pentrace {Design} | R Documentation

## Trace AIC and BIC vs. Penalty

### Description

For an ordinary unpenalized fit from `lrm` or `ols` and for a vector or list of penalties, fits a series of logistic or linear models using penalized maximum likelihood estimation, and saves the effective degrees of freedom, Akaike Information Criterion (AIC), Schwarz Bayesian Information Criterion (BIC), and Hurvich and Tsai's corrected AIC (AIC_c). Optionally `pentrace` can use the `nlminb` function to solve for the optimum penalty factor or combination of factors penalizing different kinds of terms in the model. The `effective.df` function prints the original and effective degrees of freedom for a penalized fit or for an unpenalized fit and the best penalization determined from a previous invocation of `pentrace` if `method="grid"` (the default). The effective d.f. is computed separately for each class of terms in the model (e.g., interaction, nonlinear). A `plot` method exists to plot the results, and a `print` method exists to print the most pertinent components. Both AIC and BIC may be plotted if there is only one penalty factor type specified in `penalty`. Otherwise, the first two types of penalty factors are plotted, showing only the AIC.

### Usage

    pentrace(fit, penalty, penalty.matrix,
             method=c('grid','optimize'),
             which=c('aic.c','aic','bic'), target.df,
             fitter, pr=FALSE, tol=1e-7,
             keep.coef=FALSE, complex.more=TRUE, verbose=FALSE, maxit=12, subset)

    effective.df(fit, object)

    ## S3 method for class 'pentrace':
    print(x, ...)

    ## S3 method for class 'pentrace':
    plot(x, method=c('points','image'),
         which=c('effective.df','aic','aic.c','bic'), pch=2, add=FALSE,
         ylim, ...)

### Arguments

`fit` | a result from `lrm` or `ols` with `x=TRUE, y=TRUE` and without using `penalty` or `penalty.matrix` (or optionally using penalization in the case of `effective.df`)
`penalty` | can be a vector or a list. If it is a vector, all types of terms in the model will be penalized by the same amount, specified by elements in `penalty`, with a penalty of zero automatically added. `penalty` can also be a list in the format documented in the `lrm` function, except that elements of the list can be vectors. The `expand.grid` function is invoked by `pentrace` to generate all possible combinations of penalties. For example, specifying `penalty=list(simple=1:2, nonlinear=1:3)` will generate 6 combinations to try, so that the analyst can attempt to determine whether penalizing more complex terms in the model more than the linear or categorical variable terms will be beneficial. If `complex.more=TRUE`, it is assumed that the variables given in `penalty` are listed in order from less complex to more complex. With `method="optimize"` `penalty` specifies an initial guess for the penalty or penalties. If all term types are to be equally penalized, `penalty` should be a single number, otherwise it should be a list containing single numbers as elements, e.g., `penalty=list(simple=1, nonlinear=2)`. Experience has shown that the optimization algorithm is more likely to find a reasonable solution when the starting value specified in `penalty` is too large rather than too small.
`object` | an object returned by `pentrace`. For `effective.df`, `object` can be omitted if the `fit` was penalized.
`penalty.matrix` | see `lrm`
`method` | The default is `method="grid"` to print various indexes for all combinations of penalty parameters given by the user. Specify `method="optimize"` to have `pentrace` use `nlminb` to solve for the combination of penalty parameters that gives the maximum value of the objective named in `which`, or, if `target.df` is given, to find the combination that yields `target.df` effective total degrees of freedom for the model. When `target.df` is specified, `method` is set to `"optimize"` automatically. For `plot.pentrace` this parameter applies only if more than one penalty term-type was used. The default is to use open triangles whose sizes are proportional to the ranks of the AICs, plotting the first two penalty factors respectively on the x and y axes. Use `method="image"` to plot an image plot.
`which` | the objective to maximize for either `method`. Default is `"aic.c"` (corrected AIC). For `plot.pentrace`, `which` is a vector of names of criteria to show; default is to plot all 4 types, with effective d.f. in its own separate plot
`target.df` | applies only to `method="optimize"`. See `method`. `target.df` makes sense mainly when a single type of penalty factor is specified.
`fitter` | a fitting function. Default is `lrm.fit` (`lm.pfit` is always used for `ols`).
`pr` | set to `TRUE` to print intermediate results
`tol` | tolerance for declaring a matrix singular (see `lrm.fit, solvet`)
`keep.coef` | set to `TRUE` to store matrix of regression coefficients for all the fits (corresponding to increasing values of `penalty`) in object `Coefficients` in the returned list. Rows correspond to penalties, columns to regression parameters.
`complex.more` | By default if `penalty` is a list, combinations of penalties for which complex terms are penalized less than less complex terms will be dropped after `expand.grid` is invoked. Set `complex.more=FALSE` to allow more complex terms to be penalized less. Currently this option is ignored for `method="optimize"`.
`verbose` | set to `TRUE` to print number of intercepts and sum of effective degrees of freedom
`maxit` | maximum number of iterations to allow in a model fit (default=12). This is passed to the appropriate fitter function with the correct argument name. Increase `maxit` if you had to when fitting the original unpenalized model.
`subset` | a logical or integer vector specifying rows of the design and response matrices to subset in fitting models. This is most useful for bootstrapping `pentrace` to see if the best penalty can be estimated with little error so that variation due to selecting the optimal penalty can be safely ignored when bootstrapping standard errors of regression coefficients and measures of predictive accuracy. See an example below.
`x` | a result from `pentrace`
`pch` | used for `method="points"`
`add` | set to `TRUE` to add to an existing plot. In that case, the effective d.f. plot is not re-drawn, but the AIC/BIC plot is added to.
`ylim` | 2-vector of y-axis limits for plots other than effective d.f.
`...` | other arguments passed to `plot`, `lines`, or `image`

### Value

a list of class `"pentrace"` with elements `penalty, df, objective, fit, var.adj, diag, results.all`, and optionally `Coefficients`. The first 6 elements correspond to the fit that had the best objective as named in the `which` argument, from the sequence of fits tried. Here `fit` is the fit object from `fitter` which was a penalized fit, `diag` is the diagonal of the matrix used to compute the effective d.f., and `var.adj` is Gray (1992) Equation 2.9, which is an improved covariance matrix for the penalized beta. `results.all` is a data frame whose first few variables are the components of `penalty` and whose other columns are `df, aic, bic, aic.c`. `results.all` thus contains a summary of results for all fits attempted. When `method="optimize"`, only two components are returned: `penalty` and `objective`, and the object does not have a class.

### Author(s)

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

### References

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

Hurvich CM, Tsai, CL: Regression and time series model selection in small samples. Biometrika 76:297–307, 1989.

### See Also

`lrm`, `ols`, `solvet`, `Design.Misc`, `image`

### Examples

    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))
    # 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)

    f <- lrm(y ~ blood.pressure + sex * (age + rcs(cholesterol,4)),
             x=TRUE, y=TRUE)
    p <- pentrace(f, seq(.2,1,by=.05))
    plot(p)
    p$diag      # may learn something about fractional effective d.f.
                # for each original parameter
    pentrace(f, list(simple=c(0,.2,.4), nonlinear=c(0,.2,.4,.8,1)))

    # Bootstrap pentrace 5 times, making a plot of corrected AIC plot with 5 reps
    n <- nrow(f$x)
    plot(pentrace(f, seq(.2,1,by=.05)), which='aic.c',
         col=1, ylim=c(30,120)) #original in black
    for(j in 1:5)
      plot(pentrace(f, seq(.2,1,by=.05), subset=sample(n,n,TRUE)),
           which='aic.c', col=j+1, add=TRUE)

    # Find penalty giving optimum corrected AIC.  Initial guess is 1.0
    if(!.R.) pentrace(f, 1, method='optimize')

    # Find penalty reducing total regression d.f. effectively to 5
    if(!.R.) pentrace(f, 1, target.df=5)

    # Re-fit with penalty giving best aic.c without differential penalization
    f <- update(f, penalty=p$penalty)
    effective.df(f)

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