Analysis of Variance (Wald and F Statistics)

Description

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.

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.

Usage

## 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'))

## 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, table.env=TRUE,
      caption=NULL, ...)

## S3 method for class 'anova.rms'
html(object, ...)

Arguments

Details

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.

Value

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.

Side Effects

print prints, latex creates a file with a name of the form "title.tex" (see the title argument above).

Author(s)

Frank Harrell
Department of Biostatistics, Vanderbilt University
f.harrell@vanderbilt.edu

See Also

rms, rmsMisc, lrtest, rms.trans, summary.rms, plot.Predict, ggplot.Predict, solvet, locator, dotchart2, latex, xYplot, anova.lm, contrast.rms, pantext

Examples

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')