Multiple Linear Regression (continued)
- Page ID
- 233
\[ Y_i = \beta_0 + \beta_1 X_i^{(1)} + \cdots + \beta_{p-1} X_i^{(p-1)} + \varepsilon_i, \qquad i=1,\ldots,n, \tag{1}\]
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