chapter fifteen chi-square and other nonparametric procedures
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Copyright © Houghton Mifflin Company. All rights reserved.Chapter One-Way Chi Square: The Goodness of Fit TestTRANSCRIPT
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 3
One-Way Chi Square:The Goodness of Fit Test
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 4
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 5
kNfe categoryeachin
Statistical Hypotheses
• H0: all frequencies in the population are equal
• Ha: all frequencies in the population are not equal
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 6
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 7
• 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
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Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 8
The Two-Way Chi Square:The Test of Independence
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 9
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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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 10
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
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fff 2
2 )(
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 11
Nonparametric Statistics
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 12
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212
111
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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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 13
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 14
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obt
Rank Sums Test
• Used to test two independent samples of ranks and either n is greater than 20
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 15
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 16
• 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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 17
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 18
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 19
730.21125.43
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Example 1
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Example 1
2crit for df = (2 -1)(2 - 1) = 1 is 3.84
Since 2obt > 2
crit, we reject the null hypothesis
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 21
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 22
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 23
RanksGroup 1
RanksGroup 2
4 9
6 7
3 12
5 11
1 8
10 2
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RnnnnU
RnnnnU
Example 2
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 15 - 24
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.