\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{graphicx} \usepackage{verbatim} \allowdisplaybreaks \jot=.2in \pagestyle{plain} \setlength{\topmargin}{-0.5in} \setlength{\textheight}{9.5in} \setlength{\oddsidemargin}{-0.1in} \setlength{\evensidemargin}{-0.1in} \setlength{\textwidth}{6.7in} \font\heada=cmbx10 scaled\magstep3 \font\headb=cmsl10 scaled\magstep1 \font\headc=cmr8 \pretolerance=10000 \setlength{\parindent}{2 em} \begin{document} \noindent {\heada Chapter 5 Revisited - Bivariate Numerical Data} \subsection*{5.1 Correlation} \noindent \underline{\bf Scatterplot} - a graphical display of the relationship between two numerical variables. \bigskip \noindent \underline{\bf How to Describe a Relationship}: \begin{enumerate} \item {\bf Form} - linear, non-linear (curved), clustered, etc. \item {\bf Direction} positive or negative. \item {\bf Strength} - strong, moderate, or weak \end{enumerate} \noindent \underline{\bf Sample Correlation Coefficient} - A statistic which estimates the {\it direction} and {\it strength} of a linear relationship between two variables. $$r = \frac{1}{n-1}\sum_{i=1}^n \left(\frac{x_i - \bar x}{s_x}\right) \left(\frac{y_i - \bar y}{s_y}\right)$$ \subsection*{5.2 Least Squares Regression} \begin{itemize} \item When there is a linear relationship between two numerical variables, one is interested in determining the equation of the line which describes the relationship. \item The true line which describes the relationship, called the {\bf Population Regression Line}, is $$y=\beta_0 + \beta_1x.$$ \begin{itemize} \item The parameter $\beta_0$ is the true {\em y-intercept} of the line. \item The parameter $\beta_1=\frac{{\rm change~in~}y}{{\rm change~in~}x}$ is the true {\em slope} of the line. The slope can be interpreted as the average change in $y$ for each unit increase in $x$. \item Your textbook uses the notation $y=\alpha + \beta x$. \end{itemize} \item From a SRS of bivariate data, the {\bf least-squares regression line}, which estimates the true line, can be calculated: $$\hat y = b_0 + b_1x$$ \begin{itemize} \item $\hat y$ is the {\em predicted} or {\em fitted} value. \item The statistic $b_1=r\frac{s_y}{s_x}$, the estimated slope, is an unbiased estimator of $\beta_1$. \item The statistic $b_0=\bar y-b_1\bar x$, the estimated $y$-intercept, is an unbiased estimator of $\beta_0$. \item $x$ is the explanatory, or {\em predictor}, variable. \item Your textbook uses the notation $\hat y=a + b x$. \vspace{0.1in} \end{itemize} \item The least-squares regression line is the one which is ``closest to the data" when compared to any other line. \newpage \noindent \underline{EXAMPLE}: \begin{enumerate} \item From $n=5$ skeletons for a particular species of dinosaur, femur measurements and humerus measurements (in inches) are taken. \bigskip \begin{tabular}{c|c} % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... $X$=femur & $Y$=humerus\\\hline 38 & 41 \\ 50 & 63 \\ 59 & 70 \\ 64 & 72 \\ 74 & 84 \\ \end{tabular} \bigskip Calculate the least-squares regression line which describes the relationship between $X$ and $Y$. Use the statistics $\bar x=57$, $s_x=13.71$, $\bar y=66$, $s_y=15.89$, and $r=.9776$. \vspace{2in} \item Sketch this line over the scatterplot for this data. \bigskip \item For a dinosaur with a 55" femur, what is the predicted humerus length? \vspace{1in} \item Interpret the slope of the least-squares regression line. \vspace{.5in} \end{enumerate} \item {\bf Extrapolation} - the use of a regression line to predict $y$ at an $x$ value outside the range of $x$'s used to calculate the line. DO NOT EXTRAPOLATE! Extrapolation often results in very inaccurate predictions. \bigskip \noindent \underline{EXAMPLE}: \end{itemize} \newpage \subsection*{5.3 Assessing the Fit of a Line} \begin{itemize} \item The vertical distance of a line to a data point $(x_i,y_i)$ is the residual (just as in ANOVA) $$e_i = y_i - \hat y_i.$$ \item The least-squares regression line is the line that minimizes the sum of the squared residuals or $SSE$ (just as in an ANOVA) when compared to any other line $$SSE = \sum_i e_i^2 = \sum_i (y_i - \hat y_i)^2.$$ Your textbook calls $SSResid=SSE$. Thus, a line with the minimal $SSE$ is closest to the data when compared to any other line. \vspace{0.1in} \noindent \underline{EXAMPLE}: Calculate the residuals for each data point and calculate the $SSE$. \bigskip \begin{tabular}{ccccc} $i$ &\hspace{0.2in} $x_i$ \hspace{0.2in} & \hspace{0.2in} $y_i$ \hspace{0.2in} & \hspace{1in} $\hat y_i$ \hspace{1in} & $e_i = y_i - \hat y_i$ \\ \hline & & & \\ 1& 38 & 41 \\ 2& 50 & 63 \\ 3& 59 & 70 \\ 4& 64 & 72 \\ 5& 74 & 84 \\ \end{tabular} \vspace{1in} $SSE =$ \underline{\hspace{1in}}. \bigskip \bigskip \end{itemize} \noindent \underline{\bf Partition the Variability}: \smallskip \noindent Just as in ANOVA, the total sample variability of the response $y$ can be partitioned into two pieces, $$SSTo = SSM + SSE$$ \noindent \begin{itemize} \item SSTo, the {\em Total Sum of Squares}, is $SSTo = \sum(y_i - \bar y)^2.$ $SSTo$ represents the total variability of the response $y$, $$s^2_y = \frac{1}{n-1}\sum_i(y_i-\bar y)^2=\frac{SSTo}{n-1}.$$ \item SSM, the {\em Model Sum of Squares}, is $$SSM = \sum (\hat y - \bar y)^2.$$ $SSM$ is represents the part of the variability of the response which is explained by the linear-regression line. \item SSE, the {\em Residual Sum of Squares} (or {\em Sum of Squares Error}), is $$SSE = \sum (y_i - \hat y)^2.$$ $SSE$ represents the part of the variability of the response which is \underline{not} explained by the linear-regression line. \item The variance of the residuals around the least-squares regression line is $$\sigma^2\approx S^2 = MSE = \frac{SSE}{n-2}.$$ \vspace{0.1in} \end{itemize} \noindent \underline{\bf Tools to Assess the Fit of a Line} \begin{itemize} \item Plot the residuals versus the fitted values, $(\hat y_i,ei)$. The line is a good fit if the points form no pattern on the plot. \item The \underline{\bf Coefficient of Determination} is the proportion of the total sample variability of $y$ that is explained by the linear-regression line between $x$ and $y$: $$r^2 = \frac{SSM}{SSTo}.$$ \begin{itemize} \item The notation $r^2$ is used because the coefficient of determination is the correlation coefficient squared. \item $r^2$ is a proportion, and so is a value between 0 and 1. \item The line is a good model if $r^2$ is large. \end{itemize} \end{itemize} \noindent\underline{EXAMPLE}: \begin{enumerate} \item Create the residual versus fits graph for the dinosaur data. \vspace{2.5in} \item Are there any patterns which indicate that the linear model is not appropriate? \vspace{.5in} \item Calculate $r^2$. \vspace{.5in} \item Interpret $r^2$. \end{enumerate} \vspace{1in} \section*{R code} \verbatiminput{Chapter5Rcode.txt} \section*{Exercises} 5.1 on p194: 1-11 odd, 15 \noindent 5.2 on p205: 17*, 19*, 21*, 23, 25* \noindent 5.3 on p219: 33, 35*, 39-43 odd \bigskip \noindent A ``*" indicates that you may want to use R to assist with the calculations. \section*{Reading} Sections 5.1-5.3 \end{document}