Slides to accompany
Weathington, Cunningham &
Pittenger (2010),
Chapter 16: Research with
Categorical Data
1
Objectives
• Goodness-of-Fit test
• χ2 test of Independence
• χ2 test of Homogeneity
• Reporting χ2
• Assumptions of χ2
• Follow-up tests for χ2
• McNemar Test
2
Background
• Sometimes we want to know how people fit
into categories
– Typically involves nominal and ordinal scales
• Person only fits one classification
• The DV in this type of research is a frequency
or count
3
Goodness-of-Fit Test
• Do frequencies of different categories
match (fit) what would be hypothesized in
a broader population?
• χ2 will be large if nonrandom difference
between Oi and Ei
• If χ2 < critical value, distributions match
4
Figure 16.1
5
Table 16.2
Month
Observed
frequency
Expected
proportions
Expected frequency = p x T
January
O1 = 26
0.04
E1 = .04 x 600 = 24
February
O2 = 41
0.07
E2 = .07 x 600 = 42
March
O3 = 36
0.06
E3 = .06 x 600 = 36
April
O4 = 41
0.07
E4 = .07 x 600 = 42
May
O5 = 62
0.10
E5 = .10 x 600 = 60
June
O6 = 75
0.12
E6 = .12 x 600 = 72
July
O7 = 60
0.10
E7 = .10 x 600 = 60
August
O8 = 67
0.11
E8 = .11 x 600 = 66
September
O9 = 58
0.10
E9 = .10 x 600 = 60
October
O10 = 52
0.09
E10 = .09 x 600 = 54
November
O11 = 41
0.08
E11 = .08 x 600 = 48
December
O12 = 41
0.06
E12 = .06 x 600 = 36
600
1.00
600
Totals
6
Calculation Example
(Oi –
(Oi – Ei)2
Ei
Month
O
E
Oi – Ei
January
O1 = 26
E1 = 24
2
4
0.1667
February
O2 = 41
E2 = 42
-1
1
0.0238
March
O3 = 36
E3 = 36
0
0
0.0000
April
O4 = 41
E4 = 42
-1
1
0.0238
May
O5 = 62
E5 = 60
2
4
0.0667
June
O6 = 75
E6 = 72
3
9
0.1250
July
O7 = 60
E7 = 60
0
0
0.0000
August
O8 = 67
E8 = 66
1
1
0.0152
September
O9 = 58
E9 = 60
-2
4
0.0667
O10 = 52 E10 = 54
-2
4
0.0741
November O11 = 41 E11 = 48
-7
49
1.0208
December
5
25
0.6944
October
Totals
O12 = 41 E12 = 36
600
600
Ei)2
χ2 = 2.2771
7
Another Example – Table 16.4
Season
O
E
Oi – Ei
(Oi – Ei)2
(Oi – Ei)2
Ei
Spring
495
517.5
-22.5
506.25
0.9783
Summer
503
517.5
-14.5
210.25
0.4063
Autumn
491
517.5
-26.5
702.25
1.3570
Winter
581
517.5
63.5
4032.25
7.7918
2070
2070
0.0
Totals
χ2 = 10.5334
8
Goodness-of-Fit Test
• χ2 is nondirectional (like F)
• Assumptions:
– Categories are mutually exclusive
– Conditions are exhaustive
– Observations are independent
– N is large enough
9
χ2 Test of Independence
• Are two categorical variables independent
of each other?
• If so, Oij for one variable should have
nothing to do with Eij for other variable
and the difference between them will be 0.
10
Table 16.5
Childhood sexual abuse
Abused
Not abused
Row total
Attempted suicide
16
23
39
No suicide attempts
24
108
132
Column total
40
131
171
11
Table 16.6
Childhood sexual abuse
Abused
Not abused
Row total
Attempted suicide
R1 = 39
No suicide attempts
R2 = 132
Column total
C1 = 40
C2 = 131
T = 171
12
Computing χ2 Test Statistic
13
Interpreting χ2 Test of Independence
• Primary purpose is to identify
independence
– If Ho retained, then we cannot assume the two
variables are related (independence)
– If Ho rejected, the two variables are somehow
related, but not necessarily cause-and-effect
14
χ2 Test of Homogeneity
• Can be used to test cause-effect
relationships
• Categories indicate level of change and χ2
statistic tests whether pattern of Oi
deviates from chance levels
• If significant χ2, can assume c-e relation
15
χ2 Test of Homogeneity Example
Psychotherapy condition
Control
Informative
Individual:
Type A
Individual
Type B
Row total
No change
O11 = 19
E11 = 14
O12 = 15
E12 = 14
O13 = 7
E13 = 14
O14 = 15
E14 = 14
56
Moderate
O21 = 21
E21 = 17
O22 = 22
E22 = 17
O23 = 9
E23 = 17
O24 = 16
E24 = 17
68
Good
O31 = 20
E31 = 29
O32 = 23
E32 = 29
O33 = 44
E33 = 29
O34 = 29
E34 = 29
116
Column total
60
60
60
60
240
16
Reporting χ2 Results
• Typical standard is to include the statistic,
df, sample size, and significance levels at a
minimum:
χ2 (df, N = n) = #, p < α
χ2(6, N = 240) = 23.46, p < .05
17
Follow-up Tests to χ2
• Cramér’s coefficient phi (Φ)
– Indicates degree of association between two
variables analyzed with χ2
– Values between 0 and 1
– Does not assume linear relationship between
the variables
18
Post-Hoc Tests to χ2
• Standardized residual, e
– Converts differences between Oi and Ei to a
statistic
• Shows relative difference between
frequencies
• Highlights which cells represent
statistically significant differences and
which show chance findings
19
Follow-up Tests to χ2
• McNemar Test
– For comparing correlated samples in a 2 x 2
table
– Table 16.9 illustrates  special form of χ2 test
– Ho: differences between groups are due to
chance
– Example presented in text and Table 16.10
provides an application
20
What is Next?
• **instructor to provide details
21