objectives
DESCRIPTION
Slides to accompany Weathington, Cunningham & Pittenger (2010), Chapter 16: Research with Categorical Data. Objectives. Goodness-of-Fit test χ 2 test of Independence χ 2 test of Homogeneity Reporting χ 2 Assumptions of χ 2 Follow-up tests for χ 2 McNemar Test. Background. - PowerPoint PPT PresentationTRANSCRIPT
Slides to accompany Weathington, Cunningham & Pittenger (2010),
Chapter 16: Research with Categorical Data
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Objectives
• Goodness-of-Fit test
• χ2 test of Independence
• χ2 test of Homogeneity
• Reporting χ2
• Assumptions of χ2
• Follow-up tests for χ2
• McNemar Test2
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
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Table 16.2Month
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
Totals 600 1.00 600 6
Calculation ExampleMonth O E Oi – Ei (Oi – Ei)2
(Oi – Ei)2
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
October O10 = 52 E10 = 54 -2 4 0.0741
November O11 = 41 E11 = 48 -7 49 1.0208
December O12 = 41 E12 = 36 5 25 0.6944
Totals 600 600 χ2 = 2.27717
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
Totals 2070 2070 0.0 χ2 = 10.5334
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Goodness-of-Fit Test
• χ2 is nondirectional (like F)
• Assumptions:
– Categories are mutually exclusive
– Conditions are exhaustive
– Observations are independent
– N is large enough
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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.
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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
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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
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Computing χ2 Test Statistic
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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
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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
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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
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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
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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
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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
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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
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What is Next?
• **instructor to provide details
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