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• The names of the nonparametric procedures

with ordinal scores

Parametric VersusNonparametric Statistics

Nonparametric Statistics

Nonparametric statistics are inferential

procedures used with either nominal or ordinal

data.

Chi Square Procedures

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 2

One-Way Chi Square:The Goodness of Fit Test

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

H0: all frequencies in the population are equal

Ha: all frequencies in the population are not equal

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

• 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

k

Nfe categoryeachin

Assumptions of the One-Way Chi Square

1. Participants are categorized along one variable having two or more categories, and we count the frequency in each category

2. Each participant can be in only one category

3. Category membership is independent

4. We include the responses of all participants in the study

5. The fe must be at least 5 per category

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

e

eoobt f

ff 22 )(

The 2 Distribution

“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

The Two-Way Chi Square:The Test of Independence

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.

N

fff ooe

)totalcolumnscell')(totalrowscell'(

Two-Way Chi Square

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)

e

eoobt f

ff 22 )(

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 ( ) to describe the strength of the relationship

Nobt2

Nonparametric Statistics

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 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

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 24 11

Cats 15 54

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

Males Females

Dogs 13.125 21.875

Cats 25.875 43.125

Example

730.21

125.43

)125.4354(

875.25

)875.2515(

875.21

)875.2111(

125.13

)125.1324(

22

222