Fitting a Multi-Parameter Regression (MPR) model.
Fits a Multi-Parameter Regression (MPR) model using a Newton-type algorithm via the
the name of the parametric distribution to be used in the model. See
an optional vector of initial values for the optimisation routine. If missing, default values
are used. One may also set
a positive integer specifying the maximum number of iterations to be performed before the
optimisation procedure is terminated. This is supplied to
additional arguments to be passed to
Multi-Parameter Regression (MPR) models are generated by allowing multiple distributional parameters to depend on covariates, for example, both the scale and shape parameters. This is in contrast to the more typical approach where covariates enter a model only through one distributional parameter. As these standard models have a single regression component, we may refer to them as Single Parameter Regression (SPR) models and, clearly, they are special cases of MPR models. The parameter through which covariates enter such SPR models may be referred to as the “interest” parameter since it generally has some specific subject-matter importance. However, this standard approach neglects other parameters which may also be important in describing the phenomenon at hand. The MPR approach generalises the standard SPR approach by viewing all distributional parameters as interest parameters in which covariate effects can be investigated.
In the context of survival analysis (currently the focus of the
mpr package), the Weibull model
is one of the most popular parametric models. Its hazard function is given by
h(t) = λ γ t^(γ - 1)
where λ > 0, the scale parameter, controls the overall magnitude of h(t) and γ > 0, the shape parameter, controls its time evolution. In the standard SPR Weibull model, λ depends on covariates via log λ = x' β leading to a proportional hazards (PH) model. The MPR model generalises this by allowing both parameters to depend on covariates as follows
log λ = x' β
log γ = z' α
where x and z are the scale and shape covariate vectors (which may or may not contain covariates in common) and β and α are the corresponding regression coefficients.
Note that the log-link is used above to ensure positivity of the parameters. More generally, we may have
g1(λ) = x' β
g2(γ) = z' α
where g1(.) and g2(.) are appropriate link functions.
mpr function does not
allow the user to alter these link functions but, rather, uses the following default link functions:
log-link (for parameters which must be positive) and identity-link (for parameters
which are unconstrained). Although the two-parameter Weibull distribution is discussed here
(due to its popularity), other distributions may have additional shape parameters, for example,
g3(ρ) = w' τ
where w and τ are the vectors of covariates and regression coefficients for this
additional shape component. See
distributions for further details on the distributions
The struture of the
formula within the
mpr function is, for example,
Surv(time, status) ~ list(~ x1 + x2, ~ x1) which clearly generalises the typical
formula used in
standard models (i.e., those with only one regression component) in the sense that the right hand side is a
list of one-sided
formula objects. Note the requirement that the
precedes each element within the
list. Specifically, the example shown here represents the case
where the covariates
x2 appear in the first regression component, λ,
and the covariate
x2 appears in the second regression component, γ. If there was a
third regression component, ρ, then there would be an additional component in the
Surv(time, status) ~ list(~ x1 + x2, ~ x1, ~ x1). The
mpr function also accepts
more typical two-sided
formula objects, such as
Surv(time, status) ~ x1 + x2, which imply that
the terms on the right hand side appear in each of the regression components.
mpr returns an object of class “
summary.mpr) can be used to obtain and print a summary
of the results. The the generic accessor function
coefficients extracts the
regression coefficient vectors. One can also apply
predict various quantites from the fitted
mpr model. A stepwise variable selection procedure has
been implemented for
mpr models - see
An object of class
mpr is a
list containing the following components:
the variance-covariance matrix for the estimates.
the values of the (negative) score functions from
the number of regression components in the model, i.e., the number of distributional parameters in the
a record of the names of all variables (i.e., covariates) used in fitting.
a record of the levels of any
the matched call.
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# Veterans' administration lung cancer data data(veteran, package="survival") head(veteran) # treatment variable, "trt", in scale (lambda) and shape (gamma) # components of a Weibull model mpr(Surv(time, status) ~ list(~ trt, ~ trt), data=veteran, family="Weibull") # same as first model mpr(Surv(time, status) ~ trt, data=veteran, family="Weibull") # now with "celltype" also appearing in the scale mpr(Surv(time, status) ~ list(~ trt + celltype, ~ trt), data=veteran, family="Weibull") # trt in scale only (this is a PH Weibull model) mpr(Surv(time, status) ~ list(~ trt, ~ 1), data=veteran, family="Weibull") # trt in all three components (scale and two shape parameters) of a Burr model mpr(Surv(time, status) ~ list(~ trt, ~ trt, ~ trt), data=veteran, family="Burr") # use of summary mod1 <- mpr(Surv(time, status) ~ list(~ trt, ~ trt), data=veteran) summary(mod1)
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