Dependent t-test

Student's t-test for paired samples (dependent)

Comparison of two means from dependent samples.


Examples for dependence:

Repeated measure, parallelized samples, virtual dependence (e.g. twins)


Requirements:

Dependent variable is at least interval scaled and normal distributed.

Hypothesis:


Nondirectional:

H0:non directional H0    H1:non directional H1       

Directional (example):

H0:directional H0      H1:directional H1


The dependence of two measures from the two samples is gathered from the differences of the measurement pairs.


difference i of pair i


Mean and standard deviation of these differences difference iare defined as follows:


mean of differences


standard deviation of differences

Whereas n is the number of pairs.


The decision between the hypothesis is made according to the sample distribution of mean of differences.

If both populations are normal distributed, the sample distribution is normal, too, with mean

 population mean of differences


and standard deviation

population standard deviation of differences


If sigmais estimated from the sample, we get the following t-distributed equation:


formula for paired samples t test

with degree of freedom df = n-1.

 


Example of a paired samples t-test

A psychologist is interested in his new learning program. 15 Subjects learn a list of 50 words. Learning performance is measured using a recall test. After the first test all subjects are instructed how to use the learning program and then learn a second list of 50 words. Learning performance is again measured with the recall test. In the following table the number of correct remembered words are listed for both tests.

 

Subject  Score 1  Score 2  differences squared residuals
1 24
26
-2
0.36
2 17
24
-7
19.36
3
32
31
1
12.96
4
14
17
-3
0.16
5
16
17
-1
2.56
6
22
25
-3
0.16
7
26
25
1
12.96
8
19
24
-5
5.76
9
19
22
-3
0.16
10
22
23
-1
2.56
11
21
26
-5
5.76
12
25
28
-3
0.16
13
16
19
-3
0.16
14
24
23
1
12.96
15
18
22
-4
1.96
Σ     -37
78


n=15

sum of differences

 

mean of differences

 

sum of squared residuals

 

estimated population standard deviation of differences

 

estimated population standard deviation of the mean of differences

 

formula of paired samples t test

 

degrees of freedom of paired samples t test

 

critical t value 95 percent one sided

 

critical t value 99 percent one sided

 

The observed t-value is smaller than the critical t-value (1%, one-tailed). The learning program has a positive effect. Subjects remember more words after the training.


BrightStat output of paired samples t-test example



This is a fictitious example.


Wiki link paired samples t-test




 

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