Analysis of Variance (Wald and F Statistics)
The anova function automatically tests most meaningful hypotheses
in a design. For example, suppose that age and cholesterol are
predictors, and that a general interaction is modeled using a restricted
spline surface. anova prints Wald statistics (F statistics
for an ols fit) for testing linearity of age, linearity of
cholesterol, age effect (age + age by cholesterol interaction),
cholesterol effect (cholesterol + age by cholesterol interaction),
linearity of the age by cholesterol interaction (i.e., adequacy of the
simple age * cholesterol 1 d.f. product), linearity of the interaction
in age alone, and linearity of the interaction in cholesterol
alone. Joint tests of all interaction terms in the model and all
nonlinear terms in the model are also performed. For any multiple
d.f. effects for continuous variables that were not modeled through
rcs, pol, lsp, etc., tests of linearity will be
omitted. This applies to matrix predictors produced by e.g.
poly or ns. print.anova.rms is the printing
method. plot.anova.rms draws dot charts depicting the importance
of variables in the model, as measured by Wald chi-square,
chi-square minus d.f., AIC, P-values, partial
R^2, R^2 for the whole model after deleting the effects in
question, or proportion of overall model R^2 that is due to each
predictor. latex.anova.rms is the latex method. It
substitutes Greek/math symbols in column headings, uses boldface for
TOTAL lines, and constructs a caption. Then it passes the result
to latex.default for conversion to LaTeX.
For Bayesian models such as blrm, anova computes relative
explained variation indexes (REV) based on approximate Wald statistics.
This uses the variance-covariance matrix of all of the posterior draws,
and the individual draws of betas, plus an overall summary from the
posterior mode/mean/median beta. Wald chi-squares assuming multivariate
normality of betas are computed just as with frequentist models, and for
each draw (or for the summary) the ratio of the partial Wald chi-square
to the total Wald statistic for the model is computed as REV.
The print method calls latex or html methods
depending on options(prType=), and output is to the console. For
latex a table environment is not used and an ordinary
tabular is produced.
html.anova.rms just calls latex.anova.rms.
## S3 method for class 'rms'
anova(object, ..., main.effect=FALSE, tol=1e-9,
test=c('F','Chisq'), india=TRUE, indnl=TRUE, ss=TRUE,
vnames=c('names','labels'),
posterior.summary=c('mean', 'median', 'mode'), ns=500, cint=0.95)
## S3 method for class 'anova.rms'
print(x,
which=c('none','subscripts','names','dots'),
table.env=FALSE, ...)
## S3 method for class 'anova.rms'
plot(x,
what=c("chisqminusdf","chisq","aic","P","partial R2","remaining R2",
"proportion R2", "proportion chisq"),
xlab=NULL, pch=16,
rm.totals=TRUE, rm.ia=FALSE, rm.other=NULL, newnames,
sort=c("descending","ascending","none"), margin=c('chisq','P'),
pl=TRUE, trans=NULL, ntrans=40, height=NULL, width=NULL, ...)
## S3 method for class 'anova.rms'
latex(object, title, dec.chisq=2,
dec.F=2, dec.ss=NA, dec.ms=NA, dec.P=4, dec.REV=3, table.env=TRUE,
caption=NULL, ...)
## S3 method for class 'anova.rms'
html(object, ...)object |
a |
... |
If omitted, all variables are tested, yielding tests for individual factors
and for pooled effects. Specify a subset of the variables to obtain tests
for only those factors, with a pooled Wald tests for the combined effects
of all factors listed. Names may be abbreviated. For example, specify
Can be optional graphical parameters to send to
For |
main.effect |
Set to |
tol |
singularity criterion for use in matrix inversion |
test |
For an |
india |
set to |
indnl |
set to |
ss |
For an |
vnames |
set to |
posterior.summary |
specifies whether the posterior mode/mean/median beta are to be used as a measure of central tendence of the posterior distribution, for use in relative explained variation from Bayesian models |
ns |
number of random samples from the posterior draws to use for REV highest posterior density intervals |
cint |
HPD interval probability |
x |
for |
which |
If |
what |
what type of statistic to plot. The default is the Wald
chi-square
statistic for each factor (adding in the effect of higher-ordered
factors containing that factor) minus its degrees of freedom. The
R2 choices for |
xlab |
x-axis label, default is constructed according to |
pch |
character for plotting dots in dot charts. Default is 16 (solid dot). |
rm.totals |
set to |
rm.ia |
set to |
rm.other |
a list of other predictor names to omit from the chart |
newnames |
a list of substitute predictor names to use, after omitting any. |
sort |
default is to sort bars in descending order of the summary statistic |
margin |
set to a vector of character strings to write text for
selected statistics in the right margin of the dot chart. The
character strings can be any combination of |
pl |
set to |
trans |
set to a function to apply that transformation to the statistics
being plotted, and to truncate negative values at zero. A good choice
is |
ntrans |
|
height,width |
height and width of |
title |
title to pass to |
dec.chisq |
number of places to the right of the decimal place for typesetting
chi-square values (default is |
dec.F |
digits to the right for F statistics (default is |
dec.ss |
digits to the right for sums of squares (default is |
dec.ms |
digits to the right for mean squares (default is |
dec.P |
digits to the right for P-values |
dec.REV |
digits to the right for REV |
table.env |
see |
caption |
caption for table if |
If the statistics being plotted with plot.anova.rms are few in
number and one of them is negative or zero, plot.anova.rms
will quit because of an error in dotchart2.
The latex method requires LaTeX packages relsize and
needspace.
anova.rms returns a matrix of class anova.rms containing factors
as rows and chi-square, d.f., and P-values as
columns (or d.f., partial SS, MS, F, P). An attribute
vinfo provides list of variables involved in each row and the
type of test done.
plot.anova.rms invisibly returns the vector of quantities
plotted. This vector has a names attribute describing the terms for
which the statistics in the vector are calculated.
print prints, latex creates a
file with a name of the form "title.tex" (see the title argument above).
Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com
n <- 1000 # define sample size
set.seed(17) # so can reproduce the results
treat <- factor(sample(c('a','b','c'), n,TRUE))
num.diseases <- sample(0:4, n,TRUE)
age <- rnorm(n, 50, 10)
cholesterol <- rnorm(n, 200, 25)
weight <- rnorm(n, 150, 20)
sex <- factor(sample(c('female','male'), n,TRUE))
label(age) <- 'Age' # label is in Hmisc
label(num.diseases) <- 'Number of Comorbid Diseases'
label(cholesterol) <- 'Total Cholesterol'
label(weight) <- 'Weight, lbs.'
label(sex) <- 'Sex'
units(cholesterol) <- 'mg/dl' # uses units.default in Hmisc
# Specify population model for log odds that Y=1
L <- .1*(num.diseases-2) + .045*(age-50) +
(log(cholesterol - 10)-5.2)*(-2*(treat=='a') +
3.5*(treat=='b')+2*(treat=='c'))
# Simulate binary y to have Prob(y=1) = 1/[1+exp(-L)]
y <- ifelse(runif(n) < plogis(L), 1, 0)
fit <- lrm(y ~ treat + scored(num.diseases) + rcs(age) +
log(cholesterol+10) + treat:log(cholesterol+10))
a <- anova(fit) # Test all factors
b <- anova(fit, treat, cholesterol) # Test these 2 by themselves
# to get their pooled effects
a
b
# Add a new line to the plot with combined effects
s <- rbind(a, 'treat+cholesterol'=b['TOTAL',])
class(s) <- 'anova.rms'
plot(s, margin=c('chisq', 'proportion chisq'))
g <- lrm(y ~ treat*rcs(age))
dd <- datadist(treat, num.diseases, age, cholesterol)
options(datadist='dd')
p <- Predict(g, age, treat="b")
s <- anova(g)
# Usually omit fontfamily to default to 'Courier'
# It's specified here to make R pass its package-building checks
plot(p, addpanel=pantext(s, 28, 1.9, fontfamily='Helvetica'))
plot(s, margin=c('chisq', 'proportion chisq'))
# new plot - dot chart of chisq-d.f. with 2 other stats in right margin
# latex(s) # nice printout - creates anova.g.tex
options(datadist=NULL)
# Simulate data with from a given model, and display exactly which
# hypotheses are being tested
set.seed(123)
age <- rnorm(500, 50, 15)
treat <- factor(sample(c('a','b','c'), 500, TRUE))
bp <- rnorm(500, 120, 10)
y <- ifelse(treat=='a', (age-50)*.05, abs(age-50)*.08) + 3*(treat=='c') +
pmax(bp, 100)*.09 + rnorm(500)
f <- ols(y ~ treat*lsp(age,50) + rcs(bp,4))
print(names(coef(f)), quote=FALSE)
specs(f)
anova(f)
an <- anova(f)
options(digits=3)
print(an, 'subscripts')
print(an, 'dots')
an <- anova(f, test='Chisq', ss=FALSE)
plot(0:1) # make some plot
tab <- pantext(an, 1.2, .6, lattice=FALSE, fontfamily='Helvetica')
# create function to write table; usually omit fontfamily
tab() # execute it; could do tab(cex=.65)
plot(an) # new plot - dot chart of chisq-d.f.
# Specify plot(an, trans=sqrt) to use a square root scale for this plot
# latex(an) # nice printout - creates anova.f.tex
## Example to save partial R^2 for all predictors, along with overall
## R^2, from two separate fits, and to combine them with a lattice plot
require(lattice)
set.seed(1)
n <- 100
x1 <- runif(n)
x2 <- runif(n)
y <- (x1-.5)^2 + x2 + runif(n)
group <- c(rep('a', n/2), rep('b', n/2))
A <- NULL
for(g in c('a','b')) {
f <- ols(y ~ pol(x1,2) + pol(x2,2) + pol(x1,2) %ia% pol(x2,2),
subset=group==g)
a <- plot(anova(f),
what='partial R2', pl=FALSE, rm.totals=FALSE, sort='none')
a <- a[-grep('NONLINEAR', names(a))]
d <- data.frame(group=g, Variable=factor(names(a), names(a)),
partialR2=unname(a))
A <- rbind(A, d)
}
dotplot(Variable ~ partialR2 | group, data=A,
xlab=ex <- expression(partial~R^2))
dotplot(group ~ partialR2 | Variable, data=A, xlab=ex)
dotplot(Variable ~ partialR2, groups=group, data=A, xlab=ex,
auto.key=list(corner=c(.5,.5)))
# Suppose that a researcher wants to make a big deal about a variable
# because it has the highest adjusted chi-square. We use the
# bootstrap to derive 0.95 confidence intervals for the ranks of all
# the effects in the model. We use the plot method for anova, with
# pl=FALSE to suppress actual plotting of chi-square - d.f. for each
# bootstrap repetition.
# It is important to tell plot.anova.rms not to sort the results, or
# every bootstrap replication would have ranks of 1,2,3,... for the stats.
n <- 300
set.seed(1)
d <- data.frame(x1=runif(n), x2=runif(n), x3=runif(n),
x4=runif(n), x5=runif(n), x6=runif(n), x7=runif(n),
x8=runif(n), x9=runif(n), x10=runif(n), x11=runif(n),
x12=runif(n))
d$y <- with(d, 1*x1 + 2*x2 + 3*x3 + 4*x4 + 5*x5 + 6*x6 +
7*x7 + 8*x8 + 9*x9 + 10*x10 + 11*x11 +
12*x12 + 9*rnorm(n))
f <- ols(y ~ x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12, data=d)
B <- 20 # actually use B=1000
ranks <- matrix(NA, nrow=B, ncol=12)
rankvars <- function(fit)
rank(plot(anova(fit), sort='none', pl=FALSE))
Rank <- rankvars(f)
for(i in 1:B) {
j <- sample(1:n, n, TRUE)
bootfit <- update(f, data=d, subset=j)
ranks[i,] <- rankvars(bootfit)
}
lim <- t(apply(ranks, 2, quantile, probs=c(.025,.975)))
predictor <- factor(names(Rank), names(Rank))
Dotplot(predictor ~ Cbind(Rank, lim), pch=3, xlab='Rank')Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.