Home page for accesible maths 7.1.1 Examples 7.1.1 Examples 7.2.1 Estimation of residual variance

\sigma^{2}

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7.2 Predicted values

Once we have estimated the regression coefficients $\hat{\beta}$ , we can estimate predicted values of the response variable. The predicted value for individual $i$ is defined as

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\displaystyle\hat{\mu}_{i}=\hat{\beta}_{1}x_{i,1}+\hat{\beta}_{2}x_{i,2}+% \ldots+\hat{\beta}_{p}x_{i,p}.

(7.4)

This equation can also be used to obtain predicted values for combinations of explanatory variables unobserved in the sample (see example 7.2.1). However, care should be taken not to extrapolate too far outside of the observed ranges of the explanatory variables.

The predicted value is interpreted as the expected value of the response variable for a given set of explanatory variable values. Predicted values are useful for checking model fit, calculating residuals and as model output.

TheoremExample 7.2.1 Birthweights cont.

Recall the simple linear regression example on birth weights,

\displaystyle\mathbb{E}[Y_{i}]=\beta_{1}+\beta_{2}x_{i}

where $x_{i}$ is gestational age at birth. We obtained $\hat{\beta}=(\hat{\beta}_{1},\hat{\beta}_{2})=(-1485,116)$ .

Can you predict the birth weight of a child at 37.5 weeks?

\displaystyle\hat{y}=\hat{\beta}_{1}+\hat{\beta}_{2}\times 37.5=-1485+116% \times 37.5=2865\text{~{}grams}.

7.2.1 Estimation of residual variance $\sigma^{2}$