German Breast Cancer Study Group

Description

A data frame containing the observations from the GBSG study.

Usage

data(GBSG)

Format

This data frame contains the observations of 686 women:

id

patient id 1...686.

htreat

hormonal therapy, a factor at two levels 0 (no) and 1 (yes).

age

of the patients in years.

menostat

menopausal status, a factor at two levels 1 (premenopausal) and 2 (postmenopausal).

tumsize

tumor size (in mm).

tumgrad

tumor grade, a ordered factor at levels 1 < 2 < 3.

posnodal

number of positive nodes.

prm

progesterone receptor (in fmol).

esm

estrogen receptor (in fmol).

rfst

recurrence free survival time (in days).

cens

censoring indicator (0 censored, 1 event).

References

M. Schumacher, G. Basert, H. Bojar, K. Huebner, M. Olschewski, W. Sauerbrei, C. Schmoor, C. Beyerle, R.L.A. Neumann and H.F. Rauschecker for the German Breast Cancer Study Group (1994). Randomized 2 \times 2 trial evaluating hormonal treatment and the duration of chemotherapy in node-positive breast cancer patients. Journal of Clinical Oncology, 12, 2086–2093.

W. Sauerbrei and P. Royston (1999). Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. Journal of the Royal Statistics Society Series A, Volume 162(1), 71–94.

Examples

    data(GBSG)
    mfp(Surv(rfst, cens) ~ fp(age, df = 2, select = 0.05)
                 + fp(prm, df = 4, select = 0.05), family = cox, data = GBSG)

percentage of body fat determined by underwater weighing

Description

A data frame containing the estimates of the percentage of body fat determined by underwater weighing and various body circumference measurements for 252 men.

Source: Roger W. Johnson (1996), ”Fitting Percentage of Body Fat to Simple Body Measurements”, Journal of Statistics Education. Original data are from K. Penrose, A. Nelson, and A. Fisher (1985), ”Generalized Body Composition Prediction Equation for Men Using Simple Measurement Techniques” (abstract), Medicine and Science in Sports and Exercise, 17(2), 189.

Data were supplied by Dr. A. Garth Fisher, Human Performance Research Center, Brigham Young University, who gave permission to freely distribute the data and use them for non-commercial purposes. Note, however, that there seem to be a few errors. For instance, body densities for cases 48, 76, and 96 seem to have one digit in error in the two body fat percentage values. Also note a man (case 42) over 200 pounds in weight who is less than 3 feet tall (the height should presumably be 69.5 inches, not 29.5 inches). Percent body fat estimates are truncated to zero when negative (case 182).

Usage

data(bodyfat)

Format

This data frame contains the observations of 252 men:

case

Case number.

brozek

Percent body fat using Brozek's equation: 457/Density - 414.2

siri

Percent body fat using Siri's equation: 495/Density - 450

density

Density determined from underwater weighing (gm/cm**3).

age

Age (years).

weight

Weight (lbs).

height

Height (inches).

neck

Neck circumference (cm).

chest

Chest circumference (cm).

abdomen

Abdomen circumference (cm) ”at the umbilicus and level with the iliac crest”.

hip

Hip circumference (cm).

thigh

Thigh circumference (cm).

knee

Knee circumference (cm).

ankle

Ankle circumference (cm).

biceps

Biceps (extended) circumference (cm).

forearm

Forearm circumference (cm).

wrist

Wrist circumference (cm) ”distal to the styloid processes”.

Source

e.g. http://lib.stat.cmu.edu/datasets/bodyfat

References

R.W. Johnson (1996). Fitting percentage of body fat to simple body measurements. Journal of Statistics Education [Online], 4(1).

K.W. Penrose, A.G. Nelson, A.G. Fisher (1985). Generalized body composition prediction equation for men using simple measurement techniques. Medicine and Science in Sports and Exercise, 17, 189.

P. Royston, W. Sauerbrei (2004). Improving the robustness of fractional polynomial models by preliminary covariate transformation. Submitted.

Examples

    data(bodyfat)
    bodyfat$height[bodyfat$case==42] <- 69.5   # apparent error
    bodyfat <- bodyfat[-which(bodyfat$case==39),]  # cp. Royston $\amp$ Sauerbrei, 2004
    mfp(siri ~ fp(age, df = 4, select = 0.1) + fp(weight, df = 4, select = 0.1)
                 + fp(height, df = 4, select = 0.1), family = gaussian, data = bodyfat)

Family Objects for Cox Proportional Regression Models

Description

Family objects provide a convenient way to specify the details of the models used by functions such as 'glm'. See the documentation for 'glm' for details.

Usage

cox()

Fractional Polynomial Transformation

Description

This function defines a fractional polynomial object for a quantitative input variable x.

Usage

fp(x, df = 4, select = NA, alpha = NA, scale=TRUE)

Arguments

Examples

## Not run: 
	fp(x, df = 4, select = 0.05, scale = FALSE)

## End(Not run)

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

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

Multiple Fractional Polynomial Model Object

Description

Objects returned by fitting fractional polynomial model objects.

These are objects representing fitted mfp models. Class mfp inherits from either glm or coxph depending on the type of model fitted.

Value

In addition to the standard glm/coxph components the following components are included in a mfp object.

See Also

mfp, glm.object

Predict method for mfp fits

Description

Obtains predictions from an "mfp" object

Usage

  ## S3 method for class 'mfp'
predict(object, newdata,
  type = c("link", "response", "lp", "risk", "expected", "terms"), terms,
  ref = NULL, seq = NULL, se.fit = FALSE, dispersion = NULL,
  na.action = na.pass, collapse, safe = FALSE, ...)

Arguments

Value

If type = "terms", a list with term predictions for each variable including the contrast to the reference value, standard errors, and contrasts of the first and second derivative.

Otherwise, a prediction object based on predict.glm or predict.coxph.

See Also

predict.glm, predict.coxph