Cesar Chew at 17:31, 25 November 2014

2014-11-25T17:31:11Z

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{{Cat|Testing Means| 6}}
== Learning Objectives ==
# Determine whether to use the formula for correlated comparisons or independent-groups comparisons
# Compute t for a comparison for repeated-measures data

== Specific Comparisons (Correlated Observations) ==
* In the Weapons and Aggression case study, subjects were asked to read words presented on a computer screen as quickly as they could
* Some of the words were aggressive words such as injure or shatter
* Others were control words such as relocate or consider
* These two types of words were preceded by words that were either the names of weapons such as shot gun and grenade or non-weapon words such as rabbit or fish
* For each subject, the mean reading time across words was computed for these four conditions
* The four conditions are labeled as shown in Table 1. Table 2 shows the data for five subjects.

{| class="wikitable"
|+Description of Conditions
! Variable
! Description
|-
| aw
| align="left" | The time in milliseconds (msec) to name aggressive word following a weapon word prime.
|-
| an
| align="left" | The time in milliseconds (msec) to name aggressive word following a non-weapon word prime.
|-
| cw
| align="left" | The time in milliseconds (msec) to name a control word following a weapon word prime.
|-
| cn
| align="left" | The time in milliseconds (msec) to name a control word following a non-weapon word prime.
|}

{| class="wikitable"
|+Data from Five Subjects
! Subject
! aw
! an
! cw
! cn
|-
| 1
| 447
| 440
| 432
| 452
|-
| 2
| 427
| 437
| 469
| 451
|-
| 3
| 417
| 418
| 445
| 434
|-
| 4
| 348
| 371
| 353
| 344
|-
| 5
| 471
| 443
| 462
| 463
|}

One question was whether reading times would be shorter when the preceding word was a weapon word (aw and cw conditions) than when it was a non-weapon word (an and cn conditions). In other words, is
L1 = (an + cn) - (aw + cw)
greater than 0?

This is tested for significance by computing L1 for each subject and then testing whether the mean value of L1 is significantly different from 0.

Table 3 shows L1 for the first five subjects. L1 for Subject 1 was computed by
L1 = (440 + 452) - (447 + 432) = 892 - 885 = 13

{| class="wikitable"
|+L1 for Five Subjects
! Subject
! aw
! an
! cw
! cn
! L1
|-
| 1
| 447
| 440
| 432
| 452
| 13
|-
| 2
| 427
| 437
| 469
| 451
| -8
|-
| 3
| 417
| 418
| 445
| 434
| -10
|-
| 4
| 348
| 371
| 353
| 344
| 14
|-
| 5
| 471
| 443
| 462
| 463
| -27
|}

* Once L1 is computed for each subject, the significance test described in the section [[Testing a Single Mean]] can be used
* First we compute the mean and the standard error of the mean for L1
* There were 32 subjects in the experiment
* Computing L1 for the 32 subjects, we find that the mean and standard error of the mean are 5.875 and 4.2646 respectively.

We then compute:
[[File:T_mean.gif]]
M is the sample mean
μ is the hypothesized value of the population mean (0 in this case)
and sM is the estimated standard error of the mean

* The calculations show that t = 1.378
* Since there were 32 subjects, the degrees of freedom is 32 - 1 = 31
* The t distribution calculator shows that the two-tailed probability is 0.1782

== Priming Effect ==
* A more interesting question is whether the priming effect (the difference between words preceded with a non-weapon word and words preceded by a weapon word) is different for aggressive words than it is for non-aggressive words
* That is, do weapon words prime aggressive words more than they prime non-aggressive words?
* The priming of aggressive words is (an - aw)
* The priming of non-aggressive words is (cn - cw)
* The comparison is the difference:
L2 = (an - aw) - (cn - cw)

Table 4 shows L2 for five of the 32 subjects.

{| class="wikitable"
|+ L2 for Five Subjects
! Subject
! aw
! an
! cw
! cn
! L2
|-
| 1
| 447
| 440
| 432
| 452
| -27
|-
| 2
| 427
| 437
| 469
| 451
| 28
|-
| 3
| 417
| 418
| 445
| 434
| 12
|-
| 4
| 348
| 371
| 353
| 344
| 32
|-
| 5
| 471
| 443
| 462
| 463
| -29
|}

* The mean and standard error of the mean for all 32 subjects are 8.4375 and 3.9128 respectively
* Therefore, t = 2.156 and p = 0.039.

== Multiple Comparisons ==
Issues associated with doing multiple comparisons are the same for related observations as they are for multiple comparisons among independent groups.

== Orthogonal Comparisons ==
* The most straightforward way to assess the degree of dependence between two comparisons is to correlate them directly
* For the weapons and aggression data, the comparisons L1 and L2 are correlated 0.24
* Of course, this is a sample correlation and only estimates what the correlation would be if L1 and L2 were correlated in the whole population
* Although mathematically possible, orthogonal comparisons with correlated observations are very rare.

== Questions ==

<quiz display=simple >
{Here you see taste ratings for 4 cola types: Cola A (A), generic Cola A (GA), Cola B (B), and generic Cola B (GB). L1 is the difference in taste scores between non-generic and generic Colas (nongeneric - generic). What is L1 for the fourth subject? 
<pre>A GA B GB
9 6 8 5
10 6 9 7
10 5 8 7
8 4 7 5
10 5 8 5
9 7 7 6
</pre>
|type="{}"}
{ 6 _10 }

{
{{Show Answer|
(A+B) - (GA + GB) specifically compares generic and nongeneric cola. It is (8 + 7) - (4 + 5).
}}
}

{Calculate the standard error for L1 (L1 = A + B - GA - GB). Keep in mind this is a repeated-measures design. 
<pre>A GA B GB
9 6 8 5
10 6 9 7
10 5 8 7
8 4 7 5
10 5 8 5
9 7 7 6
</pre>
|type="{}"}
{ 0.654 _10 }

{
{{Show Answer|
Compute L1 for each subject: (6, 6, 6, 6, 8, 3) and then compute the standard deviation (1.602) which you then divide by the square root of n (n = 6) to get 0.654.
}}
}

{Calculate t for L1 (L1 = A + B - GA - GB) 
<pre>A GA B GB
9 6 8 5
10 6 9 7
10 5 8 7
8 4 7 5
10 5 8 5
9 7 7 6
</pre>
|type="{}"}
{ 8.919 _10 }

{
{{Show Answer|
Compute L1 for each subject: (6, 6, 6, 6, 8, 3) and then compute the mean (5.833) and standard deviation (1.602) which you then divide by the square root of n (n = 6) to get 0.654 Then divide the mean by the standard error of 0.654 to get 8.919.
}}
}

</quiz>

Specific Comparisons (Correlated Observations) - Revision history

Cesar Chew at 17:31, 25 November 2014