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\begin{document}

%% Title page

\title{A Handbook of Statistical Analyses Using \R{} --- 3rd Edition}

\author{Torsten Hothorn and Brian S. Everitt}

\maketitle
%%\VignetteIndexEntry{Chapter Generalized Additive Models}
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<<packages, echo = FALSE, results = hide>>=
library("mgcv")
library("mboost")
library("rpart")
library("wordcloud")
@

\chapter[Scatterplot Smoothers and Additive Models]{Scatterplot 
  Smoothers and Generalized Additive Models: The Men's Olympic 1500m, 
  Air Pollution in the US, Risk Factors for Kyphosis, and Women's Role in %'
  Society \label{GAM}}

\section{Introduction}


\section{Scatterplot Smoothers and Generalized Additive Models}


\section{Analysis Using \R{}}

\subsection{Olympic 1500m Times}

To begin we will construct a scatterplot of winning time 
against the year the games were held. The \R{} code and the resulting plot 
are shown in Figure~\ref{GAM-men1500m-plot}.  There is a very clear downward trend in 
the times over the years, and, in addition there is a very clear 
outlier namely the winning time for 1896. We shall remove this 
time from the data set and now concentrate on the remaining times. 
First we will fit a simple linear regression to the data and 
plot the fit onto the scatterplot. The code and the resulting 
plot are shown in Figure~\ref{GAM-men1500m-lm}. Clearly the linear regression 
model captures in general terms the downward trend in the times. 
Now we can add the fits given by the lowess smoother and by a 
cubic spline smoother; the resulting graph and the extra \R{} code 
needed are shown in Figure~\ref{GAM-men1500m-smooth}. 


Both non-parametric fits suggest some distinct departure from 
linearity, and clearly point to a quadratic model being more 
sensible than a linear model here. And fitting a parametric model 
that includes both a linear and a quadratic effect for the year gives a 
prediction curve very similar to the non-parametric curves; see Figure~\ref{GAM-men1500m-quad}.

Here use of the non-parametric smoothers has effectively diagnosed 
our linear model and pointed the way to using a more suitable 
parametric model; this is often how such non-parametric models can be used most 
effectively. For these data, of course, it is clear that the simple linear 
model cannot be suitable if the investigator is interested in predicting 
future times since even the most basic knowledge of human physiology 
will tell us that times cannot continue to go down. There must be some 
lower limit to the time man can run 1500m. But in other situations 
use of the non-parametric smoothers may point to a parametric model 
that could not have been identified \emph{a priori}. 

\begin{figure}
\begin{center}
<<GAM-men1500m-plot, echo = TRUE, fig = TRUE, height = 5>>=
plot(time ~ year, data = men1500m, xlab = "Year",
     ylab = "Winning time (sec)")
@
\caption{Scatterplot of year and winning time. \label{GAM-men1500m-plot}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
<<GAM-men1500m-lm, echo = TRUE, fig = TRUE, height = 5>>=
men1500m1900 <- subset(men1500m, year >= 1900)
men1500m_lm <- lm(time ~ year, data = men1500m1900)
plot(time ~ year, data = men1500m1900, xlab = "Year",
     ylab = "Winning time (sec)")
abline(men1500m_lm)
@
\caption{Scatterplot of year and winning time with fitted values from
         a simple linear model. \label{GAM-men1500m-lm}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
<<GAM-men1500m-smooth, echo = TRUE, fig = TRUE, height = 5>>=
x <- men1500m1900$year
y <- men1500m1900$time
men1500m_lowess <- lowess(x, y)
plot(time ~ year, data = men1500m1900, xlab = "Year",
     ylab = "Winning time (sec)")
lines(men1500m_lowess, lty = 2)
men1500m_cubic <- gam(y ~ s(x, bs = "cr"))
lines(x, predict(men1500m_cubic), lty = 3)
@
\caption{Scatterplot of year and winning time with fitted values from a smooth
         non-parametric model. \label{GAM-men1500m-smooth}}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
<<GAM-men1500m-quad, echo = TRUE, fig = TRUE, height = 5>>=
men1500m_lm2 <- lm(time ~ year + I(year^2), 
                   data = men1500m1900)
plot(time ~ year, data = men1500m1900, xlab = "Year",
     ylab = "Winning time (sec)")
lines(men1500m1900$year, predict(men1500m_lm2))
@
\caption{Scatterplot of year and winning time with fitted values from a 
         quadratic model. \label{GAM-men1500m-quad}}
\end{center}
\end{figure}


It is of some interest to look at the predictions of winning 
times in future Olympics from both the linear and quadratic models. 
For example, for 2008 and 2012 the predicted times and their $95\%$ 
confidence intervals can be found using the following code

\newpage

<<GAM-men1500m-pred, echo = TRUE>>=
predict(men1500m_lm, 
        newdata = data.frame(year = c(2008, 2012)), 
        interval = "confidence")
predict(men1500m_lm2, 
        newdata = data.frame(year = c(2008, 2012)), 
        interval = "confidence")
@

\newpage

For predictions far into the future both 
the quadratic and the linear model fail; we leave readers to get 
some more predictions to see what happens. We can compare
the first prediction with the time actually recorded by the winner
of the men's 1500m in Beijing 2008, 
Rashid Ramzi from Brunei, who won the event in $212.94$ seconds. The confidence
interval obtained from the simple linear model does not include this value
but the confidence interval for the prediction 
derived from the quadratic model does.


\subsection{Air Pollution in US Cities}

Unfortunately, we cannot fit an additive model for
describing the $\text{SO}_2$ concentration based on all six
covariates because this leads to more parameters than cities, 
i.e., more parameters than observations when using the
default parameterization of \Rpackage{mgcv}. Thus, before we can apply
the \Rcmd{gam} function from package \Rpackage{mgcv}, we have
to decide which covariates should enter the model and which
subset of these covariates should be allowed to deviate from
a linear regression relationship.

As briefly discussed in Section~\ref{GAM:VS}, we can
fit an additive model using the iterative boosting algorithm
as described by \cite{HSAUR:BuehlmannHothorn2007}. The complexity
of the model is determined by an AIC criterion, which can also be used
to determine an appropriate number of boosting iterations to choose.
The methodology is available
from package \Rpackage{mboost} \citep{PKG:mboost}. We start 
with a small number of boosting iterations ($100$ by default) and
compute the AIC of the corresponding $100$ models:
<<GAM-USairpollution-boost, echo = TRUE>>=
library("mboost")
USair_boost <- gamboost(SO2 ~ ., data = USairpollution)
USair_aic <- AIC(USair_boost)
USair_aic
@
The AIC suggests that the boosting algorithm should be stopped
after $\Sexpr{mstop(USair_aic)}$ iterations. The partial contributions of each
covariate to the predicted $\text{SO}_2$ concentration
are given in Figure~\ref{GAM-USairpollution-boostplot}.
The plot indicates that all six covariates enter the model and
the selection of a subset of covariates for modeling 
isn't appropriate in this case. However, the number of manufacturing
enterprises seems to add linearly to the $\text{SO}_2$ concentration, which
simplifies the model. Moreover, the average annual precipitation
contribution seems to deviate from zero only for some extreme observations
and one might refrain from using the covariate at all.

\begin{figure}
\begin{center}
<<GAM-USairpollution-boostplot, echo = TRUE, fig = TRUE, height = 4>>=
USair_gam <- USair_boost[mstop(USair_aic)]
layout(matrix(1:6, ncol = 3))
plot(USair_gam, ask = FALSE)
@
\caption{Partial contributions of six exploratory covariates
         to the predicted $\text{SO}_2$ concentration. \label{GAM-USairpollution-boostplot}}
\end{center}
\end{figure}

As always, an inspection of the model fit via a residual plot
is worth the effort. Here, we plot the fitted values against the residuals
and label the points with the name of the corresponding city using the \Rcmd{textplot} function from package \Rpackage{wordcloud}.
Figure~\ref{GAM-USairpollution-residplot} shows at least two
extreme observations. Chicago has a very large observed and fitted 
$\text{SO}_2$ concentration, which is due to the huge number of inhabitants and
manufacturing plants (see Figure~\ref{GAM-USairpollution-boostplot} also).
One smaller city, Providence, is associated with a rather large
positive residual indicating that the actual $\text{SO}_2$ concentration
is underestimated by the model. In fact, this small town has
a rather high $\text{SO}_2$ concentration which is hardly explained
by our model. Overall, the model doesn't fit the data very well,
so we should avoid overinterpreting the model structure 
too much. In addition, since each of the six covariates
contributes to the model, we aren't able to select a smaller subset
of the covariates for modeling and thus fitting a model using
\Rcmd{gam} is still complicated (and will not add much knowledge anyway).

\begin{figure}
\begin{center}
<<GAM-USairpollution-residplot, echo = TRUE, fig = TRUE>>=
SO2hat <- predict(USair_gam)
SO2 <- USairpollution$SO2
plot(SO2hat, SO2 - SO2hat, type = "n",         
     xlim = c(-20, max(SO2hat) * 1.1),
     ylim = range(SO2 - SO2hat) * c(2, 1))
textplot(SO2hat, SO2 - SO2hat, rownames(USairpollution), 
         show.lines = FALSE, new = FALSE)
abline(h = 0, lty = 2, col = "grey")
@
\caption{Residual plot of $\text{SO}_2$ concentration. 
         \label{GAM-USairpollution-residplot}}
\end{center}
\end{figure}


\subsection{Risk Factors for Kyphosis}

\index{Spinogram}
Before modeling the relationship between kyphosis and
the three exploratory variables age, starting vertebral level of
the surgery, and number of vertebrae involved, we investigate the
partial associations by so-called \stress{spinograms}, as
introduced in \Sexpr{ch("DAGD")}. The
numeric exploratory covariates are discretized and their
empirical relative frequencies are plotted against the
conditional frequency of kyphosis in the corresponding
group. Figure~\ref{GAM-kyphosis-plot} shows that kyphosis
is absent in very young or very old children, children
with a small starting vertebral level, and high number of vertebrae 
involved.

\begin{figure}
\begin{center}
<<GAM-kyphosis-plot, echo = TRUE, fig = TRUE, height = 4, width = 7>>=
layout(matrix(1:3, nrow = 1))
spineplot(Kyphosis ~ Age, data = kyphosis, 
          ylevels = c("present", "absent"))
spineplot(Kyphosis ~ Number, data = kyphosis, 
          ylevels = c("present", "absent"))
spineplot(Kyphosis ~ Start, data = kyphosis, 
         ylevels = c("present", "absent"))
@
\caption{Spinograms of the three exploratory variables and response variable
         \Robject{kyphosis}. \label{GAM-kyphosis-plot}}
\end{center}
\end{figure}

The logistic additive model needed to describe the conditional
probability of kyphosis given the exploratory variables
can be fitted using function \Rcmd{gam}. Here, the dimension
of the basis ($k$) has to be modified for \Robject{Number} and
\Robject{Start} since these variables are heavily tied. As for
generalized linear models, the \Robject{family} argument determines
the type of model to be fitted, a logistic model in our case:
<<GAM-kyphosis-gam, echo = TRUE>>=
(kyphosis_gam <- gam(Kyphosis ~ s(Age, bs = "cr") + 
    s(Number, bs = "cr", k = 3) + s(Start, bs = "cr", k = 3),
    family = binomial, data = kyphosis))
@
The partial contributions of each covariate to the conditional
probability of kyphosis with confidence bands are shown in 
Figure~\ref{GAM-kyphosis-gamplot}. In essence, the same conclusions
as drawn from Figure~\ref{GAM-kyphosis-plot} can be stated here.
The risk of kyphosis being present decreases with higher
starting vertebral level and lower number of vertebrae involved.
Children about $100$ months old are under higher risk compared
to younger or older children.

\begin{figure}
\begin{center}
<<GAM-kyphosis-gamplot, echo = TRUE, fig = TRUE, height = 4, width = 7>>=
trans <- function(x) 
    binomial()$linkinv(x)
layout(matrix(1:3, nrow = 1))
plot(kyphosis_gam, select = 1, shade = TRUE, trans = trans)
plot(kyphosis_gam, select = 2, shade = TRUE, trans = trans)
plot(kyphosis_gam, select = 3, shade = TRUE, trans = trans)
@
\caption{Partial contributions of three exploratory variables
         with confidence bands. \label{GAM-kyphosis-gamplot}}
\end{center}
\end{figure}

\subsection{Women's Role in Society} %'

In Chapter~\ref{GLM}, we saw that a logistic regression with an interaction between
gender and level of education described the data better than a main-effects only
model. Using an additive logistic regression model, we can fit separate, possibly
nonlinear, functions of levels of education to both genders:
<<GAM-womensrole-gam, echo = TRUE>>= 
data("womensrole", package = "HSAUR3")
fm1 <- cbind(agree, disagree) ~ s(education, by = gender)
womensrole_gam <- gam(fm1, data = womensrole, 
                      family = binomial())
@
The resulting model is best inspected by a plot
(Figure~\ref{GAM-womensrole-gamplot}).  For males, the log-odds of agreement
decreases linearly with each additional year of education.  For females,
the log-odds is more or less constant up to five years of education and only
then begins to decrease.  After 15 years, there seems to be no further
impact on the log-odds.  When we plot the resulting fitted probabilities in
a way similar to Figure~\ref{GLM-role2plot}, we see that the interaction is
even more pronounced in the additive compared to the linear model.  The flat
curve for women with less than five years of education can be explained by
the rather large variability of the answers in this area but the plateau to 
the right is due to two groups of highly educated women with a rather 
large proportion of agreement.


\begin{figure}
\begin{center}
<<GAM-womensrole-gamplot, echo = TRUE, fig = TRUE, height = 4, width = 7>>=
layout(matrix(1:2, nrow = 1))
plot(womensrole_gam, select = 1, shade = TRUE)
plot(womensrole_gam, select = 1, shade = TRUE)
@
\caption{Effects of level of education for males (right) and females (left)
         on the log-odds scale derived from an additive logistic regression model.
         The shaded area denotes confidence bands. \label{GAM-womensrole-gamplot}}
\end{center}
\end{figure}

<<GAM-plot-setup, echo = FALSE>>=  
myplot <- function(role.fitted)  {    
    f <- womensrole$gender == "Female"
    plot(womensrole$education, role.fitted, type = "n",
         ylab = "Probability of agreeing",
         xlab = "Education", ylim = c(0,1))
    lines(womensrole$education[!f], role.fitted[!f], lty = 1)
    lines(womensrole$education[f], role.fitted[f], lty = 2)
    lgtxt <- c("Fitted (Males)", "Fitted (Females)")
    legend("topright", lgtxt, lty = 1:2, bty = "n")
    y <-  womensrole$agree / (womensrole$agree +
                              womensrole$disagree)
    size <- womensrole$agree + womensrole$disagree
    size <- size - min(size)
    size <- (size / max(size)) * 3 + 1
    text(womensrole$education, y, ifelse(f, "\\VE", "\\MA"),
         family = "HersheySerif", cex = size)
} 
@


\begin{figure}
\begin{center}
<<GAM-womensrole-probplot, echo = TRUE, fig = TRUE>>=
myplot(predict(womensrole_gam, type = "response"))
@
\caption{Effects of level of education for males (right) and females (left)
         on the log-odds scale derived from an additive logistic regression model.
         The shaded area denotes confidence bands. \label{GAM-womensrole-probplot}}
\end{center}
\end{figure}


\section{Summary of Findings}

\begin{description}
\item[Olympic 1500m times]
Here the use of a generalized additive model suggested that a quadratic
model might best describe the data.  When such a model was fitted it made
reasonable predictions of the winner's times in the Olympic Games of 2008
and 2012.
\item[Air pollution data]
Finding a suitable model for these data was problematic because of the
outliers in the data and the high correlations between some pairs of
explanatory variables.  Except for wind, the fitted partial contributions
are well approximated by a linear function for most of the observations and
it might be questioned if the more complex additive model is really needed.
\item[Kyphosis]
The risk of kyphosis being present decreases with higher starting vertebral
level and lower number of vertebrae involved.  Children about 100 months old
are under higher risk compared to younger or older children.
\item[Women's role in society]
For males, the log-odds of agreement decreases linearly with each
additional year of education.  For females, the log-odds is more or less
constant up to five years of education and only then begins to decrease. 
After $15$ years, there seems to be no further impact on the log-odds.
\end{description}


\section{Final Comments}

Additive models offer flexible modeling tools for regression
problems. They stand between generalized linear models, where
the regression relationship is assumed to be linear, and
more complex models like random forests (see \Sexpr{ch("RP")})
where the regression relationship remains unspecified. Smooth
functions describing the influence of covariates on the response
can be easily interpreted. Variable
selection is a technically difficult problem in this class
of models; boosting methods are one possibility to deal with this
problem.

\section*{Exercises}

\begin{description}

\exercise 
Consider the body fat data introduced in \Sexpr{ch("RP")}, 
Table~\ref{RP-bodyfat-tab}. First fit a generalized additive
model assuming normal errors using function \Rcmd{gam}. Are 
all potential covariates informative? Check the results
against a generalized additive model that underwent
AIC-based variable selection (fitted using function \Rcmd{gamboost}).

\exercise
Again fit an additive model to the body fat data, but this time for
a log-transformed response. Compare the two models, which one is more appropriate?

\exercise
Try to fit a logistic additive model to the glaucoma data
discussed in \Sexpr{ch("RP")}. Which covariates should enter the
model and how is their influence on the probability of 
suffering from glaucoma?

\exercise
Investigate the use of different types of scatterplot smoothers on the
Hubble data in Table~\ref{MLR-hubble-tab} in Chapter~\ref{MLR-hubble-tab}.

\end{description}


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