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Chapter 13. Chi-Square and Nonparametric Procedures. Going Forward. Your goals in this chapter are to learn: When to use nonparametric statistics The logic and use of the one-way chi square The logic and use of the two-way chi square

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chapter 13

Chapter 13

Chi-Square and Nonparametric Procedures

going forward
Going Forward

Your goals in this chapter are to learn:

  • When to use nonparametric statistics
  • The logic and use of the one-way chi square
  • The logic and use of the two-way chi square
  • The names of the nonparametric procedures with ordinal scores
slide3

Parametric Versus

Nonparametric Statistics

nonparametric statistics
Nonparametric Statistics

Nonparametric statisticsare inferential procedures used with either nominal or ordinal data.

chi square
Chi Square
  • The chi square procedure is the nonparametric procedure for testing whether the frequencies in each category in sample data represent specified frequencies in the population
  • The symbol for the chi square statistic is c2
one way chi square
One-Way Chi Square

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

statistical hypotheses
Statistical Hypotheses

H0: all frequencies in the population are equal

Ha: all frequencies in the population are not equal

observed frequency
Observed Frequency
  • The observed frequency is the frequency with which participants fall into a category
  • It is symbolized by fo
  • The sum of the fos from all categories equals N
formula for expected frequencies
Formula for Expected Frequencies
  • The expected frequency is the frequency we expect in a category if the sample data perfectly represent the distribution of frequencies in the population described by H0
  • The symbol is fe
assumptions of the one way chi square
Assumptions of the One-Way Chi Square
  • Participants are categorized along one variable having two or more categories, and we count the frequency in each category
  • Each participant can be in only one category
  • Category membership is independent
  • We include the responses of all participants in the study
  • The fe must be at least 5 per category
computing one way chi square statistic
Computing One-WayChi-Square Statistic
  • Where fo are the observed frequencies and fe are the expected frequencies
  • df = k – 1 where k is the number of categories
goodness of fit test
“Goodness of Fit” Test
  • The one-way chi square procedure is also called the goodness of fit test
  • That is, how “good” is the “fit” between the data and the frequencies we expect if H0 is true
two way chi square
Two-Way Chi Square

The two-way chi square procedure is used for testing whether category membership on one variable is independent of category membership on the other variable.

computing two way chi square statistic
Computing Two-WayChi Square Statistic
  • Where fo are the observed frequencies and fe are the expected frequencies
  • df = (number of rows – 1)(number of columns – 1)
two way chi square1
Two-Way Chi Square
  • A significant two-way chi square indicates the sample data are likely to represent variables that are dependent (correlated) in the population
  • When a 2 x 2 chi square test is significant, we compute the phi coefficient ( f ) to describe the strength of the relationship
nonparametric tests
Nonparametric Tests
  • Spearman correlation coefficient is analogous to the Pearson correlation coefficient for ranked data
  • Mann-Whitney test is analogous to the independent samples t-test
  • Wilcoxon test is analogous to the related-samples t-test
nonparametric tests1
Nonparametric Tests
  • Kruskal-Wallis test is analogous to a one-way between-subjects ANOVA
  • Friedman test is analogous to a one-way within-subjects ANOVA
example
Example

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.

example1
Example
  • 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