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Fox/Levin/Forde, Elementary Statistics in Social Research, 12e. Chapter 9: Nonparametric Tests of Significance. HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D. 4/28/2014 , Spring 2014. CHAPTER OBJECTIVES. 9 .1. Understand the logic of nonparametric tests. 9 .2.

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

  • Chapter 9: Nonparametric Tests of Significance

HLTH 300 Biostatistics for Public Health Practice,Raul Cruz-Cano, Ph.D.

4/28/2014, Spring 2014


CHAPTER OBJECTIVES

9.1

  • Understand the logic of nonparametric tests

9.2

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

9.3

  • Perform the median test

9.4

  • Perform the Mann-Whitney U and Kruskal-Wallis tests


  • Learning Objectives

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

  • 9.1

Understand the logic of nonparametric tests


9.1

Nonparametric Tests

t tests and F ratios require:

  • Normality (or especially large samples)

  • Interval level data

    What 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


  • Learning Objectives

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

  • 9.2

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


9.2

The One-Way Chi-Square Test

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


Examples

Box 9.1, page 324

Problem 13


9.2

The Two-Way Chi-Square Test

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

  • Two-way chi-square test

  • This involves cross-tabulations

    The 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

For each cell:

df=(# of rows -1 )(# of columns -1)


  • 9.2

Table 9.2


Examples

Box 9.2, page 331

Problem 15 (2 x 2)

Problem 22 (more than 2 groups)


9.2

Correcting for Small Expected Frequencies

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


Example

Page 329


9.2

Requirements for the Use of Two-Way Chi-Square

  • A Comparison between Two or More Samples

  • Nominal Data

  • Random Sampling

  • The Expected Cell Frequencies Should Not Be Too Small


  • Learning Objectives

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

  • 9.3

Perform the median test


9.3

The Median Test

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


Example

Box 9.4, page 341

Problem 36


9.3

Requirements for the Use of the Median Test

  • A Comparison between Two or More Medians

  • Ordinal Data

  • Random Sampling


  • Learning Objectives

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

  • 9.4

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


9.4

The Mann-Whitney U Test

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

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


9.4

The Kruskal-Wallis Test

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

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


Homework

Problem 14, 19, 28, 35


CHAPTER SUMMARY

  • Nonparametric tests of significance can be used to analyze data that are not normally distributed or are not measured at the interval level

9.1

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

9.2

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

9.3

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

9.4


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