chapter9-2 - Chapter 9 Analysis of Two-Way Tables 9.1...

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Chapter 9
Analysis of Two-Way Tables
9.1 Inference for Two-Way Tables
9.2 Goodness of Fit

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9.1 Inference for Two-Way Tables

Two-way tables

Expected cell counts

The chi-square statistic

The chi-square distributions

The chi-square test

Computations

Computing conditional distributions

Models for two-way tables

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Two-Way Tables
When the data are obtained from random sampling, two-way
tables of counts can be used to formally test the hypothesis
that the two categorical variables are independent in the
population from which the data were obtained.
In Chapter 2, we discussed how two-way tables can be used to
describe the joint distribution of two categorical variables. Such
tables can be used to describe the relationship between two
categorical variables. A relationship exists when the distribution of
one variable depends on the value of the other variable.

Two-way tables
 We call education the row variable and age group the column
variable .
 Each combination of values for these two variables is called a cell .
 For each cell, we can compute a proportion by dividing the cell entry
by the total sample size. The collection of these proportions would be
the joint distribution of the two variables.

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Gender
Ongoing Fright
Symptoms Male Female
Yes 7 29
No 31 50
Total 38 79
Gender
Ongoing Fright
Symptoms Male Female
Yes 18% 37%
No 82% 63%
Total 100% 100%
If the null hypothesis of no relation
between gender and ongoing fright is
true, we expect this overall percent to
apply to both men and women
The Hypothesis: No Association

Expected Cell Counts
The expected count in any cell of a two-way table when H
0
is true is:
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The rows of a two-way table are the values of one categorical
variable and the columns are the values of the other variable. The
count in any particular cell of the table equals the number of subjects
who fall into that cell. We want to test the hypothesis ( H
0
) that there is
no relationship between the two categorical variables.
To test this hypothesis, we compare actual counts from the sample
data with expected counts , where the latter counts are those
expected when there is no relationship between the two variables.

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The Chi-Square Statistic
Assume there are r rows in the two-way table and c columns, which
means there are
r  c cells.
The test statistic that makes the comparison is the chi-square statistic .
The chi-square statistic is a measure of how far the observed counts
are from the expected counts. The formula for the statistic is:
where “ observed ” represents an observed cell count, “ expected ”
represents the expected count for the same cell, and the sum is over all
r  c cells in the table.
 2 
(Observed - Expected)
2
Expected

To see if the data give convincing evidence against the null
hypothesis, we compare the observed counts from our sample with
the expected counts assuming H
0
is true.

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The Chi-Square Distributions
When the observed counts are very different from the expected counts, a large value
of  2 will result, providing evidence against the null hypothesis.