chapter fifteen chi-square and other nonparametric procedures

Chapter Fifteen

Chi-Square and Other

Nonparametric Procedures

Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 2

Nonparametric Statistics

Nonparametric statistics are used when dependent scores form skewed or otherwise nonnormal distributions, when the population variance is not homogeneous, or when scores are measured using ordinal or nominal scales.


One-Way Chi Square:The Goodness of Fit Test


One-Way Chi Square

The one-way chi square test is used when data consist of the frequencies with which participants belong to the different categories of one variable


kNfe categoryeachin

Statistical Hypotheses

• H0: all frequencies in the population are equal

• Ha: all frequencies in the population are not equal


Assumptions of the One-Way Chi Square

1.Participants are categorized along one variable having two or more categories counting the frequency in each category

2.Each participant can be in only one category

3.Category membership is independent

4.The computations include the responses of all participants in the study

5.The fe in any category must be at least 5


• Where fo are the observed frequencies and fe are the expected frequencies

• df = k - 1 where k is the number of categories

Computing One-Way2

e

eo

fff 2

2 )(


The Two-Way Chi Square:The Test of Independence


Nfff oo

e)totalcolumnscell')(totalrowscell'(

Two-Way Chi Square:the Test of Independence

• The two-way chi square procedure is used when you count the frequency of category membership along two variables


Computing Two-Way

• Where fo are the observed frequencies and fe are the expected frequencies

• df = (number of rows - 1)(number of columns - 1)

2

e

eo

fff 2

2 )(


Nonparametric Statistics


222

212

111

211

2)1())((

2)1())((

RnnnnU

RnnnnU

The Mann-Whitney U test

• Used to test two independent samples of ranks when the n in each condition is equal to or less than 20


The Mann-Whitney U test

1.Assign ranks to all scores in the experiment

2.Compute the sum of the ranks for each group

3.Compute U1 and U2

4.Determine the Uobt

5.Find the critical value of U

6.Compare Uobt to Ucrit. Uobt is significant if it is equal to or less than Ucrit.


12)1)()((

2)1(

21

exp

exp

NnnRR

z

NnR

obt

Rank Sums Test

• Used to test two independent samples of ranks and either n is greater than 20


Rank Sums Test

1. Assign ranks to the scores in the experiment

2. Choose one group and compute the sum of the ranks

3. Compute the expected sum of ranks for the chosen group (Rexp)

4. Compute the rank sums statistic zobt

5. Find the critical value of z

6. Compare zobt to zcrit

7. Describe a significant relationship using eta squared


• Wilcoxon T test is used to test two related samples of ranked data

• Kruskal-Wallis H test is used to study one factor involving at least three conditions where each is tested using independent samples and at least five participants in each sample

• Friedman test is used to study one factor involving at least three conditions where the samples in each are related (either matching or repeated measures)

2

Other Nonparametric Tests


Males Females

Dogs 24 11

Cats 15 54

Example 1

• A survey is conducted where respondents are asked to indicate (a) their sex and (b) their preference in pets between dogs and cats. The frequency of males and females making each pet selection is given below. Perform a two-way chi square test.


Males Females

Dogs 13.125 21.875

Cats 25.875 43.125

Example 1

• The expected values for each cell are:– (39)(35)/104 = 13.125– (65)(39)/104 = 21.875– (39)(69)/104 = 25.875– (65)(69)/104 = 43.125


730.21125.43

)125.4354(875.25

)875.2515(

875.21)875.2111(

125.13)125.1324(

22

222

Example 1


Example 1

2crit for df = (2 -1)(2 - 1) = 1 is 3.84

Since 2obt > 2

crit, we reject the null hypothesis


Group 1 Group 25 11

7 9

4 20

6 17

2 10

12 3

Example 2

• Using the following data set, conduct a two-tailed Mann-Whitney U test with = 0.05


Group 1 Group 2 RanksGroup 1

RanksGroup 2

5 11 4 9

7 9 6 7

4 20 3 12

6 17 5 11

2 10 1 8

12 3 10 2

Example 2


RanksGroup 1

RanksGroup 2

4 9

6 7

3 12

5 11

1 8

10 2

8492)7(6)6)(6(

2)1())((

28292)7(6)6)(6(

2)1())((

222

212

111

211

RnnnnU

RnnnnU

Example 2


Example 2

• Since this is a two-tailed test, Uobt is the smaller of U1 and U2. Then Uobt = 8

• For a two-tailed test with n1 = 6 and n2 = 6, Ucrit = 5

• In the Mann-Whitney U test, to be significant Uobt must be equal to or less than the critical value. Here, the test is not significant.

chapter fifteen chi-square and other nonparametric procedures

Documents