lrm {Design} | R Documentation

## Logistic Regression Model

### Description

Fit binary and proportional odds ordinal logistic regression models using maximum likelihood estimation or penalized maximum likelihood estimation. See `cr.setup` for how to fit forward continuation ratio models with `lrm`.

### Usage

    lrm(formula, data, subset, na.action=na.delete, method="lrm.fit",
        model=FALSE, x=FALSE, y=FALSE, linear.predictors=TRUE, se.fit=FALSE,
        penalty=0, penalty.matrix, tol=1e-7,
        strata.penalty=0, var.penalty=c('simple','sandwich'),
        weights, normwt, ...)

### Arguments

`formula` | a formula object. An `offset` term can be included. The offset causes fitting of a model such as logit(Y=1) = Xβ + W, where W is the offset variable having no estimated coefficient. The response variable can be any data type; `lrm` converts it in alphabetic or numeric order to an S factor variable and recodes it 0,1,2,... internally.
`data` | data frame to use. Default is the current frame.
`subset` | logical expression or vector of subscripts defining a subset of observations to analyze
`na.action` | function to handle `NA`s in the data. Default is `na.delete`, which deletes any observation having response or predictor missing, while preserving the attributes of the predictors and maintaining frequencies of deletions due to each variable in the model. This is usually specified using `options(na.action="na.delete")`.
`method` | name of fitting function. Only allowable choice at present is `lrm.fit`.
`model` | causes the model frame to be returned in the fit object
`x` | causes the expanded design matrix (with missings excluded) to be returned under the name `x`.
`y` | causes the response variable (with missings excluded) to be returned under the name `y`.
`linear.predictors` | causes the predicted X beta (with missings excluded) to be returned under the name `linear.predictors`. When the response variable has more than two levels, only the first intercept is used.
`se.fit` | causes the standard errors of the fitted values to be returned under the name `se.fit`.
`penalty` | The penalty factor subtracted from the log likelihood is 0.5 β' P β, where β is the vector of regression coefficients other than intercept(s), and P is `penalty factors * penalty.matrix` and `penalty.matrix` is defined below. The default is `penalty=0` implying that ordinary unpenalized maximum likelihood estimation is used. If `penalty` is a scalar, it is assumed to be a penalty factor that applies to all non-intercept parameters in the model. Alternatively, specify a list to penalize different types of model terms by differing amounts. The elements in this list are named `simple, nonlinear, interaction` and `nonlinear.interaction`. If you omit elements on the right of this series, values are inherited from elements on the left. Examples: `penalty=list(simple=5, nonlinear=10)` uses a penalty factor of 10 for nonlinear or interaction terms. `penalty=list(simple=0, nonlinear=2, nonlinear.interaction=4)` does not penalize linear main effects, uses a penalty factor of 2 for nonlinear or interaction effects (that are not both), and 4 for nonlinear interaction effects.
`penalty.matrix` | specifies the symmetric penalty matrix for non-intercept terms. The default matrix for continuous predictors has the variance of the columns of the design matrix in its diagonal elements so that the penalty to the log likelhood is unitless. For main effects for categorical predictors with c categories, the rows and columns of the matrix contain a c-1 times c-1 sub-matrix that is used to compute the sum of squares about the mean of the c parameter values (setting the parameter to zero for the reference cell) as the penalty component for that predictor. This makes the penalty independent of the choice of the reference cell. If you specify `penalty.matrix`, you may set the rows and columns for certain parameters to zero so as to not penalize those parameters. Depending on `penalty`, some elements of `penalty.matrix` may be overridden automatically by setting them to zero. The penalty matrix that is used in the actual fit is penalty times diag(pf) times penalty.matrix times diag(pf), where pf is the vector of square roots of penalty factors computed from `penalty` by `Penalty.setup` in `Design.Misc`. If you specify `penalty.matrix` you must specify a nonzero value of `penalty` or no penalization will be done.
`tol` | singularity criterion (see `lrm.fit`)
`strata.penalty` | scalar penalty factor for the stratification factor, for the experimental `strat` variable
`var.penalty` | the type of variance-covariance matrix to be stored in the `var` component of the fit when penalization is used. The default is the inverse of the penalized information matrix. Specify `var.penalty="sandwich"` to use the sandwich estimator (see below under `var`), which limited simulation studies have shown yields variances estimates that are too low.
`weights` | a vector (same length as `y`) of possibly fractional case weights
`normwt` | set to code{TRUE} to scale `weights` so they sum to the length of `y`; useful for sample surveys as opposed to the default of frequency weighting
`...` | arguments that are passed to `lrm.fit`.

### Value

The returned fit object of `lrm` contains the following components in addition to the ones mentioned under the optional arguments.

`call` | calling expression
`freq` | table of frequencies for `Y` in order of increasing `Y`
`stats` | vector with the following elements: number of observations used in the fit, maximum absolute value of first derivative of log likelihood, model likelihood ratio chi-square, d.f., P-value, c index (area under ROC curve), Somers' D_{xy}, Goodman-Kruskal gamma, Kendall's tau-a rank correlations between predicted probabilities and observed response, the Nagelkerke R^2 index, and the Brier score computed with respect to Y > its lowest level. Probabilities are rounded to the nearest 0.002 in the computations or rank correlation indexes. In the case of penalized estimation, the `"Model L.R."` is computed without the penalty factor, and `"d.f."` is the effective d.f. from Gray's (1992) Equation 2.9. The P-value uses this corrected model L.R. chi-square and corrected d.f. The score chi-square statistic uses first derivatives which contain penalty components.
`fail` | set to `TRUE` if convergence failed (and `maxiter>1`)
`coefficients` | estimated parameters
`var` | estimated variance-covariance matrix (inverse of information matrix). If `penalty>0`, `var` is either the inverse of the penalized information matrix (the default, if `var.penalty="simple"`) or the sandwich-type variance - covariance matrix estimate (Gray Eq. 2.6) if `var.penalty="sandwich"`. For the latter case the simple information-matrix - based variance matrix is returned under the name `var.from.info.matrix`.
`effective.df.diagonal` | is returned if `penalty>0`. It is the vector whose sum is the effective d.f. of the model (counting intercept terms).
`u` | vector of first derivatives of log-likelihood
`deviance` | -2 log likelihoods (counting penalty components) When an offset variable is present, three deviances are computed: for intercept(s) only, for intercepts+offset, and for intercepts+offset+predictors. When there is no offset variable, the vector contains deviances for the intercept(s)-only model and the model with intercept(s) and predictors.
`est` | vector of column numbers of `X` fitted (intercepts are not counted)
`non.slopes` | number of intercepts in model
`penalty` | see above
`penalty.matrix` | the penalty matrix actually used in the estimation

### Author(s)

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

### References

Le Cessie S, Van Houwelingen JC: Ridge estimators in logistic regression. Applied Statistics 41:191–201, 1992.

Verweij PJM, Van Houwelingen JC: Penalized likelihood in Cox regression. Stat in Med 13:2427–2436, 1994.

Gray RJ: Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. JASA 87:942–951, 1992.

Shao J: Linear model selection by cross-validation. JASA 88:486–494, 1993.

Verweij PJM, Van Houwelingen JC: Crossvalidation in survival analysis. Stat in Med 12:2305–2314, 1993.

Harrell FE: Model uncertainty, penalization, and parsimony. ISCB Presentation on UVa Web page, 1998.

### See Also

`lrm.fit`, `predict.lrm`, `Design.trans`, `Design`, `glm`, `latex.lrm`, `residuals.lrm`, `na.delete`, `na.detail.response`, `naresid`, `pentrace`, `Design.Misc`, `vif`, `cr.setup`, `predab.resample`, `validate.lrm`, `calibrate`

### Examples

    #Fit a logistic model containing predictors age, blood.pressure, sex
    #and cholesterol, with age fitted with a smooth 5-knot restricted cubic
    #spline function and a different shape of the age relationship for males
    #and females.
    #
    n <- 1000    # define sample size
    set.seed(17) # so can reproduce the results
    age            <- rnorm(n, 50, 10)
    blood.pressure <- rnorm(n, 120, 15)
    cholesterol    <- rnorm(n, 200, 25)
    sex            <- factor(sample(c('female','male'), n,TRUE))
    label(age)            <- 'Age'      # label is in Hmisc
    label(cholesterol)    <- 'Total Cholesterol'
    label(blood.pressure) <- 'Systolic Blood Pressure'
    label(sex)            <- 'Sex'
    units(cholesterol)    <- 'mg/dl'   # uses units.default in Hmisc
    units(blood.pressure) <- 'mmHg'

    # Specify population model for log odds that Y=1
    L <- .4*(sex=='male') + .045*(age-50) +
      (log(cholesterol - 10)-5.2)*(-2*(sex=='female') + 2*(sex=='male'))
    # Simulate binary y to have Prob(y=1) = 1/[1+exp(-L)]
    y <- ifelse(runif(n) < plogis(L), 1, 0)
    cholesterol[1:3] <- NA   # 3 missings, at random

    ddist <- datadist(age, blood.pressure, cholesterol, sex)
    options(datadist='ddist')

    fit <- lrm(y ~ blood.pressure + sex * (age + rcs(cholesterol,4)),
                   x=TRUE, y=TRUE)
    #      x=TRUE, y=TRUE allows use of resid(), which.influence below
    #      could define d <- datadist(fit) after lrm(), but data distribution
    #      summary would not be stored with fit, so later uses of plot.Design
    #      or summary.Design would require access to the original dataset or
    #      d or specifying all variable values to summary, plot, nomogram
    anova(fit)
    plot(fit, age=NA, sex=NA)
    plot(fit, age=20:70, sex="male")   # need if datadist not used
    print(cbind(resid(fit,"dfbetas"), resid(fit,"dffits"))[1:20,])
    which.influence(fit, .3)
    # latex(fit)                       #print nice statement of fitted model
    #
    #Repeat this fit using penalized MLE, penalizing complex terms
    #(for nonlinear or interaction effects)
    #
    fitp <- update(fit, penalty=list(simple=0,nonlinear=10), x=TRUE, y=TRUE)
    effective.df(fitp)
    # or lrm(y ~ ..., penalty=...)

    #Get fits for a variety of penalties and assess predictive accuracy
    #in a new data set.  Program efficiently so that complex design
    #matrices are only created once.

    set.seed(201)
    x1 <- rnorm(500)
    x2 <- rnorm(500)
    x3 <- sample(0:1,500,rep=TRUE)
    L  <- x1+abs(x2)+x3
    y  <- ifelse(runif(500)<=plogis(L), 1, 0)
    new.data <- data.frame(x1,x2,x3,y)[301:500,]
    #
    for(penlty in seq(0,.15,by=.005)) {
      if(penlty==0) {
        f <- lrm(y ~ rcs(x1,4)+rcs(x2,6)*x3, subset=1:300, x=TRUE, y=TRUE)
        # True model is linear in x1 and has no interaction
        X <- f$x    # saves time for future runs - don't have to use rcs etc.
        Y <- f$y    # this also deletes rows with NAs (if there were any)
        penalty.matrix <- diag(diag(var(X)))
        Xnew <- predict(f, new.data, type="x", incl.non.slopes=FALSE)
        # expand design matrix for new data
        Ynew <- new.data$y
      } else f <- lrm.fit(X,Y, penalty.matrix=penlty*penalty.matrix)
    #
      cat("\nPenalty :",penlty,"\n")
      pred.logit <- f$coef[1] + (Xnew %*% f$coef[-1])
      pred <- plogis(pred.logit)
      C.index <- somers2(pred, Ynew)["C"]
      Brier   <- mean((pred-Ynew)^2)
      Deviance<- -2*sum( Ynew*log(pred) + (1-Ynew)*log(1-pred) )
      cat("ROC area:",format(C.index),"   Brier score:",format(Brier),
          "   -2 Log L:",format(Deviance),"\n")
    }
    #penalty=0.045 gave lowest -2 Log L, Brier, ROC in test sample for S+
    #
    #Use bootstrap validation to estimate predictive accuracy of
    #logistic models with various penalties
    #To see how noisy cross-validation estimates can be, change the
    #validate(f, ...) to validate(f, method="cross", B=10) for example.
    #You will see tremendous variation in accuracy with minute changes in
    #the penalty.  This comes from the error inherent in using 10-fold
    #cross validation but also because we are not fixing the splits.
    #20-fold cross validation was even worse for some
    #indexes because of the small test sample size.  Stability would be
    #obtained by using the same sample splits for all penalty values
    #(see above), but then we wouldn't be sure that the choice of the
    #best penalty is not specific to how the sample was split.  This
    #problem is addressed in the last example.
    #
    penalties <- seq(0,.7,by=.1)   # really use by=.02
    index <- matrix(NA, nrow=length(penalties), ncol=9,
                    dimnames=list(format(penalties),
              c("Dxy","R2","Intercept","Slope","Emax","D","U","Q","B")))
    i <- 0
    for(penlty in penalties) {
      cat(penlty, "")
      i <- i+1
      if(penlty==0) {
        f <- lrm(y ~ rcs(x1,4)+rcs(x2,6)*x3, x=TRUE, y=TRUE)  # fit whole sample
        X <- f$x
        Y <- f$y
        penalty.matrix <- diag(diag(var(X)))   # save time - only do once
      } else f <- lrm(Y ~ X, penalty=penlty,
                      penalty.matrix=penalty.matrix, x=TRUE,y=TRUE)
      val <- validate(f, method="boot", B=20)  # use larger B in practice
      index[i,] <- val[,"index.corrected"]
    }
    par(mfrow=c(3,3))
    for(i in 1:9) {
      plot(penalties, index[,i],
           xlab="Penalty", ylab=dimnames(index)[[2]][i])
      lines(lowess(penalties, index[,i]))
    }
    options(datadist=NULL)

    # Example of weighted analysis
    x <- 1:5
    y <- c(0,1,0,1,0)
    reps <- c(1,2,3,2,1)
    lrm(y ~ x, weights=reps)
    x <- rep(x, reps)
    y <- rep(y, reps)
    lrm(y ~ x)   # same as above

    #
    #Study performance of a modified AIC which uses the effective d.f.
    #See Verweij and Van Houwelingen (1994) Eq. (6).  Here AIC=chisq-2*df.
    #Also try as effective d.f. equation (4) of the previous reference.
    #Also study performance of Shao's cross-validation technique (which was
    #designed to pick the "right" set of variables, and uses a much smaller
    #training sample than most methods).  Compare cross-validated deviance
    #vs. penalty to the gold standard accuracy on a 7500 observation dataset.
    #Note that if you only want to get AIC or Schwarz Bayesian information
    #criterion, all you need is to invoke the pentrace function.
    #NOTE: the effective.df( ) function is used in practice
    #
    ## Not run:
    for(seed in c(339,777,22,111,3)){
    # study performance for several datasets
      set.seed(seed)
      n <- 175; p <- 8
      X <- matrix(rnorm(n*p), ncol=p) # p normal(0,1) predictors
      Coef <- c(-.1,.2,-.3,.4,-.5,.6,-.65,.7)  # true population coefficients
      L <- X %*% Coef                 # intercept is zero
      Y <- ifelse(runif(n)<=plogis(L), 1, 0)
      pm <- diag(diag(var(X)))
      #Generate a large validation sample to use as a gold standard
      n.val <- 7500
      X.val <- matrix(rnorm(n.val*p), ncol=p)
      L.val <- X.val %*% Coef
      Y.val <- ifelse(runif(n.val)<=plogis(L.val), 1, 0)
      #
      Penalty <- seq(0,30,by=1)
      reps <- length(Penalty)
      effective.df <- effective.df2 <- aic <- aic2 <- deviance.val <-
        Lpenalty <- single(reps)
      n.t <- round(n^.75)
      ncv <- c(10,20,30,40)     # try various no. of reps in cross-val.
      deviance <- matrix(NA,nrow=reps,ncol=length(ncv))
      #If model were complex, could have started things off by getting X, Y
      #penalty.matrix from an initial lrm fit to save time
      #
      for(i in 1:reps) {
        pen <- Penalty[i]
        cat(format(pen),"")
        f.full <- lrm.fit(X, Y, penalty.matrix=pen*pm)
        Lpenalty[i] <- pen* t(f.full$coef[-1]) %*% pm %*% f.full$coef[-1]
        f.full.nopenalty <- lrm.fit(X, Y, initial=f.full$coef, maxit=1)
        info.matrix.unpenalized <- solve(f.full.nopenalty$var)
        effective.df[i] <- sum(diag(info.matrix.unpenalized %*% f.full$var)) - 1
        lrchisq <- f.full.nopenalty$stats["Model L.R."]
        # lrm does all this penalty adjustment automatically (for var, d.f.,
        # chi-square)
        aic[i] <- lrchisq - 2*effective.df[i]
        #
        pred <- plogis(f.full$linear.predictors)
        score.matrix <- cbind(1,X) * (Y - pred)
        sum.u.uprime <- t(score.matrix) %*% score.matrix
        effective.df2[i] <- sum(diag(f.full$var %*% sum.u.uprime))
        aic2[i] <- lrchisq - 2*effective.df2[i]
        #
        #Shao suggested averaging 2*n cross-validations, but let's do only 40
        #and stop along the way to see if fewer is OK
        dev <- 0
        for(j in 1:max(ncv)) {
          s    <- sample(1:n, n.t)
          cof  <- lrm.fit(X[s,],Y[s],
                          penalty.matrix=pen*pm)$coef
          pred <- cof[1] + (X[-s,] %*% cof[-1])
          dev <- dev -2*sum(Y[-s]*pred + log(1-plogis(pred)))
          for(k in 1:length(ncv)) if(j==ncv[k]) deviance[i,k] <- dev/j
        }
        #
        pred.val <- f.full$coef[1] + (X.val %*% f.full$coef[-1])
        prob.val <- plogis(pred.val)
        deviance.val[i] <- -2*sum(Y.val*pred.val + log(1-prob.val))
      }
      postscript(hor=TRUE)   # along with graphics.off() below, allow plots
      par(mfrow=c(2,4))   # to be printed as they are finished
      plot(Penalty, effective.df, type="l")
      lines(Penalty, effective.df2, lty=2)
      plot(Penalty, Lpenalty, type="l")
      title("Penalty on -2 log L")
      plot(Penalty, aic, type="l")
      lines(Penalty, aic2, lty=2)
      for(k in 1:length(ncv)) {
        plot(Penalty, deviance[,k], ylab="deviance")
        title(paste(ncv[k],"reps"))
        lines(supsmu(Penalty, deviance[,k]))
      }
      plot(Penalty, deviance.val, type="l")
      title("Gold Standard (n=7500)")
      title(sub=format(seed),adj=1,cex=.5)
      graphics.off()
    }
    ## End(Not run)
    #The results showed that to obtain a clear picture of the penalty-
    #accuracy relationship one needs 30 or 40 reps in the cross-validation.
    #For 4 of 5 samples, though, the super smoother was able to detect
    #an accurate penalty giving the best (lowest) deviance using 10-fold
    #cross-validation.  Cross-validation would have worked better had
    #the same splits been used for all penalties.
    #The AIC methods worked just as well and are much quicker to compute.
    #The first AIC based on the effective d.f. in Gray's Eq. 2.9
    #(Verweij and Van Houwelingen (1994) Eq. 5 (note typo)) worked best.

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