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%\VignetteIndexEntry{Comparing Three or More Groups}
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\title{Comparing Three or More Groups}

\author{Dave Lorenz}

\maketitle

\begin{abstract}
These examples demonstrate some of the functions and statistical methods for comparing three or more groups that are available in the \texttt{smwrQW} package.
\end{abstract}

\tableofcontents

\eject
\section{Introduction}

The examples in this vignette use the TSE dataset from the \texttt{NADA} package. The examples in this vignette use the function \texttt{as.lcens} to convert those data to a form used by the functions demonstrated; the class ''lcens'' is most appropriate for these data as they are only left-censored and have only the value and an indicator of censoring. The functions demonstrated in these examples will also accept data of class ''qw.'' The R code following this paragraph gets the data and creates a column named ''TCE'' of class ''lcens.'' With the exception of the binary method, only censored data techniques are included in this vignette. Techniques that apply to single reporting limits and can require recensoring and simple substitution are not included, as the censored techniques can be used directly by the functions in \texttt{smwrQW}.


<<echo=TRUE>>=
# Load the smwrQW package
library(smwrQW)
# And the data
data(TCE, package="NADA")
# Convert the data to column TCE
TCE <- transform(TCE, TCE=as.lcens(TCEConc, censor.codes=TCECen))
@

\eject
\section{Binary Method}

The binary method simply recodes values as 0 or 1 depending on whether the value is less than or greater than or equal to a specified criterion. The recoded values can then be tabulated and tested for the equality of proportions.

The example below illustrates the recoding of values and used the \texttt{prop.test} to test for the equality of proportions between the ''Low,'' ''Medium,'' and ''High'' population density residential land use. The \texttt{code01} function returns a data frame of values with missing values removed from the input arguments. The data must be tabulated with the rows representing the groups, the first argument to \texttt{table} and the columns the counts of the 0/1 data. The printed output from the \texttt{prop.test} indicates that all of the Low density land use are 0, with decreasing percentages for Medium and High density land uses. The p-value, 0.009864, indicates that the null hypothesis of equal proportions should be rejected at the 0.05 significance level.

<<echo=TRUE>>=
# Create Density as a factor ordered Low-Medium-High
TCE <- transform(TCE, Density=factor(Density, levels=c("Low", "Medium", "High")))
# Append a column of 0/1 values to the data
TCE <- cbind(TCE, with(TCE, code01(TCE01=TCE)))
# Tabulate the 0/1 data and print it
TCETbl <- with(TCE, table(Density, TCE01))
print(TCETbl)
# And the test
prop.test(TCETbl)
@


\eject
\section{Maximum Likelihood Estimation Method} 

Comparing multiple groups using maximum likelihood estimation method extends censored regression as ANOVA extends ordinary least squares--the test is constructed by relating a censored response variable to a factor. 

An important first step in any parametric statistical analysis is to plot the data. Figure 1 shows 2 graphs, the log-transformed q-normal plot for Medium and High density land use. The Low density graph is not shown because there are very few uncensored values. Only the log-transformed is shown because the log-transformed is the most likely to produce nearly normal distributions. 

<<echo=TRUE>>=
setSweave("graph01", 3 ,6)
# Set layout for 2 graphs
AA.lo <- setLayout(num.cols=2, num.rows=1)
# Create the graphs
setGraph(1, AA.lo)
with(subset(TCE, Density=="Medium"), qqPlot(TCE,
  ytitle="TCE in Medium", yaxis.log=TRUE))
setGraph(2, AA.lo)
with(subset(TCE, Density=="High"), qqPlot(TCE,
  ytitle="TCE in High", yaxis.log=TRUE))
graphics.off()
@

\includegraphics{graph01.pdf}
\paragraph{}

\textbf{Figure 1.} Q-normal plots to check log-normal distribution assumption.

The \texttt{censReg} function is used for regression and comparing groups. It functions much like any modeling function, like \texttt{lm} in R---it constructs the model from a formula and data and has other options similar to \texttt{lm}. Its use for the censored equivalent of the two-sample t-test is shown below. Because the data are only left-censored, it uses adjusted maximum likelihood estimation (AMLE), which eliminates first-order bias from the maximum likelihood estimate. The p-value of the overall test result is 0.0005, suggesting the the null hypothesis of no difference among the three land use densities should be rejected.

<<echo=TRUE>>=
# The ANOVA analogue test:
censReg(TCE ~ Density, data=TCE, dist="lognormal")
@

\eject
\section{Nonparametric Methods} 

The nonparametric method in the \texttt{smwrQW} package for comparing three or more groups is a test that compares the flipped survival curves. The details are described by Helsel (2012).

The test that compares flipped survival curves can be used for two or more groups and is executed by the \texttt{censKSample.test}. There are two types of the test, ''Peto'' and ''log-rank''; both are described by Helsel (2012). For these data, the both types return p-value substantially less than 0.05, suggesting that the null hypothesis of no difference be rejected. 

<<echo=TRUE>>=
# The Peto type two-sample test
with(TCE, censKSample.test(TCE, Density, type="Peto"))
# The log-rank type two-sample test
with(TCE, censKSample.test(TCE, Density, type="log-rank"))
@

\eject
\section{Multiple Comparison Tests} 

Multiple comparison tests for censored data are preformed by the \texttt{censMulticomp.test} in the \texttt{smwrQW} package. It runs repeated generalized Wilcoxon tests among all of the groups and uses \texttt{p.adjust} to modify the p-value to account for the multiple comparisons. The details are described by Helsel (2012).

The \texttt{censMulticomp.test} is demonstrated in the code below, using the default method for adjusting the p-values (''holm''). The table of paired comparisons in the print out indicates that Low density is not significantly different from Medium density and that High density is significantly different from both Low and Medium density land use areas.

<<echo=TRUE>>=
# The Peto type two-sample test
with(TCE, censMulticomp.test(TCE, Density))
@

\begin{thebibliography}{9}

\bibitem{H12}
Helsel, D.R. 2012, Statistics for Censored Environmental Data Using Minitab and R: New York, Wiley, 324 p.

\bibitem{HC}
Helsel, D.R. and Cohn, T.A., 1988, Estimation of descriptive statistics for multiply censored water quality data: Water Resources Research v. 24, n. 12, p.1997--2004

\bibitem{HH}
Helsel, D.R., and Hirsch, R.M., 2002, Statistical methods in water resources: U.S. Geological Survey Techniques of Water-Resources Investigations, book 4, chap. A3, 522 p.

\bibitem{DL}
Lorenz, D.L., 2016, smwrQW--an R package for managing and analyzing water-quality data, version 1.0.0: U.S. Geological Survey Open File Report 2016-XXXX.

\bibitem{S+}
Lorenz, D.L., Ahearn, E.A., Carter, J.M., Cohn, T.A., Danchuk, W.J., Frey, J.W., Helsel, D.R., Lee, K.E., Leeth, D.C., Martin, J.D., McGuire, V.L., Neitzert, K.M., Robertson, D.M., Slack, J.R., Starn, J., Vecchia, A.V.,Wilkison, D.H., and Williamson, J.E., 2011, USGS library for S-PLUS for Windows---Release 4.0: U.S. Geological Survey Open-File Report 2011-1130.

\end{thebibliography}

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