chapter 13: categorical data analysis statistics

Chapter 13: Categorical Data Analysis

Statistics

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

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Where We’ve Been Presented methods for making inferences

about the population proportion associated with a two-level qualitative variable (i.e., a binomial variable)

Presented methods for making inferences about the difference between two binomial proportions

Where We’re Going


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Discuss qualitative (categorical) data with more than two outcomes

Present a chi-square hypothesis test for comparing the category proportions associated with a single qualitative variable – called a one-way analysis

Present a chi-square hypothesis test relating two qualitative variables – called a two-way analysis

13.1: Categorical Data and the Multinomial Experiment

Properties of the Multinomial Experiment

1. The experiment consists of n identical trials.

2. There are k possible outcomes (called classes, categories or cells) to each trial.

3. The probabilities of the k outcomes, denoted by p1, p2, …, pk, where p1+ p2+ … + pk = 1, remain the same from trial to trial.

4. The trials are independent.

5. The random variables of interest are the cell counts n1, n2, …, nk of the number of observations that fall into each of the k categories.


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13.2: Testing Categorical Probabilities: One-Way Table

Suppose three candidates are running for office, and 150 voters are asked their preferences. Candidate 1 is the choice of 61 voters. Candidate 2 is the choice of 53 voters. Candidate 3 is the choice of 36 voters.

Do these data suggest the population may prefer one candidate over the others?


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Candidate 1 is the

choice of 61 voters.

Candidate 2 is the


Candidate 3 is the


n =150


6

130 1 2 3

13

0

1 2 3

20

2 22 1 1 2 2

1 2

: No preference

: At least one of the proprtions exceeds

150(Number of votes for each candidate| ) 50350

A chi-square ( ) test is used to test .

[ ] [ ]

a

H p p p

H

E H

E E E

H

n E n E

E E

23 3

3

2 2 22

2.05, 2

[ ]

[61 50] [53 50] [36 50]6.52

50 50 50

5.99147df

n E

E



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Reject the null hypothesis

Test of a Hypothesis about Multinomial Probabilities: One-Way Table

H0: p1 = p1,0, p2 = p2,0, … , pk = pk,0

where p1,0, p2,0, …, pk,0 represent the hypothesized values of the multinomial probabilities

Ha: At least one of the multinomial probabilities does not equal its hypothesized value

where Ei = np1,0, is the expected cell count given the null hypothesis.



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22 [ ]

Test statistic: i i

i

n E

E

2 2Rejection region: ,

with (k-1) df.

Conditions Required for a Valid 2 Test:One-Way Table

1. A multinomial experiment has been conducted. 2. The sample size n will be large enough so that, for every cell,

the expected cell count E(ni) will be equal to 5 or more.



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Legalization Decriminalization Existing Law No Opinion

7% 18% 65% 10%



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Example 13.2: Distribution of Opinions About MarijuanaPossession Before Television Series has Aired

Table 13.2: Distribution of Opinions About MarijuanaPossession After Television Series has Aired


39 99 336 26


11McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis



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Expected Distribution of 500 Opinions About MarijuanaPossession After Television Series has Aired


500(.07)=35 500(.18)=90 500(.65)=325 500(.10)=50

0 1 2 3 4

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2 2.01, 3

: .07, .18, .65, .10

: At least one of the proportions differs

from its null hypothesis value.

[ ]Test statistic:

Rejection region: 11.3449

a

i i

i

df

H p p p p

H

n E

E



13



500(.07)=35 500(.18)=90 500(.65)=325 500(.10)=50

2 2.01, 3

2 2 2 22

2


(39 35) (99 90) (336 325) (26 50)

35 90 325 50

13.249

df



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500(.07)=35 500(.18)=90 500(.65)=325 500(.10)=50

2 2.01, 3

2 2 2 22

2


(39 35) (99 90) (336 325) (26 50)

35 90 325 50

13.249

df

Reject the null hypothesis


Inferences can be made on any single proportion as well: 95% confidence interval on the proportion of citizens in the

viewing area with no opinion is


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4

4

4

ˆ4

44

4 4ˆ

ˆ4

ˆ 1.96

26ˆwhere .052

500

ˆ ˆ(1 ) .052(.948)and .0099

500ˆ 1.96 .052 1.96(.0099) .052 .019

p

p

p

p

np

n

p p

np

13.3: Testing Categorical Probabilities: Two-Way Table

Chi-square analysis can also be used to investigate studies based on qualitative factors. Does having one characteristic make it

more/less likely to exhibit another characteristic?


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Column

1 2 c Row Totals

1 n11 n12 n1c R1

Row 2 n21 n22 n2c R2

r nr1 nr2 nrc Rr

Column Totals C1 C1 C1 n

The columns are divided according to the subcategories for one qualitative variable and the rows for the other qualitative variable.



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0

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General Form of a Two-way (Contigency) Table Analysis:

A Test for Independence

: The two classifications are independent

: The two classifications are dependent

[ ]Test statistic:

where

a

ij ij

ij

H

H

n E

E

2 2

and total for row , total for row , sample size

Rejection region: , df = ( 1)( 1)

i jij

i j

RCE

nR i C j n

r c



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The results of a survey regarding marital status and religious affiliation are reported below (Example 13.3 in the text).

A B C D None Totals

Divorced 39 19 12 28 18 116

Married, never divorced

172 61 44 70 37 384

Totals 211 80 56 98 55 500

MaritalStatus

Religious Affiliation

H0: Marital status and religious affiliation are independentHa: Marital status and religious affiliation are dependent



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The expected frequencies (see Figure 13.4) are included below:

A B C D None Totals

Divorced 39(48.95)

19(18.56)

12(12.99)

28(27.74)

18(12.76)

116

Married, never divorced

172(162.05)

61(61.44)

44(43.01)

70(75.26)

37(42.24)

384

Totals 211 80 56 98 55 500

MaritalStatus

Religious Affiliation

The chi-square value computed with SAS is 7.1355, with p-value = .1289.Even at the = .10 level, we cannot reject the null hypothesis.


21McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.4: A Word of Caution About Chi-Square Tests


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13.4: A Word of Caution About Chi-Square Tests


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Be sure

chapter 13: categorical data analysis statistics

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