anova.Design {Design} | R Documentation

## 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.Design` is the printing method. `text.anova.Design` is the `text` method for inserting anova tables on graphs. `plot.anova.Design` 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.Design` 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.

### Usage

    ## S3 method for class 'Design':
    anova(object, ..., main.effect=FALSE, tol=1e-9,
          test=c('F','Chisq'), ss=TRUE)

    ## S3 method for class 'anova.Design':
    print(x, which=c('none','subscripts','names','dots'), ...)

    ## S3 method for class 'anova.Design':
    plot(x,
         what=c("chisqminusdf","chisq","aic","P","partial R2","remaining R2",
                "proportion R2"),
         xlab=NULL, pch=16,
         rm.totals=TRUE, rm.ia=FALSE, rm.other=NULL, newnames,
         sort=c("descending","ascending","none"), pl=TRUE, ...)

    ## S3 method for class 'anova.Design':
    text(x, at, cex=.5, font=2, ...)

    ## S3 method for class 'anova.Design':
    latex(object, title, psmall=TRUE,
          dec.chisq=2, dec.F=2, dec.ss=NA, dec.ms=NA, dec.P=4, ...)

### Arguments

`object` | a `Design` fit object. `object` must allow `Varcov` to return the variance-covariance matrix. For `latex`, is the result of `anova`.
`...` | 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 `anova(fit,age,cholesterol)` to get a Wald statistic for testing the joint importance of age, cholesterol, and any factor interacting with them. Can be optional graphical parameters to send to `text` or `dotchart2`, or other parameters to send to `latex.default`. Ignored for `print`.
`main.effect` | Set to `TRUE` to print the (usually meaningless) main effect tests even when the factor is involved in an interaction. The default is `FALSE`, to print only the effect of the main effect combined with all interactions involving that factor.
`tol` | singularity criterion for use in matrix inversion
`test` | For an `ols` fit, set `test="Chisq"` to use Wald chi^2 tests rather than F-tests.
`ss` | For an `ols` fit, set `ss=FALSE` to suppress printing partial sums of squares, mean squares, and the Error SS and MS.
`x` | for `print,plot,text` is the result of `anova`.
`which` | If `which` is not `"none"` (the default), `print.anova.Design` will add to the rightmost column of the output the list of parameters being tested by the hypothesis being tested in the current row. Specifying `which="subscripts"` causes the subscripts of the regression coefficients being tested to be printed (with a subscript of one for the first non-intercept term). `which="names"` prints the names of the terms being tested, and `which="dots"` prints dots for terms being tested and blanks for those just being adjusted for.
`at` | for `text` is a list containing the x- and y-coordinates for the upper left corner of the anova table to be drawn on an existing plot, e.g. `at=locator(1)`
`cex` | character expansion size for `text.anova.Design`
`font` | font for `text.anova.Design`. Default is 2 (usually Courier).
`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 last three choice for `what` only apply to `ols` models.
`xlab` | x-axis label, default is constructed according to `what`. `plotmath` symbols are used for R, by default.
`pch` | character for plotting dots in dot charts. Default is 16 (solid dot).
`rm.totals` | set to `FALSE` to keep total chi-squares (overall, nonlinear, interaction totals) in the chart.
`rm.ia` | set to `TRUE` to omit any effect that has `"*"` in its name
`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
`pl` | set to `FALSE` to suppress plotting. This is useful when you only wish to analyze the vector of statistics returned.
`title` | title to pass to `latex`, default is name of fit object passed to `anova` prefixed with `"anova."`. For Windows, the default is `"ano"` followed by the first 5 letters of the name of the fit object.
`psmall` | The default is `psmall=TRUE`, which causes `P<0.00005` to print as `<0.0001`. Set to `FALSE` to print as `0.0000`.
`dec.chisq` | number of places to the right of the decimal place for typesetting chi-square values (default is `2`). Use zero for integer, `NA` for floating point.
`dec.F` | digits to the right for F statistics (default is `2`)
`dec.ss` | digits to the right for sums of squares (default is `NA`, indicating floating point)
`dec.ms` | digits to the right for mean squares (default is `NA`)
`dec.P` | digits to the right for P-values

### Details

If the statistics being plotted with `plot.anova.Design` are few in number and one of them is negative or zero, `plot.anova.Design` will quit because of an error in `dotchart2`.

### Value

`anova.Design` returns a matrix of class `anova.Design` containing factors as rows and chi-square, d.f., and P-values as columns (or d.f., partial SS, MS, F, P). `plot.anova.Design` 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, `text` uses `tempfile` to get a temporary Unix file name, `sink`, and `unix` (to remove the temporary file). `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

`Design`, `Design.Misc`, `lrtest`, `Design.trans`, `summary.Design`, `solvet`, `text`, `locator`, `dotchart2`, `latex`, `Dotplot`, `anova.lm`, `contrast.Design`

### 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))
    anova(fit)                       # Test all factors
    anova(fit, treat, cholesterol)   # Test these 2 by themselves
                                     # to get their pooled effects
    g <- lrm(y ~ treat*rcs(age))
    dd <- datadist(treat, num.diseases, age, cholesterol)
    options(datadist='dd')
    plot(g, age=NA, treat="b")
    s <- anova(g)
    print(s)
    #p <- locator(1)                  # click mouse at upper left corner of table
    p <- list(x=32,y=2.1)
    text(s, at=p)                    # add anova table to regression plot
    plot(s)                          # new plot - dot chart of chisq-d.f.
    # latex(s)                       # nice printout - creates anova.g.tex
    options(datdist=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
    text(an, at=list(x=1.5,y=.6))    # add anova table to plot
    plot(an)                         # new plot - dot chart of chisq-d.f.
    # latex(an)                      # nice printout - creates anova.f.tex

    # 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.  We rank the negative of the adjusted
    # chi-squares so that a rank of 1 is assigned to the highest.
    # It is important to tell plot.anova.Design not to sort the results,
    # or every bootstrap replication would have ranks of 1,2,3 for the stats.

    mydata <- data.frame(x1=runif(200), x2=runif(200),
                         sex=factor(sample(c('female','male'),200,TRUE)))
    set.seed(9)  # so can reproduce example
    mydata$y <- ifelse(runif(200)<=plogis(mydata$x1-.5 + .5*(mydata$x2-.5) +
                       .5*(mydata$sex=='male')),1,0)

    if(.R.) {
    library(boot)
    b <- boot(mydata, function(data, i, ...) rank(-plot(anova(
                    lrm(y ~ rcs(x1,4)+pol(x2,2)+sex,data,subset=i)),
                    sort='none', pl=FALSE)),
                    R=25)  # should really do R=500 but will take a while
    Rank <- b$t0
    lim <- t(apply(b$t, 2, quantile, probs=c(.025,.975)))
    } else {
    b <- bootstrap(mydata, rank(-plot(anova(
                    lrm(y ~ rcs(x1,4)+pol(x2,2)+sex,mydata)), sort='none', pl=FALSE)),
                   B=25)  # should really do B=500 but will take a while
    Rank <- b$observed
    lim <- limits.emp(b)[,c(1,4)]  # get 0.025 and 0.975 quantiles
    }

    # Use the Hmisc Dotplot function to display ranks and their confidence
    # intervals.  Sort the categories by descending adj. chi-square, for ranks
    original.chisq <- plot(anova(lrm(y ~ rcs(x1,4)+pol(x2,2)+sex,data=mydata)),
                           sort='none', pl=FALSE)
    predictor <- as.factor(names(original.chisq))
    predictor <- reorder.factor(predictor, -original.chisq)

    Dotplot(predictor ~ Cbind(Rank, lim), pch=3, xlab='Rank',
                    main=if(.R.) expression(paste(
    'Ranks and 0.95 Confidence Limits for ',chi^2,' - d.f.')) else
    'Ranks and 0.95 Confidence Limits for Chi-square - d.f.')

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