Relationships Between Two Variables: Cross-Tabulation

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Relationships Between Two Variables
Cross-Tabulation

• Independent and Dependent Variables
• Constructing a Bivariate Table
• Computing Percentages in a Bivariate Table
• Dealing with Ambiguous Relationships Between
  Variables
• Reading the Research Literature
• Properties of a Bivariate Relationship
• Elaboration
• Statistics in Practice

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Introduction

• Bivariate Analysis A statistical method designed
  to detect and describe the relationship between
  two variables.
• Cross-Tabulation A technique for analyzing the
  relationship between two variables that have been
  organized in a table.

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Understanding Independent and Dependent Variables

• Example If we hypothesize that income varies
  by the level of education a person has, what is
  the independent variable, and what is the
  dependent variable?
• Independent Education
• Dependent Income

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Constructing a Bivariate Table

• Bivariate table A table that displays the
  distribution of one variable across the
  categories of another variable.
• 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.

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Percentages Can Be Computed in Different Ways

• Column Percentages column totals as base
• Row Percentages row totals as base

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Absolute Frequencies

• Support for Abortion by Job Security
• Abortion Job Find Easy Job Find Not Easy Row
  Total
• Yes 24 25 49
• No 20 26 46
  
• Column Total 44 51 95

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Column Percentages

• Support for Abortion by Job Security
• Abortion Job Find Easy Job Find Not Easy Row
  Total
• Yes 55 49 52
• No 45 51 48
• Column Total 100 100
  100 (44)
  (51) (95)

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Row Percentages

• Support for Abortion by Job Security
• Abortion Job Find Easy Job Find Not Easy Row
  Total
• Yes 49 51 100 (49)
• No 43 57 100 (46)
  
• Column Total 46 54
  100
  (95)

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Properties of a Bivariate Relationship

1. Does there appear to be a relationship?
2. How strong is it?
3. What is the direction of the relationship?

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Existence of a Relationship

• IV Number of Traumas
• DV Support for Abortion
• If the number of traumas were unrelated to
  attitudes toward abortion among women, then we
  would expect to find equal percentages of women
  who are pro-choice (or anti-choice), regardless
  of the number of traumas experienced.

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Existence of the Relationship
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Determining the Strength of the Relationship

• 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.
• We rarely see a situation with either a 0 percent
  or a 100 percent difference.

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Direction of the Relationship

• 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.

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A Positive Relationship
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A Negative Relationship
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Elaboration

• Elaboration is a process designed to further
  explore a bivariate relationship it involves the
  introduction of control variables.
• A control variable is 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.

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Three Goals of Elaboration

• Elaboration allows us to test for
  nonspuriousness.
• Elaboration clarifies the causal sequence of
  bivariate relationships by introducing variables
  hypothesized to intervene between the IV and DV.
• Elaboration specifies the different conditions
  under which the original bivariate relationship
  might hold.

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Testing for Nonspuriousness

• Direct causal relationship a bivariate
  relationship that cannot be accounted for by
  other theoretically relevant variables.
• Spurious relationship a relationship in which
  both the IV and DV are influenced by a causally
  prior control variable and there is no causal
  link between them. The relationship between the
  IV and DV is said to be explained away by the
  control variable.

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The Bivariate Relationship Between Number of
Firefighters and Property Damage

• Number of Firefighters ? Property Damage
• (IV) (DV)

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Process of Elaboration

• Partial tables bivariate tables that display the
  relationship between the IV and DV while
  controlling for a third variable.
• Partial relationship the relationship between
  the IV and DV shown in a partial table.

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The Process of Elaboration

1. 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.
2. Reexamine the relationship between the original
   two variables separately for the control variable
   subgroups.
3. Compare the partial relationships with the
   original bivariate relationship for the total
   group.

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Intervening Relationship

• 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.

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Intervening RelationshipExample

• Religion ? Preferred Family Size ? Support for
  Abortion
• (IV) (Intervening Control Variable)
  (DV)

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Conditional Relationships

• Conditional relationship a relationship in which
  the control variables 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.

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Conditional Relationships

• Another way to describe a conditional
  relationship is to say that there is a
  statistical interaction between the control
  variable and the independent variable.