Question

how other important characteristics of multiple regressions analysis such as interpretation of the coefficient values of the estimated regression model, performing a multicollinearity test and interpretation of the results, interpretation of variance and the relationships of the overall significance of the regression model.

Answer 1

Ans:

Multiple regression is an extension of simple linear regression. It is used when we want to predict the value of a variable based on the value of two or more other variables. The variable we want to predict is called the dependent variable (or sometimes, the outcome, target or criterion variable). The variables we are using to predict the value of the dependent variable are called the independent variables (or sometimes, the predictor, explanatory or regressor variables)

Interpreting and Reporting the Output of Multiple Regression Analysis

Determining how well the model fits

The R² and adjusted R² can be used to determine how well a regression model fits the data:

The "R-squared" row represents the R² value (also called the coefficient of determination), which is the proportion of variance in the dependent variable that can be explained by the independent variables (technically, it is the proportion of variation accounted for by the regression model above and beyond the mean model). You can see from our value of 0.577 that our independent variables explain 57.7% of the variability of our dependent variable, VO₂max. However, you also need to be able to interpret "Adj R-squared" (adj. R²) to accurately report your data.

Statistical significance

The F-ratio tests whether the overall regression model is a good fit for the data. The output shows that the independent variables statistically significantly predict the dependent variable, F(4, 95) = 32.39, p < .0005 (i.e., the regression model is a good fit of the data).

Estimated model coefficients

The general form of the equation to predict VO₂max from age, weight, heart_rate and gender is:

predicted VO₂max = 87.83 – (0.165 x age) – (0.385 x weight) – (0.118 x heart_rate) + (13.208 x gender)

This is obtained from the "Coef." column, as shown below:

Unstandardized coefficients indicate how much the dependent variable varies with an independent variable, when all other independent variables are held constant. Consider the effect of age in this example. The unstandardized coefficient, B₁, for age is equal to -0.165 (see the first row of the Coef. column). This means that for each 1 year increase in age, there is a decrease in VO₂max of 0.165 ml/min/kg.

Statistical significance of the independent variables

You can test for the statistical significance of each of the independent variables. This tests whether the unstandardized (or standardized) coefficients are equal to 0 (zero) in the population. If p < .05, you can conclude that the coefficients are statistically significantly different to 0 (zero). The t-value and corresponding p-value are located in the "t" and "P>|t|" columns, respectively, as highlighted below: