ppm: Fitting a Proportional Probability Model

Description

ppm provides the maximum likelihood estimate for ordinal outcomes (J>2 categories) and a Generalized Linear Model with the log link with the assumption of proportionality. That is, ppm determines the MLE for log[P(y <= j)]= cut_j + X beta subject to [cut_j-1 <= cut_j ] and [cut_j + X beta <=0]. This implementation uses constrOptim to determine the MLE and so the results should correctly account for the restricted parameter space. A proposed test for proportionality is included in lcpm.

Usage

ppm(
  formula.linear,
  data,
  conf.level = 0.95,
  y.order = NULL,
  startval = NULL,
  less.than.0 = TRUE,
  control.list = NULL,
  eps.outer = NULL,
  ...
)

Arguments

formula.linear

an object of class "formula": a symbolic description of the linear model to be fitted.

data

dataframe containing the data in linear model.

conf.level

optional confidence level (1-alpha) defaulted to 0.95.

y.order

optional if y contains ordered integer categories 1:J. If y is not ordered integer 1:J then this is a vector with the ordinal values for y ranging from the lowest to largest ordinal outcome. See Examples below.

startval

optional vector of the starting values.

less.than.0

optional logical for constraint cut_j <= 0 for all j=1:(J-1). Default is TRUE.

control.list

optional list of controls for constrOptim.

eps.outer

option for constrOptim.

...

Additional arguments for built in functions.

Value

list of class "ppm" is returned containing:

coefficients

vector of the estimate of cut_j and beta

vector of the estimate of standard errors

vcov

matrix of the inverse of the negative Hessian

fitted.values

matrix of unique covariates and the corresponding estimate of the cumulative probabilities: exp(X %*% coefficients)

loglik

numerical value of the log-likelihood at the maximum likelihood estimate

barrier.value

value of mu in the log-barrier algorithm

outer.iterations

value of the number of outer iterations

formula

formula in the call of ppm

startvalues

vector of the starting values for constrained optimization algorithm

References

Singh, G; Fick, G.H. (accepted) Ordinal outcomes: a cumulative probability model with the log link and an assumption of proportionality. Statistics in Medicine.

Examples

Run this code

# NOT RUN {
# 2 examples below showing the use of y.order if outcome are not integers 1:J.

# Example 1:

var_a <- c(rep(0,60),rep(1,60))
var_b <- c(rep(0,90),rep(1,30))
y1<-c(rep(2,5),rep(3,10),rep(5,5),rep(10,10),
rep(2,5),rep(3,10),rep(5,10),rep(10,5),
rep(2,10),rep(3,5),rep(5,5),rep(10,10),
rep(2,10),rep(3,5),rep(5,10),rep(10,5))

testdata<-data.frame(y=y1,var_a=var_a,var_b=var_b)

# PPM estimates for proportional model
test1<-ppm( y ~ var_a + var_b, data=testdata, y.order=c(2,3,5,10))
summary(test1)

# Example 2:

y2<-c(rep("a",5),rep("b",10),rep("c",5),rep("d",10),
rep("a",5),rep("b",10),rep("c",10),rep("d",5),
rep("a",10),rep("b",5),rep("c",5),rep("d",10),
rep("a",10),rep("b",5),rep("c",10),rep("d",5))
testdata2<-data.frame(y=y2,var_a=var_a,var_b=var_b)
test2<-ppm(y~var_a + var_b , data=testdata2, y.order=c("a","b","c","d"))
summary(test2)
# }