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Learn R R is a free software environment for statistical computing and

graphics.

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Tom's site | Other Courses For more information, please contact Paul Geissler

([email protected]). Previous page

Topic 10 - Introduction to R

Graphics Contents:

graphics packages help & documentation

graphics in base package plot

scatterplot hist

lattice graphics package univariate

bivariate: trivariate:

hypervariate:

arguments multipanel displays

panel functions gplots package

Graphic Packages There are many graphics packages available for R. We will only look

at a few of them. • agsemisc - Miscellaneous plotting and utility functions - High-

featured panel functions for bwplot and xyplot, various plot

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http://www.nps.gov/

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http://www.usgs.gov/ask/

http://search.usgs.gov/

http://biology.usgs.gov/status_trends/

http://science.nature.nps.gov/im/

http://science.nature.nps.gov/im/

http://www.r-project.org/

http://www.fort.usgs.gov/brdscience/LearnR.htm

http://www.fort.usgs.gov/brdscience/LearnR10-01.htm

http://www.fort.usgs.gov/brdscience/LearnR.htm#schedule

http://www.fort.usgs.gov/brdscience/LearnR.htm#References

http://www.fort.usgs.gov/brdscience/LearnR.htm#FAQ

http://www.fort.usgs.gov/brdscience/LearnR.htm#Discussion

http://science.nature.nps.gov/im/monitor/stats/R/

http://www.fort.usgs.gov/brdscience/Courses.htm

mailto:[email protected]


http://www.fort.usgs.gov/brdscience/LearnR10-11.htm#packages

http://www.fort.usgs.gov/brdscience/LearnR10-11.htm#help

http://www.fort.usgs.gov/brdscience/LearnR10-11.htm#base

http://www.fort.usgs.gov/brdscience/LearnR10-11.htm#plot

http://www.fort.usgs.gov/brdscience/LearnR10-11.htm#scatterplot

http://www.fort.usgs.gov/brdscience/LearnR10-11.htm#hist

http://www.fort.usgs.gov/brdscience/LearnR5a.htm#lattice

http://www.fort.usgs.gov/brdscience/LearnR5a.htm#Univariate

http://www.fort.usgs.gov/brdscience/LearnR5a.htm#Bivariate

http://www.fort.usgs.gov/brdscience/LearnR5a.htm#Trivariate

http://www.fort.usgs.gov/brdscience/LearnR5a.htm#Hypervariate

http://www.fort.usgs.gov/brdscience/LearnR5a.htm#arguments

http://www.fort.usgs.gov/brdscience/LearnR5b.htm#Multipanel

http://www.fort.usgs.gov/brdscience/LearnR5b.htm#Panel

http://www.fort.usgs.gov/brdscience/LearnR5b.htm#gplots

management helpers,

some other utility functions • aplpack - Another Plot PACKage - A set of functions for drawing

some special plots: stem.leaf plots a stem and leaf plot bagplot plots

a bagplot faces plots chernoff faces spin3R enables you an inspection of a 3-dim point cloud

• base graphics - built into R • dynamicGraph - Interactive graphical tool for manipulating graphs

- • gclus - Clustering Graphics - Orders panels in scatterplot matrices

and parallel coordinate displays by some merit index. Package contains various indices of merit, ordering functions,

and enhanced versions of pairs and parcoord which color panels according to their merit level.

• ggplot - An implementation of the Grammar of Graphics in R - An implementation of the grammar of graphics in R. It combines the

advantages of both base and lattice graphics: conditioning and shared axes are handled automatically, and you can still build up a

plot step by step from multiple data sources. It also implements a

more sophisticated multidimensional conditioning system and a consistent interface to map data to aesthetic attributes. See

http://had.co.nz/ggplot/ for more information, documentation and examples.

• gplots - Various R programming tools for plotting data • gridBase - Integration of base and grid graphics

• iplots - interactive graphics for R - Interactive plots for R • lattice - Implementation of Trellis Graphics.

• latticeExtra - Extra Graphical Displays based on lattice - Generic function and standard methods for Trellis-based displays

• misc3d - Miscellaneous 3D Plots - A collection of miscellaneous 3d plots, including isos

Help & Documentation Reference Manuals: To view the reference manuals for packages go

to CRAN (Comprehansive R Archive Network, http://www.r-project.org/ ) and pick a mirror site (e.g.,

http://lib.stat.cmu.edu/R/CRAN/), select "Packages" on the left and then select the package you are interested in. The reference manual

will be available as a PDF file. These manual are more complete than the help files. Consider the R Reference Card.

Help Files: From the R Commander script window or the R console

enter the command ?lattice for example for the lattice package. The help files are very useful but somewhat terse. They are intended

more for reference than for learning about a package or command. Search: To search for help, go to the CRAN site (http://www.r-

project.org/ ) and click on "Search" on the left. Notation: I will use italics to indicate S commands to be submitted

to R.



http://lib.stat.cmu.edu/R/CRAN/

http://lib.stat.cmu.edu/R/CRAN/

http://cran.r-project.org/doc/contrib/Short-refcard.pdf



References:

• J H Maindonald, 2008, Using R for Data Analysis and Graphics, http://cran.r-project.org/doc/contrib/usingR.pdf

• Nicholas Lewin-Koh, 2010, CRAN Task View: Graphic Displays &

Dynamic Graphics & Graphic Devices & Visualization, http://cran.r-project.org/web/views/Graphics.html

• M.J. Crawley, 2007. The R Book, Wiley, Chapter 5 • Paul Murrell, 2006. R Graphics, Chapman & Hall. This is an essential

reference to customize graphics.

graphics in base package References: • An Introduction to R: Software for Statistical Modeling & Computing

by Petra Kuhnert and Bill Venables ( http://cran.r-project.org/doc/contrib/Kuhnert+Venables-R_Course_Notes.zip )

Free. Good introduction. • Michael J. Crawley 2007 (The R Book, Wiley) Chapter 5.

• Paul Murrell, 2006 ( R Graphics) Chapman & Hall • John Fox 2002. (An R and S-Plus companion to applied Regression.

Sage Publications, http://socserv.mcmaster.ca/jfox/Books/Companion/index.html)

provides an excellent introduction to the graphics in the base package, as well as writing commands and scripts.

The R file for this topic is available at ftp://ftpext.usgs.gov/pub/cr/co/fort.collins/Geissler/LearnR/LearnR10

-11.R . You can copy and paste this link into the Tinn-R open file

command. plot

Help is available by submitting the command ?plot (i.e., entering it in the R Commander script window, highlighting it and clicking submit).

Click on "par" to see a listing of the parameters. Some common parameters are:

x vector with x coordinates

y vector with x coordinates

type

"p" for points,

"l" for lines, "b" for both,

"c" for the lines part alone of "b" , "o" for both 'overplotted',

"h" for histogram like (or 'high-density') vertical lines, "s" for stair steps,

"S" for other steps, see Details below, "n" for no plotting.

main main title on top

sub sub title on bottom

xlab x axis label

http://cran.r-project.org/doc/contrib/usingR.pdf

http://cran.r-project.org/web/views/Graphics.html

http://cran.r-project.org/web/views/Graphics.html

http://cran.r-project.org/doc/contrib/Kuhnert+Venables-R_Course_Notes.zip

http://cran.r-project.org/doc/contrib/Kuhnert+Venables-R_Course_Notes.zip

http://socserv.mcmaster.ca/jfox/Books/Companion/index.html

ylab y axis label

asp aspect ratio

xlim limits on x axis, e.g., c(0,15)

ylim limits on y axis, e.g., c(0,15)

log logarithmic axes: 'x', 'y', or 'xy'

axes=F suspresses drawing of axes and the box

cex character expansion specifies the size of points, default is 1

mex margin expansion specifies the size of the margins

col color

lty line type (0=blank, 1=solid, 2=dashed, 3=dotted,

4=dotdash, 5=longdash, 6=twodash)

lwd line width, default 1

Examples: Submit from R Commander script window.

?plot data(austpop, package="DAAG")

attach(austpop) ?austpop

# default plot(year,ACT) # vector of x values, vector of y values, can also be

written in model form as plot(y ~ x) plot(ACT~year)

# you can idtintify point with the curser. Press Escape when you are

finished. #!!!! submit the following line from the R Console because identify()

locks up R Commander # To recover, press alt-ctrl-delete and select the task manager. Then

end R. plot(ACT~year); identify(ACT~year)

# options plot(year,ACT,xlim=c(1910,2000),type="b",cex=2,main="main",sub

="sub",xlab="xlab",ylab="ylab",col="red",asp=0.5,log="y") # Colors can be referenced by name

("black","red","green3","blue","cyan","magenta","yellow","gray") or by RGB values (e.g., red="#FF0000").

plot(year,ACT,col="#FF0000") # color as RGB value

pie(rep(1,8),col=palette(),labels=palette())

pie(rep(1,50),col=rainbow(50))

pie(rep(1,50),col=gray(0:50/50))

plot(year,ACT,col=rainbow(8)[3]) detach(austpop)

# PLOT TEXT data(primates, package="DAAG")

attach(primates)

plot(Bodywt, Brainwt,xlim=c(0,300),xlab="Body weight

(kg)",ylab="Brain weight (g)",main="Brian Weight Versus Body Weight")

# highlight the main line and all the indented lines and submit them

together # xlim provides more space on right for labels

text(x=Bodywt, y=Brainwt, labels=row.names(primates), pos=4) # submit together with the plot statement

# pos= 1=below, 2=left, 3=above and 4=right detach(primates)

# ADD POINTS & LINES TO A PLOT

plot(1:25,xlab="Symbol Number",ylab="",type="n") for (pch in 1:25) points(pch,pch,pch=pch) # submit with the above

line lines(1:25, type="h",lty=2) # submit with the above line

lines(1:25, type="h",lty="dotted") # alternate to above line

# RUG PLOTS

data(milk) # From the DAAG package

xyrange = range(milk) plot(four ~ one, data = milk, xlim = xyrange, ylim = xyrange, pch =

16) rug(milk$one) #submit with the above line

rug(milk$four, side = 2) #submit with the above line abline(0, 1) # draw line with intercept & slope #submit with the

above line

# IDENTIFICATION & LOCATION

attach(primates)

plot(Bodywt, Brainwt) identify(Bodywt, Brainwt ) # click with mouse to identify points. Right

click to stop. #submit with the above line text(locator(n=1),labels="Where") # click with mouse to locate label

on plot #submit with the above line detach(primates)

histogram ?hist

data(possum, package="DAAG") attach(possum)

hist(totlngth)

par(mfrow=c(1,2)) # plots more than one graph in 1 row and 2 columns

hist(totlngth) hist(totlngth,breaks=seq(70,100,5)) # breaks at 70, 75, 80, ..., 100

par(mfrow=c(1,1)) # resets

hist(totlngth,breaks=10,col="black") # 10 bins

box() # submit with above line detach(possum)


graphics.















Topic 11 - Introduction to R Graphics - continued Other Packages Each of these packages must be installed before being used for the first time. Then before each use submit a library command.

library(lattice) library(gplots)

library(car) # Companion to Applied Regression

library(sciplot) lattice Graphics Package

Lattice in the open source version of the S-Plus Trellis Graphics Package. Lattice has functions that parallel the functions in the base

graphics package, but lattice has many more options and can place the plots in a multi-panel display, like a lattice or trellis.

Documentation: • Becker, R. A. and W. S. Cleveland. 1996. S-Plus Trellis Graphics

User's Manual http://cm.bell-labs.com/stat/doc/trellis.user.pdf • Sarkar, D. Lattice user's manual

http://lib.stat.cmu.edu/R/CRAN/web/packages/lattice/lattice.pdf • Enter command ?lattice for help

This presentation will follow Becker and Cleveland (1996). The high level plotting functions are:

Univariate:

barchart - bar plots bwplot - box and whisker plots

densityplot - kernel density plots dotplot - dot plots

histogram - histograms qqmath - quantile plots against mathematical distributions

stripplot - 1-dimensional scatterplot Bivariate:

qq - quantile-quantile plot for comparing two distributions xyplot - scatter plot (and possibly a lot more)

Trivariate: levelplot - level plots (similar to image plots in R)

contourplot - contour plots cloud - 3-D scatter plots

wireframe - 3-D surfaces (similar to persp plots in R)

Hypervariate: splom - scatterplot matrix

parallel - parallel coordinate plots Miscellaneous:

rfs - residual and fitted value plot (also see oneway) tmd - Tukey Mean-Difference plot

http://www.fort.usgs.gov/brdscience/LearnR1.htm#Installing

http://cm.bell-labs.com/stat/doc/trellis.user.pdf

http://lib.stat.cmu.edu/R/CRAN/web/packages/lattice/lattice.pdf

http://www.fort.usgs.gov/brdscience/LearnR10-11a.htm#Univariate

http://www.fort.usgs.gov/brdscience/LearnR10-11a.htm#Bivariate

http://www.fort.usgs.gov/brdscience/LearnR10-11a.htm#Trivariate

http://www.fort.usgs.gov/brdscience/LearnR10-11a.htm#Hypervariate

Univariate

library(MASS) data(Cars93)

attach(Cars93)

names(Cars93) mileage.means=tapply(MPG.city,Type,mean)

# for tapply see http://cran.r-project.org/doc/contrib/Kuhnert+Venables-R_Course_Notes.zip

# Bar Charts

barplot(mileage.means,names.arg=names(mileage.means),horiz=T) # base package

barchart(names(mileage.means) ~ mileage.means) # lattice package

bargraph.CI(Type, MPG.city) # in sciplot package provides se error

bars

boxplot(MPG.city~Type) # base package

bwplot(Type ~ MPG.city) # box plot lattice package

plot(density(MPG.city)) # base package

densityplot( ~ MPG.city) # lattice package

dotchart(MPG.city,groups=Type) # base package Cars93[MPG.city>35,] # to find out which ones.

dotplot(Type ~ MPG.city) # lattice package, similar to stripplot

stripplot(Type ~ MPG.city) # lattice package, similar to dotplot

hist(MPG.city) # base package

histogram(MPG.city) # lattice package

Bivariate Scatter Plots

plot(MPG.city ~ Weight) # base package

xyplot(MPG.city ~ Weight) # lattice package

scatterplot(totlngth~age | sex, reg.line=lm, smooth=TRUE,

labels=rownames(possum), boxplots='xy', span=0.5, by.groups=TRUE, data=possum) # cars package

Quantile-Quantile Plots

qq.plot(MPG.city, distribution="norm") # base package

qqmath( ~ MPG.city, distribution=qnorm) # quantile-quantile plot

against a distribution, lattice package

qq(Type ~ MPG.city,subset=(Type=="Compact" | Type=="Small")) #

quantile-quantile plot for 2 data sets - lattice package detach(Cars93)

Trivariate x=rep(seq(-1.5,1.5,length=50),50)

y=rep(seq(-1.5,1.5,length=50),rep(50,50)) z=exp(-(x^2+y^2+x*y))

# surface is proportional to bivariate normal levelplot(z~x*y) # lattice package

contourplot(z~x*y) # lattice package

xx<-seq(-1.5,1.5,length=50) yy<-xx

zz<-matrix(nrow=50,ncol=50) for (i in 1:50) {

for (j in 1:50) { zz[i,j]<-exp(-(xx[i]^2+yy[j]^2+xx[i]*yy[j]))

} } contour(xx,yy,zz) # base package, data input is different

wireframe(z~x*y) # lattice package

cloud(MPG.city~Weight*EngineSize) # lattice package

Hypervariate cars=data.frame(Weight,EngineSize,MPG.city,Fuel.tank.capacity,Type

) splom(cars)

parallel(cars)

Arguments

formula= is the first argument, and you can omit "formula=". The general format is response variable ~ predictor variables |

conditioning variables

data= specifies the data frame so you do not need to prefix each varible name with the frame name (frame$variable). Attaching the

data frame is an alternative.

subset= specifies subset of data frame you wish to plot.

data(Cars93, package="MASS")

attach(Cars93) levels(Type)

dotplot(Type ~ MPG.city, data=Cars93,subset=Type=="Small" | Type=="Compact")

aspect= aspect ratio. "xy" sets the aspect ratio to bank to 45° which is often optimal.

data(sunspot.year, package="datasets")

xyplot(sunspot.year~ 1:289, type="l") # from 1849 to 1924

xyplot(sunspot.year~ 1:289, type="l", aspect="xy") # shows that sunspots rise more rapidly than they fall

xyplot(sunspot.year~ 1:289, type="l", aspect=1/2)

Topic 11 - Introduction to R Graphics - continued

Displays

A multipanel conditioning dispay is a three-way array of panels laid out into columns, rows, and pages, with each panel containing a

graph. In the array, we move fastest through rows and slowest through pages. The formula has the format: response variable ~

predictor variables | conditioning variable, where the conditioning variable control the panels.

layout= specifies the numbers of columns, rows, and pages.

data(barley, package="lattice") head(barley)

dotplot(site ~ yield | year * variety, data=barley)

dotplot(site ~ yield | year * variety, data=barley, layout=c(2,4,3)) This command writes three pages to the graphics device, but you can

only see the last page. After submitting this command, click on "history" in the R Console and select "Recording". Then resubmit the

command, and you will be able to use PgUp and PgDn keys to see the other pages.

reorder(factor,data,function) changes the order of a conditioning

factor to facilitate perceptions.

factor = factor to be reordered data = data upon which the reordering is to be based

function = function applied to the data to provide the reordering. barley$variety = reorder(barley$variety,barley$yield, median)

dotplot(site ~ yield | year * variety, data=barley, layout=c(2,4,3))

equal.count() is used to condition on intervals of a numeric variable. Conditioning on a numeric varible normally uses each

unique value, but with a continuous variable there may be too many

unique values for a useful plot. The equal.count() and shingle() functions can be used to define subsets of numeric conditioning

variables for plotting. equal. The resul is an object of the class shingle, so named because the bins overlap like shingles on a roof.

count() produces bins with approximately equal numbers of

observationsl. The arguments: number = number of bins

overlap = proportion of observations in common with adjactent bins.

data(ethanol, package="lattice") sE=equal.count(ethanol$E, number=9, overlap=1/4)

levels(sE) sE

xyplot(NOx ~ C | sE, data=ethanol)

shingle() also produces a shingle object for conditioning on intervals

of a numeric variable, using user supplied intervals.

endpoints=seq(min(ethanol$E), max(ethanol$E), length=6); endpoints

lev=cbind(endpoints[-6],endpoints[-1]); lev sE=shingle(ethanol$E, intervals=lev)

xyplot(NOx ~ C | sE, data=ethanol)

Titles and axis labels are the same as for plot() above, including

xlab=, ylab=, main=, sub=., xlim=, ylim=

Each of these four label arguments can also be a list. The first component of the list is a new character string for the text of the

label. The other components specify the size, font, and color of the text. The component cex specifies the size; font, a positive integer,

specifies the font; and col, a positive integer, specifies the color. xyplot(NOx ~ E, data=ethanol, xlab="Equivalence Ratio",

ylab="Oxides of Nitrogen", main=list("Air Pollution", cex=2), sub=list("Single-Cylinder

Engine", cex=1.25))

scales= controls the axis lables and tick marks. xyplot(NOx ~ E, data=ethanol, scales = list(cex = 2,x =

list(tick.number = 4),y = list(tick.number = 10)))

Strip labels You can change the strip labels by changing the factor level names.

data(barley, package="lattice") levels(barley$site)

levels(barley$site)[3]="Univ.Farm" dotplot(variety ~ yield | year * site, data=barley, layout=c(2,3,2))

The size, font, and color of the text in the strip labels can by changed by the argument par.strip.text=, a list whose components are the

parameters cex for size, font for the font, and col for the color. dotplot(variety ~ yield | year * site, data=barley, layout=c(2,3,2),

par.strip.text = list(col = 2))

Panel Functions A panel function draws the graph in each panel. You can contol the

graph by supplying arguments to the panel function or by providing your own panel function, using built in components. Panel functions

names include the names of the high level functions using the

format: panel.xyplot(). for example to specify "+" as the plot character:

data(ethanol, package="lattice") xyplot(NOx ~ E, data=ethanol)

xyplot(NOx ~ E, data=ethanol, pch="+")

# plot largest point with "M" others with "+". Not == (2 = signs) not

= is equality operator.

newPanel=function(x,y) { largest=y==max(y);

panel.points(x[!largest],y[!largest],pch="+");

panel.points(x[largest],y[largest],pch="M"); }

xyplot(NOx ~ E, data=ethanol, panel=newPanel)

# To overlay a smooth curvey on the plots, combine two panel

functions.

sE=equal.count(ethanol$E, number=9, overlap=1/4) newPanel=function(x,y) {

panel.xyplot(x,y);

panel.loess(x,y); } xyplot(NOx ~ C | sE, data=ethanol, panel=newPanel)

# You can also plot the subscripts to identify the points.

xyplot(NOx ~ C | sE, data=ethanol, panel=function(x,y,subscripts){ panel.text(x,y,subscripts, ces=0.5); })

Superposition of graph elements, such as using different symbols for groups.

data(Cars93, package="MASS") attach(Cars93)

xyplot(MPG.city ~ Weight, data=Cars93, groups=Type, auto.key=T)

# You can also use it to plot symboles for groups

levels(Type)

psymbols=c("C","L","M","P","S","V") # two S so use P (peewee) for Small

xyplot(MPG.city ~ Weight, data=Cars93, groups=Type, pch=psymbols, col="black")

# another example data(barley, package="lattice")

head(barley) dotplot(variety ~ yield | site, data=barley, groups=year, auto.key=T)

# What is wrong with the data?

gplots Package

Confidence Intervals - barplot2 I was asked how to plot error bars on a bar chart. This is a somple

question, but I could not find options to add error bars. After some

searching, I fund that a function was available in the Harrell Miscellaneous (Hmisc) package. However, it did not produce as good

charts as I would like. On the call, someone suggested that I look at the gplots package. That package has many useful plots, as well as

an extension to barplot with error bars. gplots needs to be installed by clicking on "Packages" from the R console. There a number of

steps, but you can write a function to combine them. This experience demonstrats both the weakness and strength of R. It is hard to find

where to find the function you are looking for, but with the large number of functions it is probably there somewhere. Also, it is easy

to extend R by writing functions, but it takes some knowledge of the S statistical language.

data(possum, package="DAAG") attach(possum)

head(possum)

library(gplots) conf=0.95 # set argument values so you can step through the

function by submitting individual statements resp=totlngth

cond=sex confInt = function(resp,cond,conf=0.95)

{ x=data.frame(resp,cond);

x=na.omit(x); means=tapply(x$resp,x$cond,mean);

sd=tapply(x$resp,x$cond,sd); n=tapply(x$resp,x$cond,length);

se=sd/sqrt(n); delta=se*qt((1+conf)/2,df=n-1);

data.frame(means=means,lower=means-

delta,upper=means+delta); }

ci=confInt(totlngth,sex); ci ?barplot2

barplot2(ci$means,names.arg=c("Female","Male"),plot.ci=T,ci.l=ci$lower,ci.u=ci$upper)

#note also bargraph.CI(sex,totlngth) # in sciplot package provides se error bars

plotmeans data(state)

plotmeans(state.area ~ state.region)

plotmeans(state.area ~ state.region, mean.labels=TRUE, digits=-3,

connect=FALSE)

balloonplot library(MASS)

data(Cars93) attach(Cars93)

head(Cars93) balloonplot(Type,Passengers,MPG.highway,fun=mean)

two-dimensional histogram - hits2d

library(DAAG) attach(possum)

head(possum)

hist2d(skullw,totlngth,nbins=5,xlab="skullw",ylab="totlngth")

pie(rep(1,16),col=heat.colors(16)) # key, black is zero frequency.


graphics.




Topic 12 - Generalized Additive

Models and Mixed-effects Models, Crawley (2007)

Chapters 18, and 19 Chapter 18, Generalized Additive Models Generalized Additive Models (GAMs) provide nonparametric

smoothing. They allow you to view the shape of the relationship, without prejudging the particular parametric form.

Nonparametric smooths like lowess (locally weighted scatterplot smoothing) fit a smooth curve to data by fitting simple models to

localized subsets of the data. #### nonparametric smoothers #####################################

page 612

rm(list = ls()) # removes previous variables

d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/so

aysheep.txt",header=T); attach(d); d

# Year Population Delta # Population N(t) and Delta =

log(N(t+1)/N(t))

# 1 1955 710 0.087598059

# . . .

# 44 1998 1968 -0.746367877

# 45 1999 933 NA # Delta is not defined for last

year.

m=loess(Delta~Population); summary(m) # Wilipedia

# Number of Observations: 44

# Equivalent Number of Parameters: 4.66

# Residual Standard Error: 0.2616 # residual variance = 0.2616^2 =

0.068

# Trace of smoother matrix: 5.11

# Control settings:

# normalize: TRUE

# span : 0.75

# degree : 2

# family : gaussian

# surface : interpolate cell = 0.2

xv=seq(min(Population),max(Population),1);

yv=predict(m,data.frame(Population=xv))

plot(Population,Delta); lines(xv,yv)











http://www.fort.usgs.gov/brdscience/LearnR10-11b.htm

http://en.wikipedia.org/wiki/Generalized_additive_model

http://en.wikipedia.org/wiki/Lowess

http://en.wikipedia.org/wiki/Local_regression

# Looks like a step function, so use tree to find the split.

library(tree)

m1=tree(Delta~Population); print(m1)

# node), split, n, deviance, yval

# * denotes terminal node

# 1) root 44 5.2870 0.006208

# 2) Population < 1289.5 25 0.8596 0.226500

# 4) Population < 1009.5 13 0.2364 0.277600 *

# 5) Population > 1009.5 12 0.5525 0.171200

# 10) Population < 1059.5 5 0.1631 0.072120 *

# 11) Population > 1059.5 7 0.3053 0.241900 *

# 3) Population > 1289.5 19 1.6180 -0.283700

# 6) Population < 1459 9 0.7917 -0.349500 *

# 7) Population > 1459 10 0.7519 -0.224400 *

th=1289.5; m2=aov(Delta~(Population>th)); summary(m2)

# Df Sum Sq Mean Sq F value Pr(>F)

# Population > 1289.5 1 2.80977 2.80977 47.636 2.008e-08 ***

# Residuals 42 2.47736 0.05898 # loess RMS = 0.068

m=tapply(Delta[!is.na(Delta)],(Population[!is.na(Delta)]>th),mean); m

# FALSE TRUE

# 0.2265084 -0.2836616

plot(Population,Delta); lines(xv,yv)

lines (c(min(Population),th),c(m[1],m[1]),lty=2)

lines (c(th,max(Population)),c(m[2],m[2]),lty=2)

lines (c(th,th),c(m[1],m[2]),lty=2)

#### generalized additive models ############################## page

614


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/oz

one.data.txt",header=T); attach(d); d

# rad temp wind ozone

# 1 190 67 7.4 41

# . . .

pairs(d,panel=function(x,y){ points(x,y);lines(lowess(x,y)) })

library(mgcv)

m1=gam(ozone~s(rad)+s(temp)+s(wind)); summary(m1)

# Family: gaussian

# Link function: identity

# Parametric coefficients:

# Estimate Std. Error t value Pr(>|t|)

# (Intercept) 42.10 1.66 25.36 <2e-16 ***

# Approximate significance of smooth terms:

# edf Ref.df F p-value

# s(rad) 2.763 3.263 4.106 0.00699 **

# s(temp) 3.841 4.341 12.785 7.31e-09 ***

# s(wind) 2.918 3.418 14.687 1.21e-08 ***

# R-sq.(adj) = 0.724 Deviance explained = 74.8%

# GCV score = 338 Scale est. = 305.96 n = 111

m2=gam(ozone~s(temp)+s(wind)); summary(m2) # without s(rad)

anova(m1,m2,test="F")

# Analysis of Deviance Table

# Model 1: ozone ~ s(rad) + s(temp) + s(wind)

# Model 2: ozone ~ s(temp) + s(wind)

# Resid. Df Resid. Dev Df Deviance F Pr(>F)

# 1 100.4779 30742

# 2 102.8450 34885 -2.3672 -4142 5.7192 0.002696 ** # s(rad)

should stay in the model

m3=gam(ozone~s(rad)+s(temp)+s(wind)+s(rad,temp)+s(rad,wind)+s(temp,win

d)); summary(m3)



# (Intercept) 42.099 1.286 32.73 <2e-16 ***



# s(rad) 1.000e+00 1.500 0.001 0.996495

# s(temp) 1.000e+00 1.500 0.010 0.971831

# s(wind) 5.222e+00 5.722 2.115 0.063953 .

# s(rad,temp) 7.963e+00 8.463 1.219 0.298032

# s(rad,wind) 4.144e-10 0.500 1.21e-11 0.998548

# s(temp,wind) 1.830e+01 18.801 2.935 0.000478 ***

m4=gam(ozone~s(rad)+s(temp)+s(wind)+s(temp,wind)); summary(m4)



# (Intercept) 42.099 1.361 30.92 <2e-16 ***



# s(rad) 1.389 1.889 4.669 0.013368 *

# s(temp) 1.000 1.500 0.000122 0.998982

# s(wind) 5.613 6.113 2.658 0.020054 *

# s(temp,wind) 18.246 18.746 3.210 0.000131 ***

anova(m3,m4,test="F")


# Model 1: ozone ~ s(rad) + s(temp) + s(wind) + s(rad, temp) + s(rad,

wind) + s(temp, wind)

# Model 2: ozone ~ s(rad) + s(temp) + s(wind) + s(temp, wind)

# Resid. Df Resid. Dev Df Deviance F Pr(>F)

# 1 76.5127 14051.9

# 2 83.7516 17229.7 -7.2389 -3177.8 2.3903 0.02746 *

# Indicates that some other interactions are important, but we will

stay with Crawley's model.

par(mfrow=c(2,2));plot(m4,residuals=T,pch=16);par(mfrow=c(1,1)) #

Rress return in R console (not graph) after each plot.

#### an example with strongly humped data

####################################### page 620


library(SemiPar)

data(ethanol); d=ethanol; attach(d); d

# NOx C E # C = compression ratio of engine, E =

equivalance ratio (richness of mixture)

# 1 3.741 12.0 0.907

# . . .

pairs(d,panel=function(x,y){ points(x,y);lines(lowess(x,y)) })

m=gam(NOx~s(E)+C); summary(m) # C looks like a straight line, so use a

parametric fit.


# # Estimate Std. Error t value Pr(>|t|)

# (Intercept) 1.291342 0.088898 14.526 < 2e-16 ***

# C 0.055345 0.007062 7.837 1.88e-11 ***


# # edf Ref.df F p-value

# s(E) 7.553 8.053 219.6 <2e-16 ***


# GCV score = 0.067206 Scale est. = 0.05991 n = 88

par(mfrow=c(1,2));

plot.gam(m,residuals=T,pch=16,all.terms=T);par(mfrow=c(1,1)) # Rress

return in R console (not graph) after each plot.

coplot(NOx~C|E,panel=panel.smooth) # The order of the panel plots is

from the bottom and from the left.

CE=C*E; m2=gam(NOx~s(E)+s(CE)); summary(m2)



# (Intercept) 1.95738 0.02126 92.07 <2e-16 ***



# s(E) 7.636 8.136 282.52 < 2e-16 ***

# s(CE) 4.261 4.761 27.71 2.02e-15 ***



par(mfrow=c(1,2));

plot.gam(m,residuals=T,pch=16,all.terms=T);par(mfrow=c(1,1)) # Rress


#### generalized additive models with binary data

########################### page 623


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/is

olation.txt",header=T); attach(d); d

# incidence area isolation # incidence =1 if island occupied by a

bird spedies, =0 if not.

# 1 1 7.928 3.317 # area of island (km2). isolation is

distance from mainland (km)

# 2 0 1.925 7.554

# . . .

m1=gam(incidence~s(area)+s(isolation),binomial); summary(m1)


# Estimate Std. Error z value Pr(>|z|)

# (Intercept) 1.6371 0.9898 1.654 0.0981 .


# edf Ref.df Chi.sq p-value

# s(area) 2.429 2.929 3.57 0.3009

# s(isolation) 1.000 1.500 7.48 0.0132 *


# UBRE score = -0.32096 Scale est. = 1 n = 50

par(mfrow=c(1,2));

plot.gam(m1,residuals=T,pch=16,all.terms=T);par(mfrow=c(1,1)) # Rress


# Although area in not significant it appears to have a strong effect.

m2=gam(incidence~s(isolation),binomial); anova(m1,m2,test="Chi") #

without area


# Model 1: incidence ~ s(area) + s(isolation)

# Model 2: incidence ~ s(isolation)

# Resid. Df Resid. Dev Df Deviance P(>|Chi|)

# 1 45.5710 25.094

# 2 48.0000 36.640 -2.4290 -11.546 0.005 # significant -

leave s(area) in model

# Note that s(area) was not significant by itself, but made a

significant contribution to the model!

m3=gam(incidence~area+s(isolation),binomial); summary(m3) # fit

parametric area


# Estimate Std. Error z value Pr(>|z|)

# (Intercept) -1.3928 0.9002 -1.547 0.1218

# area 0.5807 0.2478 2.344 0.0191 * # highly

significant as parametric, but not as smooth!


# edf Ref.df Chi.sq p-value

# s(isolation) 1 1.5 8.275 0.0087 **


# UBRE score = -0.31196 Scale est. = 1 n = 50

Chapter 19, Mixed-Effects Models

Fixed Effects Random Effects

All levels of interest are

studied.

Levels are a random sample from a

larger population.

Influence only the response mean.

Influence only the response variance.

Informative factor levels. Levels not informative.

Examples Examples

age group litter

sex sample plot

treatment individual animals

J. Neter, M. H. Kutner, C. J. Nachtsheim and W. Wasserman (1996, Applied Linear Statistical Models, Irwin, page 959) have pointed out

that for example if a company has five stores and all stores are

included in the sample that stores would be a fixed effect. However, if the company had hundreds of stores and a random sample of five

stores were included in the sample, then stores would be a random effect. Thus it is the nature of the sample and the inferences one

wants to draw that determine whether an effect is fixed or random. Assumptions:

• Within-group defined by the fixed effect, errors are independent with mean 0 and variance σ2.

• Within-group defined by the fixed effect, errors are independent of the random effects.

• Random effects are normally distributed with mean 0 and covariance matrix Ψ

• The random effects are independent in different groups. • The covariance matrix does not depend on the group defined by the

fixed effects.

Replicates: • must be independent

• must not be repeated measurements or time series (temporal

pseudoreplication) • must not be grouped together in one place (spatial

pseudoreplication)

When you have a hierarchical model (pseudoreplication), you can: • Remove the pseudoreplication by analyzing the mean or other

function of the dependent observations. • Analyze each group with pseudoreplication separately.

• Use mixed effects models or time series analysis. #### split plots ##################################### page 632


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/sp

lityield.txt",header=T); attach(d); d

# yield block irrigation density fertilizer

# 1 90 A control low N

# . . .

# block: Blocks (whole fields) are the largest areas

# irrigation: Blocks were split in half and irrigation treatments were

applies to half of each field.

# density: Irrigation plots were split into thirds and seeds sown at

three densities (low, medium and high).

# fertilizer: Density plots were split into thirds and fertilizer

treatments applied (N, P and NP).

library(nlme) # You may need to install nlme

?lme

m=lme(yield~irrigation*density*fertilizer,random=~1|block/irrigation/d

ensity); summary(m)

# Linear mixed-effects model fit by REML

# Data: NULL

# AIC BIC logLik

# 481.6212 525.3789 -218.8106

# Random effects:

# Formula: ~1 | block

# (Intercept)

# StdDev: 0.0006600339

# Formula: ~1 | irrigation %in% block

# (Intercept)

# StdDev: 1.982461

# Formula: ~1 | density %in% irrigation %in% block

# (Intercept) Residual

# StdDev: 6.975554 9.292805

# Fixed effects: yield ~ irrigation * density * fertilizer

# Value

Std.Error DF t-value p-value

# (Intercept) 80.50

5.893741 36 13.658558 0.0000

# irrigation[T.irrigated] 31.75

8.335008 3 3.809234 0.0318

# density[T.low] 5.50

8.216282 12 0.669403 0.5159

# density[T.medium] 14.75

8.216282 12 1.795216 0.0978

# fertilizer[T.NP] 5.50

6.571005 36 0.837010 0.4081

# fertilizer[T.P] 4.50

6.571005 36 0.684827 0.4978

# irrigation[T.irrigated]:density[T.low] -39.00

11.619577 12 -3.356404 0.0057

# irrigation[T.irrigated]:density[T.medium] -22.25

11.619577 12 -1.914872 0.0796

# irrigation[T.irrigated]:fertilizer[T.NP] 13.00

9.292805 36 1.398932 0.1704

# irrigation[T.irrigated]:fertilizer[T.P] 5.50

9.292805 36 0.591856 0.5576

# density[T.low]:fertilizer[T.NP] 3.25

9.292805 36 0.349733 0.7286

# density[T.medium]:fertilizer[T.NP] -6.75

9.292805 36 -0.726368 0.4723

# density[T.low]:fertilizer[T.P] -5.25

9.292805 36 -0.564953 0.5756

# density[T.medium]:fertilizer[T.P] -5.50

9.292805 36 -0.591856 0.5576

# irrigation[T.irrigated]:density[T.low]:fertilizer[T.NP] 7.75

13.142011 36 0.589712 0.5591

# irrigation[T.irrigated]:density[T.medium]:fertilizer[T.NP] 3.75

13.142011 36 0.285344 0.7770

# irrigation[T.irrigated]:density[T.low]:fertilizer[T.P] 20.00

13.142011 36 1.521837 0.1368

# irrigation[T.irrigated]:density[T.medium]:fertilizer[T.P] 4.00

13.142011 36 0.304367 0.7626

# Correlation: omitted because they are too wide

# Standardized Within-Group Residuals:

# Min Q1 Med Q3 Max

# -2.12362041 -0.37841447 -0.03057733 0.41805004 1.90433189


# Number of Groups:

# block irrigation %in%

block density %in% irrigation %in% block

# 4

8 24

## use maximum likelihood (ML) instead of default restricted maximum

likelihood (REML) so we can use anova

## Restricted maximum likelihood (REML) allows for degrees of freedom

to be used up in estimating fixed effects,

## unlike maximum likelihood (ML).

## Thus variance components are estimated without being affected by

the fixed effects.

## REML estimators are less sensitive to outliers than ML estimators.

m1=lme(yield~irrigation*density*fertilizer,random=~1|block/irrigation/

density,method="ML");summary(m1);anova(m1)

# summary gives the same results

# numDF denDF F-value p-value

# (Intercept) 1 36 2674.6630 <.0001

# irrigation 1 3 30.9211 0.0115

# density 2 12 3.7842 0.0532

# fertilizer 2 36 11.4493 0.0001

# irrigation:density 2 12 5.9119 0.0163

# irrigation:fertilizer 2 36 5.5204 0.0081

# density:fertilizer 4 36 0.8826 0.4841

# irrigation:density:fertilizer 4 36 0.6795 0.6107

m2=lme(yield~(irrigation+density+fertilizer)^2,random=~1|block/irrigat

ion/density,method="ML")# remove higher order interactions

anova(m2)


# (Intercept) 1 40 2872.7394 <.0001

# irrigation 1 3 33.2110 0.0104

# density 2 12 4.0645 0.0449

# fertilizer 2 40 11.4341 0.0001



# density:fertilizer 4 40 0.8815 0.4837

anova(m1,m2)

# Model df AIC BIC logLik Test L.Ratio p-value

# m1 1 22 573.5108 623.5974 -264.7554

# m2 2 18 569.0046 609.9845 -266.5023 1 vs 2 3.493788 0.4788 # m2

better

m3=update(m2,~.-density:fertilizer); anova(m3)


# (Intercept) 1 44 3070.8771 <.0001

# irrigation 1 3 35.5016 0.0095

# density 2 12 4.3448 0.0381

# fertilizer 2 44 11.2013 0.0001



anova(m2,m3)


# m2 1 18 569.0046 609.9845 -266.5023

# m3 2 14 565.1933 597.0667 -268.5967 1 vs 2 4.188774 0.3811

## m3 not significantly different, and gives lower AIC and BIC so use

m3

m4=update(m3,~.-irrigation:fertilizer); anova(m4)


# (Intercept) 1 46 3169.893 <.0001

# irrigation 1 3 36.646 0.0090

# density 2 12 4.485 0.0351

# fertilizer 2 46 9.167 0.0004


anova(m3,m4)


# m3 1 14 565.1933 597.0667 -268.5967

# m4 2 12 572.3373 599.6573 -274.1687 1 vs 2 11.14397 0.0038

## m4 is significantly different and gives a larger AIC and BIC, so

keep m3

m5=update(m3,~.-irrigation:density); anova(m5)


# (Intercept) 1 44 2138.9678 <.0001

# irrigation 1 3 24.7281 0.0156

# density 2 14 2.6264 0.1075

# fertilizer 2 44 11.5626 0.0001


anova(m3,m5)


# m3 1 14 565.1933 597.0667 -268.5967

# m5 2 12 572.9022 600.2221 -274.4511 1 vs 2 11.70883 0.0029

## m5 is significantly different and gives a larger AIC and BIC, so

keep m3

summary(m3); anova(m3)

# Linear mixed-effects model fit by maximum likelihood

# Data: NULL

# AIC BIC logLik

# 565.1933 597.0667 -268.5967

# Random effects:

# Formula: ~1 | block

# (Intercept)

# StdDev: 0.0005260787

# Formula: ~1 | irrigation %in% block

# (Intercept)

# StdDev: 1.716888

# Formula: ~1 | density %in% irrigation %in% block


# StdDev: 5.722413 8.718327

# Fixed effects: yield ~ irrigation + density + fertilizer +

irrigation:density + irrigation:fertilizer

# Value Std.Error DF

t-value p-value

# (Intercept) 82.08333 4.756285 44

17.257867 0.0000

# irrigation[T.irrigated] 27.80556 6.726403 3

4.133793 0.0257

# density[T.low] 4.83333 5.807347 12

0.832279 0.4215

# density[T.medium] 10.66667 5.807347 12

1.836754 0.0911

# fertilizer[T.NP] 4.33333 3.835552 44

1.129781 0.2647 # fertilizer p= 0.0001 in anova below

# fertilizer[T.P] 0.91667 3.835552 44

0.238992 0.8122

# irrigation[T.irrigated]:density[T.low] -29.75000 8.212829 12 -

3.622382 0.0035

# irrigation[T.irrigated]:density[T.medium] -19.66667 8.212829 12 -

2.394628 0.0338

# irrigation[T.irrigated]:fertilizer[T.NP] 16.83333 5.424290 44

3.103325 0.0033

# irrigation[T.irrigated]:fertilizer[T.P] 13.50000 5.424290 44

2.488805 0.0167

# Correlation:

# (Intr) ir[T.] dnsty[T.l]

dnsty[T.m] f[T.NP f[T.P] irrgtn[T.rrgtd]:dnsty[T.l]

# irrigation[T.irrigated] -0.707

# density[T.low] -0.610 0.432

# density[T.medium] -0.610 0.432 0.500

# fertilizer[T.NP] -0.403 0.285 0.000

0.000

# fertilizer[T.P] -0.403 0.285 0.000

0.000 0.500

# irrigation[T.irrigated]:density[T.low] 0.432 -0.610 -0.707 -

0.354 0.000 0.000

# irrigation[T.irrigated]:density[T.medium] 0.432 -0.610 -0.354 -

0.707 0.000 0.000 0.500

# irrigation[T.irrigated]:fertilizer[T.NP] 0.285 -0.403 0.000

0.000 -0.707 -0.354 0.000

# irrigation[T.irrigated]:fertilizer[T.P] 0.285 -0.403 0.000

0.000 -0.354 -0.707 0.000

# irrgtn[T.rrgtd]:dnsty[T.m]

i[T.]:[T.N

# irrigation[T.irrigated]

# density[T.low]

# density[T.medium]

# fertilizer[T.NP]

# fertilizer[T.P]

# irrigation[T.irrigated]:density[T.low]

# irrigation[T.irrigated]:density[T.medium]

# irrigation[T.irrigated]:fertilizer[T.NP] 0.000

# irrigation[T.irrigated]:fertilizer[T.P] 0.000

0.500


# Min Q1 Med Q3 Max

# -2.58166961 -0.51480885 0.07893406 0.60157076 2.19570825


# Number of Groups:

# block irrigation %in%

block density %in% irrigation %in% block

# 4

8 24

anova(m3)


# (Intercept) 1 44 3070.8771 <.0001 # note

differences in denDF

# irrigation 1 3 35.5016 0.0095 # due to split

plots

# density 2 12 4.3448 0.0381

# fertilizer 2 44 11.2013 0.0001



## In R console set history > recording after plot

plot(m3);plot(m3,yield~fitted(.)); qqnorm(m3,~resid(.)|block)

## When an experiment is balanced and there are no missing values,

aov() can be used as in Topic 6.

## If it is not balanced, then lme() must be used.

#### hierarchical sampling and variance components

##################### page 638


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/hr

e.txt",header=T); attach(d); d

# subject town district street family gender replicate

# 1 0.66198060 A d1 s1 f1 male 1

# . . .

## epidemiological study of childhood diseases, with blood samples

taken from

## individual children, families, streets, districts and towns at

different spatial scales.

m1=lme(subject~1,random=~1|town/district/street/family/gender);

summary(m1)


# Data: NULL

# AIC BIC logLik

# 3351.294 3383.339 -1668.647

# Random effects:

# Formula: ~1 | town

# (Intercept)

# StdDev: 1.150604

# Formula: ~1 | district %in% town

# (Intercept)

# StdDev: 1.131932

# Formula: ~1 | street %in% district %in% town

# (Intercept)

# StdDev: 1.489864

http://www.fort.usgs.gov/brdscience/learnR10-06.htm

# Formula: ~1 | family %in% street %in% district %in% town

# (Intercept)

# StdDev: 1.923191

# Formula: ~1 | gender %in% family %in% street %in% district %in%

town


# StdDev: 3.917264 0.9245321

# Fixed effects: subject ~ 1

# Value Std.Error DF t-value p-value

# (Intercept) 8.010941 0.6719753 360 11.92148 0


# Min Q1 Med Q3 Max

# -2.64600654 -0.47626815 -0.06009422 0.47531635 2.35647504


# Number of Groups:

# town

district %in% town

# 5

15

# street %in% district %in% town

family %in% street %in% district %in% town

# 60

180

# gender %in% family %in% street %in% district %in% town

# 360

## variance components from StdDev above. I can't get them from m1 or

s1 using statements.

v=c(1.150604, 1.131932, 1.489864, 1.923191, 3.917264, 0.9245321)^2

names(v)=c("town","district","street","family","gender","residual");v

# town district street family gender residual

# 1.3238896 1.2812701 2.2196947 3.6986636 15.3449572 0.8547596

v/sum(v)*100 # percent

# town district street family gender residual

# 5.354840 5.182453 8.978173 14.960274 62.066948 3.457313

## Restricted maximum likelihood (REML) allows for degrees of freedom

to be used up in estimating fixed effects,

## unlike maximum likelihood (ML).

## Thus variance components are estimated without being affected by

the fixed effects.

## REML estimators are less sensitive to outliers than ML estimators.

#### using lmer

library(lme4)

?lmer

m2=lmer(subject~1 + (1|town/district/street/family/gender));

summary(m2)

# Linear mixed model fit by REML

# Formula: subject ~ 1 + (1 | town/district/street/family/gender)

# AIC BIC logLik deviance REMLdev

# 3351 3383 -1669 3338 3337

# Random effects: ## gives variance

components ##

# Groups Name Variance

Std.Dev.

# gender:(family:(street:(district:town))) (Intercept) 15.34509

3.91728

# family:(street:(district:town)) (Intercept) 3.69852

1.92315

# street:(district:town) (Intercept) 2.21970

1.48987

# district:town (Intercept) 1.28123

1.13191

# town (Intercept) 1.32386

1.15059

# Residual 0.85476

0.92453

# Number of obs: 720, groups:

gender:(family:(street:(district:town))), 360;

family:(street:(district:town)), 180;

# street:(district:town), 60; district:town, 15; town, 5

# Fixed effects:

# Estimate Std. Error t value

# (Intercept) 8.0109 0.6718 11.93

#### model simplification in hierarchical sampling ################

page 640

## Test the effect of leaving out the effect of towns.

## You need to recode factor levels because, for example, town A in

district d1 is not the same

## town as town A in district d2 or d3. Combine town and district

names to make the town identity unique.

## This step would not be necessary if towns had unique names.

newDistrict=factor(paste(town,district,sep="")); levels(newDistrict)

# [1] "Ad1" "Ad2" "Ad3" "Bd1" "Bd2" "Bd3" "Cd1" "Cd2" "Cd3" "Dd1"

"Dd2" "Dd3" "Ed1" "Ed2" "Ed3"

m3=lme(subject~1,random=~1|newDistrict/street/family/gender);

anova(m1,m3)


# m1 1 7 3351.294 3383.339 -1668.647

# m3 2 6 3350.524 3377.991 -1669.262 1 vs 2 1.229803 0.2674

# m3 is not significantly different and has a lower AIC and BIC so use

m3

## now remove streets

newStreet=factor(paste(newDistrict,street,sep="")); levels(newStreet)

# [1] "Ad1s1" "Ad1s2" "Ad1s3" "Ad1s4" "Ad2s1" "Ad2s2" "Ad2s3" "Ad2s4"

"Ad3s1" "Ad3s2" "Ad3s3" "Ad3s4" "Bd1s1" "Bd1s2"

# [15] "Bd1s3" "Bd1s4" "Bd2s1" "Bd2s2" "Bd2s3" "Bd2s4" "Bd3s1" "Bd3s2"

"Bd3s3" "Bd3s4" "Cd1s1" "Cd1s2" "Cd1s3" "Cd1s4"

# [29] "Cd2s1" "Cd2s2" "Cd2s3" "Cd2s4" "Cd3s1" "Cd3s2" "Cd3s3" "Cd3s4"

"Dd1s1" "Dd1s2" "Dd1s3" "Dd1s4" "Dd2s1" "Dd2s2"

# [43] "Dd2s3" "Dd2s4" "Dd3s1" "Dd3s2" "Dd3s3" "Dd3s4" "Ed1s1" "Ed1s2"

"Ed1s3" "Ed1s4" "Ed2s1" "Ed2s2" "Ed2s3" "Ed2s4"

# [57] "Ed3s1" "Ed3s2" "Ed3s3" "Ed3s4"

m4=lme(subject~1,random=~1|newStreet/family/gender); anova(m3,m4)


# m3 1 6 3350.524 3377.991 -1669.262

# m4 2 5 3354.084 3376.973 -1672.042 1 vs 2 5.559587 0.0184

## Now there is a significance difference between the models and AIC

increases (stay with m3), but BIC decreases (use m4).

#### mixed-effects models with temporal pseudoreplication (repeated

measurements) ########################## page 641


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/fe

rtilizer.txt",header=T); attach(d); d

# root week plant fertilizer

# 1 1.30 2 ID1 added

# . . .

library(nlme);library(lattice)

gd=groupedData(root~week|plant,outer=~fertilizer,d);gd

# Grouped Data: root ~ week | plant

# root week plant fertilizer

# 1 1.30 2 ID1 added

# . . .

## Several modeling and plotting functions can use the formula stored

with a groupedData object to construct default plots and models.

plot(gd);plot(gd,outer=T)

m=lme(root~fertilizer,random=~week|plant);summary(m)


# Data: NULL

# AIC BIC logLik

# 171.0236 183.3863 -79.51181

# Random effects:

# Formula: ~week | plant

# Structure: General positive-definite, Log-Cholesky parametrization

# StdDev Corr

# (Intercept) 2.8639831 (Intr)

# week 0.9369412 -0.999

# Residual 0.4966308

# Fixed effects: root ~ fertilizer


# (Intercept) 2.799709 0.1438367 48 19.464499 0e+00

# fertilizer[T.control] -1.039383 0.2034158 10 -5.109645 5e-04

# Correlation:

# (Intr)

# fertilizer[T.control] -0.707


# Min Q1 Med Q3 Max

# -1.9928119 -0.6586835 -0.1004301 0.6949713 2.0225381


# Number of Groups: 12

anova(m)


# (Intercept) 1 48 502.5360 <.0001

# fertilizer 1 10 26.1085 5e-04

## Two treatments (fertilizers) and 12 plants (6 in each treatment)

## so there are 2(6-1)=10 df for testing fertilizer.

## Without using mixed models, you can test the data for week 10

without repeated measures.

m1=aov(root~fertilizer,subset=(week==10)); summary(m1)


# fertilizer 1 4.9408 4.9408 11.486 0.006897 **

# Residuals 10 4.3017 0.4302

## Mixed models uses more data and has more power to detect

differences.

#### time series analyses in mixed models

################################ page 645


library(nlme);library(lattice)

data(Ovary);d=Ovary; attach(d); d

# Grouped Data: follicles ~ Time | Mare # already a groupedData

object

# Mare Time follicles

# 1 1 -0.13636360 20

# . . .

plot(d) # mares 1 through 11, mare 4 has least, and mare 8 the most

follicles.

m1=lme(follicles~sin(2*pi*Time)+cos(2*pi*Time),random=~1|Mare);summary

(m1)

## No allowance for correlation structure.


# Data: NULL

# AIC BIC logLik

# 1669.360 1687.962 -829.6802

# Random effects:

# Formula: ~1 | Mare


# StdDev: 3.041344 3.400466

# Fixed effects: follicles ~ sin(2 * pi * Time) + cos(2 * pi * Time)


# (Intercept) 12.182244 0.9390009 295 12.973623 0.0000

# sin(2 * pi * Time) -3.339612 0.2894013 295 -11.539727 0.0000

# cos(2 * pi * Time) -0.862422 0.2715987 295 -3.175353 0.0017

# Correlation:

# (Intr) s(*p*T

# sin(2 * pi * Time) 0.00

# cos(2 * pi * Time) -0.06 0.00


# Min Q1 Med Q3 Max

# -2.4500138 -0.6721813 -0.1349236 0.5922957 3.5506618


# Number of Groups: 11 # Mares

plot(ACF(m1),alpha=0.05) # ACF is autocorrelation function

## highly signification autocorrelation at lags 1 and 2; and

marginally significant ones at lags 3 and 4

m2=update(m1,correlation=corARMA(q=2)); anova(m1,m2) # moving average

model with first two lags.


# m1 1 5 1669.360 1687.962 -829.6802

# m2 2 7 1574.895 1600.937 -780.4476 1 vs 2 98.4652 <.0001

## m2 has lower AIC and BIC and so is preferred.

m3=update(m2,correlation=corAR1()); anova(m2,m3) # First order

autoregressive model


# m2 1 7 1574.895 1600.937 -780.4476

# m3 2 6 1562.447 1584.769 -775.2233 1 vs 2 10.44840 0.0012

## p value is very different from text but AIC and BIC are the same.

## m3 has lower AIC and BIC and so is preferred.

## Time series analysis is covered in Chapter 22.

plot(m3,resid(.,type="p")~fitted(.)|Mare); qqnorm(m3,~resid(.)|Mare)

#### random effects in designed experiments ################# page

648


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ra

ts.txt",header=T); attach(d); d

# Glycogen Treatment Rat Liver

# 1 131 1 1 1

# 2 130 1 1 1

# 3 131 1 1 2

# . . .

# Each rat's liver is cut into 3 pieces, and 2 readings made on each

piece.

# Rats are numbered 1 and 2 within each treatment.

Treatment=factor(Treatment); levels(Treatment)

# [1] "1" "2" "3"

Liver=factor(Liver); levels(Liver)

# [1] "1" "2" "3"

Rat=factor(Rat); levels(Rat)

# [1] "1" "2"

library(lme4)

m=lmer(Glycogen~Treatment+(1|Treatment/Rat/Liver)); summary(m)

## Note that Treatment is both a fixed effect and one level of random

effect hierarchy

# Linear mixed model fit by REML

# Formula: Glycogen ~ Treatment + (1 | Treatment/Rat/Liver)

# AIC BIC logLik deviance REMLdev

# 233.6 244.7 -109.8 234.9 219.6

# Random effects:

# Groups Name Variance Std.Dev.

# Liver:(Rat:Treatment) (Intercept) 14.1668 3.7639

# Rat:Treatment (Intercept) 36.0651 6.0054

# Treatment (Intercept) 4.7035 2.1688

# Residual 21.1666 4.6007

# Number of obs: 36, groups: Liver:(Rat:Treatment), 18; Rat:Treatment,

6; Treatment, 3

# Fixed effects:

# Estimate Std. Error t value

# (Intercept) 140.500 5.182 27.112

# Treatment[T.2] 10.500 7.329 1.433

# Treatment[T.3] -5.333 7.329 -0.728

# Correlation of Fixed Effects:

# (Intr) T[T.2]

#Trtmnt[T.2] -0.707

#Trtmnt[T.3] -0.707 0.500

anova(m)

# Analysis of Variance Table

# Df Sum Sq Mean Sq F value

# Treatment 2 101.943 50.971 2.4081

v=c(14.1668,36.0651,21.1666); # Treatment is a fixed effect

names(v)=c("liver","rats","readings"); v

# liver rats readings

# 14.1668 36.0651 21.1666

100*v/sum(v) # percent

# liver rats readings

# 19.84187 50.51241 29.64572

#### regression in mixed-effects models ####################### page

650


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/fa

rms.txt",header=T); attach(d); d

# N size farm

# 1 18.18014 96.48147 1

# 2 20.47343 98.64003 1

# . . .

## regression of plant size against local point measurement of soil

nitrogen (N)

## at five places within each of 24 farms.

plot(size~N,pch=16,col=farm)

## fit a separate regressions for each farm

m=lmList(size~N|farm,d); c=coef(m); c

# (Intercept) N

# 1 67.46260 1.5153805

# 2 118.52443 -0.5550273

# 3 91.58055 0.5551292

# 4 87.92259 0.9212662

# 5 92.12023 0.5380276

# 6 97.01996 0.3845431

# 7 68.52117 0.9339957

# 8 91.54383 0.8220482

# 9 92.04667 0.8842662

# 10 85.08964 1.4676459

# 11 114.93449 -0.2689370

# 12 82.56263 1.0138488

# 13 78.60940 0.1324811

# 14 80.97221 0.6551149

# 15 84.85382 0.9809902

# 16 87.12280 0.3699154

# 17 52.31711 1.7555136

# 18 83.40400 0.8715070

# 19 88.91675 0.2043755

# 20 93.08216 0.8567066

# 21 90.24868 0.7830692

# 22 78.30970 1.1441291

# 23 59.88093 0.9536750

# 24 89.07963 0.1091016

range(c[,"(Intercept)"])

# [1] 52.31711 118.52443

range(c[,"N"])

# [1] -0.5550273 1.7555136

## Now fit a single mixed model, taking into account the differences

between farms

## in their contributions to the variance.

m1=lme(size~1,random=~N|farm); summary(m1);


# Data: NULL

# AIC BIC logLik

# 643.4823 657.3779 -316.7411

# Random effects:

# Formula: ~N | farm


# StdDev Corr

# (Intercept) 12.3857402 (Intr)

# N 0.6215039 -0.735


# Fixed effects: size ~ 1


# (Intercept) 97.95195 1.810111 96 54.11378 0

#


# Min Q1 Med Q3 Max

# -2.19364865 -0.56777008 0.04701894 0.64022046 2.01476221



v=c(12.3857402,0.6215039,1.9826698)^2;

names(v)=c("(Intercept)","N","Residual"); v

# (Intercept) N Residual

# 153.4065603 0.3862671 3.9309795

100*v/sum(v) # percent of variance

# (Intercept) N Residual

# 97.2627806 0.2449009 2.4923184

c1=coef(m1); c1

# (Intercept) N

# 1 85.98140 0.574205232

# 2 104.67366 -0.045401462

# 3 95.03442 0.331080899

# 4 98.62679 0.463579764

# 5 95.00270 0.407906188

# 6 99.82294 0.207203693

# 7 85.57345 0.285520337

# 8 96.09461 0.520896445

# 9 95.22186 0.672262902

# 10 93.14157 1.017995666

# 11 108.27200 0.015213757

# 12 87.36387 0.689406363

# 13 80.83933 0.003616946

# 14 89.84309 0.306402229

# 15 93.37050 0.636778651

# 16 92.10914 0.145772142

# 17 94.93395 0.084935465

# 18 85.90160 0.709943233

# 19 92.00628 0.052485978

# 20 95.26296 0.738029377

# 21 93.35069 0.591151930

# 22 87.66161 0.673119211

# 23 70.57827 0.432993864

# 24 90.29151 0.036747095

par(mfrow=c(1,2));

plot(c[,"(Intercept)"],c1[,"(Intercept)"],main="Intercept",xlab="separ

ate regressions",ylab="mixed model")

abline(0,1)

plot(c[,"N"],c1[,"N"],main="Slope",xlab="separate

regressions",ylab="mixed model")

abline(0,1)

par(mfrow=c(1,1));

farm=factor(farm)

## N and farm as fixed effects

## Use ML to compare models with anova()

m2=lme(size~N*farm,random=~1|farm,method="ML") # full model

m3=lme(size~N+farm,random=~1|farm,method="ML") # common slope,

different intercepts

m4=lme(size~N,random=~1|farm,method="ML") # common slope and

intercept

m5=lme(size~1,random=~1|farm,method="ML") # no effect of N

anova(m2,m3,m4,m5)


# m2 1 50 542.9035 682.2781 -221.4518

# m3 2 27 524.2971 599.5594 -235.1486 1 vs 2 27.39359 0.2396

# m4 3 4 614.3769 625.5269 -303.1885 2 vs 3 136.07981 <.0001

# m5 4 3 658.0058 666.3683 -326.0029 3 vs 4 45.62892 <.0001

## m3 has lowest AIC and BIC and is not significantly different from

m2

summary(m3)

# Linear mixed-effects model fit by maximum likelihood

# Data: NULL

# AIC BIC logLik

# 524.2971 599.5594 -235.1486

# Random effects:

# Formula: ~1 | farm


# StdDev: 3.939764e-05 1.717093

# Fixed effects: size ~ N + farm


# (Intercept) 82.89803 2.056033 95 40.31941 0

# N 0.72923 0.095045 95 7.67243 0

# farm[T.2] 0.89264 1.409247 0 0.63342 NaN

# farm[T.3] 5.98197 1.281886 0 4.66654 NaN

# farm[T.4] 9.55083 1.276565 0 7.48166 NaN

# farm[T.5] 4.93723 1.248755 0 3.95372 NaN

# farm[T.6] 8.56774 1.265568 0 6.76988 NaN

# farm[T.7] -9.02108 1.368892 0 -6.59006 NaN

# farm[T.8] 10.06828 1.287429 0 7.82046 NaN

# farm[T.9] 11.52867 1.286639 0 8.96030 NaN

# farm[T.10] 15.59936 1.228585 0 12.69701 NaN

# farm[T.11] 9.04516 1.262585 0 7.16400 NaN

# farm[T.12] 3.87177 1.304774 0 2.96739 NaN

# farm[T.13] -13.73477 1.272983 0 -10.78944 NaN

# farm[T.14] -3.80255 1.334955 0 -2.84845 NaN

# farm[T.15] 8.22376 1.319036 0 6.23467 NaN

# farm[T.16] -3.70231 1.242163 0 -2.98053 NaN

# farm[T.17] -4.41222 1.341786 0 -3.28832 NaN

# farm[T.18] 2.68927 1.286822 0 2.08985 NaN

# farm[T.19] -4.45777 1.220937 0 -3.65110 NaN

# farm[T.20] 12.62388 1.221451 0 10.33515 NaN

# farm[T.21] 8.23361 1.258682 0 6.54146 NaN

# farm[T.22] 3.64534 1.220706 0 2.98626 NaN

# farm[T.23] -18.50683 1.221327 0 -15.15305 NaN

# farm[T.24] -3.52487 1.277863 0 -2.75841 NaN

# Correlation: omitted


# Min Q1 Med Q3 Max

# -3.15278548 -0.68522977 0.03259033 0.74036886 2.49262804



anova(m3)


# (Intercept) 1 95 320006.5 # .0001

# N 1 95 79.3 # .0001

# farm 23 0 94.6 NaN

#### using lm() without random effects

m6=lm(size~N*farm) # full model

m7=lm(size~N+farm) # common slope, different intercepts

m8=lm(size~N) # common slope and intercept

m9=lm(size~1) # no effect of N

anova(m6,m7,m8,m9)


# Model 1: size ~ N * farm

# Model 2: size ~ N + farm

# Model 3: size ~ N

# Model 4: size ~ 1

# Res.Df RSS Df Sum of Sq F Pr(>F)

# 1 72 281.6

# 2 95 353.8 -23 -72.2 0.8028 0.717

# 3 118 8454.9 -23 -8101.1 90.0575 < 2.2e-16 ***

# 4 119 8750.4 -1 -295.5 75.5424 7.846e-13 ***

## same conclusion - common slope but different intercepts

#### lme() is vastly superior to lm() when there is unequal

replication.

#### error plots from a hierarchical analysis

################################# page 657


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/hr

e.txt",header=T); attach(d); d

# subject town district street family gender replicate

# 1 0.66198060 A d1 s1 f1 male 1

# . . .

## epidemiological study of childhood diseases, with blood samples

taken from

## individual children, families, streets, districts and towns at

different spatial scales.

library(nlme); library(lattice); trellis.par.set(col.whitebg())

gd=groupedData(subject~gender|town/district/street/family/gender/repli

cate,outer=~gender,data=d)

## More comprehensive model checking is available with grouped data.

m=lme(subject~gender,random=~1|town/district/street/family/gender,data

=gd); anova(m)


# (Intercept) 1 360 142.11589 <.0001

# gender 1 179 23.98874 <.0001

plot(m,gender~resid(.))

plot(m,resid(.,type="p")~fitted(.)|town)

qqnorm(m,~resid(.)|gender); qqnorm(m,~resid(.)|town)

qqnorm(m,~ranef(.,level=3)) # street

plot(m,subject~fitted(.))


graphics.




Topic 13 - Non-linear

Regression, Tree Methods, and Time Series Analysis, Crawley

(2007) Chapters 20, 21 and 22 Contents:

Non-linear Regression Tree Methods

Time Series Analysis The R code is available at

ftp://ftpext.usgs.gov/pub/cr/co/fort.collins/Geissler/LearnR/LearnR10-13.R

Chapter 20, Non-linear Regression Michael Crawley, 2007,The R Book, Chapter 20.

See the text for descriptions. You can copy and paste the statements below into the R Commander script window and execute them.

Anything after # on a line is a comment. I have added annotations as comments and shown the output as comments following each

command. Non-linear regression is used for relationships cannot be transformed

so that they are linear in the parameters. Many curved lines such as polynomials can be transformed to be

linear in the parameters and then fit by lm(). Example of jaw bone length (y) as a function of deer age (x).

Theory suggests the relationship y=a - b*exp(-cx) rm(list = ls()) # removes previous variables

d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ja

ws.txt",header=T); attach(d); d

# age bone # age deer with jaw bone length

# 1 0.000000 0.00000

# 2 5.112000 20.22000

# . . .

plot(age,bone)












http://www.fort.usgs.gov/brdscience/LearnR10-13.htm#20






http://en.wikipedia.org/wiki/Non-linear_regression

# We need initial estimates of the parameters to start the search for

the best estimates.

# From the graph a ~= 120 asymptote

# Intercept ~= 10 so b = 120 - 10 = 110

# The curve raises most steeply at y ~= 40 where x=5

# c = -log((a-y)/b)/x

-log((120-40)/110)/5 # 0.06369075

m1=nls(bone ~ a - b * exp(-c * age), start=list(a=120,b=110,c=0.064));

summary(m1)

## need to explicitly enter equation and provide starting values from

graph.

# Formula: bone ~ a - b * exp(-c * age)

# Parameters:


# a 115.2528 2.9139 39.55 < 2e-16 ***

# b 118.6875 7.8925 15.04 < 2e-16 ***

# c 0.1235 0.0171 7.22 2.44e-09 ***

# Residual standard error: 13.21 on 51 degrees of freedom

# Number of iterations to convergence: 5

# Achieved convergence tolerance: 2.383e-06

## Try starting with naive estimates

m0=nls(bone~a-b * exp(-c * age), start=list(a=1,b=1,c=1))

# ERROR: Missing value or an infinity produced when evaluating the

model

## You need reasonable starting values.

m2=nls(bone ~ a * (1-exp(-c * age)), start=list(a=120,c=0.064));

anova(m1,m2)


# Model 1: bone ~ a - b * exp(-c * age)

# Model 2: bone ~ a * (1 - exp(-c * age))

# Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)

# 1 51 8897.3

# 2 52 8929.1 -1 -31.8 0.1825 0.671

## m2 is not significantly different, so use simplifier m2 model.

summary(m2)

# Formula: bone ~ a * (1 - exp(-c * age))

# Parameters:


# a 115.58056 2.84365 40.645 < 2e-16 ***

# c 0.11882 0.01233 9.635 3.69e-13 ***




xv=seq(0,50,0.1); yv=predict(m1,list(age=xv))

plot(bone~age);lines(xv,yv)

## Try a Michaelis-Menten curve

m3=nls(bone~a*age/(1+b*age),start=list(a=8,b=0.08)); summary(m3)

# Formula: bone ~ a * age/(1 + b * age)

# Parameters:


# a 18.72539 2.52587 7.413 1.09e-09 ***

# b 0.13596 0.02339 5.814 3.79e-07 ***




## Residual standard error is slightly larger.

yv3=predict(m3,list(age=xv))

plot(bone~age);lines(xv,yv);lines(xv,yv3,lty=2)

#### generalized additive models ################################ page

665

## GAMs are useful when you don't know the functional form of the

relationship.


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/hu

mp.txt",header=T); attach(d); d

# y x

# 1 3.741 0.907

# . . .

library(mgcv)

m=gam(y~s(x)) # s(x) is the smooth of x

xv=seq(min(x),max(x),0.001); yv=predict(m,list(x=xv))

plot(x,y); lines(xv,yv)

http://www.fort.usgs.gov/brdscience/learnR10-12.htm

summary(m)

# Family: gaussian

# Link function: identity

# Formula:

# y ~ s(x)



# (Intercept) 1.95737 0.03446 56.8 <2e-16 ***



# s(x) 7.452 7.952 123.3 <2e-16 ***



#### grouped data for non-linear estimation

############################## page 667


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/re

action.txt",header=T); attach(d); d

# strain enzyme rate # reaction rates as a function of enzyme

concentration for 10 bacteria strains

# 1 A 0.0 11.91119

# . . .

plot(enzyme,rate,pch=as.numeric(strain))

library(nlme)

## fit separate regressions for each strain

m1=nlsList(rate~c+a*enzyme/(1+b*enzyme)|strain,data=d,start=c(a=20,b=0

.25,c=10)); summary(m1)

# Call:

# Model: rate ~ c + a * enzyme/(1 + b * enzyme) | strain

# Data: d

# Coefficients:

# a


# A 51.79746 4.093791 12.652686 1.943005e-06

# B 26.05893 3.063474 8.506335 2.800344e-05

# C 51.86774 5.086678 10.196781 7.842353e-05

# D 94.46245 5.813975 16.247482 2.973297e-06

# E 37.50984 4.840749 7.748767 6.462817e-06

# b


# A 0.4238572 0.04971637 8.525506 2.728565e-05

# B 0.2802433 0.05761532 4.864041 9.173722e-04

# C 0.5584898 0.07412453 7.534479 5.150210e-04

# D 0.6560539 0.05207361 12.598587 1.634553e-05

# E 0.5253479 0.09354863 5.615774 5.412405e-05

# c


# A 11.46498 1.194155 9.600916 1.244488e-05

# B 11.73312 1.120452 10.471780 7.049415e-06

# C 10.53219 1.254928 8.392663 2.671651e-04

# D 10.40964 1.294447 8.041768 2.909373e-04

# E 10.30139 1.240664 8.303123 4.059887e-06


gd=groupedData(rate~enzyme|strain,data=d)

plot(gd)

m2=nlme(rate~c+a*enzyme/(1+b*enzyme),fixed=a+b+c~1,random=a+b+c~1|stra

in,data=gd,start=c(a=20,b=0.25,c=10)); summary(m2)

# Nonlinear mixed-effects model fit by maximum likelihood

# Model: rate ~ c + a * enzyme/(1 + b * enzyme)

# Data: gd

# AIC BIC logLik

# 253.4806 272.6008 -116.7403

# Random effects:

# Formula: list(a ~ 1, b ~ 1, c ~ 1)

# Level: strain


# StdDev Corr

# a 22.9153193 a b

# b 0.1132367 0.876

# c 0.4229784 -0.537 -0.875


# Fixed effects: a + b + c ~ 1

##### Fixed effects are means of parameter values #####


# a 51.59881 10.741441 43 4.803714 0

# b 0.47665 0.058786 43 8.108293 0

# c 10.98537 0.556448 43 19.741930 0

# Correlation:

# a b

# b 0.843

# c -0.314 -0.543


# Min Q1 Med Q3 Max

# -1.7918584 -0.6563331 0.0568836 0.7426879 2.0272251



coef(m2) # same order as plot - ranked by asymptote

# a b c

# E 34.09031 0.4533430 10.81731

# B 28.01280 0.3238698 11.54809

# C 49.63874 0.5193754 10.67196

# A 53.20483 0.4426258 11.23607

# D 93.04738 0.6440399 10.65341

plot(augPred(m2))

## This plot shows the model fit, whereas the last one connected the

dots.

## Non-linear time series models (temporal pseudoreplication) page 671


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/no

nlinear.txt",header=T); attach(d); head(d)

# Growth curves, diam is response variable, time indicates a repeated

measure on each dish.

gd=groupedData(diam~time|dish,data=d)

m1=nlme(diam~a+b*time/(1+c*time),fixed=a+b+c~1,data=gd,correlation=cor

AR1(),start=c(a=0.5,b=5,c=0.5))

summary(m1)


# Model: diam ~ a + b * time/(1 + c * time)

# Data: gd

# AIC BIC logLik

# 129.7694 158.3157 -53.88469

# Random effects:

# Formula: list(a ~ 1, b ~ 1, c ~ 1)

# Level: dish


# StdDev Corr

# a 0.1014472 a b

# b 1.2060357 -0.557

# c 0.1095790 -0.958 0.772


#

# Correlation Structure: AR(1)

# Formula: ~1 | dish

# Parameter estimate(s):

# Phi

# -0.03344977



# a 1.288262 0.1086390 88 11.85819 0

# b 5.215250 0.4741948 88 10.99812 0

# c 0.498221 0.0450643 88 11.05578 0

# Correlation:

# a b

# b -0.506

# c -0.542 0.823


# Min Q1 Med Q3 Max

# -1.74222962 -0.64713559 -0.03349711 0.70298828 2.24686664



plot(augPred(m1))

range(time) # 0 10

xv=seq(0,10,0.1)

plot(time,diam,pch=as.numeric(dish),col=as.numeric(dish))

sapply(1:9,function(i)lines(xv,predict(m1,list(dish=i,time=xv)),lty=2)

)

#### self-starting functions ################################ page

674

## Models will fail if the starting values are too far off.

## The most used self-starting functions are:

## SSasymp asymptotic regression model y=a-

b*exp(-c*x)

## SSasympOff asymptotic regression model with an offset y=a-

b*exp(-c*(x-d))

## SSasympOrig asymptotic regression model through the origin

y=a*(1-exp(-b*x))

## SSbiexp biexponential model

y=a*exp(b*x)-c*exp(-d*x)

## SSfol first order compartment model

y=k*exp(-exp(a)*x)-exp(-exp(b)*x)

## SSfpl four-parameter logistic model

y=a+(b-a)/(1+exp(c-x)/d)

## SSgompertz Gompertz growth model

y=a*exp(b*exp(-c*x))

## SSlogis logistic model

y=a/(1+b*exp(-c*x))

## SSmicmen Michaelis-Menten model

y=a*b/(b+x)

## SSweibull Weibull growth model y=a-

b*exp(c*x^d)

## self-startimg Michaelis-Menten model

## y=a*b/(b+x) a=asymptote, b=x for which y=a/2


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/mm

.txt",header=T); attach(d); d

# conc rate

# 1 0.02 76

# . . .

m=nls(rate~SSmicmen(conc,a,b)); summary(m)

# Formula: rate ~ SSmicmen(conc, a, b)

# Parameters:


# a 2.127e+02 6.947e+00 30.615 3.24e-11 ***

# b 6.412e-02 8.281e-03 7.743 1.57e-05 ***




xv=seq(min(conc),max(conc),0.01); yv=predict(m,list(conc=xv))

plot(rate~conc); lines(xv,yv)

## Non-linear time series models (temporal pseudoreplication) page 671


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/no

nlinear.txt",header=T); attach(d); head(d)

# Growth curves, diam is response variable, time indicates a repeated

measure on each dish.

gd=groupedData(diam~time|dish,data=d)

m1=nlme(diam~a+b*time/(1+c*time),fixed=a+b+c~1,data=gd,correlation=cor

AR1(),start=c(a=0.5,b=5,c=0.5))

summary(m1)

plot(augPred(m1))

range(time) # 0 10

xv=seq(0,10,0.1)

plot(time,diam,pch=as.numeric(dish),col=as.numeric(dish))

sapply(1:9,function(i)lines(xv,predict(m1,list(dish=i,time=xv)),lty=2)

)


# Model: diam ~ a + b * time/(1 + c * time)

# Data: gd

# AIC BIC logLik

# 129.7694 158.3157 -53.88469

# Random effects:

# Formula: list(a ~ 1, b ~ 1, c ~ 1)

# Level: dish


# StdDev Corr

# a 0.1014472 a b

# b 1.2060357 -0.557

# c 0.1095790 -0.958 0.772


# Correlation Structure: AR(1)

# Formula: ~1 | dish

# Parameter estimate(s):

# Phi

# -0.03344977



# a 1.288262 0.1086390 88 11.85819 0

# b 5.215250 0.4741948 88 10.99812 0

# c 0.498221 0.0450643 88 11.05578 0

# Correlation:

# a b

# b -0.506

# c -0.542 0.823


# Min Q1 Med Q3 Max

# -1.74222962 -0.64713559 -0.03349711 0.70298828 2.24686664



#### self-starting functions ################################ page

674

## Models will fail if the starting values are too far off.

## The most used self-starting functions are:

## SSasymp asymptotic regression model y=a-

b*exp(-c*x)

## SSasympOff asymptotic regression model with an offset y=a-

b*exp(-c*(x-d))

## SSasympOrig asymptotic regression model through the origin

y=a*(1-exp(-b*x))

## SSbiexp biexponential model

y=a*exp(b*x)-c*exp(-d*x)

## SSfol first order compartment model

y=k*exp(-exp(a)*x)-exp(-exp(b)*x)

## SSfpl four-parameter logistic model

y=a+(b-a)/(1+exp(c-x)/d)

## SSgompertz Gompertz growth model

y=a*exp(b*exp(-c*x))

## SSlogis logistic model

y=a/(1+b*exp(-c*x))

## SSmicmen Michaelis-Menten model

y=a*b/(b+x)

## SSweibull Weibull growth model y=a-

b*exp(c*x^d)

## self-startimg Michaelis-Menten model

## y=a*b/(b+x) a=asymptote, b=x for which y=a/2


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/mm

.txt",header=T); attach(d); d

# conc rate

# 1 0.02 76

# . . .

m=nls(rate~SSmicmen(conc,a,b)); summary(m)

# Formula: rate ~ SSmicmen(conc, a, b)

# Parameters:


# a 2.127e+02 6.947e+00 30.615 3.24e-11 ***

# b 6.412e-02 8.281e-03 7.743 1.57e-05 ***




xv=seq(min(conc),max(conc),0.01); yv=predict(m,list(conc=xv))

plot(rate~conc); lines(xv,yv)

## self-startimg asymptotic exponential model

## y=a-b*exp(-c*x) a=asymptote, b=a-intercept, c=rate constant


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ja

ws.txt",header=T); attach(d); d

# age deer using length of jaw bones

# Example of jaw bone length (y) as a function of deer age (x).

# age bone

# 1 0.000000 0.00000

# 2 5.112000 20.22000

# . . .

m=nls(bone~SSasymp(age,a,b,c)); summary(m)

# Formula: bone ~ SSasymp(age, a, b, c)

# Parameters:


# a 115.2527 2.9139 39.553 <2e-16 ***

# b -3.4348 8.1961 -0.419 0.677

# c -2.0915 0.1385 -15.101 <2e-16 ***




xv=seq(min(age),max(age),0.02); yv=predict(m,list(age=xv))

plot(bone~age); lines(xv,yv)

par(mfrow=c(2,2)); plot(profile(m)); par(mfrow=c(1,1))

## Investigates the behavior of the objective function (log-likelihood

for nls) p. 676

## near the solution (fitted value).

## self-startimg logistic model p. 676

## y=a/(1+b*exp(-c*x))


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ss

logistic.txt",header=T); attach(d); head(d)

# density concentration

# 1 0.017 0.04882812

# . . .

m=nls(density~SSlogis(log(concentration),a,b,c)); summary(m)

# Formula: density ~ SSlogis(log(concentration), a, b, c)

# Parameters:


# a 2.34518 0.07815 30.01 2.17e-13 ***

# b 1.48309 0.08135 18.23 1.22e-10 ***

# c 1.04146 0.03227 32.27 8.51e-14 ***




xv=seq(log(min(concentration)),log(max(concentration)),0.01);

yv=predict(m,list(concentration=exp(xv)))

plot(log(concentration),density); lines(xv,yv)

## self-startimg four-parameter logistic model p. 678

## y=a+(b-a)/(1+exp(c-x)/d) has lower as well as upper asymptote.


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ch

icks.txt",header=T); attach(d); d

# weight Time

# 1 42 0

# . . .

m=nls(weight~SSfpl(Time,a,b,c,d)); summary(m)

# Formula: weight ~ SSfpl(Time, a, b, c, d)

# Parameters:


# a 27.453 6.601 4.159 0.003169 **

# b 348.971 57.899 6.027 0.000314 ***

# c 19.391 2.194 8.836 2.12e-05 ***

# d 6.673 1.002 6.662 0.000159 ***




xv=seq(min(Time),max(Time),(max(Time)-min(Time))/100);

yv=predict(m,list(Time=xv))

plot(weight~Time); lines(xv,yv)

## self-startimg Weibull growth function p. 679

## y=a-b*exp(c*x^d)


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/we

ibull.growth.txt",header=T); attach(d); head(d)

# weight time

# 1 49 2

# . . .

m=nls(weight~SSweibull(time,a,b,c,d)); summary(m)

# Formula: weight ~ SSweibull(time, a, b, c, d)

# Parameters:


# a 158.5012 1.1769 134.67 3.28e-13 ***

# b 110.9971 2.6330 42.16 1.10e-09 ***

# c -5.9934 0.3733 -16.05 8.83e-07 ***

# d 2.6461 0.1613 16.41 7.62e-07 ***




xv=seq(min(time),max(time),(max(time)-min(time))/100);

yv=predict(m,list(time=xv))

plot(weight~time); lines(xv,yv)

## self-startimg first-order compartment function p. 680

## y=k*exp(-exp(a)*x) - exp(-exp(b)*x)

## where k=Dose * x * exp(a+b+c)/(exp(b)-exp(a)) x=Time below.


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/fo

l.txt",header=T); attach(d); head(d)

# Wt Dose Time conc

# 1 79.6 4.02 0.00 0.74

# . . .

m=nls(conc~SSfol(Dose,Time,a,b,c)); summary(m)

# Formula: conc ~ SSfol(Dose, Time, a, b, c)

# Parameters:


# a -2.9196 0.1709 -17.085 1.40e-07 ***

# b 0.5752 0.1728 3.328 0.0104 *

# c -3.9159 0.1273 -30.768 1.35e-09 ***




xv=seq(min(Time),max(Time),(max(Time)-min(Time))/100);

yv=predict(m,list(Time=xv))

plot(conc~Time); lines(xv,yv)

Chapter 21, Tree Models Tree models are computationally intensive methods that are used in situations where there are many explanatory variables, and we would

like guidance about which of them to include in the model. Tree models are particularly good at tasks that might be regarded as

appropriate for multivariate statistics, such as classification problems.

Their advantages are that they: • are very simple

• are excellent for initial data inspection • give a very clear picture of the structure of the data

• provide a highly intuitive insight into the interactions

The model is fit using binary recursive partitioning, where the data are successively split so that at any node, the split that maximally

distinguishes the response variable is selected. Each explanatory variable is assessed turn and the variable explaining the greatest

amount of deviance in the response variable is selected. Deviance is defined as D=Σ(yi-u|i|)

2 where u|i| is the mean of all the response

values assigned to node i and the sum of squares add over all nodes. The value of any split is defined as the reduction in this residual sum

of squares. The procedure is:

• Select a threshold value of an explanatory variable. • Calculate the mean value of the response variable above and below

this threshold value of the explanatory variable. • Use these two means u|i| to calculate the deviance D.

• Loop through all possible values of the threshold for all the explanatory variable.

• Determine which value of the threshold gives the lowest deviance.

• Split the data into high and low subsets on the basis of the

threshold for this response variable. • Repeat the procedure on each subset of the data on either side of

the threshold.

• Keep going until no further reduction id deviance is obtained, or there are too few data points to merit further subdivision.

If the explanatory variable are categorical, it is a classification tree.

If the explanatory variable are continuous, it is a regression tree. regression trees library(tree)


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/Po

llute.txt",header=T); attach(d); head(d)

# Pollution Temp Industry Population Wind Rain Wet.days

# 1 24 61.5 368 497 9.1 48.34 115

# . . .

m1=tree(Pollution~.,d); plot(m1); text(m1)

m1



# 1) root 41 22040 30.05

# 2) Industry < 748 36 11260 24.92

# 4) Population < 190 7 4096 43.43 *

# 5) Population > 190 29 4187 20.45

# 10) Wet.days < 108 11 96 12.00 *

# 11) Wet.days > 108 18 2826 25.61

# 22) Temp < 59.35 13 1895 29.69

# 44) Wind < 9.65 8 1213 33.88 *

# 45) Wind > 9.65 5 318 23.00 *

# 23) Temp > 59.35 5 152 15.00 *

# 3) Industry > 748 5 3002 67.00 *

## in the line "2) Industry < 748 36 11260 24.92"

## "2)" labels the node

## "Industry < 748" is the split criterion

## "36" is the number of cases going into the split

## "11260" is the deviance at the node

## "24.92" is the mean of the response variable at the node.

## * indicates a terminal node (estimate)

d1=data.frame(d,node=as.numeric(m1$where),predicted=predict(m1));

attach(d1)

d1[order(node),]

## Nodes are numbered differently than above.

## Node 3: Industry < 748 and Population < 190

# Pollution Temp Industry Population Wind Rain Wet.days node

predicted

# 4 28 51.0 137 176 8.7 15.17 89 3

43.42857

# 6 46 47.6 44 116 8.8 33.36 135 3

43.42857

# 18 13 61.0 91 132 8.2 48.52 100 3

43.42857

# 19 31 55.2 35 71 6.6 40.75 148 3

43.42857

# 23 56 49.1 412 158 9.0 43.37 127 3

43.42857

# 27 36 54.0 80 80 9.0 40.25 114 3

43.42857

# 41 94 50.0 343 179 10.6 42.75 125 3

43.42857

## Node 5: Industry < 748 and Population > 190 and Wet.days < 108


predicted

# 7 9 66.2 641 844 10.9 35.94 78 5

12.00000

# 13 14 51.5 181 347 10.9 30.18 98 5

12.00000

# 15 17 51.9 454 515 9.0 12.95 86 5

12.00000

# 20 12 56.7 453 716 8.7 20.66 67 5

12.00000

# 21 10 70.3 213 582 6.0 7.05 36 5

12.00000

# 24 10 68.9 721 1233 10.8 48.19 103 5

12.00000

# 26 8 56.6 125 277 12.7 30.58 82 5

12.00000

# 36 10 61.6 337 624 9.2 49.10 105 5

12.00000

# 38 14 54.5 381 507 10.0 37.00 99 5

12.00000

# 39 17 49.0 104 201 11.2 30.85 103 5

12.00000

# 40 11 56.8 46 244 8.9 7.77 58 5

12.00000

## Node 8: Industry < 748 and Population > 190 and Wet.days > 108

## and Temp < 59.35 and Wind < 9.65


predicted

# 2 30 55.6 291 593 8.3 43.11 123 8

33.87500

# 9 26 57.8 197 299 7.6 42.59 115 8

33.87500

# 10 61 50.4 347 520 9.4 36.22 147 8

33.87500

# 11 29 57.3 434 757 9.3 38.98 111 8

33.87500

# 16 23 54.0 462 453 7.1 39.04 132 8

33.87500

# 17 47 55.0 625 905 9.6 41.31 111 8

33.87500

# 29 29 51.1 379 531 9.4 38.79 164 8

33.87500

# 34 26 51.5 266 540 8.6 37.01 134 8

33.87500


## and Temp < 59.35 and Wind > 9.65


predicted

# 12 28 52.3 361 746 9.7 38.74 121 9

23.00000

# 28 16 45.7 569 717 11.8 29.07 123 9

23.00000

# 30 29 43.5 669 744 10.6 25.94 137 9

23.00000

# 35 31 59.3 96 308 10.6 44.68 116 9

23.00000

# 37 11 47.1 391 463 12.4 36.11 166 9

23.00000


## and Temp > 59.35


predicted

# 1 24 61.5 368 497 9.1 48.34 115 10

15.00000

# 5 14 68.4 136 529 8.8 54.47 116 10

15.00000

# 14 18 59.4 275 448 7.9 46.00 119 10

15.00000

# 32 9 68.3 204 361 8.4 56.77 113 10

15.00000

# 33 10 75.5 207 335 9.0 59.80 128 10

15.00000

## Node 11: Industry > 748


predicted

# 3 56 55.9 775 622 9.5 35.89 105 11

67.00000

# 8 35 49.9 1064 1513 10.1 30.96 129 11

67.00000

# 22 110 50.6 3344 3369 10.4 34.44 122 11

67.00000

# 25 69 54.6 1692 1950 9.6 39.93 115 11

67.00000

# 31 65 49.7 1007 751 10.9 34.99 155 11

67.00000

plot(node,Pollution)

#### tree models as regressions

########################################## page 689

d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ca

r.test.frame.txt",header=T); attach(d); head(d)

# Price Country Reliability Mileage Type Weight Disp. HP

# 1 8895 USA 4 33 Small 2560 97 113

# . . .

plot(Weight,Mileage)

m2=tree(Mileage~Weight); plot(m2); text(m2)

xv=seq(min(Weight),max(Weight),10); yv=predict(m2,list(Weight=xv))

plot(Weight,Mileage); lines(xv,yv)

#### model simplification ############################### page 690

## Simplification is a comprise between fit and explanatory power,

## because a model with perfect fit would have as many parameters and

data points.

## prune.tree() returns a nested sequence of sub-trees by recursively

'snipping' off the least

## important splits, based on a cost-complexity measure.

pt=prune.tree(m1); pt # pollution example

# $size # number of terminal nodes

# [1] 6 5 4 3 2 1

# $dev # total deviance of each subtree

# [1] 8876.589 9240.484 10019.992 11284.887 14262.750 22037.902

# $k # cpst-complexity pruning parameter.

# [1] -Inf 363.8942 779.5085 1264.8946 2977.8633 7775.1524

# $method

# [1] "deviance"

# attr(,"class")

# [1] "prune" "tree.sequence"

m3=prune.tree(m1,best=4); m3 # best tree with 4 terminal nodes



# 1) root 41 22040 30.05

# 2) Industry < 748 36 11260 24.92

# 4) Population < 190 7 4096 43.43 *

# 5) Population > 190 29 4187 20.45

# 10) Wet.days < 108 11 96 12.00 *

# 11) Wet.days > 108 18 2826 25.61 *

# 3) Industry > 748 5 3002 67.00 *

plot(m3); text(m3)

#### classification trees with categorical explanatory variables

########### page 693

## Tree models are very useful for developing efficient and effective

taxonomic keys.

## First split where it explains the most variability.


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ep

ilobium.txt",header=T); attach(d); d

# species stigma stem.hairs glandular.hairs seeds pappilose

stolons petals base

# 1 hirsutum lobed spreading absent none uniform

absent <9mm rounded

# 2 parviflorum lobed spreading absent none uniform

absent >10mm rounded

# 3 montanum lobed spreading present none uniform


# 4 lanceolatum lobed spreading present none uniform

absent >10mm cuneate

# 5 tetragonum clavate appressed present none uniform


# 6 obscurum clavate appressed present none uniform

stolons >10mm rounded

# 7 roseum clavate spreading present none uniform

absent >10mm cuneate

# 8 palustre clavate spreading present appendage uniform


# 9 ciliatum clavate spreading present appendage ridged


m=tree(species~.,d);m # only one node because only 1 entry for each

species

# node), split, n, deviance, yval, (yprob)


# 1) root 9 39.55 ciliatum ( 0.1111 0.1111 0.1111 0.1111 0.1111 0.1111

0.1111 0.1111 0.1111 ) *

m=tree(species~.,d,minsize=2,mindev=1e-6);m



# 1) root 9 39.550 ciliatum ( 0.1111 0.1111 0.1111 0.1111 0.1111

0.1111 0.1111 0.1111 0.1111 )

# 2) stigma: clavate 5 16.090 ciliatum ( 0.2000 0.0000 0.0000

0.0000 0.2000 0.2000 0.0000 0.2000 0.2000 )

# 4) stem.hairs: appressed 2 2.773 obscurum ( 0.0000 0.0000

0.0000 0.0000 0.5000 0.0000 0.0000 0.0000 0.5000 )

# 8) stolons: absent 1 0.000 tetragonum ( 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 ) *

# 9) stolons: stolons 1 0.000 obscurum ( 0.0000 0.0000 0.0000

0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 ) *

# 5) stem.hairs: spreading 3 6.592 ciliatum ( 0.3333 0.0000

0.0000 0.0000 0.0000 0.3333 0.0000 0.3333 0.0000 )

# 10) seeds: appendage 2 2.773 ciliatum ( 0.5000 0.0000 0.0000

0.0000 0.0000 0.5000 0.0000 0.0000 0.0000 )

# 20) pappilose: ridged 1 0.000 ciliatum ( 1.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ) *

# 21) pappilose: uniform 1 0.000 palustre ( 0.0000 0.0000

0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 ) *

# 11) seeds: none 1 0.000 roseum ( 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 1.0000 0.0000 ) *

# 3) stigma: lobed 4 11.090 hirsutum ( 0.0000 0.2500 0.2500 0.2500

0.0000 0.0000 0.2500 0.0000 0.0000 )

# 6) glandular.hairs: absent 2 2.773 hirsutum ( 0.0000 0.5000

0.0000 0.0000 0.0000 0.0000 0.5000 0.0000 0.0000 )

# 12) petals: <0mm 1 0.000 parviflorum ( 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 ) *

# 13) petals: >9mm 1 0.000 hirsutum ( 0.0000 1.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ) *

# 7) glandular.hairs: present 2 2.773 lanceolatum ( 0.0000

0.0000 0.5000 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 )

# 14) base: cuneate 1 0.000 lanceolatum ( 0.0000 0.0000 1.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ) *

# 15) base: rounded 1 0.000 montanum ( 0.0000 0.0000 0.0000

1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ) *

plot(m); text(m)

#### classification trees for replicated data ######################

page 695


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ta

xonomy.txt",header=T); attach(d); head(d)

## Construct the key for the four taxa with the smallest error rate.

# Taxon Petals Internode Sepal Bract Petiole Leaf

Fruit

# 1 I 5.621498 29.48060 2.462107 18.20341 11.279097 1.128033

7.876151

# . . .

m1=tree(Taxon~.,d); m1



# 1) root 120 332.70 I ( 0.2500 0.2500 0.2500 0.2500 )

# 2) Sepal < 3.53232 90 197.80 I ( 0.3333 0.3333 0.3333 0.0000 )

# 4) Leaf < 2.00426 60 83.18 I ( 0.5000 0.5000 0.0000 0.0000 )

# 8) Petiole < 9.91246 30 0.00 II ( 0.0000 1.0000 0.0000

0.0000 ) *

# 9) Petiole > 9.91246 30 0.00 I ( 1.0000 0.0000 0.0000 0.0000

) *

# 5) Leaf > 2.00426 30 0.00 III ( 0.0000 0.0000 1.0000 0.0000 )

*

# 3) Sepal > 3.53232 30 0.00 IV ( 0.0000 0.0000 0.0000 1.0000 ) *

plot(m1); text(m1)

summary(m1)

# Classification tree:

# tree(formula = Taxon ~ ., data = d)

# Variables actually used in tree construction:

# [1] "Sepal" "Leaf" "Petiole"

# Number of terminal nodes: 4

# Residual mean deviance: 0 = 0 / 116

# Misclassification error rate: 0 = 0 / 120 # impressive

Chapter 22, Time Series Analysis Time series data are vectors of numbers, typically regularly spaced in time. rm(list = ls()) # removes previous variables

d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/bl

owfly.txt",header=T); attach(d); head(d)

# flies # A. J. Nicholson reared blowfly larvae in a laboratory

# 1 948 # and kept the experiment running for almost seven years.

# 2 942

# . . .

# 361 6803

# 362 \032

## The last point seems to be bad, so I will remove it.

d=d[1:361,]

flies=ts(d); plot(flies) # time series object

#

## You can see the pattern better by changes the aspect ratio.

library(lattice)

xyplot(flies~1:length(flies), type="l" ,aspect=0.3, xlab="")

#

http://en.wikipedia.org/wiki/Time_series

## The process seems to have changed around week 200.

n=length(flies); n

par(mfrow=c(2,2))

## Plot lags, such as flies[1] vs flies[2], and flies[1] vs flies[3],

etc.

## Need to make the vectors the same length, so they match.

sapply(1:4, function(x) plot(as.numeric(flies[-c(n-

x+1:n)]),as.numeric(flies[-c(1:x)])) )

par(mfrow=c(1,1))

#

## The correlation drops off quickly with increasing lags.

## Autocorrelation is the correlation between successive observations

over time.

## Partial autocorrelations adjust for the effects of correlations

with fewer lags.

par(mfrow=c(1,2))

acf(flies,main="autocorrelation")

acf(flies, type="p",main="partial autocorrelation")

par(mfrow=c(1,1))

http://en.wikipedia.org/wiki/Auto-correlation

#

## At lag 0, autocorrelation with itself is 1.

## Partial autocorrelations starts with lag 1, which is the same as

the autocorrelation.

## The autocorrelation drops off with time, but becomes negative with

the next down slope of the cycle.

## The negative partial autocorrelation at lags 2 and 3 reflects the

length of the

## larval (1 week) and pupal (2 weeks) periods.

## Cycles are caused by over compensating density dependence,

resulting form larval competition for food.

## It looks like the process is different after week 200.

period=numeric(n); period[1:200]=0; period[201:n]=1; weeks=1:n

m1=lm(flies~weeks*period); anova(m1)


# Response: flies


# weeks 1 8091 8091 0.8299 0.3629

# period 1 424 424 0.0435 0.8349

# weeks:period 1 258019 258019 26.4662 4.434e-07 ***

# Residuals 357 3480395 9749

m2=lm(flies~weeks+period); anova(m1,m2)


# Model 1: flies ~ weeks * period

# Model 2: flies ~ weeks + period

# Res.Df RSS Df Sum of Sq F Pr(>F)

# 1 357 3480395

# 2 358 3738414 -1 -258019 26.466 4.434e-07 ***

## Both the intercept and slope changes after week 200,

##so we should analyze these periods separately.

flies1=flies[1:200]; weeks1=1:200; flies2=flies[201:n];

weeks2=1:length(flies2)

m1=lm(flies1~weeks1);summary(m1)

# Coefficients:


# (Intercept) 204.4682 15.3166 13.349 <2e-16 ***

# weeks1 -0.3012 0.1322 -2.279 0.0237 *

m2=lm(flies2~weeks2);summary(m2)

# Coefficients:


# (Intercept) 119.6196 13.6147 8.786 2.37e-15 ***

# weeks2 0.7613 0.1458 5.222 5.47e-07 ***

## Decrease in period 1, but an increase in period 2.

## Tests do not consider autocorrelations (spatial pseudoreplication).

dt1=flies1-predict(m1); dt2=flies2-predict(m2) # detrended

par(mfrow=c(1,3))

ts.plot(dt1)

acf(dt1, main="autocorrelation")

acf(dt1, type="p", main="partial autocorrelation")

#

ts.plot(dt2)

acf(dt2, main="autocorrelation")

acf(dt2, type="p", main="partial autocorrelation")

par(mfrow=c(1,1))

#

#### moving average ##############################################

page 708


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/te

mp.txt",header=T); attach(d); head(d)

# temps

# 1 3.170968

# . . .

## A three point moving average removes much of the local noise.

## yi = (xi-1 + xi + xi+1)/3

ma=function(x,d) {

d=as.integer(d/2); n=length(x); y=numeric(n); y[1:d]=NA; y[(n-

d+1):n]=NA

for (i in (d+1):(n-d)) y[i]=mean(x[(i-d):(i+d)])

y

}

par(mfrow=c(3,1))

plot(temps, main="points"); lines(temps);

mavg=ma(temps,3); plot(temps,main="moving average 3");

lines(ma(temps,3))

mavg=ma(temps,7); plot(temps,main="moving average 7");

lines(ma(temps,7))

par(mfrow=c(1,1))

#

## Note the seasonal pattern.

#### seasonal data ###############################################

page 708


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/Si

lwoodWeather.txt",header=T); attach(d); head(d)

# upper lower rain month yr # daily weather from Silwood Park,

England 1987-2005

# 1 10.8 6.5 12.2 1 1987

# 2 10.5 4.5 1.3 1 1987

# . . .

plot(upper,type="l")

#

n=length(upper); n; yrLen=n/19; yrLen # n=6940, yr=365.2632 days

t=(1:n)/yrLen; head(t) # time in years

m=lm(upper~sin(t*2*pi)+cos(t*2*pi)); summary(m)

# Coefficients:


# (Intercept) 14.95647 0.04088 365.86 <2e-16 ***

# sin(t * 2 * pi) -2.53888 0.05781 -43.91 <2e-16 ***

# cos(t * 2 * pi) -7.24015 0.05781 -125.23 <2e-16 ***

plot(t,upper,pch="."); lines(t,predict(m))

#

plot(m$resid,pch=".")

#

## some periodicity, but no obvious trend

par(mfrow=c(1,2)); acf(m$resid); acf(m$resid,type="p");

par(mfrow=c(1,1));

#

## strong autocorrelation, which drop off quickly

## Only the partial autocorrelation at lag 1 is large,

## suggesting an autoregressive model with lag 1, AR(1).

#### pattern in monthly means ############################### page

713

moMeans=tapply(upper,list(month,yr),mean) # monthly means

temp=ts(as.vector(moMeans))

par(mfrow=c(1,2)); plot(temp); acf(temp); par(mfrow=c(1,1))

#

## cycle with period of 12 months

yrMeans=tapply(upper,yr,mean) # monthly means

ytemp=ts(as.vector(yrMeans))

par(mfrow=c(1,2)); plot(ytemp); acf(ytemp); par(mfrow=c(1,1))

#

## No evidence for autocorrelation among years.

#### built-in time series functions ######################### page

714

high=ts(upper,start=c(1987,1),frequency=365); plot(high) # time series

object

#

## decomposition

up=stl(high,"periodic"); plot(up) # Seasonal Decomposition

#

## testing for a trend in time series

tapply(upper, factor(yr>1996), mean)

# FALSE TRUE # mean after 1966 is greater

# 14.62056 15.32978

yr=factor(yr); n=length(upper); ix=1:n; yrLen=n/19; t=(1:n)/yrLen #

time in years

library(lme4)

m1=lmer(upper~ix+sin(t*2*pi)+cos(t*2*pi) + (1|factor(yr)), REML=F)

m2=lmer(upper~sin(t*2*pi)+cos(t*2*pi) + (1|factor(yr)), REML=F) #

remove ix index

anova(m1,m2)

# Models:

# m2: upper ~ sin(t * 2 * pi) + cos(t * 2 * pi) + (1 | yr)

# m1: upper ~ ix + sin(t * 2 * pi) + cos(t * 2 * pi) + (1 | yr)

# Df AIC BIC logLik Chisq Chi Df Pr(>Chisq)

# m2 5 36452 36486 -18221

# m1 6 36458 36499 -18223 0 1 1

## m2, without ix, is preferred. No suggestion of global warming.

m3=lm(yrMeans~I(1:length(yrMeans))); summary(m3)

## Analyzing yearly means removes the spatial pseudoreplication,

because they are uncorrelated.

# Coefficients:


# (Intercept) 14.27105 0.32220 44.293 <2e-16 ***

# I(1:length(yrMeans)) 0.06858 0.02826 2.427 0.0266 *

## Significantly increasing trend.

m4=lm(yrMeans[-1]~I(1:(length(yrMeans)-1))); summary(m4)

# Coefficients:


# (Intercept) 14.59826 0.30901 47.243 <2e-16 ***

# I(1:(length(yrMeans) - 1)) 0.04761 0.02855 1.668 0.115

## Not significant if drop first year.

#### spectral analysis ##################################### Page 717

## Spectral analysis is an alternative approach that is based on the

analysis of

## frequencies rather than the fluctuations of numbers.


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ly

nx.txt",header=T); attach(d); head(d)

# Lynx # annual number of lynx pelts bought by Hudson's Bay

Company

# 1 269

# . . .

plot.ts(Lynx)

#

spectrum(lynx)

#

## The graph is interpreted as showing strong cycles with a frequency

of about 0.1.

## The vertical blue bar shows the 95% confidence interval.

## A frequency of 0.1 indicates a period of 1/0.1 = 10 years.

#### multiple time series ########################################

Page 718


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/tw

oseries.txt",header=T); attach(d); head(d)

# x y

# 1 101 121

# . . .

plot.ts(cbind(x,y))

#

par(mfrow=c(1,2)); acf(x,type="p"); acf(y,type="p");

par(mfrow=c(1,1));

#

## The evidence for periodicity is stronger in x than in y.

## The partial autocorrelation for x is significant at a lag of 2.

acf(cbind(x,y))

#

## x & y is the cross correlation of the two time series

## The lag is reversed for y & x.

acf(cbind(x,y),type="p")

#

## Partial autocorrelations control for both x and y.

plot(diff(x),diff(y))

#

## Plotting the differences shows a negative correlation, with two

outliers.

#### simulated time series ################################## Page

722

## Shows how a first-order autoregressive process [AR(1)] appears in

the plots.


n=250; e=rnorm(n,0,2)# error term

y=e # no autocorrelation.

par(mfrow=c(1,3)); plot.ts(y); acf(y); acf(y,type="p")

#

a=-0.5; y[1]=e[1]

for (i in 2:n) y[i]=a*y[i-1]+e[i]


#

a=0.5; y[1]=e[1]



#

a=1; y[1]=e[1] # random walk



#

par(mfrow=c(1,1))

#### time series models ############################ Page 726

• Moving average (AV)

• Autoregressive (AR)

• Autoregressive moving average (ARMA)


d=read.table("http://www.bio.ic.ac.uk/research/mjcraw/therbook/data/ly

nx.txt",header=T); attach(d); head(d)

# Lynx # annual number of lynx pelts bought by Hudson's Bay

Company

# 1 269

# . . .

par(mfrow=c(1,2)); acf(Lynx); acf(Lynx,type="p"); par(mfrow=c(1,1))

http://en.wikipedia.org/wiki/Moving_average_model

http://en.wikipedia.org/wiki/Autoregressive

http://en.wikipedia.org/wiki/Autoregressive_moving_average

#

## Crawley comments that "the population is very clearly cyclic, with

a period

## of 10 years. The dynamics appear to be driven by strong, negative

density dependence

##(a partial autocorrelation of =0.588) at lag 2. There are other

significant partials

## at lags 1 and 8 (positive) and lag 4 (negative). Of course you

cannot infer the

## mechanism by observing the dynamics, but the lags associated with

significant negative

## and positive feedbacks are extremely interesting and highly

suggestive.

## The main prey species of the lynx is the snowshoe hare and the

negative feedback at

## lag 2 may reflect the timescale of this predator-prey interaction.

The hares are

## known to cause medium-term induced reductions in the quality of

their food plants as

## a result of heavy browsing pressure when the hares [are] at high

density, and this

## could map through to lynx populations with lag 4."

## arima (autoregressive moving average models) order=c(p,d,q) where

## p = autoregressive order

## d = degree of differencing

## q = moving average order

m100=arima(Lynx,order=c(1,0,0))












AIC(m100,m200,m300,m400,m500,m600,m001,m002,m003,m004,m005,m006)

# df AIC

# m100 3 1926.991

# m200 4 1878.032

# m300 5 1879.957

# m400 6 1874.222 # min for AR models

# m500 7 1875.276

# m600 8 1876.858

# m001 3 1917.947

# m002 4 1890.061

# m003 5 1887.770

# m004 6 1888.279

# m005 7 1885.698

# m006 8 1885.230 # min for MA models

AIC(m100,m200,m300,m400,m500,m600,m001,m002,m003,m004,m005,m006,k=log(

length(Lynx))) # BIC

# df AIC

# m100 3 1935.199

# m200 4 1888.977 # min for AR models

# m300 5 1893.638

# m400 6 1890.639

# m500 7 1894.429

# m600 8 1898.748

# m001 3 1926.155

# m002 4 1901.006 # min for MA models

# m003 5 1901.451

# m004 6 1904.696

# m005 7 1904.851

# m006 8 1907.119



m206=arima(Lynx,order=c(2,0,6)) # no estimate



m406=arima(Lynx,order=c(4,0,6)) # no estimate

AIC(m200,m201,m202,m400,m401,m402)

# df AIC

# m200 4 1878.032

# m201 5 1879.459

# m202 6 1876.167

# m400 6 1874.222

# m401 7 1875.351

# m402 8 1862.435 # min AR 4, dif 0, MA 2

AIC(m200,m201,m202,m400,m401,m402,k=log(length(Lynx))) # BIC

# df AIC

# m200 4 1888.977

# m201 5 1893.140

# m202 6 1892.585

# m400 6 1890.639

# m401 7 1894.504

# m402 8 1884.325 # min AR 4, dif 0, MA 2

m412=arima(Lynx,order=c(4,1,2)) # Add difference



AIC(m402,m412,m422,m432,m200,m210,m220,m230)

# df AIC

# m402 8 1862.435

# m412 7 1863.830

# m422 7 1859.194 # min AR 4, dif 2, MA 2

# m432 7 1878.830

AIC(m402,m412,m422,m432,m200,m210,m220,m230,k=log(length(Lynx))) # BIC

# df AIC

# m402 8 1884.325

# m412 7 1882.984

# m422 7 1878.348 # min AR 4, dif 2, MA 2

# m432 7 1897.984

# m200 4 1888.977

# m210 3 1903.359

# m220 3 1928.970

# m230 3 1969.465

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