Analysis in Multiple Regression

Correlation models differ from regression models in that each variable (y,s and XiS) plays a symmetrical role, with neither variable designated as a response or predictor variable. They are viewed as relational, instead of predictive in this process. Correlation models can be very useful for making inferences about any one variable relative to another, or to a group of variables. We use the correlation models in terms of y and single or multiple x,s. 

Multiple regression s use of the correlation coefficient, r, and the coefficient of determination are direct extensions of simple linear regression correlation models already discussed. The difference is, in multiple regression, that multiple Xt predictor variables, as a group, are correlated with the response variable, y. Recall that the correlation coefficient, r, by itself, has no exact interpretation, except that the closer the value of r is to 0, weaker the linear relationship between y and x,s, whereas the closer to 1, stronger the linear relationship. On the other hand, can be interpreted more directly. The coefficient of determination, say = 0.80, means the multiple x,- predictor variables in the model explain 80% of the y term s variability. As given in Equation 5.1, r and are very much related to the sum of squares in the analysis of variance (ANOVA) models that were used to evaluate the relationship of SSr to SSe in Chapter 4  

Like the Fc value, increases as predictor variables, x,s, are introduced into the regression model, regardless of the actual contribution of the added x,s. Note, 

In Example 4.1 of the previous chapter, we looked at the data recovered from a stability study in which the mg/mL of a drug product was predicted over time based on two x, predictor variables, Xi, the week and X2, the humidity. Table 4.2 provided the basic regression analysis data, including and 7 (adj), via MiniTab 

R in the multiple linear regression model that predicts y from two independent predictor variables, Xi and X2, explains 75.3% or when adjusted, 73.9% of the variability in the model. The other 1 — 0.753 = 0.247 is xmexplained error. In addition, note that a fit of = 50% would infer that the prediction of y based on x and X2 is no better than y. 

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