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Chapter 6 – 1
Chapter 10: Relationships Between Two Variables:
CrossTabulation• 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
Chapter 6 – 2
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.
Chapter 6 – 3
Understanding Independent and Dependent Variables
• Example: If we hypothesize that English proficiency varies by whether person is native born or foreign born, what is the independent variable, and what is the dependent variable?
• Independent: nativity
• Dependent: English proficiency
Chapter 6 – 4
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.
Chapter 6 – 5
Chapter 6 – 6
Chapter 6 – 7
Chapter 6 – 8
Percentages Can Be Computed in Different Ways:
1. Column Percentages: column totals as base
2. Row Percentages: row totals as base
Chapter 6 – 9
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
Chapter 6 – 10
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)
Chapter 6 – 11
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)
Chapter 6 – 12
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?
Chapter 6 – 13
Existence of a Relationship
IV: Number of TraumasDV: 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.
Chapter 6 – 14
Existence of the Relationship
Chapter 6 – 15
# of Traumas
Abortion 0 1 2+ Total
Yes 46% 38% 22% 35%
No 54% 62% 78% 65%
Total(N)
100%(27)
100%(44)
100%(33)
100%(104)
# of Traumas
Abortion 0 1 2+ Total
Yes 46% 46% 46% 46%
No 54% 54% 54% 54%
Total(N)
100%(27)
100%(44)
100%(33)
100%(104)
Chapter 6 – 16
Education
Income Less than HS HS College Total
$0-$2,0000 54% 28% 12% 32%
$2,0000-$3,0000 36% 62% 10% 36%
$3,0000+ 10% 10% 78% 32%
Total 100% 100% 100% 100%
Example
Chapter 6 – 17
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.
Chapter 6 – 18
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.
Chapter 6 – 19
A Positive Relationship
Chapter 6 – 20
A Negative Relationship
Chapter 6 – 21
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.
Chapter 6 – 22
Three Goals of Elaboration
1. Elaboration allows us to test for non-spuriousness.
2. Elaboration clarifies the causal sequence of bivariate relationships by introducing variables hypothesized to intervene between the IV and DV.
3. Elaboration specifies the different conditions under which the original bivariate relationship might hold.
Chapter 6 – 23
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.
Chapter 6 – 24
The Bivariate Relationship Between Number of Firefighters
and Property Damage
Number of Firefighters Property Damage
(IV) (DV)
Chapter 6 – 25
Chapter 6 – 26
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.
Chapter 6 – 27
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. Re-examine 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.
Chapter 6 – 28
Chapter 6 – 29
Chapter 6 – 30
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.
Chapter 6 – 31
Intervening Relationship:Example
• Renzi's Hypothesis(1975)
Religion Preferred Family Size Support for Abortion
(IV) (Intervening Control Variable) (DV)
Renzi, Mario.1975. Ideal Family Size as an Intervening Variable between Religion and Attitudes Towards
Abortion, Journal for the Scientific Study of Religion, Vol. 14, No. 1,pp. 23-27
Chapter 6 – 32
Conditional Relationships
• 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.
Chapter 6 – 33
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.
Chapter 6 – 34
Stance on Legal Abortion
Pro-choice Pre-life
Always wrong 37 98
Not Wrong 63 2
Conditional Relationships: example
Chapter 6 – 35
Conditional RelationshipsMale Stance on Legal Abortion
Pro-choice Pre-life
Always wrong 29 96
Not Wrong 71 4
Female Stance on Legal Abortion
Pro-choice Pre-life
Always wrong 46 100
Not Wrong 54 0
Chapter 6 – 36
Conditional Relationships
Chapter 6 – 37
A more complex example
FrequencyCol Pct
Defendant’s race
Black White
Death Penalty 14989.76
14188.13No
Yes 1710.24
1911.88
Data: 326 defendants were recorded as being Data: 326 defendants were recorded as being indicted for homicide in 20 Florida counties during indicted for homicide in 20 Florida counties during 1976-19771976-1977..
Chapter 6 – 38
Partial Tables: ExampleDefendant’s race
Black White
Victim’s race Victim’s race
Black White Black White
Death Penalty
97 52 9 132No
Yes 6 11 0 19
Chapter 6 – 39
Partial Tables: ExampleVictim’s race=Black
Death Penalty Defendant race
Total
FrequencyCol Pct Black White
No 9794.17
9100.00
106
Yes 65.83
00.00
6
Total 103 9 112
Chapter 6 – 40
Partial Tables: ExampleVictim’s race=White
Death Penalty Defendant race
TotalFrequency
Col Pct Black White
No 5282.54
13287.42
184
Yes 1117.46
1912.58
30
Total 63 151 214
