# Multiple Linear Regression (continued)

A response variable $$Y$$ is linearly related to $$p-1$$ different explanatory variables $$X^{(1)},\ldots,X^{(p-1)}$$. The regression model is given by
$Y_i = \beta_0 + \beta_1 X_i^{(1)} + \cdots + \beta_{p-1} X_i^{(p-1)} + \varepsilon_i, \qquad i=1,\ldots,n, \tag{1}$
where $$\varepsilon_i$$ have mean zero, variance $$\sigma^2$$ and are independent with a normal distribution (working assumption). The equation (1) can be expressed in matrix notations as
$Y = \mathbf{X} \beta + \varepsilon, \qquad \mbox{where} \qquad Y = \begin{bmatrix} Y_1 \\Y_2 \\ \cdot\\Y_n\end{bmatrix}, \qquad \varepsilon = \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \cdot\\ \varepsilon_n \end{bmatrix}.$

## Fitted values and residuals

The fitted value for the $$i$$-th observation is $$\widehat{Y}_i = b_0 + b_1 X_i^{(1)} + . . . + b_{p-1} X_i^{(p-1)}$$, and the residual is $$e_i = Y_i - \widehat{Y}_i.$$ Using matrix notations, the vector of fitted values, $$\widehat{Y}$$, can be expressed as
$$\widehat{Y} = X b = X \widehat{\beta} = X ( X^T X)^{-1} X^T Y$$
The $$n \times n$$ matrix $$X ( X^T X)^{-1} X^T Y$$ is called the hat matrix and is denoted by H. Thus $$\widehat{Y}$$ = HY. The vector of residuals, to be denoted by $$\mathbf{e}$$ (with $$i$$-th coordinate $$e_i$$, for $$i=1,\ldots,n$$) can therefore be expressed as
$$e = Y - \widehat{Y}$$ = Y - HY = ($$I_n$$ - H) Y = $$( I_n - X (X^T X)^{-1} X^T ) Y.$$
• Hat matrix: check that the matrix H has the property that HH = H and ($$I_n$$ - H)(($$I_n$$ - H) = ($$I_n$$ - H). A square matrix A having the property that AA = A is called an indempotent matrix. So both H and $$I_n$$ - H are indempotent matrices. The important implication of the equation $$\widehat Y = \mathbf{H} Y$$ is that the matrix $$\mathbf{H}$$ the response vector $$\mathbf{Y}$$ as a linear combination of the columns of the matrix $$\mathbf{X}$$ to obtain the vector of fitted values, $$\widehat{Y}$$. Similarly, the matrix $$I_n - \mathbf{H}$$ applied to $$\mathbf{Y}$$ gives the residual vector $$\mathbf{e}$$.
• Properties of Residuals: Many of the properties of residual can be deduced by studying the properties of the matrix $$\mathbf{H}$$. Some of them are listed below. $$\sum_i e_i = 0 and \sum_i X_i^{(j)}e_i = 0 , for j=1,\ldots,p-1$$. These are results of the following: $$\mathbf{X}^T\mathbf{e} \mathbf{X}^T(I_n - \mathbf{H})Y = \mathbf{X}^TY - \mathbf{X}^T\mathbf{X} (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T Y = \mathbf{X}^T Y - \mathbf{X}^T Y = 0.$$ Also note that $$\widehat{Y} = \mathbf{X}(\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T Y$$, and hence $$\sum_i \widehat Y_i e_i = \widehat Y^T \mathbf{e} = Y^T \mathbf{X} (\mathbf{X}^T\mathbf{X})^{-1} \mathbf{X}^T\mathbf{e} = 0.$$

## ANOVA

The matrix viewpoint gives a coherent way of representing the different components of the analysis of variance of the response in regression. As before, we need to deal with the objects
$$SSTO = \sum_i (Y_i - \overline{Y})^2, \qquad SSE = \sum_i(Y_i - \widehat Y_i)^2 = \sum_i e_i^2, \qquad \mbox{and}~~SSR = SSTO - SSE.$$
The degrees of freedom of $$SSR$$ is $$\mathbf{p} - 1$$. The degrees of freedom of $$SSTO$$ is $$\mathbf{n} - 1$$ and d.f.$$(SSE)$$ = d.f.$$(SSTO)$$ - d.f.$$(SSR)$$ = $$\mathbf{n} - 1 - (\mathbf{p}-1) = \mathbf{n}-\mathbf{p}$$. Moreover,
$$\overline{Y} = \frac{1}{n} \sum_i Y_i = (\frac{1}{n}) Y^T \mathbf{1}$$
$$SSTO = \sum_i Y_i^2 - \frac{1}{n}(\sum_i Y_i)^2 = Y^T Y - (\frac{1}{n}) Y^T \mathbf{J} Y$$
$$SSE = \mathbf{e}^T \mathbf{e} = \mathbf{Y}^T(I-\mathbf{H}) (I-\mathbf{H})\mathbf{Y} = \mathbf{Y}^T (I-\mathbf{H}) \mathbf{Y}\\ SSE = Y^T Y - \widehat \beta^T \mathbf{X}^T Y where \mathbf{J} = \mathbf{1}\mathbf{1}^T$$.
• We can use the ANOVA decomposition to test $$H_0 : \beta_1 = \beta_2 = \cdots = \beta_{p-1} = 0$$ (no regression effect), against $$H_1$$ : not all $$\beta_j$$ are equal to zero. The test statistic is $$F^* = \frac{\frac{SSR}{\mbox{d.f.}(SSR)}}{\frac{SSE}{\mbox{d.f.}(SSE)}} = \frac{SSR/(p-1)}{SSE/(n-p)}.$$ Under $$H_0$$ and assumption of normal errors, $$F^*$$ has $$F_{p-1, n-p}$$ distribution. So, reject $$H_0$$ in favor of $$H_1$$, at level $$\alpha$$ if $$F^* > F(1-\alpha;p-1,n-p)$$.

## Inference on Multiple Linear Regression

We can ask the same questions regarding estimation of various parameters as we did in the case of regression with one predictor variable.
• Mean and standard error of estimates: We already checked that (with $$\mathbf{b} \equiv \widehat \beta$$) $$E(\mathbf{b}) = \beta$$ and Var$$(\mathbf{b}) = \sigma^2 (\mathbf{X}^T \mathbf{X})^{-1}$$. And hence the estimated variance-covariance matrix of $$\mathbf{b}$$ is $$\widehat{\mbox{Var}}(\mathbf{b}) = MSE(\mathbf{X}^T \mathbf{X})^{-1}$$. Denote by $$s(b_j)$$ the standard error of $$b_j = \widehat \beta_j$$. Then $$s^2(b_j)$$ is the $$(j+1)$$-th diagonal entry of the $$p \times p$$ matrix $$\widehat{\mbox{Var}}(\mathbf{b})$$.
• Note that $$\mbox{Var}(\mathbf{b}) = \sigma^2 (\mathbf{X}^T \mathbf{X})^{-1} ~~\mbox{so that}~~ \widehat{\mbox{Var}}(\mathbf{b}) = \mbox{MSE} ~ (\mathbf{X}^T \mathbf{X})^{-1}.$$
• ANOVA : Under $$H_0 : \beta_1=\beta_2=\cdots=\beta_{p-1} =0$$, the F-ratio $$F^* = MSR/MSE$$ has an $$F_{p-1,n-p}$$ distribution. So, reject $$H_0$$ in favor of $$H_1$$: at least one $$j \in\{1,\ldots,p-1\}, \beta \neq$$ 0 , at level $$\alpha$$ if $$F^* > F(1-\alpha;p-1,n-p)$$.
• Hypothesis tests for individual parameters : Under $$H_0 : \beta_j = \beta_j^0$$, for a given $$j \in \{1,\ldots,p-1\}$$, $$t^* = \frac{b_j-\beta_j^0}{s(b_j)} \sim t_{n-p}.$$ So, if $$H_1 : \beta_j \neq \beta_j^0$$, then reject $$H_0$$ in favor of $$H_1$$ at level $$\alpha$$ if $$|t^*| > t(1-\alpha/2;n-p)$$.
• Confidence intervals for individual parameters : Based on the result above, 100(1-$$\alpha$$) % two-sided confidence interval for $$\beta_j$$ is given by $$b_j \pm t(1-\alpha/2;n-p)s(b_j).$$
• Estimation of mean response : Since $$E(Y|X_h) = \beta^T X_h, where X_h = \begin{bmatrix}1 \\X_h^{(1)}\\ \cdot \\ \cdot \\X_h^{(p-1)} \end{bmatrix},$$ an unbiased point estimate of $$E(Y|X_h)$$ is $$\widehat Y_h = \mathbf{b}^T X_h = b_0 + b_1X_h^{(1)} + \cdots + b_{p-1}X_h^{(p-1)}$$. Using the Working-Hotelling procedure, an $$100(1-\alpha)$$ % confidence region for the entire regression surface (that is, confidence region for $$E(Y|X_h)$$ for all possible values of $$X_h$$), is given by $$\widehat Y_h \pm \sqrt{p F(1-\alpha;p,n-p)} \hspace{.05in} s (\widehat Y_h),$$ where $$s(\widehat Y_h)$$ is the estimated standard error of $$\widehat Y_h$$ and is given by $$s^2(\widehat Y_h) = (MSE) \cdot X_h^T (\mathbf{X}^T \mathbf{X})^{-1}X_h.$$ The last formula can be deduced from the fact that $$\mbox{Var}(\widehat Y_h) = \mbox{Var}(X_h^T \mathbf{b}) = X_h^T \mbox{Var}(\mathbf{b}) X_h = \sigma^2 X_h^T (\mathbf{X}^T\mathbf{X})^{-1} X_h.$$ Also, using the fact that $$(\widehat Y_h - X_h^T \beta)/s(\widehat Y_h) \sim t_{n-p}$$, a pointwise, $$100(1-\alpha)$$ % two-sided confidence interval for $$E(Y|X_h) = X_h^T \beta$$ is given by $$\widehat Y_h \pm t(1-\alpha/2;n-p) s(\widehat Y_h).$$ Extensions to the case where we want to simultaneously estimate $$E(Y|X_h)$$ for $$g$$ different values of $$X_h$$ can be achieved using either the Bonferroni procedure, or the Working-Hotelling procedure.
• Simultaneous prediction of new observations : Analogous to the one variable regression case, we consider the simultaneous prediction of new observations $$Y_{h(new)} = \beta^T X_h + \varepsilon_{h(new)}$$ for $$g$$ different values of $$X_h$$. Use $$s(Y_{h(new)} - \widehat Y_{h(new)})$$ to denote the estimated standard deviation of prediction error when $$X=X_h$$. We have $$s^2(Y_{h(new)} - \widehat Y_{h(new)}) = (MSE) (1+X_h^T (\mathbf{X}^T\mathbf{X})^{-1}X_h).$$ Bonferroni procedure yields simultaneous 100(1-$$\alpha$$) % prediction intervals of the form $$\widehat Y_h \pm t(1-\alpha/2g;n-p) s(Y_{h(new)} - \widehat Y_{h(new)}).$$ Scheff'{e}'s procedure gives the following simultaneous confidence intervals $$\widehat Y_h \pm \sqrt{gF(1-\alpha;g,n-p)} \hspace{.06in} s(Y_{h(new)} - \widehat Y_{h(new)}).$$
• Coefficient of multiple determination : The quantity $$R^2 = 1 - \frac{SSE}{SSTO} = \frac{SSR}{SSTO}$$ is a measure of association between the response $$Y$$ and the predictors $$X^{(1)},\ldots,X^{(p-1)}$$. This has the interpretation that $$R^2$$ is the proportion of variability in the response explained by the predictors. Another interpretation is that $$R^2$$ is the maximum squared correlation between $$Y$$ and any linear function of $$X^{(1)},\ldots,X^{(p-1)}$$.
• Adjusted $$R^2$$ : If one increases number of predictor variables in the regression model, then $$R^2$$ increases. To take into account the number of predictors, the measure called adjusted multiple $$R$$-squared, or, $$R_a^2 = 1-\frac{MSE}{MSTO} = 1 - \frac{SSE/(n-p)}{SSTO/(n-1)} = 1- \left(\frac{n-1}{n-p}\right) \frac{SSE}{SSTO},$$ is used. Notice that $$R_a^2 < R^2$$, and when the number of observationsis not too large, $$R_a^2$$ can be substantially smaller than $$R^2$$. Even though $$R_a^2$$ does not have as nice an interpretation as $$R^2$$, in multiple linear regression, this considered to be a better measure of association.

## Contributors:

• Valerie Regalia
• Debashis Paul