© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

HLTH 300 Biostatistics for Public Health Practice,

Raul Cruz-Cano, Ph.D.4/28/2014, Spring 2014

Fox/Levin/Forde, Elementary Statistics in Social Research, 12e

Chapter 9: Nonparametric Tests of Significance

1

© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

Understand the logic of nonparametric tests

Conduct one-way and two-way chi-square tests

Perform the median test

Perform the Mann-Whitney U and Kruskal-Wallis tests

CHAPTER OBJECTIVES

9.1

9.2

9.3

9.4

Understand the logic of nonparametric tests

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

9.1

4

9.1

t tests and F ratios require:• Normality (or especially large samples)• Interval level dataWhat if these requirements cannot be met?• We must use nonparametric tests

– Chi-square– The median test– Mann-Whitney U test– Kruskal-Wallis test

Nonparametric tests are less powerful than parametric

• Power = the probability of rejecting the null hypothesis when it is actually false and should be rejected

Nonparametric Tests

Conduct one-way and two-way chi-square tests

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

9.2

6

9.2

Observed frequency: the set of frequencies obtained in an actual frequency distribution

Expected frequency: the frequencies that are expected to occur under the terms of the null hypothesis

• In general, this is found by dividing N by the number of categories

Chi-square allows us to test the significance of differences between observed and expected frequencies

The One-Way Chi-Square Test

22 o e

e

f ff

2 chi-square value

expected frequency in any categoryobserved frequency in any category

e

o

ff

df 1k

7

Examples

Box 9.1, page 324Problem 13

8

9.2

How can we compare observed and expected frequencies for more than one variable?

• Two-way chi-square test • This involves cross-tabulationsThe methods for calculating one-way and two-

way chi-squares are very similar• In fact, the same formula is used• The only major difference is in how we calculate expected

frequencies

The Two-Way Chi-Square Test

row marginal total column marginal totalef N

For each cell:

df=(# of rows -1 )(# of columns -1) 22 o e

e

f ff

9.2

Table 9.2

402025

401520

402025

401520

10

Examples

Box 9.2, page 331Problem 15 (2 x 2)Problem 22 (more than 2 groups)

11

9.2

One of the few demands on the chi-square test is that the sample size should not be too small

• Be wary of expected frequencies that are less than 5– In this case, it might be best to collapse categories

• When expected frequencies are greater than 5 but less than 10, use Yate’s correction– Reduces the size of the chi-square value– Only used for 2 X 2 tables, hence df= 1

Correcting for Small Expected Frequencies

2

2 .5o e

e

f ff

12

Example

Page 329

Requirements for the Use of Two-Way Chi-Square9.2

A Comparison between Two or More Samples

Nominal Data

Random Sampling

The Expected Cell Frequencies Should Not Be Too Small

Perform the median test

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

9.3

15

9.3

Used when dealing with ordinal data• Determines the likelihood that two or more random samples

have been taken from populations with the same median

First, determine the median of the two groups combined

Then, create a cross-tabulation with the two categories and the scores that fall above the median and the scores that do not fall above the median

Finally, conduct a chi-square test • Using Yate’s corrections if there are any expected frequencies

that are less than 10

The Median Test

16

Example

Box 9.4, page 341Problem 36

Requirements for the Use of the Median Test9.3

A Comparison between Two or More Medians

Ordinal Data

Random Sampling

Perform the Mann-Whitney U Test and the Kruskal-Wallis Test

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

9.4

19

9.4

The median test ignores the specific rank-order of cases

This test examines the rank-ordering of all cases

• It determines whether the rank values for a variable are equally distributed throughout two samples

The smaller of the two U values is used for testing the differences between groups

• This value is compared against the critical U value found in Table G in Appendix C

The Mann-Whitney U Test

1 11 2 1

2 21 2 2

12

12

a

b

N NU N N R

N NU N N R

We won’t study but be aware of its existence when comparing your work vs. answers in the back of the book

20

9.4

Can be used to compare several independent samples

• Requires only ordinal-level data

The H statistic is compared to the critical values of chi-square found in Table F in Appendix C

The Kruskal-Wallis Test

212 3 1

1i

i

RH N

N N n

We won’t study but be aware of its existence when comparing your work vs. answers in the back of the book

21

Homework

Problem 14, 19, 28, 35

© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

Nonparametric tests of significance can be used to analyze data that are not normally distributed or are

not measured at the interval level

One-way and two-way chi-square statistics can be calculated for variables measured at the nominal level

The median test can be used to examine data measured at the ordinal level

The Mann-Whitney U and Kruskal Wallis tests are more powerful than the median test and can also be used to

examine ordinal data

CHAPTER SUMMARY

9.1

9.2

9.3

9.4