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Bivariate Analysis

Once single variables have been summarized and their pattern of distribution described, the researcher confront the next task in the analysis of data: examining the pattern of relationship between the variables under investigation.

The Concept of Relationship

To state a relationship between x and y is to say that certain categories of the variable x call with certain categories of the variable y. This principle of co variation is basic to the notion of association and relation, and it refers to the idea that observations can be placed in several categories simultaneously. For example, it can be said that highly educated individuals have higher incomes.

The first step in examining the relationship between two variables is the construction of a bivariate table: two cross-classified variables. Usually, bivariate tables are set up with the independent variable as columns and the dependent variable as row variable.

Median and Mean as Covatiation Measures

When the variables of a bivariate distribution are ordinal, the medians of the various univariate distributions can be used as measures of covariation. With interval variables, the arithmetic mean can be used as a comparative measure.

The Measurement of Relationship

Measures of relationship, often referred to as correlation coefficients, reflect the strength and the direction of association betweenthe variables and the degree to which one variable can be predicted from the other.

Proportional Reduction of Error Principle

The strength of the association between two variables can be assessed by calculating the proportional reduction in prediction error when using one variable to predict the other. The proportional reduction of error is defined as follows:

where
b=original number of errors (before employing the independent variable as a predictor)
a=new number of errors (after employing the independent variable as a predictor)

If the proportional reduction of error is absolute, as reflected in the magnitude of the coefficient 1,there is a perfect relationship between the two variables. The coefficient 0 expresses the absence of any association between the two variables. Any measure of association can be developed along similar lines, provided it is based on two kinds of rules: (a) rules that allow the prediction of the dependent variable on the basis of an independent the variable and (b) rules that allow the prediction of the dependent variable independently of an independent variables.

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