Multiple Regression: Covariates in Multiple Regression

Describing Multiple Regression

Multiple regression provides an estimate of using a combined set of predictor variables in estimating some dependent variable. What happens is that the process of multiple regression is the generation of an equation. Suppose that the goal was the estimation of the level of communication apprehension felt by a person in a communication setting (Y). The equation uses a combination of three predictors: (a) age of the person (X1), (b) number of other persons in the communication situation (X2), and (c) importance of the outcome of the message (X3). The equation generated by such a process would appear as follows:

$$\begin{array}{l}\mathrm{Predicted}\mathrm{value}\mathrm{of}Y=\\ {\mathrm{\beta }}_1\times X_1+{\mathrm{\beta }}_2\times X_2+{\mathrm{\beta }}_3\times X_3,\end{array}$$

where the term β indicates the standardized coefficient for each variable and the term b would indicate the unstandardized coefficient (that equation would also include an intercept, much like any normal linear equation). This equation provides the basis of evaluating the contribution of each of the variables in contributing to the prediction of the dependent variable. Associated with each standardized coefficient (bi) becomes a significance test providing information on whether the contribution of the independent variable (Xi) should be considered greater than random chance (usually p < .05). The coefficients are adjusted by the association or correlation among the predictor variables. Theoretically, if the correlation between all the dependent variables were zero, then the regression coefficient should be equal to the correlation coefficient between the dependent variable and each predictor. In most cases, the predictor variables are correlated with each other as well as the dependent variable and adjustment becomes necessary. The multiplication in the system by the covariate provides a means of removing the influence of one source of variability.

The multiple regression correlation (R) indicates the correlation between the predicted score of the dependent variable and the actual value observed in the data. The greater the multiple correlation (R), the more accurate the estimation of the actual value of the dependent variable. The multiple R usually becomes reported with accompanying significance test (usually using a p < .05 and often in the form of an F value).

...