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\cfoot{Statistical Sleuth in R: Chapter 11}
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\newcommand{\cran}{\href{http://www.R-project.org/}{CRAN}}
\title{The Statistical Sleuth in R: \\
Chapter 11}
\author{
Kate Aloisio\and Ruobing Zhang \and Nicholas J. Horton\thanks{Department of Mathematics, Amherst College, nhorton@amherst.edu}
}
\date{\today}
\begin{document}
\maketitle
\tableofcontents
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<>=
print.pval = function(pval) {
threshold = 0.0001
return(ifelse(pval < threshold, paste("p<", sprintf("%.4f", threshold), sep=""),
ifelse(pval > 0.1, paste("p=",round(pval, 2), sep=""),
paste("p=", round(pval, 3), sep=""))))
}
@
<>=
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fig.align="center",
prompt=TRUE, # show the prompts; but perhaps we should not do this
comment=NA # turn off commenting of ouput (but perhaps we should not do this either
)
@
<>=
require(Sleuth2)
require(mosaic)
trellis.par.set(theme=col.mosaic()) # get a better color scheme
set.seed(123)
# this allows for code formatting inline. Use \Sexpr{'function(x,y)'}, for exmaple.
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if (is.numeric(x)) return(knitr:::format_sci(x, 'latex'))
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h = gsub('(["\'])', '\\1{}', h)
gsub('^\\\\begin\\{alltt\\}\\s*|\\\\end\\{alltt\\}\\s*$', '', h)
})
showOriginal=FALSE
showNew=TRUE
@
\section{Introduction}
This document is intended to help describe how to undertake analyses introduced as examples in the Second Edition of the \emph{Statistical Sleuth} (2002) by Fred Ramsey and Dan Schafer.
More information about the book can be found at \url{http://www.proaxis.com/~panorama/home.htm}.
This
file as well as the associated \pkg{knitr} reproducible analysis source file can be found at
\url{http://www.amherst.edu/~nhorton/sleuth}.
This work leverages initiatives undertaken by Project MOSAIC (\url{http://www.mosaic-web.org}), an NSF-funded effort to improve the teaching of statistics, calculus, science and computing in the undergraduate curriculum. In particular, we utilize the
\pkg{mosaic} package, which was written to simplify the use of R for introductory statistics courses. A short summary of the R needed to teach introductory statistics can be found in the mosaic package vignette (\url{http://cran.r-project.org/web/packages/mosaic/vignettes/MinimalR.pdf}).
To use a package within R, it must be installed (one time), and loaded (each session). The package can be installed using the following command:
<>=
install.packages('mosaic') # note the quotation marks
@
Once this is installed, it can be loaded by running the command:
<>=
require(mosaic)
@
This
needs to be done once per session.
In addition the data files for the \emph{Sleuth} case studies can be accessed by installing the \pkg{Sleuth2} package.
<>=
install.packages('Sleuth2') # note the quotation marks
@
<>=
require(Sleuth2)
@
We also set some options to improve legibility of graphs and output.
<>=
trellis.par.set(theme=col.mosaic()) # get a better color scheme for lattice
options(digits=3, show.signif.stars=FALSE)
@
The specific goal of this document is to demonstrate how to calculate the quantities described in Chapter 11: Model Checking and Refinement using R.
\section{Alcohol metabolism in men and women}
How do men and women metabolise alcohol? This is the question addressed in case study 11.1 in the \emph{Sleuth}.
\subsection{Data coding, summary statistics and graphical display}
We begin by reading the data and summarizing the variables.
<<>>=
summary(case1101)
@
A total of \Sexpr{nrow(case1101)} volunteers were included in this data. There were \Sexpr{nrow(subset(case1101, Sex=="Female"))} females and \Sexpr{nrow(subset(case1101, Sex=="Male"))} males. As recorded in Display 9.2 (page 237 of the \emph{Sleuth}).
The following is a graphical display of the variables akin to Display 11.2 (page 306).
<>=
xyplot(Metabol ~ Gastric | Sex+Alcohol, data=case1101, auto.key=TRUE,
xlab="Gastric AD activity (mu mol/min/g of tissue)",
ylab="first pass metabolism (mmol/liter-hour)")
@
\subsection{Multiple regression}
First we can fit a full model for estimating \emph{metabolism} given a subjects \emph{gastric AD activity}, whether they are \emph{alcoholic} and \emph{gender}. This first model is summarized on page 315 (Display 11.9).
<>=
case1101 = transform(case1101, Sex = factor(Sex, levels = c("Male", "Female")))
case1101 = transform(case1101, Alcohol = factor(Alcohol,
levels = c("Non-alcoholic", "Alcoholic")))
lm1 = lm(Metabol ~ Gastric+Sex+Alcohol+Gastric*Sex+Sex*Alcohol+
Gastric*Alcohol+Gastric*Sex*Alcohol, data=case1101); summary(lm1)
@
Next we can calculate a number of model diagnostics, including
leverage, studentized resids and Cook's distance (pages 319--320).
<>=
require(MASS)
@
<<>>=
case1101 = transform(case1101, hat = hatvalues(lm1))
case1101 = transform(case1101, studres = studres(lm1))
case1101 = transform(case1101, cooks = cooks.distance(lm1))
case1101[31,]
@
The following is a residual plot for the full model akin to Display 11.7 (page 313).
<<>>=
xyplot(residuals(lm1) ~ fitted(lm1), xlab="Fitted values", ylab="Residuals",
type=c("p", "r", "smooth"))
@
From these diagnostics it appears that observations 31 and 32 may be influential points. Therefore, we next re-fit the full model excluding these two observations. The following results are found in Display 11.9 and discussed on page 315.
<<>>=
case11012 = case1101[-c(31, 32),]
lm2 = lm(Metabol ~ Gastric+Sex+Alcohol+Gastric*Sex+Sex*Alcohol+
Gastric*Alcohol+Gastric*Sex*Alcohol, data=case11012); summary(lm2)
@
\subsection{Refining the Model}
This section addresses the process of refining the model.
We first tested the lack of fit for the removal of {\tt Alcohol} as shown in Display 11.13 (page 322).
<>=
lm3 = lm(Metabol ~ Gastric+Sex+Gastric*Sex, data=case11012); summary(lm3)
anova(lm3, lm2) # page 322
@
Next we assessed a model without an intercept which is scientifically plausible as summarized in Display 11.14
(page 323).
<>=
lm4 = lm(Metabol ~ Gastric+Gastric:Sex -1 , data=case11012); summary(lm4)
anova(lm4, lm3)
@
Note that the ``Summary of Statistical Findings" section (page 306) is based on this final model.
\section{Blood brain barrier}
Neuroscientists working to better understand the blood brain barrier have infused rats with cells to induce brain tumors. This is the topic addressed in case study 11.2 in the \emph{Sleuth}.
\subsection{Data coding and summary statistics}
We begin by reading the data, performing transformations where needed and summarizing the variables.
<>=
case1102 = transform(case1102, Y = Brain/Liver)
case1102 = transform(case1102, logliver = log(Liver))
case1102 = transform(case1102, logbrain = log(Brain))
case1102 = transform(case1102, SAC = as.factor(Time))
case1102 = transform(case1102, logy = log(Brain/Liver))
case1102 = transform(case1102, logtime = log(Time))
case1102 = transform(case1102, Treat = relevel(Treat, ref="NS"))
summary(case1102)
@
A total of \Sexpr{nrow(case1102)} rats were included in this experiment. Each rat was given either the barrier solution (n = \Sexpr{nrow(subset(case1102, Treat=="BD"))}) or a normal saline solution (n = \Sexpr{nrow(subset(case1102, Treat=="NS"))}). Then variables of interest were calculated and are displayed in Display 11.4 (page 308 of the \emph{Sleuth}).
We can graphically relationships between the variables using a pairs plot.
<>=
smallds = case1102[,c("logy", "logbrain","logliver","Treat", "SAC")]
pairs(smallds)
@
\subsection{Graphical presentation}
The following displays a scatterplot of log ratio (Y) as a function of log time, akin to Display 11.5 on page 309.
<<>>=
xyplot(Y ~ Time, group=Treat, scales=list(y=list(log=TRUE),
x=list(log=TRUE)), auto.key=TRUE, data=case1102)
@
The following graphs are akin to the second and third plots in Display 11.16 on page 326.
<<>>=
case1102 = transform(case1102, female = ifelse(Sex=="F", 1, 0))
xyplot(logy ~ jitter(female), xlab="Sex", type=c("p", "r", "smooth"),
data=case1102)
@
<<>>=
xyplot(logy ~ jitter(Days), type=c("p", "r", "smooth"),
data=case1102)
@
\subsection{Multiple regression}
We first fit a model that reflects the initial investigation. This is the proposed model from page 311.
<>=
lm1 = lm(logy ~ SAC+Treat+SAC*Treat+Days+Sex+
Weight+Loss+Tumor, data=case1102); summary(lm1)
@
We can then display a residual plot to assess the fit of the above model. This is provided in
Display 11.6
(page 312).
<<>>=
xyplot(residuals(lm1) ~ fitted(lm1), xlab="Fitted values", ylab="Residuals",
type=c("p", "r", "smooth"))
@
\subsection{Refining the model}
Lastly, we fit a refined model. These results can be found in Display 11.17 (page 327).
<>=
lm2 = lm(logy ~ SAC+Treat, data=case1102); summary(lm2)
anova(lm2, lm1)
@
\end{document}