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

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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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