mfp {mfp}

Fit a Multiple Fractional Polynomial Model

Description

Selects the multiple fractional polynomial (MFP) model which best predicts the outcome. The model may be a generalized linear model or a proportional hazards (Cox) model.

Usage

	mfp(formula, data, family = gaussian, method = c("efron", "breslow"),
	    subset = NULL, na.action = na.omit, init = NULL, alpha=0.05, 
	    select = 1, maxits = 20, keep = NULL, rescale = FALSE, 
	    verbose = FALSE, x = TRUE, y = TRUE)

Arguments

formula

a formula object, with the response of the left of a ~ operator, and the terms, separated by + operators, on the right. Fractional polynomial terms are indicated by fp. If a Cox PH model is required then the outcome should be specified using the Surv() notation used by coxph.

data

a data frame containing the variables occurring in the formula. If this is missing, the variables should be on the search list.

family

a family object - a list of functions and expressions for defining the link and variance functions, initialization and iterative weights. Families supported are gaussian, binomial, poisson, Gamma, inverse.gaussian and quasi. Additionally Cox models are specified using "cox".

method

a character string specifying the method for tie handling. This argument is used for Cox models only and has no effect for other model families. See 'coxph' for details.

subset

expression saying which subset of the rows of the data should be used in the fit. All observations are included by default.

na.action

function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with na.fail) is to create an error if any missing values are found.

init

vector of initial values of the iteration (in Cox models only).

alpha

sets the FP selection level for all predictors. Values for individual predictors may be changed via the fp function in the formula.

select

sets the variable selection level for all predictors. Values for individual predictors may be changed via the fp function in the formula.

maxits

maximum number of iterations for the backfitting stage.

keep

keep one or more variables in the model. The selection level for these variables will be set to 1.

rescale

logical; uses re-scaling to show the parameters for covariates on their original scale (default TRUE). Should only be used if no non-linear terms are selected.

verbose

logical; run in verbose mode (default FALSE).

x

logical; return the design matrix in the model object?

y

logical; return the response in the model object?

Details

The estimation algorithm processes the predictors in turn. Initially, mfp silently arranges the predictors in order of increasing P-value (i.e. of decreasing statistical significance) for omitting each predictor from the model comprising all the predictors with each term linear. The aim is to model relatively important variables before unimportant ones.

At the initial cycle, the best-fitting FP function for the first predictor is determined, with all the other variables assumed linear. The FP selection procedure is described below. The functional form (but NOT the estimated regression coefficients) for this predictor is kept, and the process is repeated for the other predictors in turn. The first iteration concludes when all the variables have been processed in this way. The next cycle is similar, except that the functional forms from the initial cycle are retained for all variables excepting the one currently being processed.

A variable whose functional form is prespecified to be linear (i.e. to have 1 df) is tested only for exclusion within the above procedure when its nominal P-value (selection level) according to select() is less than 1.

Updating of FP functions and candidate variables continues until the functions and variables included in the overall model do not change (convergence). Convergence is usually achieved within 1-4 cycles.

Model Selection

mfp uses a form of backward elimination. It start from a most complex permitted FP model and attempt to simplify it by reducing the df. The selection algorithm is inspired by the so-called "closed test procedure", a sequence of tests in each of which the "familywise error rate" or P-value is maintained at a prespecified nominal value such as 0.05.

The "closed test" algorithm for choosing an FP model with maximum permitted degree m=2 (4 df) for a single continuous predictor, x, is as follows:

1. Inclusion: test the FP in x for possible omission of x (4 df test, significance level determined by select). If x is significant, continue, otherwise drop x from the model.

2. Non-linearity: test the FP in x against a straight line in x (3 df test, significance level determined by alpha). If significant, continue, otherwise the chosen model is a straight line.

3. Simplification: test the FP with m=2 (4 df) against the best FP with m=1 (2 df) (2 df test at alpha level). If significant, choose m=2, otherwise choose m=1.

All significance tests are carried out using an approximate P-value calculation based on a difference in deviances (-2 x log likelihood) having a chi-squared or F distribution, depending on the regression in use. Therefore, each of the tests in the procedure maintains a significance level only approximately equal to select. The algorithm is thus not truly a closed procedure. However, for a given significance level it does provide some protection against over-fitting, that is against choosing over-complex MFP models.

Value

an object of class mfp is returned which either inherits from both glm and lm or coxph.

Side Effects

details are produced on the screen regarding the progress of the backfitting routine. At completion of the algorithm a table is displayed showing the final powers selected for each variable along with other details.

Known Bugs

glm models should not be specified without an intercept term as the software does not yet allow for that possibility.

Author(s)

Gareth Ambler and Axel Benner

References

Ambler G, Royston P (2001) Fractional polynomial model selection procedures: investigation of Type I error rate. Journal of Statistical Simulation and Computation 69: 89–108.

Benner A (2005) mfp: Multivariable fractional polynomials. R News 5(2): 20–23.

Royston P, Altman D (1994) Regression using fractional polynomials of continuous covariates. Appl Stat. 3: 429–467.

Sauerbrei W, Royston P (1999) Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. Journal of the Royal Statistical Society (Series A) 162: 71–94.

See Also

mfp.object, fp, glm

Examples

	data(GBSG)
	f <- mfp(Surv(rfst, cens) ~ fp(age, df = 4, select = 0.05)
                 + fp(prm, df = 4, select = 0.05), family = cox, data = GBSG)
	print(f)
	survfit(f$fit)  # use proposed coxph model fit for survival curve estimation