The t-test for dependent samples can be used to examine data from within-subjects designs when two observations are made on each subject. The dependent t-test is sometimes call the t-test for repeated measures because it can be used in situations involving collectiong two measures on each subject. The same formula and logic applies to studies involving siblings or research on husbands and wives in the same family.
Hypotheses
Assumptions
The Trick to the Dependent t-test
Formulas
Example
Consider these hypothetical scores for husbands and wives regarding their attitudes towards bilingual education.
| Wives | Husbands | d |
|---|---|---|
| 107 | 102 | 5 |
| 120 | 109 | 11 |
| 100 | 111 | -11 |
| 121 | 117 | 4 |
| 116 | 121 | -5 |
| 109 | 103 | 6 |
| 120 | 111 | 9 |
| 115 | 110 | 5 |
| 117 | 109 | 8 |
| 123 | 114 | 9 |
| 108 | 109 | -1 |
| 121 | 113 | 8 |
| mean | 4 |
Stata Analysis of Example
input wife husb
107 102
120 109
100 111
121 117
116 121
109 103
120 111
115 110
117 109
123 114
108 109
121 113
end
generate diff = wife-husb
ttest wife=husb
Paired t test
------------------------------------------------------------------------------
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
wife | 12 114.75 2.067516 7.162085 110.1994 119.3006
husb | 12 110.75 1.523179 5.276449 107.3975 114.1025
---------+--------------------------------------------------------------------
diff | 12 4 1.882938 6.522688 -.144318 8.144318
------------------------------------------------------------------------------
Ho: mean(wife - husb) = mean(diff) = 0
Ha: mean(diff) < 0 Ha: mean(diff) != 0 Ha: mean(diff) > 0
t = 2.1243 t = 2.1243 t = 2.1243
P < t = 0.9714 P > |t| = 0.0571 P > t = 0.0286
ttest diff=0
One-sample t test
------------------------------------------------------------------------------
Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
---------+--------------------------------------------------------------------
diff | 12 4 1.882938 6.522688 -.144318 8.144318
------------------------------------------------------------------------------
Degrees of freedom: 11
Ho: mean(diff) = 0
Ha: mean < 0 Ha: mean != 0 Ha: mean > 0
t = 2.1243 t = 2.1243 t = 2.1243
P < t = 0.9714 P > |t| = 0.0571 P > t = 0.0286
display 2.1242^2
4.5122256
Using ANOVA
input pairid a y
1 1 107
1 2 102
2 1 120
2 2 109
3 1 100
3 2 111
4 1 121
4 2 117
5 1 116
5 2 121
6 1 109
6 2 103
7 1 120
7 2 111
8 1 115
8 2 110
9 1 117
9 2 109
10 1 123
10 2 114
11 1 108
11 2 109
12 1 121
12 2 113
end
anova y a pairid
Number of obs = 24 R-squared = 0.7579
Root MSE = 4.61224 Adj R-squared = 0.4938
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 732.5 12 61.0416667 2.87 0.0456
|
a | 96 1 96 4.51 0.0571
pairid | 636.5 11 57.8636364 2.72 0.0558
|
Residual | 234 11 21.2727273
-----------+----------------------------------------------------
Total | 966.5 23 42.0217391
display sqrt(e(F_1))
2.12434
regress y i.a i.pairid
Source | SS df MS Number of obs = 24
-------------+------------------------------ F( 12, 11) = 2.87
Model | 732.5 12 61.0416667 Prob > F = 0.0456
Residual | 234 11 21.2727273 R-squared = 0.7579
-------------+------------------------------ Adj R-squared = 0.4938
Total | 966.5 23 42.0217391 Root MSE = 4.6122
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
2.a | -4 1.882938 -2.12 0.057 -8.144318 .144318
|
pairid |
2 | 10 4.612237 2.17 0.053 -.1514645 20.15146
3 | 1 4.612237 0.22 0.832 -9.151465 11.15146
4 | 14.5 4.612237 3.14 0.009 4.348535 24.65146
5 | 14 4.612237 3.04 0.011 3.848535 24.15146
6 | 1.5 4.612237 0.33 0.751 -8.651465 11.65146
7 | 11 4.612237 2.38 0.036 .8485355 21.15146
8 | 8 4.612237 1.73 0.111 -2.151465 18.15146
9 | 8.5 4.612237 1.84 0.092 -1.651465 18.65146
10 | 14 4.612237 3.04 0.011 3.848535 24.15146
11 | 4 4.612237 0.87 0.404 -6.151465 14.15146
12 | 12.5 4.612237 2.71 0.020 2.348535 22.65146
|
_cons | 106.5 3.394514 31.37 0.000 99.02872 113.9713
------------------------------------------------------------------------------
Linear Statistical Models Course
Phil Ender, 25apr06, 12Feb98