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SDM-CHAP24.tex
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SDM-CHAP24.tex
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\title{Multiple Regression Wisdom - Chapter 24}
\author{Alan Arnholt}
\date{Last updated: May 10, 2022 at 09:35:59 AM}
\begin{document}
\maketitle
{
\setcounter{tocdepth}{2}
\tableofcontents
}
\hypertarget{read-in-data-with-read.csv}{%
\section{\texorpdfstring{Read in data with \texttt{read.csv()}}{Read in data with read.csv()}}\label{read-in-data-with-read.csv}}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{site }\OtherTok{\textless{}{-}} \StringTok{"http://statlearning.com/s/Advertising.csv"}
\NormalTok{AD }\OtherTok{\textless{}{-}} \FunctionTok{read.csv}\NormalTok{(site)}
\FunctionTok{head}\NormalTok{(AD)}
\end{Highlighting}
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\begin{verbatim}
X TV radio newspaper sales
1 1 230.1 37.8 69.2 22.1
2 2 44.5 39.3 45.1 10.4
3 3 17.2 45.9 69.3 9.3
4 4 151.5 41.3 58.5 18.5
5 5 180.8 10.8 58.4 12.9
6 6 8.7 48.9 75.0 7.2
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{dim}\NormalTok{(AD)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 200 5
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(DT)}
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\AttributeTok{caption =} \StringTok{\textquotesingle{}Table 1: This is a simple caption for the table.\textquotesingle{}}\NormalTok{) }
\end{Highlighting}
\end{Shaded}
\begin{center}\includegraphics{SDM-CHAP24_files/figure-latex/readin-1} \end{center}
\hypertarget{base-r-graph}{%
\subsection{Base R Graph}\label{base-r-graph}}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{plot}\NormalTok{(sales }\SpecialCharTok{\textasciitilde{}}\NormalTok{ TV, }\AttributeTok{data =}\NormalTok{ AD, }\AttributeTok{col =} \StringTok{"red"}\NormalTok{, }\AttributeTok{pch =} \DecValTok{19}\NormalTok{)}
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\FunctionTok{abline}\NormalTok{(mod1, }\AttributeTok{col =} \StringTok{"blue"}\NormalTok{)}
\end{Highlighting}
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\begin{figure}
{\centering \includegraphics{SDM-CHAP24_files/figure-latex/base1-1}
}
\caption{Base R scatterplot of `sales` versus `TV`}\label{fig:base1}
\end{figure}
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\begin{Highlighting}[]
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\FunctionTok{plot}\NormalTok{(sales }\SpecialCharTok{\textasciitilde{}}\NormalTok{ radio, }\AttributeTok{data =}\NormalTok{ AD, }\AttributeTok{col =} \StringTok{"red"}\NormalTok{, }\AttributeTok{pch =} \DecValTok{19}\NormalTok{)}
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\begin{figure}
{\centering \includegraphics{SDM-CHAP24_files/figure-latex/youchange-1}
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\caption{You should change this caption}\label{fig:youchange}
\end{figure}
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Change the caption in Figure \ref{fig:youchange}.
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\subsection{\texorpdfstring{Using \texttt{ggplot2}}{Using ggplot2}}\label{using-ggplot2}}
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\begin{Highlighting}[]
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\begin{figure}
{\centering \includegraphics{SDM-CHAP24_files/figure-latex/unnamed-chunk-1-1}
}
\caption{Using `grid.arrange()` with `ggplot`}\label{fig:unnamed-chunk-1}
\end{figure}
\hypertarget{using-plotly}{%
\subsection{\texorpdfstring{Using \texttt{plotly}}{Using plotly}}\label{using-plotly}}
First create a plot with \texttt{ggplot2}, then pass the \texttt{ggplot2} object to \texttt{ggplotly} from the \texttt{plotly} package.
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(plotly)}
\NormalTok{p11 }\OtherTok{\textless{}{-}} \FunctionTok{ggplotly}\NormalTok{(p)}
\NormalTok{p11}
\end{Highlighting}
\end{Shaded}
\begin{center}\includegraphics{SDM-CHAP24_files/figure-latex/plotly-1} \end{center}
\hypertarget{scatterplot-matrices}{%
\subsection{Scatterplot Matrices}\label{scatterplot-matrices}}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(car)}
\FunctionTok{scatterplotMatrix}\NormalTok{(}\SpecialCharTok{\textasciitilde{}}\NormalTok{ sales }\SpecialCharTok{+}\NormalTok{ TV }\SpecialCharTok{+}\NormalTok{ radio }\SpecialCharTok{+}\NormalTok{ newspaper, }\AttributeTok{data =}\NormalTok{ AD)}
\end{Highlighting}
\end{Shaded}
\begin{center}\includegraphics{SDM-CHAP24_files/figure-latex/SPM-1} \end{center}
\hypertarget{basic-regression}{%
\section{Basic Regression}\label{basic-regression}}
Recall \texttt{mod1}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{mod1 }\OtherTok{\textless{}{-}} \FunctionTok{lm}\NormalTok{(sales }\SpecialCharTok{\textasciitilde{}}\NormalTok{ TV, }\AttributeTok{data =}\NormalTok{ AD)}
\FunctionTok{summary}\NormalTok{(mod1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
Call:
lm(formula = sales ~ TV, data = AD)
Residuals:
Min 1Q Median 3Q Max
-8.3860 -1.9545 -0.1913 2.0671 7.2124
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.032594 0.457843 15.36 <2e-16 ***
TV 0.047537 0.002691 17.67 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.259 on 198 degrees of freedom
Multiple R-squared: 0.6119, Adjusted R-squared: 0.6099
F-statistic: 312.1 on 1 and 198 DF, p-value: < 2.2e-16
\end{verbatim}
\begin{equation}
\text{Residual}\equiv e_i = y_i - \hat{y_i}
\label{eq:resid}
\end{equation}
To obtain the residuals for \texttt{mod1} use the function \texttt{resid} on a linear model object.
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{eis }\OtherTok{\textless{}{-}} \FunctionTok{resid}\NormalTok{(mod1)}
\NormalTok{RSS }\OtherTok{\textless{}{-}} \FunctionTok{sum}\NormalTok{(eis}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{)}
\NormalTok{RSS}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 2102.531
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{RSE }\OtherTok{\textless{}{-}} \FunctionTok{sqrt}\NormalTok{(RSS}\SpecialCharTok{/}\NormalTok{(}\FunctionTok{dim}\NormalTok{(AD)[}\DecValTok{1}\NormalTok{]}\SpecialCharTok{{-}}\DecValTok{2}\NormalTok{))}
\NormalTok{RSE}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 3.258656
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Or}
\FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{sigma}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 3.258656
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Or}
\FunctionTok{library}\NormalTok{(broom)}
\NormalTok{NDF }\OtherTok{\textless{}{-}} \FunctionTok{augment}\NormalTok{(mod1)}
\FunctionTok{sum}\NormalTok{(NDF}\SpecialCharTok{$}\NormalTok{.resid}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 2102.531
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{RSE }\OtherTok{\textless{}{-}} \FunctionTok{sqrt}\NormalTok{(}\FunctionTok{sum}\NormalTok{(NDF}\SpecialCharTok{$}\NormalTok{.resid}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{)}\SpecialCharTok{/}\FunctionTok{df.residual}\NormalTok{(mod1))}
\NormalTok{RSE}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 3.258656
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(moderndive)}
\FunctionTok{get\_regression\_table}\NormalTok{(mod1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
# A tibble: 2 x 7
term estimate std_error statistic p_value lower_ci upper_ci
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 intercept 7.03 0.458 15.4 0 6.13 7.94
2 TV 0.048 0.003 17.7 0 0.042 0.053
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{MDDF }\OtherTok{\textless{}{-}} \FunctionTok{get\_regression\_points}\NormalTok{(mod1)}
\NormalTok{MDDF}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
# A tibble: 200 x 5
ID sales TV sales_hat residual
<int> <dbl> <dbl> <dbl> <dbl>
1 1 22.1 230. 18.0 4.13
2 2 10.4 44.5 9.15 1.25
3 3 9.3 17.2 7.85 1.45
4 4 18.5 152. 14.2 4.27
5 5 12.9 181. 15.6 -2.73
6 6 7.2 8.7 7.45 -0.246
7 7 11.8 57.5 9.77 2.03
8 8 13.2 120. 12.7 0.454
9 9 4.8 8.6 7.44 -2.64
10 10 10.6 200. 16.5 -5.93
# ... with 190 more rows
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(dplyr)}
\NormalTok{MDDF }\SpecialCharTok{\%\textgreater{}\%}
\FunctionTok{summarize}\NormalTok{(}\AttributeTok{RSS =} \FunctionTok{sum}\NormalTok{(residual}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{))}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
# A tibble: 1 x 1
RSS
<dbl>
1 2102.
\end{verbatim}
The least squares estimators of \(\beta_0\) and \(\beta_1\) are
\[b_0 = \hat{\beta_0} = \bar{y} - b_1\bar{x}\]
\[b_1 = \hat{\beta_1} = \frac{\sum_{i = 1}^n(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n(x_i-\bar{x})^2}\]
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{y }\OtherTok{\textless{}{-}}\NormalTok{ AD}\SpecialCharTok{$}\NormalTok{sales}
\NormalTok{x }\OtherTok{\textless{}{-}}\NormalTok{ AD}\SpecialCharTok{$}\NormalTok{TV}
\NormalTok{b1 }\OtherTok{\textless{}{-}} \FunctionTok{sum}\NormalTok{( (x }\SpecialCharTok{{-}} \FunctionTok{mean}\NormalTok{(x))}\SpecialCharTok{*}\NormalTok{(y }\SpecialCharTok{{-}} \FunctionTok{mean}\NormalTok{(y)) ) }\SpecialCharTok{/} \FunctionTok{sum}\NormalTok{((x }\SpecialCharTok{{-}} \FunctionTok{mean}\NormalTok{(x))}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{)}
\NormalTok{b0 }\OtherTok{\textless{}{-}} \FunctionTok{mean}\NormalTok{(y) }\SpecialCharTok{{-}}\NormalTok{ b1}\SpecialCharTok{*}\FunctionTok{mean}\NormalTok{(x)}
\FunctionTok{c}\NormalTok{(b0, b1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 7.03259355 0.04753664
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Or using}
\FunctionTok{coef}\NormalTok{(mod1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) TV
7.03259355 0.04753664
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{summary}\NormalTok{(mod1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
Call:
lm(formula = sales ~ TV, data = AD)
Residuals:
Min 1Q Median 3Q Max
-8.3860 -1.9545 -0.1913 2.0671 7.2124
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.032594 0.457843 15.36 <2e-16 ***
TV 0.047537 0.002691 17.67 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.259 on 198 degrees of freedom
Multiple R-squared: 0.6119, Adjusted R-squared: 0.6099
F-statistic: 312.1 on 1 and 198 DF, p-value: < 2.2e-16
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{XTXI }\OtherTok{\textless{}{-}} \FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{cov.unscaled}
\NormalTok{MSE }\OtherTok{\textless{}{-}} \FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{sigma}\SpecialCharTok{\^{}}\DecValTok{2}
\NormalTok{var.cov.b }\OtherTok{\textless{}{-}}\NormalTok{ MSE}\SpecialCharTok{*}\NormalTok{XTXI}
\NormalTok{var.cov.b}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) TV
(Intercept) 0.209620158 -1.064495e-03
TV -0.001064495 7.239367e-06
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{seb0 }\OtherTok{\textless{}{-}} \FunctionTok{sqrt}\NormalTok{(var.cov.b[}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{])}
\NormalTok{seb1 }\OtherTok{\textless{}{-}} \FunctionTok{sqrt}\NormalTok{(var.cov.b[}\DecValTok{2}\NormalTok{, }\DecValTok{2}\NormalTok{])}
\FunctionTok{c}\NormalTok{(seb0, seb1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 0.457842940 0.002690607
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{coef}\NormalTok{(}\FunctionTok{summary}\NormalTok{(mod1))}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.03259355 0.457842940 15.36028 1.40630e-35
TV 0.04753664 0.002690607 17.66763 1.46739e-42
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{coef}\NormalTok{(}\FunctionTok{summary}\NormalTok{(mod1))[}\DecValTok{1}\NormalTok{, }\DecValTok{2}\NormalTok{]}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 0.4578429
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{coef}\NormalTok{(}\FunctionTok{summary}\NormalTok{(mod1))[}\DecValTok{2}\NormalTok{, }\DecValTok{2}\NormalTok{]}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 0.002690607
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{tb0 }\OtherTok{\textless{}{-}}\NormalTok{ b0}\SpecialCharTok{/}\NormalTok{seb0}
\NormalTok{tb1 }\OtherTok{\textless{}{-}}\NormalTok{ b1}\SpecialCharTok{/}\NormalTok{seb1}
\FunctionTok{c}\NormalTok{(tb0, tb1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 15.36028 17.66763
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{pvalues }\OtherTok{\textless{}{-}} \FunctionTok{c}\NormalTok{(}\FunctionTok{pt}\NormalTok{(tb0, }\DecValTok{198}\NormalTok{, }\AttributeTok{lower =} \ConstantTok{FALSE}\NormalTok{)}\SpecialCharTok{*}\DecValTok{2}\NormalTok{, }\FunctionTok{pt}\NormalTok{(tb1, }\DecValTok{198}\NormalTok{, }\AttributeTok{lower =} \ConstantTok{FALSE}\NormalTok{)}\SpecialCharTok{*}\DecValTok{2}\NormalTok{)}
\NormalTok{pvalues}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 1.40630e-35 1.46739e-42
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{coef}\NormalTok{(}\FunctionTok{summary}\NormalTok{(mod1))}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.03259355 0.457842940 15.36028 1.40630e-35
TV 0.04753664 0.002690607 17.66763 1.46739e-42
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{TSS }\OtherTok{\textless{}{-}} \FunctionTok{sum}\NormalTok{((y }\SpecialCharTok{{-}} \FunctionTok{mean}\NormalTok{(y))}\SpecialCharTok{\^{}}\DecValTok{2}\NormalTok{)}
\FunctionTok{c}\NormalTok{(RSS, TSS)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 2102.531 5417.149
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{R2 }\OtherTok{\textless{}{-}}\NormalTok{ (TSS }\SpecialCharTok{{-}}\NormalTok{ RSS)}\SpecialCharTok{/}\NormalTok{TSS}
\NormalTok{R2}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 0.6118751
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Or}
\FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{r.squared}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 0.6118751
\end{verbatim}
\hypertarget{confidence-interval-for-beta_1}{%
\subsection{\texorpdfstring{Confidence Interval for \(\beta_1\)}{Confidence Interval for \textbackslash beta\_1}}\label{confidence-interval-for-beta_1}}
\begin{equation}
\text{CI}_{1 - \alpha}(\beta_1) = \left[b_1 - t_{1- \alpha/2, n - p + 1}SE(b1), b_1 + t_{1- \alpha/2, n - p + 1}SE(b1) \right]
\label{eq:ci}
\end{equation}
\textbf{Example:} Use Equation \eqref{eq:ci} to construct a 90\% confidence interval for \(\beta_1\).
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{alpha }\OtherTok{\textless{}{-}} \FloatTok{0.10}
\NormalTok{ct }\OtherTok{\textless{}{-}} \FunctionTok{qt}\NormalTok{(}\DecValTok{1} \SpecialCharTok{{-}}\NormalTok{ alpha}\SpecialCharTok{/}\DecValTok{2}\NormalTok{, }\FunctionTok{df.residual}\NormalTok{(mod1))}
\NormalTok{ct}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 1.652586
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{b1 }\SpecialCharTok{+} \FunctionTok{c}\NormalTok{(}\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{)}\SpecialCharTok{*}\NormalTok{ct}\SpecialCharTok{*}\NormalTok{seb1}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 0.04309018 0.05198310
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Or}
\FunctionTok{confint}\NormalTok{(mod1, }\AttributeTok{parm =} \StringTok{"TV"}\NormalTok{, }\AttributeTok{level =} \FloatTok{0.90}\NormalTok{)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
5 % 95 %
TV 0.04309018 0.0519831
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{confint}\NormalTok{(mod1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
2.5 % 97.5 %
(Intercept) 6.12971927 7.93546783
TV 0.04223072 0.05284256
\end{verbatim}
\hypertarget{linear-algebra}{%
\subsubsection{Linear Algebra}\label{linear-algebra}}
\textbf{Solution of linear systems} Find the solution(s) if any to the following linear equations.
\[2x + y - z = 8\]
\[-3x - y + 2z = -11\]
\[-2x + y + 2z = -3\]
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{A }\OtherTok{\textless{}{-}} \FunctionTok{matrix}\NormalTok{(}\FunctionTok{c}\NormalTok{(}\DecValTok{2}\NormalTok{, }\SpecialCharTok{{-}}\DecValTok{3}\NormalTok{, }\SpecialCharTok{{-}}\DecValTok{2}\NormalTok{, }\DecValTok{1}\NormalTok{, }\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{, }\SpecialCharTok{{-}}\DecValTok{1}\NormalTok{, }\DecValTok{2}\NormalTok{, }\DecValTok{2}\NormalTok{), }\AttributeTok{nrow =} \DecValTok{3}\NormalTok{)}
\NormalTok{b }\OtherTok{\textless{}{-}} \FunctionTok{matrix}\NormalTok{(}\FunctionTok{c}\NormalTok{(}\DecValTok{8}\NormalTok{, }\SpecialCharTok{{-}}\DecValTok{11}\NormalTok{, }\SpecialCharTok{{-}}\DecValTok{3}\NormalTok{), }\AttributeTok{nrow =} \DecValTok{3}\NormalTok{)}
\NormalTok{x }\OtherTok{\textless{}{-}} \FunctionTok{solve}\NormalTok{(A)}\SpecialCharTok{\%*\%}\NormalTok{b}
\NormalTok{x}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1]
[1,] 2
[2,] 3
[3,] -1
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Or}
\FunctionTok{solve}\NormalTok{(A, b)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1]
[1,] 2
[2,] 3
[3,] -1
\end{verbatim}
See \href{https://en.wikipedia.org/wiki/Matrix_multiplication}{wikipedia} for a review of matrix multiplication rules and properties.
Consider the 2 \(\times\) 2 matrix \(A\).
\[A = \begin{bmatrix}
2 & 4 \\
9 & 5 \\
\end{bmatrix}
\]
\hypertarget{linear-regression-matrix-notation}{%
\subsubsection{Linear Regression Matrix Notation}\label{linear-regression-matrix-notation}}
\begin{equation}
\hat{\mathbf{\beta}} = (\mathbf{X'X})^{-1}\mathbf{X'Y}
\label{eq:betas}
\end{equation}
\begin{equation}
\sigma^2_{\hat{\beta}} = \sigma^2(\mathbf{X'X})^{-1}
\label{eq:varcov}
\end{equation}
\begin{equation}
\hat{\sigma}^2_{\hat{\beta}} = MSE(\mathbf{X'X})^{-1}
\label{eq:varcovest}
\end{equation}
\[\hat{\mathbf{\beta}} \sim\mathcal{N}(\mathbf{\beta}, \sigma^2(\mathbf{X'X})^{-1})\]
\hypertarget{estimation-of-the-mean-response-for-new-values-x_h}{%
\subsubsection{\texorpdfstring{Estimation of the Mean Response for New Values \(X_h\)}{Estimation of the Mean Response for New Values X\_h}}\label{estimation-of-the-mean-response-for-new-values-x_h}}
Not only is it desirable to create confidence intervals on the parameters of the regression models, bit it is also common to estimate the mean response \(\left(E(Y_h)\right)\) for a particular set of \(\mathbf{X}\) values.
\[\hat{Y}_h \sim \mathcal{N}(Y_h = X_h\beta, \sigma^2\mathbf{X_h}(\mathbf{X'X})^{-1}\mathbf{X_h'})\]
For a vector of given values \((\mathbf{X_h})\), a \((1 - \alpha)\cdot 100\%\) confidence interval for the mean response \(E(Y_h)\) is
\[CI_{1-\alpha}\left[E(Y_h)\right] = \left[\hat{Y}_h - t_{1 - \alpha/2;n - p - 1}\cdot s_{\hat{Y}_h}, \hat{Y}_h + t_{1 - \alpha/2;n - p - 1}\cdot s_{\hat{Y}_h} \right]\]
The function \texttt{predict()} applied to a linear model object will compute \(\hat{Y}_h\) and \(s_{\hat{Y}_h}\) for a given \(\mathbf{X}_h\). \texttt{R} output has \(\hat{Y}_h\) labeled \texttt{fit} and \(s_{\hat{Y}_h}\) labeled \texttt{se.fit}.
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{A }\OtherTok{\textless{}{-}} \FunctionTok{matrix}\NormalTok{(}\FunctionTok{c}\NormalTok{(}\DecValTok{2}\NormalTok{, }\DecValTok{9}\NormalTok{, }\DecValTok{4}\NormalTok{, }\DecValTok{5}\NormalTok{), }\AttributeTok{nrow =} \DecValTok{2}\NormalTok{)}
\NormalTok{A}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1] [,2]
[1,] 2 4
[2,] 9 5
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{t}\NormalTok{(A) }\CommentTok{\# Transpose of A}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1] [,2]
[1,] 2 9
[2,] 4 5
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{t}\NormalTok{(A)}\SpecialCharTok{\%*\%}\NormalTok{A }\CommentTok{\# A\textquotesingle{}A}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1] [,2]
[1,] 85 53
[2,] 53 41
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{solve}\NormalTok{(A)}\SpecialCharTok{\%*\%}\NormalTok{A }\CommentTok{\# I\_2}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1] [,2]
[1,] 1.000000e+00 -1.110223e-16
[2,] 1.110223e-16 1.000000e+00
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{zapsmall}\NormalTok{(}\FunctionTok{solve}\NormalTok{(A)}\SpecialCharTok{\%*\%}\NormalTok{A) }\CommentTok{\# What you expect I\_2}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1] [,2]
[1,] 1 0
[2,] 0 1
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{X }\OtherTok{\textless{}{-}} \FunctionTok{model.matrix}\NormalTok{(mod1)}
\NormalTok{XTX }\OtherTok{\textless{}{-}} \FunctionTok{t}\NormalTok{(X)}\SpecialCharTok{\%*\%}\NormalTok{X}
\FunctionTok{dim}\NormalTok{(XTX)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[1] 2 2
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{XTXI }\OtherTok{\textless{}{-}} \FunctionTok{solve}\NormalTok{(XTX)}
\NormalTok{XTXI}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) TV
(Intercept) 0.0197403984 -1.002458e-04
TV -0.0001002458 6.817474e-07
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# But it is best to compute this quantity using}
\FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{cov.unscaled}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) TV
(Intercept) 0.0197403984 -1.002458e-04
TV -0.0001002458 6.817474e-07
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{betahat }\OtherTok{\textless{}{-}}\NormalTok{ XTXI}\SpecialCharTok{\%*\%}\FunctionTok{t}\NormalTok{(X)}\SpecialCharTok{\%*\%}\NormalTok{y}
\NormalTok{betahat}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1]
(Intercept) 7.03259355
TV 0.04753664
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{coef}\NormalTok{(mod1)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) TV
7.03259355 0.04753664
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{XTXI }\OtherTok{\textless{}{-}} \FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{cov.unscaled}
\NormalTok{MSE }\OtherTok{\textless{}{-}} \FunctionTok{summary}\NormalTok{(mod1)}\SpecialCharTok{$}\NormalTok{sigma}\SpecialCharTok{\^{}}\DecValTok{2}
\NormalTok{var\_cov\_b }\OtherTok{\textless{}{-}}\NormalTok{ MSE}\SpecialCharTok{*}\NormalTok{XTXI}
\NormalTok{var\_cov\_b}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) TV
(Intercept) 0.209620158 -1.064495e-03
TV -0.001064495 7.239367e-06
\end{verbatim}
\textbf{Example} Use the \texttt{GRADES} data set and model \texttt{gpa} as a function of \texttt{sat}. Compute the expected GPA (\texttt{gpa}) for an SAT score (\texttt{sat}) of 1300. Construct a 90\% confidence interval for the mean GPA for students scoring 1300 on the SAT.
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(PASWR2)}
\NormalTok{mod.lm }\OtherTok{\textless{}{-}} \FunctionTok{lm}\NormalTok{(gpa }\SpecialCharTok{\textasciitilde{}}\NormalTok{ sat, }\AttributeTok{data =}\NormalTok{ GRADES)}
\FunctionTok{summary}\NormalTok{(mod.lm)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
Call:
lm(formula = gpa ~ sat, data = GRADES)
Residuals:
Min 1Q Median 3Q Max
-1.04954 -0.25960 -0.00655 0.26044 1.09328
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.1920638 0.2224502 -5.359 2.32e-07 ***
sat 0.0030943 0.0001945 15.912 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3994 on 198 degrees of freedom
Multiple R-squared: 0.5612, Adjusted R-squared: 0.5589
F-statistic: 253.2 on 1 and 198 DF, p-value: < 2.2e-16
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{betahat }\OtherTok{\textless{}{-}} \FunctionTok{coef}\NormalTok{(mod.lm)}
\NormalTok{betahat}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
(Intercept) sat
-1.19206381 0.00309427
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{knitr}\SpecialCharTok{::}\FunctionTok{kable}\NormalTok{(}\FunctionTok{tidy}\NormalTok{(mod.lm))}
\end{Highlighting}
\end{Shaded}
\begin{tabular}{l|r|r|r|r}
\hline
term & estimate & std.error & statistic & p.value\\
\hline
(Intercept) & -1.1920638 & 0.2224502 & -5.35879 & 2e-07\\
\hline
sat & 0.0030943 & 0.0001945 & 15.91171 & 0e+00\\
\hline
\end{tabular}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\#}
\NormalTok{Xh }\OtherTok{\textless{}{-}} \FunctionTok{matrix}\NormalTok{(}\FunctionTok{c}\NormalTok{(}\DecValTok{1}\NormalTok{, }\DecValTok{1300}\NormalTok{), }\AttributeTok{nrow =} \DecValTok{1}\NormalTok{)}
\NormalTok{Yhath }\OtherTok{\textless{}{-}}\NormalTok{ Xh}\SpecialCharTok{\%*\%}\NormalTok{betahat}
\NormalTok{Yhath}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
[,1]
[1,] 2.830488
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{predict}\NormalTok{(mod.lm, }\AttributeTok{newdata =} \FunctionTok{data.frame}\NormalTok{(}\AttributeTok{sat =} \DecValTok{1300}\NormalTok{))}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
1
2.830488
\end{verbatim}
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\# Linear Algebra First}
\FunctionTok{anova}\NormalTok{(mod.lm)}
\end{Highlighting}
\end{Shaded}
\begin{verbatim}
Analysis of Variance Table
Response: gpa
Df Sum Sq Mean Sq F value Pr(>F)
sat 1 40.397 40.397 253.18 < 2.2e-16 ***
Residuals 198 31.592 0.160
---