The Chi-Square Statistic
G & W Chapter 17

I. Parametric vs. Non-parametric Statistics

A. Sample vs. Population

Remember statistic vs. parameter?

B. Parametric

1. distribution assumptions parametric assumption notes

2. data requirements

C. Non-parametric

II. Chi-square: Test for Goodness-of-Fit

(How well do the proportions for a sample distribution fit the corresponding population proportions?)

**Analogous to which parametric test?

A. Null Hypothesis -

1. : no preference in population/ equally divided

: population is NOT equally divided among the categories

2. : no difference from a comparison—where a specific population distribution is already known

: population distribution is NOT shaped the same as the population specified above in

B. Calculating Chi-square

1. observed frequencies,

2. expected frequencies,

C. Looking up chi-square ( table of critical values)

1. df = # of categories - 1

2. alpha level, generally in psychology we use

3. the chi-square distribution

4. Table B.8 (G&W p. 737)

D. Make a decision about

Figure 17-2 (p. 584)
Chi-square distributions are positively skewed. The critical region is placed in the extreme tail, which reflects large chi-square values.

Figure 17-3 (p. 585)
The shape of the chi-square distribution for different values of df. As the number of categories increases, the peak (mode) of the distribution has a larger chi-square value.

E. Examples

III. Chi-square: Test for Independence

(Is there a relationship between two variables?)

(Is the distribution across categories the same for all treatment groups?)

**Analogous to which 2 parametric tests?

A. Null Hypothesis

: distribution of responses across response categories is the same for all treatment groups

: distribution of responses across response categories is NOT the same for all treatment groups

B. Calculating Chi-square

1. What are the observed frequencies?

2. Calculate expected frequencies (if Ho is indeed true)

C. Looking up chi-square (critical value)

2. Table B.8 (G&W p. 737)

D. Make a decision about

Table 17-8 (p. 596)
A frequency distribution showing the level of self-esteem according to the level of academic performance for a sample of n = 150 ten-year-old children.

E. Assumptions and Restrictions for Chi-Square Tests

Each observation is independent

Expected frequency of each cell must be at least 5

IV. Effect Size

Remember that test statistics are influenced by size of the treatment effect and size of the sample(s). If enough participants are used, even the smallest effect can generate a significant test statistic. As a result, we calculated eta squared for treatment effects in ANOVA. Here are some options for the chi-square test for independence.

A. Phi Coefficient ( ø)

Measures the strength of the relationship between the two variables for a 2 x 2 matrix.

Interpreting ø

.10 is a small effect

.30 is a medium effect

.50 is a large effect

ø² represents the percentage of variance accounted for (like r²)

B. Cramér’s V

A modification of the phi coefficient. Measures the strength of the relationship between the two variables in a matrix larger than 2 x 2.

Calculated like ø, except that df* is also represented in the denominator

df* is the smaller of either (R-1) or (C-1)

Table 17-11 (p. 606)
Standards for interpreting Cramér’s V as proposed by Cohen (1988)