**Example**

use http://www.philender/com/courses/data/hsbdemo, clear regress write read math scienceSource | SS df MS Number of obs = 200 -------------+------------------------------ F( 3, 196) = 57.30 Model | 8353.98999 3 2784.66333 Prob > F = 0.0000 Residual | 9524.88501 196 48.5963521 R-squared = 0.4673 -------------+------------------------------ Adj R-squared = 0.4591 Total | 17878.875 199 89.843593 Root MSE = 6.9711 ------------------------------------------------------------------------------ write | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- read | .2356606 .0691053 3.41 0.001 .0993751 .3719461 math | .3194791 .0756752 4.22 0.000 .1702369 .4687213 science | .2016571 .0690962 2.92 0.004 .0653896 .3379246 _cons | 13.19155 3.068867 4.30 0.000 7.139308 19.24378 ------------------------------------------------------------------------------

_cons = 13.19155 -- The predicted value when all of the predictors equal zero.

_b[read] = .2356606 -- For every one unit increase in **read**, the predicted value
for **write** increases by .2356606 when all other variable in the model are held constant.

_b[math] = .3194791 -- For every one unit increase in **math**, the predicted value
for **write** increases by .3194791 when all other variable in the model are held constant.

_b[science] = .2016571 -- For every one unit increase in **science**, the predicted value
for **write** increases by .2016571 when all other variable in the model are held constant.

**Conditional Expectation**

In the multiple regression model we can write the conditional expectation as **E**(y | x1, x2), which
indicates that we are interested in in the effect of variable *x1* on
the expected value of *y* while holding the variable *x2* constant.

**Regression Equation**

**Prediction Equation**

**The Two Predictor Case**

**Squared Multiple Correlation**

**Regression Coefficients**

**Sums of Squares**

**Raw Regression Coefficient vs Standardized Regression Coefficient**

**
b vs β**

- b is affected by the scale of measurement and by the variability of the variables.

**Note**

**Prediction Equation in Standardized Form**

**Beta Coefficients**

**More on Betas**

**More on Squared Multiple Correlations**

**Even More Squared Multiple Correlation**

**Variance of Estimate/Standard Error of Estimate**

The variance of estimate is also called the mean square error in the ANOVA summary table of the regression analysis.

The standard error of estimate gives an indicatin of how far, on the average, observations fall from the regression line.

**Testing the Model**

**The Overall F-test**

**Interpreting Regression Coefficients**

**Interpreting Standardized Regression Coefficients (Betas)**

**Tests of Regression Coefficients**

**About Tests of Regression Coefficients**

**Note:**

**Comparing Variables**

**Interpreting R ^{2}**

R^{2} has several interpretations:

R^{2}/k F = ---------------- (1 - R^{2})/(N-k-1)

Linear Statistical Models Course