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Inference for Categorical Data

William P. Wattles, Ph. D.Francis Marion University

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Continuous vs. Categorical• Continuous (measurement) variables have

many values• Categorical variables have only certain

values representing different categories• Ordinal-a type of categorical with a natural

order (e.g., year of college)• Nominal-a type of categorical with no order

(e.g., brand of cola)

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Categorical Data• Tells which category an individual is in

rather than telling how much.• Sex, race, occupation naturally categorical• A quantitative variable can be grouped to

form a categorical variable. • Analyze with counts or percents.

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Describing relationships in categorical data

• No single graph portrays the relationship

• Also no similar number summarizes the relationship

• Convert counts to proportions or percents

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Prediction

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Prediction

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Moving from descriptive to Inferential

• Chi Square Inference involves a test of independence.

• If variable are independent, knowledge of one variable tells you nothing about the other.

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Moving from descriptive to Inferential

• Inference involves expected counts. – Expected count=The count that would occur if

the variables are independent

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Inference for two-way tables

• Chi Square test of independence.• For more than two groups• Cannot compare multiple groups one at a

time.

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To Analyze Categorical Data

• First obtain counts• In Excel can do this with a pivot table• Put data in a Matrix or two-way table

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Matrix or two-way table

Republican Democrat Independent

Male 18 43 14

Female 39 23 18

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Inference for two-way tables

• Expected count• The count that would occur if the variables

are independent

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Matrix or two-way table• Rows• Columns• Distribution: how often each outcome

occurred• Marginal distribution: Count for all entries

in a row or column

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Row and column totals

RepublicanDemocrat IndependentMale 18 43 14 75Female 39 23 18 80

57 66 32 155

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RepublicanDemocrat IndependentMale 75 48%Female 80 52%

57 66 32 15537% 43% 21%

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Expected counts• 37% of all subjects are Republicans• If independent 37% of females should be

Republican (expected value)• 37% of 80= 29• 37% of 75 = 28

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Expected counts rounded

Republican Democrat Independent totalMale 28 32 15 75Female 29 34 17 80total 57 66 32 155

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Observed vs. ExpectedRepublicanDemocrat Independent

Male 18 43 14 75Female 39 23 18 80

57 66 32 155

Republican Democrat Independent totalMale 28 32 15 75Female 29 34 17 80total 57 66 32 155

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Chi-Square• Chi-square A measure of how far the

observed counts are from the expected counts

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Chi-square test of independence

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Chi Square test of independence with SPSS

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Chi Square test of independence with SPSS

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Chi Square

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Chi-square test of independence

• Degrees of Freedom• df=number of rows-1 times number of

columns -1• compare the observed and expected counts.• P-value comes from comparing the Chi-

square statistic with critical values for a chi-square distribution

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Example• Have the percent of majors changed by

school?

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Data collection

http://www.fmarion.edu/about/FactBook2004/2005 Fall 2004 Graduates by Major

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Chi Square

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Marital Status, page 543

job grade single married divorced widowed1 58 874 15 82 222 3927 70 203 50 2396 34 104 7 533 7 4

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Marital Status, page 543

Test Statistics Value df p-valuePearson Chi-Square 67.491 9 0.0000

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Olive Oil, page 578

 

low medium highColon cancer 398 397 430rectal 250 241 217controls 1368 1377 1409

Olive Oil

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Olive Oil, page 578

Test Statistics Value df p-valuePearson Chi-Square 1.552 4 0.817Continuity Adjusted Chi-Square1.396 4 0.845Likelihood Ratio Chi-Square1.549 4 0.818

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Business Majors, page 563

Female MaleAccounting 68 56Administration 91 40Economics 5 6Finance 61 59

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Business Majors, page 563

Test Statistics Value df p-valuePearson Chi-Square 10.827 3 0.013

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Exam Three• 37 multiple choice

questions, 4 short answer• T-tests and chi square on

Excel• General questions about

analyzing categorical data and t-tests

• Review from earlier this term

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Inference as a decision• We must decide if the null hypothesis is

true.• We cannot know for sure.• We choose an arbitrary standard that is

conservative and set alpha at .05• Our decision will be either correct or

incorrect.

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Type I and Type II errors

Ho is really True

Ho is really False

We reject Ho

Type I Error (false alarm)

Correct Decision

We accept Ho

Correct decision Type II Error (miss)

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Type I error• If we reject Ho when in fact Ho is true, this

is a Type I error• Statistical procedures are designed to

minimize the probability of a Type I error, because they are more serious for science.

• With a Type I error we erroneously conclude that an independent variable works.

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Type II error

• If we accept Ho when in fact Ho is false this is a Type II error.

• A type two error is serious to the researcher.• The Power of a test is the probability that

Ho will be rejected when it is, in fact, false.

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Probability

Ho is really True

Ho is really False

We reject Ho

p= p=1-

We accept Ho

p=1- p=

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Power• The goal of any scientific research is to

reject Ho when Ho is false.• To increase power:

– a. increase sample size– b. increase alpha– c. decrease sample variability– d. increase the difference between the means

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Categorical data example• African-American students more likely to

register via the web.

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Table

Variable White African-AmericanStudents University-Wide n Percent n PercentRegister on the Web 447 34% 284 44%Register with other method 876 66% 356 56%Total 1323 640

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Web Registration by Race

34%

25%

44%

29%

0%

10%

20%

30%

40%

50%

60%

2000 2001Year

WhiteAfrican-American

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Categorical Data Example• African-American students university-wide

(44%) were more likely that white students (34%) to use web registration, X2(1, N = 1963) = 20.7 , p < .001.

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Smoking among French Men

• Do these data show a relationship between education and smoking in French men?

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The EndThe End

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Benford’s Law page 550• Faking data?

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Problem 20.14Digit ratio Observed

1 0.301 62 0.176 43 0.125 64 0.097 75 0.079 36 0.067 57 0.058 68 0.051 49 0.046 4

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Digit ratio Expected Observed1 0.301 13.545 62 0.176 7.92 43 0.125 5.625 64 0.097 4.365 75 0.079 3.555 36 0.067 3.015 57 0.058 2.61 68 0.051 2.295 49 0.046 2.07 4

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Expected Observed13.545 6 4.20280731

7.92 4 1.940202025.625 6 0.0254.365 7 1.590658653.555 3 0.086645573.015 5 1.306873962.61 6 4.40310345

2.295 4 1.266677562.07 4 1.7994686

16.6214371

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Significance test

chitest p = 0.03430

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Example• Survey2 Berk & Carey

page 261