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Chi-square (χ2)PSY 2114-15-08

LAST STATISTIC OF THE SEMESTER!NEXT TIME: FOOD DAY, REVIEW, WRAP-UP

A. Introduction

1. Parametric tests: Use at least one numeric rating, so scores can be placed in a frequency distribution that usually has a normal shape

Correlation/Regression: Continuous variables only

t-test/ANOVA: One categorical variable, one continuous variable

2. Non-parametric tests: Do not use numerical values, scores cannot be placed on a frequency distribution (also sometimes used for numeric variables that have non-normal distributions)

Chi-square (χ2): Categorical variables only

Pronunciation note: Chi is pronounced “khi” (from Greece), not “chai” (India), not “chee” (China)

B. Two Types of χ2

χ2 Test for Goodness of Fito Involves a single categorical variable only

χ2 Test for Independenceo Involves 2(+) categorical variables

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C. χ2 Test for Goodness of Fit

Generally just involves one categorical variable Null hypothesis specifies the proportion of the

population in each category Determines how well sample data conform to

proportions set forth by the null hypothesis Test statistic (χ2) examines whether the

proportions in the sample reliably differ from the null hypothesis

Examples:

Use logic or past research to guide the null hypothesis

H0 for Gender:

Female 50%Male 50%

H0 for Negative Childhood Emotion:

Shame 25%Anger 25%Anxiety 25%Sadness 25%

Often an equal percentage (proportion) is chosen for each group

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Alternatively, null hypothesis could be based on known proportions in a larger population

Examples:

H0 for Ethnicity:

White 75%non-White 25%

H0 for Vegetarianism:

No 97%Yes 3%

H0 for Religious Affiliation:

Christianity 84%Non-religious / Don’t care 10%Agnosticism 2%Atheism 1%Other 3%

χ2 used to examine whether the proportions in a sample reliably differ from those hypothesized

Like F…χ2 ranges from 0 to ∞Is small when the null hypothesis is likely true.Is large when the null hypothesis is rejected.

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H0 for Gender:

Female 50%Male 50%

N = 279Proportion Female = 50% = .50Hypothesized frequency Female = .50 * 279 = 139.5Proportion Male = 50% = .50Hypothesized frequency Male = .50 * 279 = 139.5

Chi-Square TestFrequencies

15. Gender

213 139.5 73.566 139.5 -73.5

279

FemaleMaleTotal

Observed N Expected N Residual

Test Statistics

77.4521

.000

Chi-Square a

dfAsymp. Sig.

15. Gender

0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 139.5.

a.

The Observed N indicates the actual frequency in each group

The Expected N indicates the frequency that is hypothesized, based on the null hypothesis.

The Residual just indicates the Observed frequency minus the Expected frequency

The Test Statistics box indicates that the χ2 value is 44.17, the degrees of freedom (df) are 1, and p < .001.

The chi-square test for goodness of fit was significant, χ2(1, N = 279) = 77.45, p < .001. The sample included an unexpectedly high number of females.

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Last Semester…

Chi-Square TestFrequencies

2. Gender

223 163.0 60.0103 163.0 -60.0326

FemaleMaleTotal

Observed N Expected N Residual

Test Statistics

44.1721

.000

Chi-Square a

dfAsymp. Sig.

2. Gender

0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 163.0.

a.

The chi-square test for goodness of fit was significant, χ2(1, N = 326) = 44.17, p < .001. The sample included an unexpectedly high number of females.

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H0 for Religious Affiliation:

Christianity 84%Non-religious / Don’t care 10%Agnosticism 2%Atheism 1%Other 3%

N = 279Hypothesized frequency Christian = .84 * 279 = 234.4… … …Hypothesized frequency Other = .03 * 279 = 8.4

Chi-Square TestFrequencies

31. Worldview or Religion

204 234.4 -30.427 27.9 -.910 5.6 4.435 2.8 32.23 8.4 -5.4

279

ChristianityAgnosticismAtheismDon't Know / Don't CareOtherTotal

Observed N Expected N Residual

Test Statistics

382.7674

.000

Chi-Square a

dfAsymp. Sig.

31. Worldviewor Religion

1 cells (20.0%) have expected frequencies less than5. The minimum expected cell frequency is 2.8.

a.

The chi-square test for goodness of fit was significant, χ2(4, N = 279) = 382.77, p < .001. There were fewer Christians than expected and more people who were apathetic.

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Last Semester…

Chi-Square TestFrequencies

23. Religion

42 32.6 9.4231 273.8 -42.819 6.5 12.516 3.3 12.718 9.8 8.2

326

Non-religiousChristianityAgnosticismAtheismOtherTotal

Observed N Expected N Residual

Test Statistics

89.9974

.000

Chi-Square a

dfAsymp. Sig.

23. Religion

1 cells (20.0%) have expected frequencies less than5. The minimum expected cell frequency is 3.3.

a.

The chi-square test for goodness of fit was significant, χ2(4, N = 326) = 90.00, p < .001. There were fewer Christians than expected and more people than expected in every other religious group.

Expected frequencies based on the expected proportions identified previously. For example, for Christianity .84 x 326 = 273.84.

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H0 for Yearly Physical:

No 50%Yes 50%

Chi-Square TestFrequencies

2. Yearly Physical

143 139.5 3.5136 139.5 -3.5279

NoYesTotal

Observed N Expected N Residual

Test Statistics

.1761

.675

Chi-Square a

dfAsymp. Sig.

2. YearlyPhysical

0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 139.5.

a.

The chi-square test for goodness of fit was not significant, χ2(1, N = 279) = 0.18, p = .68. There was about an equal number of people getting yearly physicals as those who were not.

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D. χ2 Test for Independence

Examines the relationship between two (or more) categorical variables to determine if they are independent

Two variables are said to be independent if there is no relationship between them

Two variables are said to be dependent if there is a relationship between them

Similar to the correlation coefficient, except that instead of both variables being continuous, both variables are categoricalo Can’t correlate Academic Major with

Favorite Barnyard Animal, but you can do a chi-square!

Null hypothesis: no relationship Alternative hypothesis: some relationship

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Examples:

Does Gender related to Beliefs about Human Origins?

15. Gender * 18. Belief about Human Origins Crosstabulation

124 89 213117.6 95.4 213.0

30 36 6636.4 29.6 66.0154 125 279

154.0 125.0 279.0

CountExpected CountCountExpected CountCountExpected Count

Female

Male

15. Gender

Total

ReligiousTexts

EvolutionaryTheory

18. Belief about HumanOrigins

Total

Chi-Square Tests

3.318b 1 .0692.822 1 .0933.304 1 .069

.089 .047

3.306 1 .069

279

Pearson Chi-SquareContinuity Correctiona

Likelihood RatioFisher's Exact TestLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)Exact Sig.(2-sided)

Exact Sig.(1-sided)

Computed only for a 2x2 tablea.

0 cells (.0%) have expected count less than 5. The minimum expected count is 29.57.

b.

Observed frequencies compared to those that are expected, based on the null hypothesisSample sizeChi-square value, degrees of freedom, and p-value

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MaleFemale

15. Gender

125

100

75

50

25

0

Cou

nt

Evolutionary TheoryReligious Texts

18. Belief about HumanOrigins

Bar Chart

The chi-square test for independence was non-significant, χ2(1, N = 279) = 3.32, p = .07. Gender was not reliably related to beliefs about human origins.

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Does being a parent impact one’s political views?13. Parent * 20. Top Political Issue Crosstabulation

76 58 34 11 46 22568.5 65.3 39.5 10.5 41.1 225.0

9 23 15 2 5 5416.5 15.7 9.5 2.5 9.9 54.0

85 81 49 13 51 27985.0 81.0 49.0 13.0 51.0 279.0

CountExpected CountCountExpected CountCountExpected Count

No

Yes

13. Parent

Total

Education Economy Healthcare Foreign Policy Other20. Top Political Issue

Total

Chi-Square Tests

15.516a 4 .00415.831 4 .003

.080 1 .777

279

Pearson Chi-SquareLikelihood RatioLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)

1 cells (10.0%) have expected count less than 5. Theminimum expected count is 2.52.

a.

13

YesNo

13. Parent

80

60

40

20

0

Cou

nt

OtherForeign PolicyHealthcareEconomyEducation

20. Top Political Issue

Bar Chart

The chi-square test for independence was significant, χ2(4, N = 279) = 15.52, p = .004. Non-parents were mainly concerned with education, whereas parents were mainly concerned with the economy.

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Does ethnicity relate to musical preference?

11. White * 32. Favorite Music Crosstabulation

2 2 12 10 266.0 8.8 3.6 7.6 26.062 92 27 72 253

58.0 85.2 35.4 74.4 253.064 94 39 82 279

64.0 94.0 39.0 82.0 279.0

CountExpected CountCountExpected CountCountExpected Count

No

Yes

11. White

Total

Country RockRap, hip

hop, R&B Other

32. Favorite Music

Total

Chi-Square Tests

30.695a 3 .00026.787 3 .000

9.472 1 .002

279

Pearson Chi-SquareLikelihood RatioLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)

1 cells (12.5%) have expected count less than 5. Theminimum expected count is 3.63.

a.

15

YesNo

11. White

100

80

60

40

20

0

Cou

nt

OtherRap, hip hop, R&BRockCountry

32. Favorite Music

Bar Chart

The chi-square test for independence was significant, χ2(3, N = 279) = 30.70, p < .001. Compared to other ethnic groups, white people were less likely to prefer rap, hip hop, and R&B.

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Some Examples from Last Semester…

Is Gender related to Vegetarianism?

2. Gender * 10. Vegetarian Crosstabulation

206 17 223209.3 13.7 223.0

100 3 10396.7 6.3 103.0306 20 326

306.0 20.0 326.0

CountExpected CountCountExpected CountCountExpected Count

Female

Male

2. Gender

Total

No Yes10. Vegetarian

Total

Chi-Square Tests

2.715b 1 .0991.959 1 .1623.081 1 .079

.136 .076

2.707 1 .100

326

Pearson Chi-SquareContinuity Correctiona

Likelihood RatioFisher's Exact TestLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)Exact Sig.(2-sided)

Exact Sig.(1-sided)

Computed only for a 2x2 tablea.

0 cells (.0%) have expected count less than 5. The minimum expected count is 6.32.

b.

Observed frequencies compared to those that are expected, based on the null hypothesisSample sizeChi-square value, degrees of freedom, and p-value

The chi-square test for independence was non-significant, χ2(1, N = 326) = 2.72, ns. Gender was not reliably related to vegetarianism.

The sample is 68% female, so females are expected to make up 68% of the vegetarians and 68% of the non-vegetarians.

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Does ethnicity relate to musical preference?

20. Ethnicity * 29. Music Crosstabulation

16 12 25 82 40 66 51 292

16.1 13.4 25.1 85.1 37.6 61.8 52.8 292.0

2 3 3 13 2 3 8 34

1.9 1.6 2.9 9.9 4.4 7.2 6.2 34.0

18 15 28 95 42 69 59 326

18.0 15.0 28.0 95.0 42.0 69.0 59.0 326.0

CountExpectedCountCountExpectedCountCountExpectedCount

White

Non-White

20.Ethnicity

Total

Rap R&BHipHop

Alternative

ClassicRock Country Other

29. Music

Total

Chi-Square Tests

7.354a 6 .2897.962 6 .241

.853 1 .356

326

Pearson Chi-SquareLikelihood RatioLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)

4 cells (28.6%) have expected count less than 5. Theminimum expected count is 1.56.

a.

The chi-square test for independence was non-significant, χ2(6, N = 326) = 7.35, ns. Ethnicity was not reliably related to music preference.

White people make up 90% of the sample, so they are expected to make up 90% of those who like rap, 90% of those who like R&B, etc.

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Does being an athlete related to choice of hero?

4. Athlete * 22. Hero Crosstabulation

58 20 9 17 10 8 3 3 20 148

53.6 27.7 8.6 18.2 10.9 4.5 3.6 6.4 14.5 148.0

60 41 10 23 14 2 5 11 12 178

64.4 33.3 10.4 21.8 13.1 5.5 4.4 7.6 17.5 178.0

118 61 19 40 24 10 8 14 32 326

118.0 61.0 19.0 40.0 24.0 10.0 8.0 14.0 32.0 326.0

CountExpectedCountCountExpectedCountCountExpectedCount

No

Yes

4.Athlete

Total

Mom Dad SiblingOther

Relative FriendRomanticPartner

FamousPerson

Teacher/Coach Other

22. Hero

Total

Chi-Square Tests

16.937a 8 .03117.516 8 .025

.671 1 .413

326

Pearson Chi-SquareLikelihood RatioLinear-by-LinearAssociationN of Valid Cases

Value dfAsymp. Sig.

(2-sided)

3 cells (16.7%) have expected count less than 5. Theminimum expected count is 3.63.

a.

The chi-square test for independence was statistically significant, χ2(8, N = 326) = 16.94, p = .03. Specifically, athletes were more likely than expected to indicate that their dad was their hero.