R: General Contrasts of Regression Coefficients

contrast.rms {rms} | R Documentation

## General Contrasts of Regression Coefficients

### Description

This function computes one or more contrasts of the estimated regression coefficients in a fit from one of the functions in rms, along with standard errors, confidence limits, t or Z statistics, P-values. General contrasts are handled by obtaining the design matrix for two sets of predictor settings (`a`, `b`) and subtracting the corresponding rows of the two design matrics to obtain a new contrast design matrix for testing the `a` - `b` differences. This allows for quite general contrasts (e.g., estimated differences in means between a 30 year old female and a 40 year old male). This can also be used to obtain a series of contrasts in the presence of interactions (e.g., female:male log odds ratios for several ages when the model contains age by sex interaction). Another use of `contrast` is to obtain center-weighted (Type III test) and subject-weighted (Type II test) estimates in a model containing treatment by center interactions. For the latter case, you can specify `type="average"` and an optional `weights` vector to average the within-center treatment contrasts. The design contrast matrix computed by `contrast.rms` can be used by other functions.

`contrast.rms` also allows one to specify four settings to contrast, yielding contrasts that are double differences - the difference between the first two settings (`a` - `b`) and the last two (`a2` - `b2`). This allows assessment of interactions.

If `usebootcoef=TRUE`, the fit was run through `bootcov`, and `conf.type="individual"`, the confidence intervals are bootstrap nonparametric percentile confidence intervals, basic bootstrap, or BCa intervals, obtained on contrasts evaluated on all bootstrap samples.

By omitting the `b` argument, `contrast` can be used to obtain an average or weighted average of a series of predicted values, along with a confidence interval for this average. This can be useful for "unconditioning" on one of the predictors (see the next to last example).

Specifying `type="joint"`, and specifying at least as many contrasts as needed to span the space of a complex test, one can make multiple degree of freedom tests flexibly and simply. Redundant contrasts will be ignored in the joint test. See the examples below. These include an example of an "incomplete interaction test" involving only two of three levels of a categorical variable (the test also tests the main effect).

When more than one contrast is computed, the list created by `contrast.rms` is suitable for plotting (with error bars or bands) with `xYplot` or `Dotplot` (see the last example before the `type="joint"` examples).

### Usage

    contrast(fit, ...)
    ## S3 method for class 'rms'
    contrast(fit, a, b, a2, b2, cnames=NULL,
             type=c("individual", "average", "joint"),
             conf.type=c("individual","simultaneous"), usebootcoef=TRUE,
             boot.type=c("percentile","bca","basic"),
             weights="equal", conf.int=0.95, tol=1e-7, expand=TRUE, ...)

    ## S3 method for class 'contrast.rms'
    print(x, X=FALSE, fun=function(u)u, jointonly=FALSE, ...)

### Arguments

`fit` | a fit of class `"rms"`
`a` | a list containing settings for all predictors that you do not wish to set to default (adjust-to) values. Usually you will specify two variables in this list, one set to a constant and one to a sequence of values, to obtain contrasts for the sequence of values of an interacting factor. The `gendata` function will generate the necessary combinations and default values for unspecified predictors, depending on the `expand` argument.
`b` | another list that generates the same number of observations as `a`, unless one of the two lists generates only one observation. In that case, the design matrix generated from the shorter list will have its rows replicated so that the contrasts assess several differences against the one set of predictor values. This is useful for comparing multiple treatments with control, for example. If `b` is missing, the design matrix generated from `a` is analyzed alone.
`a2` | an optional third list of settings of predictors
`b2` | an optional fourth list of settings of predictors. Mandatory if `a2` is given.
`cnames` | vector of character strings naming the contrasts when `type!="average"`. Usually `cnames` is not necessary as `contrast.rms` tries to name the contrasts by examining which predictors are varying consistently in the two lists. `cnames` will be needed when you contrast "non-comparable" settings, e.g., you compare `list(treat="drug", age=c(20,30))` with `list(treat="placebo"), age=c(40,50))`
`type` | set `type="average"` to average the individual contrasts (e.g., to obtain a Type II or III contrast). Set `type="joint"` to jointly test all non-redundant contrasts with a multiple degree of freedom test and no averaging.
`conf.type` | The default type of confidence interval computed for a given individual (1 d.f.) contrast is a pointwise confidence interval. Set `conf.type="simultaneous"` to use the `multcomp` package's `glht` and `confint` functions to compute confidence intervals with simultaneous (family-wise) coverage, thus adjusting for multiple comparisons. Note that individual P-values are not adjusted for multiplicity.
`usebootcoef` | If `fit` was the result of `bootcov` but you want to use the bootstrap covariance matrix instead of the nonparametric percentile, basic, or BCa method for confidence intervals (which uses all the bootstrap coefficients), specify `usebootcoef=FALSE`.
`boot.type` | set to `'bca'` to compute BCa confidence limits or `'basic'` to use the basic bootstrap. The default is to compute percentile intervals
`weights` | a numeric vector, used when `type="average"`, to obtain weighted contrasts
`conf.int` | confidence level for confidence intervals for the contrasts
`tol` | tolerance for `qr` function for determining which contrasts are redundant, and for inverting the covariance matrix involved in a joint test
`expand` | set to `FALSE` to have `gendata` not generate all possible combinations of predictor settings. This is useful when getting contrasts over irregular predictor settings.
`...` | passed to `print` for main output. A useful thing to pass is `digits=4`.
`x` | result of `contrast`
`X` | set `X=TRUE` to print design matrix used in computing the contrasts (or the average contrast)
`fun` | a function to transform the contrast, SE, and lower and upper confidence limits before printing. For example, specify `fun=exp` to anti-log them for logistic models.
`jointonly` | set to `FALSE` to omit printing of individual contrasts

### Value

a list of class `"contrast.rms"` containing the elements `Contrast`, `SE`, `Z`, `var`, `df.residual` `Lower`, `Upper`, `Pvalue`, `X`, `cnames`, `redundant`, which denote the contrast estimates, standard errors, Z or t-statistics, variance matrix, residual degrees of freedom (this is `NULL` if the model was not `ols`), lower and upper confidence limits, 2-sided P-value, design matrix, contrast names (or `NULL`), and a logical vector denoting which contrasts are redundant with the other contrasts. If there are any redundant contrasts, when the results of `contrast` are printed, and asterisk is printed at the start of the corresponding lines. The object also contains `ctype` indicating what method was used for compute confidence intervals.

### Author(s)

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

### See Also

`Predict`, `gendata`, `bootcov`, `summary.rms`, `anova.rms`,

### Examples

    set.seed(1)
    age <- rnorm(200,40,12)
    sex <- factor(sample(c('female','male'),200,TRUE))
    logit <- (sex=='male') + (age-40)/5
    y <- ifelse(runif(200) <= plogis(logit), 1, 0)
    f <- lrm(y ~ pol(age,2)*sex)
    anova(f)
    # Compare a 30 year old female to a 40 year old male
    # (with or without age x sex interaction in the model)
    contrast(f, list(sex='female', age=30), list(sex='male', age=40))
    # Test for interaction between age and sex, duplicating anova
    contrast(f, list(sex='female', age=30),
                list(sex='male',   age=30),
                list(sex='female', age=c(40,50)),
                list(sex='male',   age=c(40,50)), type='joint')
    # Duplicate overall sex effect in anova with 3 d.f.
    contrast(f, list(sex='female', age=c(30,40,50)),
                list(sex='male',   age=c(30,40,50)), type='joint')

    # For a model containing two treatments, centers, and treatment
    # x center interaction, get 0.95 confidence intervals separately
    # by center
    center <- factor(sample(letters[1 : 8], 500, TRUE))
    treat  <- factor(sample(c('a','b'), 500, TRUE))
    y      <- 8*(treat == 'b') + rnorm(500, 100, 20)
    f <- ols(y ~ treat*center)

    lc <- levels(center)
    contrast(f, list(treat='b', center=lc),
                list(treat='a', center=lc))

    # Get 'Type III' contrast: average b - a treatment effect over
    # centers, weighting centers equally (which is almost always
    # an unreasonable thing to do)
    contrast(f, list(treat='b', center=lc),
                list(treat='a', center=lc),
             type='average')

    # Get 'Type II' contrast, weighting centers by the number of
    # subjects per center.  Print the design contrast matrix used.
    k <- contrast(f, list(treat='b', center=lc),
                     list(treat='a', center=lc),
                  type='average', weights=table(center))
    print(k, X=TRUE)
    # Note: If other variables had interacted with either treat
    # or center, we may want to list settings for these variables
    # inside the list()'s, so as to not use default settings

    # For a 4-treatment study, get all comparisons with treatment 'a'
    treat  <- factor(sample(c('a','b','c','d'),  500, TRUE))
    y      <- 8*(treat == 'b') + rnorm(500, 100, 20)
    dd     <- datadist(treat, center); options(datadist='dd')
    f <- ols(y ~ treat*center)
    lt <- levels(treat)
    contrast(f, list(treat=lt[-1]),
                list(treat=lt[ 1]),
             cnames=paste(lt[-1], lt[1], sep=':'), conf.int=1 - .05 / 3)

    # Compare each treatment with average of all others
    for(i in 1 : length(lt)) {
      cat('Comparing with', lt[i], '\n\n')
      print(contrast(f, list(treat=lt[-i]),
                        list(treat=lt[ i]), type='average'))
    }
    options(datadist=NULL)

    # Six ways to get the same thing, for a variable that
    # appears linearly in a model and does not interact with
    # any other variables.  We estimate the change in y per
    # unit change in a predictor x1.  Methods 4, 5 also
    # provide confidence limits.  Method 6 computes nonparametric
    # bootstrap confidence limits.  Methods 2-6 can work
    # for models that are nonlinear or non-additive in x1.
    # For that case more care is needed in choice of settings
    # for x1 and the variables that interact with x1.

    ## Not run:
    coef(fit)['x1']                            # method 1
    diff(predict(fit, gendata(x1=c(0,1))))     # method 2
    g <- Function(fit)                         # method 3
    g(x1=1) - g(x1=0)
    summary(fit, x1=c(0,1))                    # method 4
    k <- contrast(fit, list(x1=1), list(x1=0)) # method 5
    print(k, X=TRUE)
    fit <- update(fit, x=TRUE, y=TRUE)         # method 6
    b <- bootcov(fit, B=500)
    contrast(fit, list(x1=1), list(x1=0))

    # In a model containing age, race, and sex,
    # compute an estimate of the mean response for a
    # 50 year old male, averaged over the races using
    # observed frequencies for the races as weights

    f <- ols(y ~ age + race + sex)
    contrast(f, list(age=50, sex='male', race=levels(race)),
             type='average', weights=table(race))

    ## End(Not run)

    # Plot the treatment effect (drug - placebo) as a function of age
    # and sex in a model in which age nonlinearly interacts with treatment
    # for females only

    set.seed(1)
    n <- 800
    treat <- factor(sample(c('drug','placebo'), n,TRUE))
    sex   <- factor(sample(c('female','male'),  n,TRUE))
    age   <- rnorm(n, 50, 10)
    y     <- .05*age + (sex=='female')*(treat=='drug')*.05*abs(age-50) + rnorm(n)
    f     <- ols(y ~ rcs(age,4)*treat*sex)
    d     <- datadist(age, treat, sex); options(datadist='d')

    # show separate estimates by treatment and sex

    ggplot(Predict(f, age, treat, sex='female'))
    ggplot(Predict(f, age, treat, sex='male'))
    ages  <- seq(35,65,by=5); sexes <- c('female','male')
    w     <- contrast(f, list(treat='drug',    age=ages, sex=sexes),
                         list(treat='placebo', age=ages, sex=sexes))
    # add conf.type="simultaneous" to adjust for having done 14 contrasts
    xYplot(Cbind(Contrast, Lower, Upper) ~ age | sex, data=w,
           ylab='Drug - Placebo')
    w <- as.data.frame(w[c('age','sex','Contrast','Lower','Upper')])
    ggplot(w, aes(x=age, y=Contrast)) + geom_point() + facet_grid(sex ~ .) +
       geom_errorbar(aes(ymin=Lower, ymax=Upper), width=0)
    xYplot(Cbind(Contrast, Lower, Upper) ~ age, groups=sex, data=w,
           ylab='Drug - Placebo', method='alt bars')
    options(datadist=NULL)

    # Examples of type='joint' contrast tests

    set.seed(1)
    x1 <- rnorm(100)
    x2 <- factor(sample(c('a','b','c'), 100, TRUE))
    dd <- datadist(x1, x2); options(datadist='dd')
    y  <- x1 + (x2=='b') + rnorm(100)

    # First replicate a test statistic from anova()

    f <- ols(y ~ x2)
    anova(f)
    contrast(f, list(x2=c('b','c')), list(x2='a'), type='joint')

    # Repeat with a redundancy; compare a vs b, a vs c, b vs c

    contrast(f, list(x2=c('a','a','b')), list(x2=c('b','c','c')), type='joint')

    # Get a test of association of a continuous predictor with y
    # First assume linearity, then cubic

    f <- lrm(y>0 ~ x1 + x2)
    anova(f)
    contrast(f, list(x1=1), list(x1=0), type='joint')  # a minimum set of contrasts
    xs <- seq(-2, 2, length=20)
    contrast(f, list(x1=0), list(x1=xs), type='joint')

    # All contrasts were redundant except for the first, because of
    # linearity assumption

    f <- lrm(y>0 ~ pol(x1,3) + x2)
    anova(f)
    contrast(f, list(x1=0), list(x1=xs), type='joint')
    print(contrast(f, list(x1=0), list(x1=xs), type='joint'), jointonly=TRUE)

    # All contrasts were redundant except for the first 3, because of
    # cubic regression assumption

    # Now do something that is difficult to do without cryptic contrast
    # matrix operations: Allow each of the three x2 groups to have a different
    # shape for the x1 effect where x1 is quadratic.  Test whether there is
    # a difference in mean levels of y for x2='b' vs. 'c' or whether
    # the shape or slope of x1 is different between x2='b' and x2='c' regardless
    # of how they differ when x2='a'.  In other words, test whether the mean
    # response differs between group b and c at any value of x1.
    # This is a 3 d.f. test (intercept, linear, quadratic effects) and is
    # a better approach than subsetting the data to remove x2='a' then
    # fitting a simpler model, as it uses a better estimate of sigma from
    # all the data.

    f <- ols(y ~ pol(x1,2) * x2)
    anova(f)
    contrast(f, list(x1=xs, x2='b'),
                list(x1=xs, x2='c'), type='joint')

    # Note: If using a spline fit, there should be at least one value of
    # x1 between any two knots and beyond the outer knots.
    options(datadist=NULL)

[Package rms version 5.1-1 Index]