Cesar Chew at 17:31, 25 November 2014

2014-11-25T17:31:28Z

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{{Cat|Testing Means| 70}}
== Learning Objectives ==
# Compute the Bonferroni correction
# Calculate pairwise comparisons using the Bonferroni correction

== Pairwise Comparisons (Correlated Observations) ==
* In the section on all pairwise comparisons among independent groups, the Tukey HSD Test was the recommended procedure
* When you have one group with several scores from the same subjects, the Tukey test makes an assumption that is unlikely to hold:
The variance of difference scores is the same for all pairwise differences between means.

* The standard practice for pairwise comparisons with correlated observations is to compare each pair of means using the method outlined in the section [[Difference between Two Means (Correlated Pairs)]] with the addition of the Bonferroni correction described in the section [[Specific Comparisons (Independent Groups)]]
* For example, suppose you were going to do all pairwise comparisons among four means and hold the familywise error rate at 0.05
* Since there are six possible pairwise comparisons among four means, you would use the 0.05/6 = 0.0083 for the per comparison error rate.

== Stroop Example ==
* There were three tasks each performed by 47 subjects:
*# "words" task, subjects read the names of 60 color words written in black ink
*# "color" task, subjects named the colors of 60 rectangles
*# "interference" task, subjects named the ink color of 60 conflicting color words

* The times to read the stimuli were recorded
* In order to do compute all pairwise comparisons, the difference in times for each pair of conditions for each subject is calculated. Table 1 shows these scores for 5 of the 42 subjects.

{| class="wikitable"
|+Pairwise Differences
! W-C
! W-I
! C-I
|-
| -3
| -24
| -21
|-
| 2
| -41
| -43
|-
| -1
| -18
| -17
|-
| -4
| -23
| -19
|-
| -2
| -17
| -15
|}

* The means, standard deviations, and standard error of the mean (Sem), t, and p for all 47 subjects are shown in Table 2
* The t's are computed by dividing the means by the standard errors of the mean. Since there are 47 subject, the degrees of freedom is 46
* Notice how different the standard deviations are
* For the Tukey test to be valid, all population values of the standard deviation would have to be the same.

{| class="wikitable"
|+Distribution of colors
! Comparison
! Mean
! Sd
! Sem
! t
! p
|-
| W-C
| -4.15
| 2.99
| 0.43
| -9.53
| <0.001
|-
| W-I
| -20.51
| 7.84
| 1.14
| -17.93
| <0.001
|-
| C-I
| -16.36
| 7.47
| 1.09
| -15.02
| <0.001
|}

* Using the Bonferroni correction for three comparisons, the p value has to be below 0.05/3 = 0.0167 for an effect to be significant at the 0.05 level
* For these data, all p values are far below that, and therefore all pairwise differences are significant.

== Questions ==

<quiz display=simple >
{Bonferonni adjustments are necessary when making the multiple comparisons because they keep the type I error rate from being inflated.
|type="()"}
+ True
- False

{
{{Show Answer|
True. the Bonferonni adjustment is designed to keep alpha the same level by reducing the the critical value for any each comparison. If each comparison had a critical value of 0.05 then the chance of making one or more type 1 errors would be higer than 0.05. The Bonferonni adjustment corrects for this.
}}
}

{You plan to test all pairwise comparisons among 4 means. What is the critical value after a Bonferonni adjustment needed to maintain an experiment-wise type 1 error rate of 0.05?<br />
|type="{}"}
{ 0.00833 _10 }

{
{{Show Answer|
6 comparisons 0.05/6 = 0.00833.
}}
}

{Calculate t for the comparison of the mean for CA with the mean for CB.<br />
<pre>CA CB CC CD
4 6 8 9
5 7 9 9
8 9 10 10
7 7 9 10
5 7 9 8
6 7 7 9
7 9 10 11
3 5 6 7
5 4 7 7
6 7 9 10
</pre>
|type="{}"}
{ 3.674 _10 }

{
{{Show Answer|
t = 3.674.
}}
}

{Calculate p for the comparison of CC to CD.<br />
<pre>CA CB CC CD
4 6 8 9
5 7 9 9
8 9 10 10
7 7 9 10
5 7 9 8
6 7 7 9
7 9 10 11
3 5 6 7
5 4 7 7
6 7 9 10
</pre>
|type="{}"}
{ 0.0510 _10 }

{
{{Show Answer|
t(9) = 2.25, p = 0.0510.
}}
}

</quiz>

Pairwise Comparisons (Correlated Observations) - Revision history

Cesar Chew at 17:31, 25 November 2014