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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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**Support for Abortion by Job Security**

Absolute Frequencies Support for Abortion by Job Security Abortion Job Find Easy Job Find Not Easy Row Total Yes No Column Total

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**Support for Abortion by Job Security**

Column Percentages Support for Abortion by Job Security Abortion Job Find Easy Job Find Not Easy Row Total Yes % % 52% No % % 48% Column Total % % % (44) (51) (95)

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**Support for Abortion by Job Security**

Row Percentages Support for Abortion by Job Security Abortion Job Find Easy Job Find Not Easy Row Total Yes % % 100% (49) No % % 100% (46) Column Total % % % (95)

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**Properties of a Bivariate Relationship**

Does there appear to be a relationship? How strong is it? 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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**Number of Firefighters Property Damage**

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**

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. Reexamine the relationship between the original two variables separately for the control variable subgroups. 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 Relationship: Example**

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 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.

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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.

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