## Multivariate analysis

In the preceding sections, simple regression and correlation have been used to determine the association between two variables. However, it is not usual for one independent variable to play such a major part in explaining the dependent variable. Where an r2 value between two variables is less than 0.8 it is necessary to search for additional independent variables, which can increase the variation explained by the regression or correlation models.

Multiple regression analysis is usually the preferred way of introducing more than one independent variable into the study. The multiple regression model is an expansion of the simple regression model:

The number of independent variables used in each model depends on how many are needed to estimate adequately the value of the dependent variable. Most equations usually have fewer than ten independent variables.

A multiple regression model for the house data just examined follows.

Stepwise regression of price on four predictors (rooms, baths, acres and area),

 with N = 20: Step 1 2 Constant 79,312 69,085 Area 32.6 30.1 i-ratio 4.91 5.24 Acres 22,800 i-ratio 2.77 S 16,231 13,859 R-SQ 57.27 70.58

We see that only areas and acres are kept in the regression equation. These two independent variables account for more than 70% of the variance in the dependent variable. Since no other independent variables have been included and the total variance accounted for does not come near to 80%, we can only conclude that there must be other factors that are relevant to the house price than those reported here.

For further information on multivariate analysis, see McKenzie et al.2 and Hair et al.3

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