R: Bootstrap Covariance and Distribution for Regression...

bootcov {rms} | R Documentation

## Bootstrap Covariance and Distribution for Regression Coefficients

### Description

`bootcov` computes a bootstrap estimate of the covariance matrix for a set of regression coefficients from `ols`, `lrm`, `cph`, `psm`, `Rq`, and any other fit where `x=TRUE, y=TRUE` was used to store the data used in making the original regression fit and where an appropriate `fitter` function is provided here. The estimates obtained are not conditional on the design matrix, but are instead unconditional estimates. For small sample sizes, this will make a difference as the unconditional variance estimates are larger. This function will also obtain bootstrap estimates corrected for cluster sampling (intra-cluster correlations) when a "working independence" model was used to fit data which were correlated within clusters. This is done by substituting cluster sampling with replacement for the usual simple sampling with replacement. `bootcov` has an option (`coef.reps`) that causes all of the regression coefficient estimates from all of the bootstrap re-samples to be saved, facilitating computation of nonparametric bootstrap confidence limits and plotting of the distributions of the coefficient estimates (using histograms and kernel smoothing estimates).

The `loglik` option facilitates the calculation of simultaneous confidence regions from quantities of interest that are functions of the regression coefficients, using the method of Tibshirani(1996). With Tibshirani's method, one computes the objective criterion (-2 log likelihood evaluated at the bootstrap estimate of beta but with respect to the original design matrix and response vector) for the original fit as well as for all of the bootstrap fits. The confidence set of the regression coefficients is the set of all coefficients that are associated with objective function values that are less than or equal to say the 0.95 quantile of the vector of `B + 1` objective function values. For the coefficients satisfying this condition, predicted values are computed at a user-specified design matrix `X`, and minima and maxima of these predicted values (over the qualifying bootstrap repetitions) are computed to derive the final simultaneous confidence band.

The `bootplot` function takes the output of `bootcov` and either plots a histogram and kernel density estimate of specified regression coefficients (or linear combinations of them through the use of a specified design matrix `X`), or a `qqnorm` plot of the quantities of interest to check for normality of the maximum likelihood estimates. `bootplot` draws vertical lines at specified quantiles of the bootstrap distribution, and returns these quantiles for possible printing by the user. Bootstrap estimates may optionally be transformed by a user-specified function `fun` before plotting.

The `confplot` function also uses the output of `bootcov` but to compute and optionally plot nonparametric bootstrap pointwise confidence limits or (by default) Tibshirani (1996) simultaneous confidence sets. A design matrix must be specified to allow `confplot` to compute quantities of interest such as predicted values across a range of values or differences in predicted values (plots of effects of changing one or more predictor variable values).

`bootplot` and `confplot` are actually generic functions, with the particular functions `bootplot.bootcov` and `confplot.bootcov` automatically invoked for `bootcov` objects.

A service function called `histdensity` is also provided (for use with `bootplot`). It runs `hist` and `density` on the same plot, using twice the number of classes than the default for `hist`, and 1.5 times the `width` than the default used by `density`.

A comprehensive example demonstrates the use of all of the functions.

### Usage

    bootcov(fit, cluster, B=200, fitter,
            coef.reps=TRUE, loglik=FALSE,
            pr=FALSE, maxit=15, eps=0.0001, group=NULL, stat=NULL)

    bootplot(obj, which=1 : ncol(Coef), X,
             conf.int=c(.9,.95,.99),
             what=c('density', 'qqnorm', 'box'),
             fun=function(x) x, labels., ...)

    confplot(obj, X, against,
             method=c('simultaneous','pointwise'),
             conf.int=0.95, fun=function(x)x,
             add=FALSE, lty.conf=2, ...)

    histdensity(y, xlab, nclass, width, mult.width=1, ...)

### Arguments

`fit` | a fit object containing components `x` and `y`. For fits from `cph`, the `"strata"` attribute of the `x` component is used to obtain the vector of stratum codes.
`obj` | an object created by `bootcov` with `coef.reps=TRUE`.
`X` | a design matrix specified to `confplot`. See `predict.rms` or `contrast.rms`. For `bootplot`, `X` is optional.
`y` | a vector to pass to `histdensity`. `NA`s are ignored.
`cluster` | a variable indicating groupings. `cluster` may be any type of vector (factor, character, integer). Unique values of `cluster` indicate possibly correlated groupings of observations. Note the data used in the fit and stored in `fit$x` and `fit$y` may have had observations containing missing values deleted. It is assumed that if there were any NAs, an `naresid` function exists for the class of `fit`. This function restores NAs so that the rows of the design matrix coincide with `cluster`.
`B` | number of bootstrap repetitions. Default is 200.
`fitter` | the name of a function with arguments `(x,y)` that will fit bootstrap samples. Default is taken from the class of `fit` if it is `ols`, `lrm`, `cph`, `psm`, `Rq`.
`coef.reps` | set to `TRUE` if you want to store a matrix of all bootstrap regression coefficient estimates in the returned component `boot.Coef`.
`loglik` | set to `TRUE` to store -2 log likelihoods for each bootstrap model, evaluated against the original `x` and `y` data. The default is to do this when `coef.reps` is specified as `TRUE`. The use of `loglik=TRUE` assumes that an `oos.loglik` method exists for the type of model being analyzed, to calculate out-of-sample -2 log likelihoods (see `rmsMisc`). After the `B` -2 log likelihoods (stored in the element named `boot.loglik` in the returned fit object), the `B+1` element is the -2 log likelihood for the original model fit.
`pr` | set to `TRUE` to print the current sample number to monitor progress.
`maxit` | maximum number of iterations, to pass to `fitter`
`eps` | argument to pass to various fitters
`group` | a grouping variable used to stratify the sample upon bootstrapping. This allows one to handle k-sample problems, i.e., each bootstrap sample will be forced to select the same number of observations from each level of group as the number appearing in the original dataset. You may specify both `group` and `cluster`.
`stat` | a single character string specifying the name of a `stats` element produced by the fitting function to save over the bootstrap repetitions. The vector of saved statistics will be in the `boot.stats` part of the list returned by `bootcov`.
`which` | one or more integers specifying which regression coefficients to plot for `bootplot`
`conf.int` | a vector (for `bootplot`, default is `c(.9,.95,.99)`) or scalar (for `confplot`, default is `.95`) confidence level.
`what` | for `bootplot`, specifies whether a density or a q-q plot is made, a `ggplot2` is used to produce a box plot of all coefficients over the bootstrap reps
`fun` | for `bootplot` or `confplot` specifies a function used to translate the quantities of interest before analysis. A common choice is `fun=exp` to compute anti-logs, e.g., odds ratios.
`labels.` | a vector of labels for labeling the axes in plots produced by `bootplot`. Default is row names of `X` if there are any, or sequential integers.
`...` | For `bootplot` these are optional arguments passed to `histdensity`. Also may be optional arguments passed to `plot` by `confplot` or optional arguments passed to `hist` from `histdensity`, such as `xlim` and `breaks`. The argument `probability=TRUE` is always passed to `hist`.
`against` | For `confplot`, specifying `against` causes a plot to be made (or added to). The `against` variable is associated with rows of `X` and is used as the x-coordinates.
`method` | specifies whether `"pointwise"` or `"simultaneous"` confidence regions are derived by `confplot`. The default is simultaneous.
`add` | set to `TRUE` to add to an existing plot, for `confplot`
`lty.conf` | line type for plotting confidence bands in `confplot`. Default is 2 for dotted lines.
`xlab` | label for x-axis for `histdensity`. Default is `label` attribute or argument name if there is no `label`.
`nclass` | passed to `hist` if present
`width` | passed to `density` if present
`mult.width` | multiplier by which to adjust the default `width` passed to `density`. Default is 1.

### Details

If the fit has a scale parameter (e.g., a fit from `psm`), the log of the individual bootstrap scale estimates are added to the vector of parameter estimates and and column and row for the log scale are added to the new covariance matrix (the old covariance matrix also has this row and column).

For `Rq` fits, the `tau`, `method`, and `hs` arguments are taken from the original fit.

### Value

a new fit object with class of the original object and with the element `orig.var` added. `orig.var` is the covariance matrix of the original fit. Also, the original `var` component is replaced with the new bootstrap estimates. The component `boot.coef` is also added. This contains the mean bootstrap estimates of regression coefficients (with a log scale element added if applicable). `boot.Coef` is added if `coef.reps=TRUE`. `boot.loglik` is added if `loglik=TRUE`. If `stat` is specified an additional vector `boot.stats` will be contained in the returned object. `B` contains the number of successfully fitted bootstrap resamples. A component `clusterInfo` is added to contain elements `name` and `n` holding the name of the `cluster` variable and the number of clusters.

`bootplot` returns a (possible matrix) of quantities of interest and the requested quantiles of them. `confplot` returns three vectors: `fitted`, `lower`, and `upper`.

### Side Effects

`bootcov` prints if `pr=TRUE`

### Author(s)

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

Bill Pikounis
Biometrics Research Department
Merck Research Laboratories
v\_bill\_pikounis@merck.com

### References

Feng Z, McLerran D, Grizzle J (1996): A comparison of statistical methods for clustered data analysis with Gaussian error. Stat in Med 15:1793–1806.

Tibshirani R, Knight K (1996): Model search and inference by bootstrap "bumping". Department of Statistics, University of Toronto. Technical report available from
http://www-stat.stanford.edu/~tibs/. Presented at the Joint Statistical Meetings, Chicago, August 1996.

### See Also

`robcov`, `sample`, `rms`, `lm.fit`, `lrm.fit`, `survival-internal`, `predab.resample`, `rmsMisc`, `Predict`, `gendata`, `contrast.rms`, `Predict`, `setPb`, `cluster.boot`

### Examples

    set.seed(191)
    x <- exp(rnorm(200))
    logit <- 1 + x/2
    y <- ifelse(runif(200) <= plogis(logit), 1, 0)
    f <- lrm(y ~ pol(x,2), x=TRUE, y=TRUE)
    g <- bootcov(f, B=50, pr=TRUE)
    anova(g)    # using bootstrap covariance estimates
    fastbw(g)   # using bootstrap covariance estimates
    beta <- g$boot.Coef[,1]
    hist(beta, nclass=15)     #look at normality of parameter estimates
    qqnorm(beta)
    # bootplot would be better than these last two commands

    # A dataset contains a variable number of observations per subject,
    # and all observations are laid out in separate rows. The responses
    # represent whether or not a given segment of the coronary arteries
    # is occluded. Segments of arteries may not operate independently
    # in the same patient.  We assume a "working independence model" to
    # get estimates of the coefficients, i.e., that estimates assuming
    # independence are reasonably efficient.  The job is then to get
    # unbiased estimates of variances and covariances of these estimates.

    set.seed(2)
    n.subjects <- 30
    ages <- rnorm(n.subjects, 50, 15)
    sexes  <- factor(sample(c('female','male'), n.subjects, TRUE))
    logit <- (ages-50)/5
    prob <- plogis(logit)  # true prob not related to sex
    id <- sample(1:n.subjects, 300, TRUE) # subjects sampled multiple times
    table(table(id))  # frequencies of number of obs/subject
    age <- ages[id]
    sex <- sexes[id]
    # In truth, observations within subject are independent:
    y   <- ifelse(runif(300) <= prob[id], 1, 0)
    f <- lrm(y ~ lsp(age,50)*sex, x=TRUE, y=TRUE)
    g <- bootcov(f, id, B=50)  # usually do B=200 or more
    diag(g$var)/diag(f$var)
    # add ,group=w to re-sample from within each level of w
    anova(g)            # cluster-adjusted Wald statistics
    # fastbw(g)         # cluster-adjusted backward elimination
    plot(Predict(g, age=30:70, sex='female'))  # cluster-adjusted confidence bands

    # Get design effects based on inflation of the variances when compared
    # with bootstrap estimates which ignore clustering
    g2 <- bootcov(f, B=50)
    diag(g$var)/diag(g2$var)

    # Get design effects based on pooled tests of factors in model
    anova(g2)[,1] / anova(g)[,1]

    # Simulate binary data where there is a strong
    # age x sex interaction with linear age effects
    # for both sexes, but where not knowing that
    # we fit a quadratic model.  Use the bootstrap
    # to get bootstrap distributions of various
    # effects, and to get pointwise and simultaneous
    # confidence limits

    set.seed(71)
    n   <- 500
    age <- rnorm(n, 50, 10)
    sex <- factor(sample(c('female','male'), n, rep=TRUE))
    L   <- ifelse(sex=='male', 0, .1*(age-50))
    y   <- ifelse(runif(n)<=plogis(L), 1, 0)

    f <- lrm(y ~ sex*pol(age,2), x=TRUE, y=TRUE)
    b <- bootcov(f, B=50, loglik=TRUE, pr=TRUE)   # better: B=500

    par(mfrow=c(2,3))
    # Assess normality of regression estimates
    bootplot(b, which=1:6, what='qq')
    # They appear somewhat non-normal

    # Plot histograms and estimated densities
    # for 6 coefficients
    w <- bootplot(b, which=1:6)
    # Print bootstrap quantiles
    w$quantiles

    # Show box plots for bootstrap reps for all coefficients
    bootplot(b, what='box')

    # Estimate regression function for females
    # for a sequence of ages
    ages <- seq(25, 75, length=100)
    label(ages) <- 'Age'

    # Plot fitted function and pointwise normal-
    # theory confidence bands
    par(mfrow=c(1,1))
    p <- Predict(f, age=ages, sex='female')
    plot(p)
    # Save curve coordinates for later automatic
    # labeling using labcurve in the Hmisc library
    curves <- vector('list',8)
    curves[[1]] <- with(p, list(x=age, y=lower))
    curves[[2]] <- with(p, list(x=age, y=upper))

    # Add pointwise normal-distribution confidence
    # bands using unconditional variance-covariance
    # matrix from the 500 bootstrap reps
    p <- Predict(b, age=ages, sex='female')
    curves[[3]] <- with(p, list(x=age, y=lower))
    curves[[4]] <- with(p, list(x=age, y=upper))

    dframe <- expand.grid(sex='female', age=ages)
    X <- predict(f, dframe, type='x')  # Full design matrix

    # Add pointwise bootstrap nonparametric
    # confidence limits
    p <- confplot(b, X=X, against=ages, method='pointwise',
                  add=TRUE, lty.conf=4)
    curves[[5]] <- list(x=ages, y=p$lower)
    curves[[6]] <- list(x=ages, y=p$upper)

    # Add simultaneous bootstrap confidence band
    p <- confplot(b, X=X, against=ages, add=TRUE, lty.conf=5)
    curves[[7]] <- list(x=ages, y=p$lower)
    curves[[8]] <- list(x=ages, y=p$upper)
    lab <- c('a','a','b','b','c','c','d','d')
    labcurve(curves, lab, pl=TRUE)

    # Now get bootstrap simultaneous confidence set for
    # female:male odds ratios for a variety of ages

    dframe <- expand.grid(age=ages, sex=c('female','male'))
    X <- predict(f, dframe, type='x')  # design matrix
    f.minus.m <- X[1:100,] - X[101:200,]
    # First 100 rows are for females.  By subtracting
    # design matrices are able to get Xf*Beta - Xm*Beta
    # = (Xf - Xm)*Beta

    confplot(b, X=f.minus.m, against=ages,
             method='pointwise', ylab='F:M Log Odds Ratio')
    confplot(b, X=f.minus.m, against=ages,
             lty.conf=3, add=TRUE)

    # contrast.rms makes it easier to compute the design matrix for use
    # in bootstrapping contrasts:

    f.minus.m <- contrast(f, list(sex='female',age=ages),
                             list(sex='male',  age=ages))$X
    confplot(b, X=f.minus.m)

    # For a quadratic binary logistic regression model use bootstrap
    # bumping to estimate coefficients under a monotonicity constraint
    set.seed(177)
    n <- 400
    x <- runif(n)
    logit <- 3*(x^2-1)
    y <- rbinom(n, size=1, prob=plogis(logit))
    f <- lrm(y ~ pol(x,2), x=TRUE, y=TRUE)
    k <- coef(f)
    k
    vertex <- -k[2]/(2*k[3])
    vertex

    # Outside [0,1] so fit satisfies monotonicity constraint within
    # x in [0,1], i.e., original fit is the constrained MLE

    g <- bootcov(f, B=50, coef.reps=TRUE, loglik=TRUE)
    bootcoef <- g$boot.Coef    # 100x3 matrix
    vertex <- -bootcoef[,2]/(2*bootcoef[,3])
    table(cut2(vertex, c(0,1)))
    mono <- !(vertex >= 0 & vertex <= 1)
    mean(mono)    # estimate of Prob{monotonicity in [0,1]}

    var(bootcoef)   # var-cov matrix for unconstrained estimates
    var(bootcoef[mono,])   # for constrained estimates

    # Find second-best vector of coefficient estimates, i.e., best
    # from among bootstrap estimates
    g$boot.Coef[order(g$boot.loglik[-length(g$boot.loglik)])[1],]
    # Note closeness to MLE

    ## Not run:
    # Get the bootstrap distribution of the difference in two ROC areas for
    # two binary logistic models fitted on the same dataset.  This analysis
    # does not adjust for the bias ROC area (C-index) due to overfitting.
    # The same random number seed is used in two runs to enforce pairing.

    set.seed(17)
    x1 <- rnorm(100)
    x2 <- rnorm(100)
    y <- sample(0:1, 100, TRUE)
    f <- lrm(y ~ x1, x=TRUE, y=TRUE)
    g <- lrm(y ~ x1 + x2, x=TRUE, y=TRUE)
    set.seed(3)
    f <- bootcov(f, stat='C')
    set.seed(3)
    g <- bootcov(g, stat='C')
    dif <- g$boot.stats - f$boot.stats
    hist(dif)
    quantile(dif, c(.025,.25,.5,.75,.975))
    # Compute a z-test statistic.  Note that comparing ROC areas is far less
    # powerful than likelihood or Brier score-based methods
    z <- (g$stats['C'] - f$stats['C'])/sd(dif)
    names(z) <- NULL
    c(z=z, P=2*pnorm(-abs(z)))

    ## End(Not run)

[Package rms version 5.1-1 Index]