Chapter 9: Analysis of Two-Way Tables
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Data Analysis for Two-Way Tables (IPS section 9.1 pages 612-620)
The Two-Way Table – A two-way table of counts organizes data about two
categorical variables. Values of the row variable label the rows that run across
the table, and values of the column variable label the columns that run down the
table. Two-way tables are often used to summarize large amounts of data by
grouping outcomes into categories. Each combination of values for these two
variables is called a cell. For each cell, a proportion is obtained by dividing the
cell entry by the total sample size. The collection of these proportions is the joint
distribution of the two categorical variables.
Marginal Distributions – The row totals and column totals in a two-way
table give the marginal distributions of the two variables separately. It is clearer
to present these distributions as percents of the table total. Marginal distributions
do not give any information about the relationships between the variables.
Describing Relations in Two-Way Tables – Relationships among
categorical variables are described by calculating appropriate percents from the
counts given.
Conditional Distributions – When we condition one the value of one
variable and calculate the distribution of the other variable, we obtain a
conditional distribution. To find the conditional distribution of the row variable for
one specific value of the column variable, look only at that one column in the
table. Find each entry in the column as a percent of the column total. There is a
conditional distribution of the row variable for each column in the table.
Comparing these conditional distributions is one way to describe the association
between the row and column variables. It is particularly useful when the column
variable is the explanatory variable. When the row variable is explanatory, find
the conditional distribution of the column variable for each row and compare
these distributions. Bar graphs are a flexible means of presenting categorical
data. There is not single best way to describe an association between two
categorical variables.
Simpson’s Paradox – An association or comparison that holds for all of
several groups can reverse direction when the data are combined to form a
single group. This reversal is called Simpson’s paradox.
III. Inference for Two-Way Tables (IPS section 9.2 pages 620-629)
A. The Hypothesis: No Association – The null hypothesis Ho of interest in a two-way
table is that there is no association between the row variable and the column
variable.
B. Expected Cell Counts – To test the null hypothesis in r x c tables, we compare
the observed cell counts with expected cell counts calculated under the
assumption that the null hypothesis is true. A numerical summary of the
comparison will be our test statistic.
row total x column total
expected cell count =
n
C. Chi-Square Statistic – The chi-square statistic is a measure of how much the
observed cell counts in a two-way table diverge from the expected cell counts.
The recipe for the statistic is
(observed count - expected count)2
expected count
where “observed” represents an observed sample count, “expected” represents
the expected count for the same cell, and the sum is over all
r x c cells in the table.
D. Chi-Square Distribution (denoted  2 ) - Like the t distribution, the  2 distributions
form a family described by a single parameter, the degrees of freedom. We use
 2 (df) to indicate a particular member of this family.  2 distributions take only
positive values and are skewed to the right.
X2  
E. Chi-Square Test for Two-Way Tables – The null hypothesis Ho is that there is no
association between the row and column variables in a two-way table. The
alternative is that these variables are related. If Ho is true, the chi-square statistic
 2 has approximated a  2 distribution with
(r-1)(c-1) degrees of freedom. The P-value for the chi-square test is
P(  2 ≥ X2)
where  2 is a random variable having the  2 (df) distribution with df = (r-1)(c-1). The
chi-square test always uses the upper tail of the  2 distribution because any deviation
from the null hypothesis makes that statistic larger. The approximation of the
distribution of X2 by  2 becomes more accurate as the cell counts increase.
F. The Chi-Square Test and the z Test – A comparison of the proportions of “successes”
in two populations leads to a 2 x 2 table. We can compute two population proportions
either by the chi-square test or by the two-sample z test from section 8.2. In fact,
these tests always give exactly the same result, because the  2 statistic is equal to
the square of the z statistic, and  2 (1) critical values are equal to the squares of the
corresponding N(0,1) critical values. The advantage of the z test is that we can test
either one-sided or two-sided alternatives. The chi-square test always tests the twosided alternative. Of course, the chi-square test can compare more than two
populations, whereas the z test compares only two.