cross-tabulatio
n A technique for analyzing the relationship between two nominal or ordinal variables that
have been organized in a table.
bivariate analysis
A statistical method designed to detect and describe the relationship between two
nominal or ordinal variables.
bivariate table
A table that displays the distribution of one variable across the categories of another variable
A bivariate table displays what and how is it obtained
the distribution of one variable across the categories of another
variable. It is obtained by classifying cases based on their joint scores on two nominal or
ordinal variables. It can be thought of as a series of frequency distributions joined to make
one table
Column variable
A variable whose categories are the columns of a bivariate table
Row variable
A variable whose categories are the rows of a bivariate table
Cell
The intersection of a row and a column in a bivariate table.
Marginals
The row and column totals in a bivariate table.
Finally, it is important to understand that ultimately what guides the construction and interpretation
of bivariate tables is
the theoretical question posed by the researcher.
In the preceding section, we saw how to establish whether an association exists in a
bivariate table. If it does, how do we determine the strength of the association between the
two variables? A quick method is to
examine the percentage difference across the different
categories of the independent variable. The larger the percentage difference across the
categories, the stronger the association.
Percentage differences are
a rough indicator of the
strength of a relationship between two variables.
Positive relationship
A bivariate relationship between two variables measured at the ordinal level or higher
in which the variables vary in the same direction.
Negative relationship
A bivariate relationship between two variables measured at the ordinal level or higher
in which the variables vary in opposite directions.
Elaboration
A process designed to further explore a bivariate relationship; it involves the introduction of
control variables.
Control variable
An additional variable considered in a bivariate relationship. The variable is controlled for
when we take into account its effect on the variables in the bivariate relationship.
The introduction of additional control variables into a bivariate relationship serves three
primary goals in data analysis.
- Elaboration allows us to test for nonspuriousness. Establishing cause-and-effect
relations requires not only showing that an independent and a dependent variable are
associated but also establishing the time order between them and providing
theoretical and empirical evidence that the association is nonspurious—that is, it
cannot be “explained away” by other variables. - Elaboration clarifies the causal sequence of bivariate relationships by introducing
variables hypothesized to intervene between the independent and dependent
variables. - Elaboration specifies the different conditions under which the original bivariate
relationship might hold.
Direct causal relationship
A bivariate relationship that cannot be accounted for by other theoretically
relevant variables.
Spurious relationship
A relationship in which both the independent and dependent variables are influenced
by a causally prior control variable, and there is no causal link between them. The relationship between the
independent and dependent variables is said to be “explained away” by the control variable.
The introduction of the control variable size of fire into the original bivariate relationship
between number of firefighters and amount of damage illustrates the process of elaboration.
These are the three steps:
- Divide the observations into subgroups on the basis of the control variable. We have
as many subgroups as there are categories in the control variable. (In our case, there
were two subgroups: small and large fires.) - Reexamine the relationship between the original two variables separately for the
control variable subgroups. The separate tables are called partial tables; they display
the partial relationship between the independent (number of firefighters) and
dependent (amount of damage) variables within each specific category of the control
variable (small vs. large fire size). - Compare the partial relationships with the original bivariate relationship for the total
group. In a direct causal pattern, the partial relationships will be very close to the
original bivariate relationship. In a spurious pattern, the partial relationship will be
much weaker than the original bivariate relationship.
Partial tables
Bivariate tables that display the relationship between the independent and dependent variables
while controlling for a third variable.
Partial relationship
The relationship between the independent and dependent variables shown in a partial
table.
Intervening variable
A control variable that follows an independent variable but precedes the dependent
variable in a causal sequence.
Intervening relationship
A relationship in which the control variable intervenes between the independent
and dependent variables.
Conditional relationship
A relationship in which the control variable’s effect on the dependent variable is
conditional on its interaction with the independent variable. The relationship between the independent and
dependent variables will change according to the different conditions of the control variable.
