A transformation can be used to reparametrize parameters for
optimization, such that the likelihoods are evaluated at
transform(parameters), where parameters are
unconstrained. This should be specified when fitting linear excess
relative risk and linear-exponential models to ensure nonnegative
odds/risk/hazard. The included function transform1 applies
an exponential transformation to the desired parameters, see below.
When supplying a function to transform, this should be a
function of the full parameter vector, returning a full transformed
parameter vector. The parameter order is the same order used by
coef() for the fitted method. For models without
subgroup-coded modifiers, this order is \(\alpha_0, \mathbf \alpha, \beta_1, \beta_2,
\mathbf \beta_{m1}, \mathbf \beta_{m2}, \sigma\), where \(\mathbf \alpha\), \(\mathbf\beta_{m1}\) and \(\mathbf \beta_{m2}\) can be vectors, with
lengths matching the covariates and modifier design columns,
respectively. \(\sigma\) is only
included for the linear model (Gaussian family), and no intercept is
included for the proportional hazards and conditional logistic models.
For multinomial models, the full parameter vector is the concatenation
of the parameter vectors for the non-reference outcome levels. For
subgroup-coded modifiers, user-specified transformations act on the
reported subgroup-effect scale before ameras maps those subgroup effects
to the internal reference-plus-contrast scale used by the
likelihood.
When no transformation is specified and the linear ERR model is used,
transform1 is used for ERR parameters \(\beta_1\) and \(\beta_2\) by default, with lower limits
\(-1/max(D)\) for linear dose-response
and \((0,-1/max(D^2))\) for
linear-quadratic dose-response, respectively (see below). For the
linear-exponential model, a lower limit of 0 is used for \(\beta_1\), and no transformation is used
for \(\beta_2\). If
reference-plus-contrast effect modifiers are specified, no default
transformation is used for modifier contrast parameters. With
subgroup-specific modifier coding, the default transformation is applied
to the subgroup dose-effect parameters. When negative RRs are obtained
during optimization, an error will be generated and a different
transformation or bounds should be used; for ERR and linear-exponential
models with effect modifiers, subgroup-specific coding can be more
numerically robust because bounds are applied directly to each subgroup
dose-effect parameter. All output is returned in the original
parametrization given in ?ameras. The Jacobian of the
transformation (transform.jacobian) is required when using
a transformation with methods other than BMA. For
transform1, the Jacobian is given by
transform1.jacobian.
The included function transform1 applies the exponential
transformation \(f(\theta_i)=\exp(\theta_i)+LB_i\) to one or
multiple components of parameter vector \(\mathbf \theta\), where \(LB_i\) are lower limits that can be
different for each component. In particular, a vector of indices of
parameters to be transformed and a vector of corresponding lower bounds
LB can be supplied to arguments index.t and
lowlimit, respectively, resulting in transformed parameters
\(f(\theta_i)=\exp(\theta_i)+\text{LB}_i\).
In particular, transform1 and
transform1.jacobian are defined as follows:
transform1 <- function(params, index.t=1:length(params), lowlimit=rep(0,length(index.t)),
boundcheck=FALSE, boundtol=1e-3, ...){
if(length(index.t)!=length(lowlimit))
stop("Length mismatch between index.t and lowlimit")
if(any(!(index.t %in% 1:length(params))))
stop("Incorrect indices for transformation specified")
params[index.t] <- exp(params[index.t]) + lowlimit
if(boundcheck){
if(any(params[index.t]-lowlimit < boundtol))
warning(paste0("WARNING: one or multiple parameter estimates within ", boundtol, " of
lower bounds. Try different bounds or starting values."))
}
return(params)
}
transform1.jacobian <- function(params, index.t=1:length(params), ...){
if(any(!(index.t %in% 1:length(params))))
stop("Incorrect indices for transformation specified")
grad <- rep(1, length(params))
grad[index.t] <- exp(params[index.t])
if(length(params)>1){
return(diag(grad))
} else{
return(matrix(grad))
}
}If you wish to supply your own transformation, it is helpful to start
from the definition of transform1. It is also important to
keep the following in mind:
params
and ..., where params is the full parameter
vector. Parameters are in a specific order (see above), in case of doubt
it is always possible to run ameras with the default
settings to verify the correct order from the result.amerasboundcheck is an argument to the function,
it should be a logical. When transforming parameters after the
optimization, the transformation is called with
boundcheck=TRUE. If boundcheck is not an
argument of the function, this is ignored.See the definition of transform1 above for an example of
how to apply the transformation only to specific parameters, and how to
use extra arguments.
As an example, suppose instead of the exponential transformation from
transform1, for the parameters \(\beta_1\) and \(\beta_2\) we wish to use the sigmoid
transformation \(f: \mathbb{R} \rightarrow
(a,b)\) given by \[f_{a_i,b_i}(\theta_i)= a_i + (b_i-a_i)
\frac{1}{1+\exp(-\theta_i)}.\] Then, using
transform1 as a starting point, we can define the
transformation as follows (note that since \(\mathbf{a}\) and \(\mathbf{b}\) act as bounds, we use an
updated bound check):
transform.sigmoid <- function(params, index.t=1:length(params), a=rep(0,length(index.t)),
b=rep(1,length(index.t)), boundcheck=FALSE, boundtol=1e-3, ...){
if(length(index.t)!=length(a) | length(index.t) != length(b))
stop("Length mismatch between index.t, a, and b")
if(any(!(index.t %in% 1:length(params))))
stop("Incorrect indices for transformation specified")
params[index.t] <- a + (b-a) * 1/(1+exp(-1*params[index.t]))
if(boundcheck){
if(any( (params[index.t]-a < boundtol) | (b-params[index.t] < boundtol)))
warning(paste0("WARNING: one or multiple parameter estimates within ", boundtol,
" of bounds. Try different bounds or starting values."))
}
return(params)
}Next, noting that \(\mathrm{d}f_{a_i,b_i}/d\theta_i =
(b_i-a_i)\exp(-\theta_i)/\{1+\exp(-\theta_i) \}^2\), we can
define the Jacobian as follows, using transform1.jacobian
as a starting point:
transform.sigmoid.jacobian <- function(params, index.t=1:length(params),
a=rep(0,length(index.t)), b=rep(1,length(index.t)), ...){
if(length(index.t)!=length(a) | length(index.t) != length(b))
stop("Length mismatch between index.t, a, and b")
if(any(!(index.t %in% 1:length(params))))
stop("Incorrect indices for transformation specified")
grad <- rep(1, length(params))
grad[index.t] <- (b-a)*exp(-1*params[index.t])/(1+exp(-1*params[index.t]))^2
if(length(params)>1){
return(diag(grad))
} else{
return(matrix(grad))
}
}Now let us try this transformation on the example data using
regression calibration for a linear-quadratic ERR model. Note that all
parameters are returned after transformation, and so there should be no
difference between using transform1 and
transform.sigmoid. First, we fit the model using the
sigmoid transformation:
data(data, package="ameras")
fit.ameras.sigmoid <- ameras(Y.binomial~dose(V1:V10, deg=2, model="ERR")+X1+X2, data=data,
family="binomial", methods="RC", transform=transform.sigmoid,
transform.jacobian=transform.sigmoid.jacobian, index.t=4:5)
#> Fitting RC
summary(fit.ameras.sigmoid)
#> Call:
#> ameras(formula = Y.binomial ~ dose(V1:V10, deg = 2, model = "ERR") +
#> X1 + X2, data = data, family = "binomial", methods = "RC",
#> transform = transform.sigmoid, transform.jacobian = transform.sigmoid.jacobian,
#> index.t = 4:5)
#>
#> Rows: 3000
#>
#> Total CPU runtime: 0.2 seconds
#>
#> CPU runtime in seconds by method:
#>
#> Method Fit CI Total
#> RC 0.191 0.0 0.191
#>
#> Summary of coefficients by method:
#>
#> Method Term Estimate SE
#> RC (Intercept) -0.87358 0.09758
#> RC X1 0.44587 0.07672
#> RC X2 -0.33552 0.09610
#> RC dose 0.04875 0.21281
#> RC dose_squared 0.28764 0.08100
#>
#> Note: confidence intervals not yet computed. Use confint() to add them.Next with default settings, using transform1:
fit.ameras.transform1 <- ameras(Y.binomial~dose(V1:V10, deg=2, model="ERR")+X1+X2, data=data,
family="binomial", methods="RC")
#> Fitting RC
summary(fit.ameras.transform1)
#> Call:
#> ameras(formula = Y.binomial ~ dose(V1:V10, deg = 2, model = "ERR") +
#> X1 + X2, data = data, family = "binomial", methods = "RC")
#>
#> Rows: 3000
#>
#> Total CPU runtime: 0.2 seconds
#>
#> CPU runtime in seconds by method:
#>
#> Method Fit CI Total
#> RC 0.165 0.0 0.165
#>
#> Summary of coefficients by method:
#>
#> Method Term Estimate SE
#> RC (Intercept) -0.87359 0.09759
#> RC X1 0.44587 0.07672
#> RC X2 -0.33552 0.09610
#> RC dose 0.04878 0.21283
#> RC dose_squared 0.28763 0.08100
#>
#> Note: confidence intervals not yet computed. Use confint() to add them.As expected, there is no difference between the results.