Newsom
USP 534 Data Analysis
Spring 2006
Simple Within-Subjects Tests
Within-subjects t-test
The within-subjects t-test, used for comparisons with a continuous dependent variable, is also
known as the paired samples t-test (the SPSS term), the dependent samples t-test, correlated
samples t-test, or the repeated measures t-test. The reason the test has all of these names is
because it is used in several different situations: the same person is in the study twice
(longitudinal or repeated measures design), pairs of individuals are linked together or “yoked”
(e.g., twins, or married couples), or because they are naturally linked or the experimenter linked
them as when they are ‘matched’ on some score (e.g., matched on age).
Example. We could have conducted the same school privatization (charter) study in a different
way—by comparing teachers’ satisfaction ratings before and after a school was converted to a
privately operated school. This design could be classified as a single-group pretest-posttest
design (Remember from our design discussion that there are some methodological problems
with this design that can be addressed with some changes, but let’s just assume this is the
design for now).
Using the same numbers as in the first between-subjects example given in class, we have two
scores for each of 5 teachers. However, notice that in this design we only are using half the
number of cases.
Teacher
Public
(Pretest)
2
4
6
8
10
1
2
3
4
5
Private
(Posttest)
7
8
10
8
12
D
DD
-5
-4
-4
0
-2
D  3
2
1
1
3
1
 D  D
 D  D
2
4
1
1
9
1
2
 16
Formulas:
s
2
D
 D  D

2
sD 
sD2
N
t
D
sD
N 1
3
16
4

 3.35


.89
4
5
4
 .89
df = N-1 = 5-1 = 4. tcrit at a = .05 with df = 4 is 2.776. The difference is significant, because the
(absolute value of the) calculated value, 3.35, exceeds the critical value of 2.776.
Statistical comment. Notice that the same data in the very first between-subjects example
(presented in class) yielded a non-significant difference (with twice as many cases!!). The
reason the within-subjects test has more powerful is that variation due to individual differences
is eliminated in the within-subjects design. Each subject serves as his/her own comparison or
control.
Newsom
USP 534 Data Analysis
Spring 2006
SPSS Menus Steps
1. Analyze Compare Means Paired Samples t-test
2. Move over the two variables (e.g., pretest score variable, posttest score variable)
3. Click “Ok.”
The Output will look something like this (note that this was generated in SPSS 11.5):
SPSS Output
Paired Samples Statistics
Mean
Pair
1
PUBLIC rating of
public school
PRIVATIZ rating of
privatized s chool
N
Std. Deviation
Std. Error
Mean
6.0000
5
3.16228
1.41421
9.0000
5
2.00000
.89443
Paired Samples Correlations
N
Pair
1
PUBLIC rating of public
school & PRIVATIZ rating
of privatized school
Correlation
5
.791
Sig.
.111
Pa ired Sa mpl es Test
Paired Differences
Mean
Pair
1
PUBLIC rating of public
sc hool - PRIVATIZ rating
of privatized sc hool
-3. 0000
St d. Deviat ion
St d. Error
Mean
2.00000
.89443
95% Confidenc e
Int erval of t he
Difference
Lower
Upper
-5. 4833
-.5167
t
-3. 354
df
Sig. (2-tailed)
4
Example write-up. Using a repeated measures t-test, public school teachers’ satisfaction
ratings prior to privatization of their school were compared to their satisfaction ratings after
privatization of their school. Satisfaction ratings were significantly lower after privatization (M =
6) than before privatization (M = 9) as indicated by a significant t-test, t(4) = 3.35, p < .05. This
finding indicates that there was a decline in satisfaction ratings over time that was not likely to
be due to chance.
.028
Newsom
USP 534 Data Analysis
Spring 2006
Chi-square for within-subjects
For a binary dependent variable, there is a form of the chi-square test for within-subjects
designs called McNemar's chi-square. As with the paired t-test or the within-subjects ANOVA,
the McNemar test is used whenever the same individuals are measured (or surveyed) twice,
matched on some variable (e.g., yoked by age), participants are paired in some way (e.g., twins
or married couples), or responses on two measures are used (e.g., favorability to gun control
compared to favorability for abolishing the second amendment).
For instance, we might examine the favorability of voters for gun control legislation in April and
in June.
June
April
No
80
10
90
No
Yes
Yes
100
110
210
180
120
300
To compute McNemar's, the following formula is used:
bc  bg

2
McNemar ' s 
2
cb
c, b, and d come from labeling the cells in the table as below.
June
April
No
Yes
No
a
c
Yes
b
d
b10  100g

2
100  10
b90g

2
110
 73.63
df in this test is 1, the critical value is 3.84 (from the chi-square table), and because calculated
value of 73.63 exceeds this value, there is a significant difference in April and June responses.
For more than 2 related groups, one can use Cochran’s Q test, which I will not detail here.
Newsom
USP 534 Data Analysis
Spring 2006
SPSS Menus Steps for McNemar’s test
1. Analyze Descriptive statistics  Crosstabs
2. Move over the two variables to the row and column boxes (I used rows for the pretest and
columns for the posttest)
3. Click on Statistics and check McNemar, then click Continue.
4. Click on Cells and then check Row and Column under Percentages, then click Continue.
5. Click OK.
SPSS Output for McNemar’s test
April Favor gun legislation in April * June Favor gun legislation in June Crosstabulation
April Favor gun
legislation in April
1 no
2 yes
Total
Count
% within April Favor
gun legislation in April
% within June Favor
gun legislation in June
Count
% within April Favor
gun legislation in April
% within June Favor
gun legislation in June
Count
% within April Favor
gun legislation in April
% within June Favor
gun legislation in June
June Favor gun
legislation in June
1 no
2 yes
80
100
Total
180
44.4%
55.6%
100.0%
88.9%
47.6%
60.0%
10
110
120
8.3%
91.7%
100.0%
11.1%
52.4%
40.0%
90
210
300
30.0%
70.0%
100.0%
100.0%
100.0%
100.0%
Chi-Square Tests
Value
McNemar Test
N of Valid Cases
Exact Sig.
(2-sided)
.000a
300
a. Binomial distribution us ed.
Voter favorability toward the gun control measure changed significantly over the two-month
period (p < .001). Voters were more likely to favor the gun control legislation in June (70%) than
in April (40%) when they were first polled. (Note that SPSS does not give the value of the
McNemar chi-square, just it’s p-value. Also, in this case, it make more sense to use the marginal
total percentages rather than the percentages within particular cells.)