Introduction to Research Design and Statistics

Testing Means (t-distribution)

t-test for Dependent Groups (Within-Subjects)

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 called the t-test for repeated measures because it can be used in situations involving collecting 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

2-tail - H₀: μ_d = 0 H₁: μ_d ≠ 0
1-tail - H₀: μ_d <= 0 H₁: μ_d > 0

1-tail - H₀: μ_d >= 0 H₁: μ_d < 0

Assumptions

Independence

Each difference score is independent of the other difference scores.

Normality

Observations are sampled from normal populations.
Observe histograms for difference score.

The Trick to the Dependent t-test

Compute the difference between the two scores.

Use the single sample t-test formula.

Formulas

Standard Deviation for the Differences

Dependent t-test

Degrees of freedom: df = n - 1, where n is the number of pairs of values.

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

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 Examples

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
 

use http://www.gseis.ucla.edu/courses/data/hsb2
  
generate diff =  write - math
  
ttest write = math
 
Paired t test
 
------------------------------------------------------------------------------
Variable |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf. Interval]
---------+--------------------------------------------------------------------
   write |     200      52.775    .6702372    9.478586    51.45332    54.09668
    math |     200      52.645    .6624493    9.368448    51.33868    53.95132
---------+--------------------------------------------------------------------
    diff |     200         .13    .5828931    8.243353    -1.01944     1.27944
------------------------------------------------------------------------------
 
                   Ho: mean(write - math) = mean(diff) = 0
 
  Ha: mean(diff) < 0         Ha: mean(diff) ~= 0        Ha: mean(diff) > 0
       t =   0.2230                t =   0.2230              t =   0.2230
   P < t =   0.5881          P > |t| =   0.8237          P > t =   0.4119
  
/*  check normality of difference scores  */
  
kdbox diff, norm mean  /*  findit kdbox  */
  
     
     
ttest write = math if female==1
 
Paired t test
  
------------------------------------------------------------------------------
Variable |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf. Interval]
---------+--------------------------------------------------------------------
   write |     109    54.99083    .7790686    8.133715    53.44658    56.53507
    math |     109     52.3945    .8765083    9.151015     50.6571    54.13189
---------+--------------------------------------------------------------------
    diff |     109     2.59633    .6734012    7.030515    1.261532    3.931128
------------------------------------------------------------------------------
  
                   Ho: mean(write - math) = mean(diff) = 0
 
  Ha: mean(diff) < 0         Ha: mean(diff) ~= 0        Ha: mean(diff) > 0
       t =   3.8555                t =   3.8555              t =   3.8555
   P < t =   0.9999          P > |t| =   0.0002          P > t =   0.0001
  
/*  check normality of difference scores  */
  
kdbox diff if female==1, norm mean

t-test Summary

Single sample t-test -- ttest write=50
Two-sample t-test for independent groups
t-test with pooled (equal) variances -- ttest write, by(female)
t-test with separate (unequal) variances -- ttest write, by(female) unequal

t-test for dependent samples (paired samples) -- ttest write=read

UCLA Department of Education

Phil Ender, 14JNov00