Curvilinear Regression

Researchers often use regression techniques to describe the relationship between two (or more) variables. In the simplest case (bivariate linear regression), it is assumed that the relationship can be described well by a straight line, Y = a + bX. One can use Student t to test hypotheses about or construct confidence intervals around the regression coefficients a (intercept or constant) and b (slope, number of units Y changes for each one-point change in X).

Often the relationship between variables can be better described with a line that is not straight. Curvilinear regression can be employed to describe some such relationships. In some cases, the researcher has good reason to expect a particular curvilinear relationship even before the data are collected. For example, an microbiologist may expect that the relationship between elapsed time and the number of bacteria in a rich medium is exponential. A psychophysicist may expect that the perceived intensity of a visual stimulus is a function of the logarithm of the physical intensity of the stimulus. A psychologist may expect that the relationship between the amount eaten by an individual and the number of persons present at the meal is a power function.

In other cases, the researcher does not expect any particular curvilinear relationship but discovers during data screening that the variables are not related in a linear fashion. One should always inspect a scatter plot of the data before conducting a regression analysis. All too often, researchers employ linear regression analysis when a much better fit would be obtained with a curvilinear analysis. Most researchers are familiar with linear regression and wish to stay within that framework when dealing with curvilinear relationships. This can be accomplished by applying a nonlinear transformation to one or both variables and then conducting linear regression analysis with the transformed data. Alternatively, one can conduct a polynomial regression (also known as a trend analysis), which is a multiple linear regression in which powers of the predictor variable(s) are included in the[Page 212] model. Less commonly, the researcher may choose to conduct a true nonlinear analysis. In such an analysis, the researcher estimates the values of the model parameters by attempting to minimize the squared residuals (differences between observed Y and predicted Y) for the actual nonlinear model rather than attempting to minimize the squared residuals for a linear regression on transformed data.

Plotting the Data

SPSS was employed to create a scatter plot with a quadratic regression line (Figure 1). Clearly, the relationship between number of persons and amount eaten can be described with a curved line better than with a straight line.

Figure 1 Scatter Plot With Quadratic Regression

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