

di = Xi1 - Xi2

This difference score ignores the range of scores for that item -- it equates items which started high and items which started low by looking to see if they changed.

It is then possible to compute a mean of these difference scores:

Md = SUMi (di)

What we then ask with the within-subjects t-test is:& Is the average difference score large compared to the variability in the difference scores?

&

t = Md / SEMd& = Md / (sd / sqrt N)

Example:

Text Complexity

Student simple complex d

1 10.4& & & 14.3 -3.9

2 16.9 & 20.4 -3.5

3 & 6.7 & & 7.5 -0.8

4 25.9 26.8 -0.9

5 12.4 14.6 -2.2

6 18.5 20.3 -1.8

7 13.5 12.3 1.2

8 & & & & & & & 8.2 10.2 -2.0

. . . .

M 14.06 15.80 -1.74

SD & & & 6.24 & 6.31 & 1.62

&

Within-subjects t-test

t = Md / SEMd & = Md / (sd / sqrt N)

& t& =& - 1.74 / (1.62 / sqrt 8)

t (7)& =& - 3.038, p < .02

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