Bivariate Linear Correlation

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Bivariate Linear Correlation

One way to describe the association between two variables is to assume that the value of the one variable is a linear function of the value of the other variable. If this relationship is perfect, then it can be described by the slope-intercept equation for a straight line, Y = a + bX. Even if the relationship is not perfect, one may be able to describe it as nonperfect linear.

Distinction Between Correlation and Regression

Correlation and regression are very closely related topics. Technically, if the X variable (often called the “independent variable, even in nonexperimental research) is fixed, that is, if it includes all of the values of X to which the researcher wants to generalize the results, and the probability distribution of the values of X matches that in the population of interest, then the analysis is a regression analysis. If both the X and the Y variable (often called the “dependent” variable, even in nonexperimental research) are random, free to vary (were the research repeated, different values and sample probability distributions of X and Y would be obtained), then the analysis is a correlation analysis. For example, suppose I decide to study the correlation between dose of alcohol (X) and reaction time. If I arbitrarily decide to use as values of X doses of 0, 1, 2, and 3 ounces of 190 proof grain alcohol and restrict X to those values, and have the equal numbers of subjects at each level of X, then I’ve fixed X and do a regression analysis. If I allow X to vary “randomly,” for example, I recruit subjects from a local bar, measure their blood alcohol (X), and then test their reaction time, then a correlation analysis is appropriate.

In actual practice, when one is using linear models to develop a way to predict Y given X, the typical behavioral researcher is likely to say she is doing regression analysis. If she is using linear models to measure the degree of association between X and Y, she says she is doing correlation analysis.

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