Package 'ameras'

Analyze Multiple Exposure Realizations in Association Studies

Description

Analyze association studies with multiple realizations of a noisy or uncertain exposure. These can be obtained from e.g. a two-dimensional Monte Carlo dosimetry system (Simon et al 2015 <doi:10.1667/RR13729.1>) to characterize exposure uncertainty. Methods include regression calibration (Carroll et al. 2006 doi:10.1201/9781420010138), extended regression calibration (Little et al. 2023 doi:10.1038/s41598-023-42283-y), Monte Carlo maximum likelihood (Stayner et al. 2007 doi:10.1667/RR0677.1), frequentist model averaging (Kwon et al. 2023 doi:10.1371/journal.pone.0290498), and Bayesian model averaging (Kwon et al. 2016 doi:10.1002/sim.6635). Supported model families are Gaussian, binomial, multinomial, Poisson, proportional hazards, and conditional logistic.

Details

The function used to fit models is ameras. To attach confidence/credible intervals, use the method confint. To visualize the exposure uncertainty in the dose realizations, use ecdfplot.

Author(s)

Sander Roberti <[email protected]>, William Wheeler <[email protected]>, Ruth Pfeiffer <[email protected]>, and Deukwoo Kwon <[email protected]

References

Roberti, S., Kwon D., Wheeler W., Pfeiffer R. (in preparation). ameras: An R Package to Analyze Multiple Exposure Realizations in Association Studies

---

Analyze Multiple Exposure Realizations

Description

Fit regression models accounting for exposure uncertainty using multiple Monte Carlo exposure realizations. Six outcome model families are supported. The first is the Gaussian family for continuous outcomes,

$Y_i \sim N(\mu_i, \sigma^2),$

with $\mu_i = \alpha_0 + \bm X_i^T \bm \alpha +\beta_1 D_i+\beta_2 D_i^2 + \bm M_i^T \bm \beta_{m1}D_i + \bm M_i^T \bm \beta_{m2}D_i^2$. Here $\bm X_i$ are covariates, $D_i$ is the exposure with measurement error, and $\bm M_i$ are binary effect modifiers. The quadratic exposure terms and effect modification are optional.

For non-Gaussian families, three relative risk models for the main exposure are supported, the usual exponential $RR_i=\exp(\beta_1 D_i+\beta_2 D_i^2+ \bm M_i^T \bm \beta_{m1}D_i + \bm M_i^T \bm \beta_{m2} D_i^2)$ and the linear excess relative risk (ERR) model $RR_i= 1+\beta_1 D_i+\beta_2 D_i^2 + \bm M_i^T \bm \beta_{m1}D_i + \bm M_i^T \bm \beta_{m2}D_i^2$, where the quadratic and effect modification terms are optional. Finally, the linear-exponential relative risk model $RR_i= 1+(\beta_1 + \bm{M}_i^T \bm{\beta}_{m1}) D_i \exp\{(\beta_2+ \bm{M}_i^T \bm{\beta}_{m2})D_i\}$ is supported.

The second supported family is logistic regression for binary outcomes, with probabilities

$p_i/(1-p_i)=RR_i\exp(\alpha_0+\bm X_i^T \bm \alpha).$

Third is Poisson regression for counts,

$Y_i \sim \text{Poisson}(\mu_i),$

where $\mu_i=RR_i \exp(\alpha_0 +\bm X_i^T \bm \alpha)\times \text{offset}_i$ with optional offset.

Fourth is proportional hazards regression for time-to-event data, with hazard function

$h(t) = h_0(t)RR_i\exp(\bm X_i^T \bm \alpha),$

with $h_0$ the baseline hazard.

Fifth is multinomial logistic regression for a categorical outcome with $Z>2$ outcome categories, with the last category as the referent category (i.e., $\alpha_{0,Z}=\bm \alpha_{Z}=\beta_{1,Z}=\beta_{2,Z}=\bm \beta_{m1,Z} = \bm \beta_{m2,Z}=0$):

$P(Y_i=z)=RR_i\exp(\alpha_{0,z}+\bm X_i^T \bm \alpha_{z})/\{1+\sum_{s=1}^{Z-1} RR_i\exp(\alpha_{0,s}+\bm X_i^T \bm \alpha_{s})\}$

Sixth is conditional logistic regression for matched case control data, for which

$P\left(Y_i = 1, Y_k = 0 \forall k \neq i \bigg| \sum_{i \in \mathcal{R}} Y_i = 1\right) = RR_i\exp(\bm X_i^T \bm \alpha)/\{\sum_{k \in \mathcal{R}}RR_k\exp(\bm X_k^T \bm \alpha)\},$

where $\mathcal{R}$ is the matched set corresponding to individual $i$.

Methods include regression calibration (Carroll et al. 2006 doi:10.1201/9781420010138), extended regression calibration (Little et al. 2023 doi:10.1038/s41598-023-42283-y), Monte Carlo maximum likelihood (Stayner et al. 2007 doi:10.1667/RR0677.1), frequentist model averaging (Kwon et al. 2023 doi:10.1371/journal.pone.0290498), and Bayesian model averaging (Kwon et al. 2016 doi:10.1002/sim.6635).

Usage

ameras(formula=NULL, data, family="gaussian", methods="RC", 
  Y=NULL, dosevars=NULL, doseRRmod=NULL, deg=NULL,
  M=NULL, X=NULL, offset=NULL, entry=NULL, exit=NULL,
  setnr=NULL,
  CI=NULL, params.profCI=NULL,
  maxit.profCI=NULL, tol.profCI=NULL,
  transform=NULL,
  transform.jacobian=NULL, inpar=NULL, loglim=1e-30, MFMA=100000, 
  prophaz.numints.BMA=10, ERRprior.BMA="doubleexponential", nburnin.BMA=5000, 
  niter.BMA=20000, nchains.BMA=2, thin.BMA=10, included.realizations.BMA=NULL, 
  included.replicates.BMA=NULL, optim.method="Nelder-Mead", control=NULL, 
  keep.data=TRUE, ... )

Arguments

formula	an object of class "formula" containing the model specification. See Details.
data	input data frame.
family	outcome model family: "gaussian", "binomial", "poisson", "prophaz", "multinomial" or "clogit" (default "gaussian").
methods	character vector of one or multiple methods to apply. Options: "RC", "ERC", "MCML", "FMA", "BMA" (default "RC").
Y	Deprecated. Use the formula interface instead. Name or column index of the outcome variable for linear, binomial, Poisson, multinomial and conditional logistic models, or event indicator variable for the proportional hazards model.
dosevars	Deprecated. Use the formula interface instead. Names or column indices of exposure realization vectors.
doseRRmod	Deprecated. Use the formula interface instead. The functional form of the dose-response relationship; options are exponential RR ("EXP"), linear ERR ("ERR"), or linear-exponential RR ("LINEXP") (default "ERR").
deg	Deprecated. Use the formula interface instead. For doseRRmod="ERR" and doseRRmod="EXP", whether to fit a linear (deg=1) or linear-quadratic (deg=2) dose-response model (default linear).
M	Deprecated. Use the formula interface instead. Names or column indices of binary effect modifying variables (optional).
X	Deprecated. Use the formula interface instead. Names or column indices of other covariates (optional).
offset	Deprecated. Use the formula interface instead. Name or column index of offset variable for Poisson regression (optional).
entry	Deprecated. Use the formula interface instead. Name or column index of left truncation time variable for proportional hazards regression (optional).
exit	Deprecated. Use the formula interface instead. Name or column index of exit time variable, required when family=prophaz.
setnr	Deprecated. Use the formula interface instead. Name or column index of integer-valued matched set variable, required when family="clogit".
CI	Deprecated. Use confint() to compute confidence intervals instead. Method for calculation of 95% confidence or credible intervals (see Details). For RC, ERC, and MCML, options are "wald.orig", "wald.transformed", "proflik" (default "proflik"). For FMA and BMA, options are "percentile" and "hpd" (default "percentile"). If methods contains at least one of RC, ERC, and MCML and at least one of FMA and BMA, CI must be length 2 and specify one method for RC, ERC, and MCML, and one for FMA and BMA (see Details).
params.profCI	Deprecated. Use confint() to compute confidence intervals instead. When CI="proflik", whether to obtain profile-likelihood CIs for all parameters ("all") or only dose-related parameters ("dose", default).
maxit.profCI	Deprecated. Use confint() to compute confidence intervals instead. Maximum iterations for determining profile-likelihood CIs; passed to uniroot (default 20).
tol.profCI	Deprecated. Use confint() to compute confidence intervals instead. Tolerance for determining profile-likelihood CIs; passed to uniroot (default 1e-2).
transform	function for internal parameter transformation (see Details).
transform.jacobian	Jacobian of the transformation function (see Details).
inpar	vector of initial values for log-likelihood optimization (optional).
loglim	parameter used in likelihood computations to avoid taking the log of very small or negative numbers via log(max(x, loglim)) (default 1e-30).
MFMA	number of samples for "FMA" to compute estimates and CIs (default 100,000).
prophaz.numints.BMA	for methods="BMA" with family="prophaz", the number of subintervals with constant baseline hazard (default 10). Cut points are determined based on quantiles of the event time distribution among cases.
ERRprior.BMA	prior for dose-related parameters when doseRRmod="ERR" or "LINEXP" and methods="BMA". Options: "truncated_normal", "truncated_horseshoe", "truncated_doubleexponential", "normal", "horseshoe", "doubleexponential", see Details (default "doubleexponential").
nburnin.BMA	number of MCMC burn-in iterations for BMA (default 5,000).
niter.BMA	number of MCMC iterations per chain for BMA (default 20,000).
nchains.BMA	number of MCMC chains for BMA (default 2).
thin.BMA	thinning rate for BMA (default 10).
included.realizations.BMA	indices of exposure realizations used in BMA (defaults to all realizations).
included.replicates.BMA	Deprecated. Use included.realizations.BMA instead. Indices of exposure realizations used in BMA (defaults to all realizations).
optim.method	method used for optimization by optim. Options are "Nelder-Mead" and "BFGS". When using Nelder-Mead, a second optimization with BFGS is run to ensure an optimal fit.
control	control list passed to optim (default list(reltol=1e-10)).
keep.data	whether to attach data to the output object (default TRUE). When the data object is large, keep.data can be set to FALSE to preserve memory. The attached data is used to compute profile likelihood confidence intervals, but can also be supplied seperately when keep.data=FALSE. See confint.
...	other arguments, passed to functions such as transform.

Details

Models are specified through formulas of the form Y~dose(dose_expression, model="ERR", deg=1, modifier=M1+M2)+X1+X2. Here dose_expression specifies the dose realization columns and is parsed by eval_select from the tidyselect package. Useful examples are D1:D1000 if the doses are in a sequence of columns with sequential names such as D1-D1000, and all_of(dosevars) where dosevars is a vector with the names of all dose columns. Further, model specifies, for non-Gaussian families, whether to use the exponential dose-response model (model="EXP"), the linear-exponential model (model="LINEXP") or the linear ERR model (model="ERR"). Next, deg is used to specify whether a quadratic dose term should (deg=2) or should not (deg=1) be estimated for the exponential or linear ERR dose-response model. The modifier term is optional and used to specify binary effect modification variables. Note that interactions in the modifier term are not allowed, e.g. M1*M2. When deg, modifier, and model are not supplied, the defaults are deg=1, no effect modifiers, and model="ERR". Finally, X1 and X2 above represent optional additional covariates, which can include factor variables and interactions such as X1*X2. The matched set variable setnr required for conditional logistic regression is specified on the right-hand side of the formula through a term strata(setnr), and an optional offset variable offset for Poisson regression similarly through a term offset(offset). For proportional hazards regression, the left-hand side of the formula should have the form Surv(exit, status) or Surv(entry, exit, status).

A transformation can be used to reparametrize parameters internally (i.e., such that the likelihoods are evaluated at transform(parameters), where parameters are unconstrained), and 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 ?transform1. When supplying a function to transform, this should be a function of the full parameter vector, returning a full (transformed) parameter vector. In particular, the full parameter vector contains parameters in the following order: $\alpha_0, \bm \alpha, \beta_1, \beta_2, \bm \beta_{m1}, \bm \beta_{m2}, \sigma$, where $\bm \alpha$, $\bm\beta_{m1}$ and $\bm \beta_{m2}$ can be vectors, with lengths matching $\bm X$ and $\bm M$, 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 the multinomial model, the full parameter vector is the concatenation of $Z-1$ parameter vectors in the order as given above, where $Z$ is the number of outcome categories, with the last category chosen as the referent category. See vignette("transformations", package="ameras") for an example of how to specify a custom transformation function.

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 $\beta_1$ in the linear dose-response and $(0,-1/max(D^2))$ for $(\beta_1,\beta_2)$ in the linear-quadratic dose-response, respectively. For the linear-exponential model, a lower limit of 0 is used for $\beta_1$, and no transformation is used for $\beta_2$. If effect modifiers M are specified, no transformation is used for those parameters. When negative RRs are obtained during optimization, an error will be generated and a different transformation or bounds should be used. All output is returned in the original parametrization. The Jacobian of the transformation (transform.jacobian) is required when using a transformation. For transform1, the Jacobian is given by transform1.jacobian. No transformations are used in BMA, and FMA is applied on the parameters using the parametrization as given in above with variances obtained using the delta method with the provided Jacobian function.

For BMA, a prior distribution for exposure-response parameters can be chosen when using linear or linear-exponential exposure-response model. The options are normal, horshoe, and double exponential priors, and the same priors truncated at 0 to yield positive values. In particular:

• Normal: $\beta_j \sim N(0,1000)$ for all exposure-response parameters $\beta_j$

• Horseshoe (shrinkage prior): $\tau \sim \text{Cauchy}(0,1)^+; \lambda_j \sim \text{Cauchy}(0,1)^+; \beta_j \sim N(0, \tau^2 \lambda_j^2)$. Here $\tau$ is shared across all parameters

• Double exponential (shrinkage prior): $\lambda_j \sim \text{Cauchy}(0,1)^+; \beta_j \sim \text{DoubleExponential}(0,\lambda_j)$

For all other parameters, and when using the exponential exposure-response model or the Gaussian outcome family, the prior is $N(0, 1000)$. For the parameter $\sigma$ in the Gaussian family, this prior is truncated at 0.

Because the proportional hazards model is not available in nimble, ameras uses a piecewise constant baseline hazard for Bayesian model averaging. The interval min(entry), max(exit)) is divided into prophaz.numints.BMA subintervals with cutpoints obtained as quantiles of the distribution of event times among cases, and a baseline hazard parameter is estimated for each subinterval.

Value

The output is an object of class amerasfit. General components are call (the function call to ameras), formula (the formula object specifying the model), num.rows (the number of rows in data), num.realizations (the number of dose realizations provided), transform (the used transformation function, if applicable), transform.jacobian (the used Jacobian function for the transformation, if applicable), other.args (any other arguments passed to ...), model (a list containing the specified model components parsed from the formula), CI.computed (logical, whether confidence intervals have been attached by confint), and data (either the data frame used for model fitting when keep.data=TRUE or NULL otherwise).

For each method supplied to methods, the output contains a list with components:

coefficients	named vector of model coefficients.
sd	named vector of standard deviations.
vcov	covariance matrix for the full parameter vector. Based on the observed information (negative second derivative of log-likelihood) for RC, ERC, and MCML, and on the obtained samples for FMA and BMA.
runtime	string with the runtime in seconds.

For RC, ERC, and MCML the following additional output is included:

optim	a list object with results returned by optim. Components are par (raw parameters before applying a transformation if applicable), hessian (Hessian matrix for par), convergence (convergence code with 0 indicating convergence and 1 indicating that the maximal number of iterations was reached), and counts (the number of likelihood function evaluations used during optimization).
loglik	log-likelihood value at the optimum.

For RC and ERC, the output additionally contains:

ERC	logical, whether the output is for ERC (ERC=TRUE) or RC (ERC=FALSE).

For BMA the output additionally contains:

samples	MCMC posterior samples, as obtained from nimble. This is a list object with nchains.BMA components, each a named matrix with the samples from one chain in its rows, with columns corresponding to model parameters.
Rhat	data frame with two columns, Rhat and n.eff. The first column contains the Gelman-Rubin statistics $\hat R \geq 1$ that can be used to assess convergence of MCMC chains. A value of 1 indicates good convergence and values $>1.05$ indicate poor convergence. The effective sample size n.eff is a measure of how many independent samples the auto-correlated MCMC samples correspond to. A low effective sample size indicates high correlations and/or poor mixing.
included.realizations	indices of realization exposures that were included to obtain the results.
prophaz.timepoints	for family="prophaz", time points defining the intervals on which the estimated baseline hazards is constant; these are prophaz.numints.BMA + 1 time points covering the interval (min(entry), max(exit)), based on quantiles among observed event times. See Details.

Finally, for FMA the output additionally contains:

samples	data frame with samples generated from the normal distributions associated with parameters estimated for each dose realization.
weigths	vector of weights corresponding to the models fit to each included realization
included.samples	the total number of samples included.
included.realizations	indices of realization exposures that were included to obtain results. Fits without a valid variance estimate (i.e., non-invertible Hessian or inverse that is not positive definite) or that reach the maximal number of iterations without convergence are filtered out and not used to obtain results.

The class amerasfit supports the methods print, coef, confint, summary, and traceplot.

References

Roberti, S., Kwon D., Wheeler W., Pfeiffer R. (in preparation). ameras: An R Package to Analyze Multiple Exposure Realizations in Association Studies

See Also

confint for computing confidence intervals, summary for a summary of the fitted model including confidence intervals if computed, coef for extracting coefficients.

Examples

data(data, package="ameras")
  ameras(Y.gaussian~dose(V1:V10, modifier=M1+M2)+X1+X2, data=data, family="gaussian")

---

Estimated Coefficients for an amerasfit Object

Description

Returns a data frame with all the parameters of a fitted amerasfit object. The resulting object has a column for every method supplied to 'methods' when calling 'ameras', with rows corresponding to parameters.

Usage

## S3 method for class 'amerasfit'
coef(object, ...)

Arguments

object	A fitted model object of class amerasfit, as returned by ameras.
...	Additional arguments, currently unused.

Value

Data frame with estimated model parameters. Column names correspond to the 'methods' used in the 'ameras' call, and row names correspond to parameter names.

See Also

ameras for model fitting, summary for a summary of the fitted model including confidence intervals if computed.

Examples

data("data", package = "ameras")
dosevars <- paste0("V", 1:10)

## Fit the model
fit <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial", 
methods = c("RC","ERC"))
## Full matrix
coef(fit)

## Vector with RC parameters
coef(fit)$RC

---

Confidence Intervals for an amerasfit Object

Description

Computes and/or prints confidence intervals for the parameters of a fitted amerasfit object. This is a separate step from model fitting, i.e., ameras fits the model and confint computes intervals, attaches them to the fitted object, and prints them. If confint is called a second time on the same object, intervals are only printed and not recomputed, unless force=TRUE.

Usage

## S3 method for class 'amerasfit'
confint(object, parm="dose", level=0.95, 
        type=c("proflik","percentile"), maxit.profCI=20, 
        tol.profCI=1e-2, data=NULL, force=FALSE, print=TRUE,
        digits = max(3, getOption("digits") - 3), ...)

Arguments

object	A fitted model object of class amerasfit, as returned by ameras.
parm	Either "dose" to compute intervals for dose-related parameters only, "all" for all parameters, or a character vector of specific parameter names. Only used when type = "proflik" since Wald intervals are cheap to compute for all parameters simultaneously. Defaults to "dose" since profile likelihood computation can be extensive.
level	The confidence level (default 0.95).
type	The type(s) of confidence intervals to determine. For RC, ERC, and MCML, this can be one of: "wald.orig" Wald intervals on the original parameter scale using the delta method variance-covariance matrix. "wald.transformed" Wald intervals computed on the transformed (reparametrized) scale and then back-transformed. Only available when a transformation was used during fitting. "proflik" Profile likelihood intervals based on the chi-squared approximation. More accurate than Wald intervals but computationally intensive, especially for large datasets or complex models. For FMA and BMA, confidence intervals are based on the generated samples and possible confidence interval types are: "percentile" Equal-tailed percentile intervals. "hpd" Highest posterior density intervals via HPDinterval from the coda package. If object contains results for at least one of RC, ERC, and MCML and at least one of FMA and BMA, type must be length 2 and specify one method for RC, ERC, and MCML, and one for FMA and BMA.
maxit.profCI	Maximum number of iterations for the root-finding algorithm used to locate profile likelihood interval bounds. Only used when type = "proflik". Defaults to 20.
tol.profCI	Tolerance for the root-finding algorithm. Only used when type = "proflik". Defaults to 1e-2. Reduce for more precise bounds at the cost of additional computation.
data	The original data frame used for fitting. Only required when type = "proflik" and the model was fitted with keep.data = FALSE
force	Logical. If TRUE, confidence intervals are recomputed even if they have already been computed for this object. Defaults to FALSE, in which case a warning is issued and the object is returned unchanged if confidence intervals are already present.
print	Logical. If TRUE (default), confidence intervals are printed to the console.
digits	Number of significant digits to be printed. Default is 'max(3, getOption("digits") - 3)'
...	Additional arguments, currently unused.

Details

For (extended) regression calibration and Monte Carlo maximum likelihood, Wald and profile likelihood intervals can be obtained. When a parameter transformation $\bm\theta = h(\bm\eta)$ is used, type="wald.transformed" yields the CI at significance level $\alpha$ of $h(\bm\eta \pm z_{1-\alpha/2} \bm V)$ where $z_{1-\alpha/2}$ is the $1-\alpha/2$-quantile of the standard normal distribution and $\bm V$ is the vector of standard deviations estimated using the inverse Hessian matrix, and type="wald.orig" uses the delta method to obtain the CI $h(\bm\eta)\pm z_{1-\alpha/2} \bm V_*$ where $\bm V_*$ is the vector of standard deviations estimated using $J H^{-1} J^T$ with $J$ the Jacobian of the transformation and $H$ is the Hessian. When no transformation is used, type="wald.orig" should be used. The third option is proflik, which uses the profile likelihood to compute confidence bounds.

For FMA and BMA, the options for confidence/credible intervals are type="percentile" which uses percentiles, and type="hpd" which computes highest posterior density intervals using HPDinterval from the coda package, both using the FMA samples or Bayesian posterior samples.

Profile likelihood intervals (type="proflik") require re-evaluating the likelihood repeatedly and can be time-consuming. The parm argument can be used to restrict computation to dose parameters only (the default) when intervals for the other parameters are not of interest.

When the model was fitted with keep.data=FALSE and type="proflik" is used for confint, the original data must be supplied via the data argument. Wald intervals do not require the data and can always be computed from the stored Hessian and parameter estimates alone.

Value

The original amerasfit object with a CI element added to each fitted method result. For RC, ERC, and MCML the CI element is a data frame with columns:

lower

Lower confidence bound.

upper

Upper confidence bound.

When type = "proflik", four additional columns are included:

pval.lower

P-value at the lower bound, should be close to $1 -$ level.

pval.upper

P-value at the upper bound, should be close to $1 -$ level.

iter.lower

Number of iterations used by the root-finding algorithm for the lower bound.

iter.upper

Number of iterations used by the root-finding algorithm for the upper bound.

For FMA and BMA the CI element is a data frame with columns lower and upper.

See Also

ameras for model fitting, summary for a summary of the fitted model including confidence intervals if computed, confint for the generic function.

Examples

data("data", package = "ameras")

## Fit the model
fit <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial",
               methods = "RC")

## Wald intervals (fast)
fit <- confint(fit, type = "wald.orig")
summary(fit)

## Profile likelihood intervals for dose parameters only (slower)

fit <- confint(fit, type = "proflik", parm = "dose")
summary(fit)


## With keep.data = FALSE, supply data explicitly for proflik

fit2 <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial",
               methods = "RC", keep.data = FALSE)
fit2 <- confint(fit2, type = "proflik", data = data)


## FMA and BMA with percentile intervals

fit3 <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial", 
               methods = c("FMA", "BMA"))
fit3 <- confint(fit3, type = "percentile")
summary(fit3)

---

Example Data

Description

Data includes outcomes of all six supported types in the appropriately named columns. For proportional hazards regression, the observed exit time is time and event status is event. For conditional logistic regression, the matched set variable is setnr. The data has 10 exposure realizations in columns V1-V10.

Examples

data(data, package="ameras")

 # Display a few rows of the data
 data[1:5, ]

---

Visualize Multiple Dose Realizations

Description

Create a descriptive figure to visualize the distribution of dose and its uncertainty.

Usage

ecdfplot(data, dosevars, xlab="Dose", 
  ylab="Cumulative distribution", show.mean=TRUE, log.xaxis=TRUE)

Arguments

data	data frame containing columns with dose vectors
dosevars	names or column indices of dose vectors.
xlab	label for the x-axis, default "Dose".
ylab	label for the y-axis, default "Cumulative distribution".
show.mean	logical, whether to plot the cumulative distribution of the mean dose across realizations and across individuals, default TRUE.
log.xaxis	logical, whether to use a log-scale for the dose axis (default TRUE). When log.xaxis=TRUE, any zeros in the doses are excluded while plotting.

Details

In the left panel, the empirical cumulative distribution function (ECDF) is plotted for each dose realization. In other words, each curve shows one distribution of dose across individuals. The spread within individual curves reflects the dose range across individuals, while the spread between curves reflects between-realization variation on the cohort level. If show.mean=TRUE, the solid black curve is the cumulative distribution of the mean dose for each individual.

In the right panel, ECDFs are plotted for each individual, showing distributions within individuals. A wide spread within individual curves is indicative of large within-individual variation, while the spread between curves reflects between-individual variation. If show.mean=TRUE, the solid black curve is the cumulative distribution of the mean for each dose realization.

When using a log-scale for the x-axis, any zero dose values are excluded before plotting.

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
   data(data, package="ameras")
   ecdfplot(data,  dosevars=paste0("V", 1:10))
  }

---

Extract Included Realizations from an amerasfit Object

Description

Extracts the indices of dose realizations included in model averaging for FMA and/or BMA results of a fitted amerasfit object. Realizations are excluded from FMA if the optimization did not converge or the Hessian was not positive definite.

Usage

included_realizations(x, ...)

## S3 method for class 'amerasfit'
included_realizations(x,
                              methods = c("FMA", "BMA"),
                              ...)

Arguments

x	A fitted model object of class amerasfit, as returned by ameras, with FMA and/or BMA results present.
methods	Character vector specifying which method(s) to extract included realizations for. One or both of "FMA" and "BMA". Defaults to both. Only methods that were run are returned.
...	Additional arguments, currently unused.

Details

For FMA, realizations are excluded if the optimization did not converge or the Hessian was not invertible or not positive definite. If more than 20% of realizations are excluded, a warning is issued during fitting.

For BMA, all realizations are included by default. A subset of realizations can be specified via the included.replicates.BMA argument to ameras, for example by passing the indices returned by included_realizations(fit, methods="FMA") from a prior FMA fit.

Value

If a single method is requested or only one method was run, an integer vector of indices of the included dose realizations. If both FMA and BMA results are present and both are requested, a named list with elements FMA and BMA, each containing an integer vector of included realization indices.

See Also

ameras, Rhat.amerasfit

Examples

data("data", package="ameras")
dosevars <- paste0("V", 1:10)

## FMA only
fit <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
              data=data, family="binomial", methods="FMA")
included_realizations(fit)
length(included_realizations(fit))

## Both FMA and BMA
fit2 <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
               data=data, family="binomial", methods=c("FMA", "BMA"))
included_realizations(fit2)
included_realizations(fit2, methods="FMA")
included_realizations(fit2, methods="BMA")

---

Diagnostic Plots for an amerasfit Object

Description

Produces diagnostic plots for a fitted amerasfit object, including residuals versus fitted values and normal Q-Q plots. Plots are produced for each estimation method present in the fitted object.

Usage

## S3 method for class 'amerasfit'
plot(x,
     methods = c("RC", "ERC", "MCML", "FMA", "BMA"), 
     which = NULL, type = NULL, dose.col = NULL, 
     add.smooth = getOption("add.smooth", TRUE), 
     qqline = TRUE, id.n = 3, ask = NULL, data = NULL, ...)

Arguments

x	A fitted model object of class amerasfit, as returned by ameras.
methods	Character vector specifying which estimation methods to produce plots for. One or more of "RC", "ERC", "MCML", "FMA", and "BMA". Defaults to all. Methods not present in x are silently skipped.
which	Character vector specifying which plots to produce. For non-"prophaz" families, one or both of "residuals-vs-fitted" and "qq". For family="prophaz", "schoenfeld" produces a scaled Schoenfeld residual plot against event time. Defaults to both for non-"prophaz" families and to "schoenfeld" for "prophaz" families.
type	The type of residuals to use. One of "pearson", "deviance", or "response" for non-"prophaz" models. For family="prophaz", this must be "schoenfeld". Defaults to "pearson" for non-"prophaz" families and to "schoenfeld" for "prophaz". See residuals.amerasfit for details.
dose.col	The dose realization to use for computation of residuals. If NULL (default), a best fitting realization is determined for each method (see Details).
add.smooth	Logical. If TRUE, a loess smooth line is added to the residuals versus fitted plot via panel.smooth. Defaults to getOption("add.smooth", TRUE). Is not used for the Schoenfeld residual plot for which a spline is always shown, mirroring plot.cox.zph.
qqline	Logical. If TRUE, a reference line is added to normal Q-Q plots via qqline. Defaults to TRUE.
id.n	Integer. The number of extreme residuals to label in residuals vs fitted and normal Q-Q plots. Labels show the row index of the observation. Defaults to 3. Set to 0 to suppress labeling.
ask	Logical. If TRUE, the user is prompted before each new plot. Defaults to NULL, in which case prompting occurs automatically when the number of plots exceeds the number of available panels and the session is interactive.
data	The original data frame used for fitting. Only required when the model was fitted with keep.data=FALSE.
...	Additional graphical arguments passed to plot.default and qqnorm.

Details

If not otherwise specified, the dose realization used to compute fitted values is selected for each estimation method as follows. For RC and ERC the mean dose across realizations is used. For MCML and FMA the realization yielding the largest likelihood at the final parameter estimates is used (which for FMA corresponds to the highest model averaging weight). For BMA the realization most frequently selected by the MCMC sampler is used. The selected dose column is shown in the plot title.

For the "residuals-vs-fitted" plot, the smooth line is added via panel.smooth using a loess smoother when add.smooth=TRUE. For the "qq" plot, the residuals are plotted against the theoretical quantiles of the normal distribution when qqline=TRUE. For both plots, the id.n most extreme residuals are labeled. For family="multinomial", one panel is produced per non-reference outcome category.

For proportional hazards models, scaled Schoenfeld residuals are drawn against the observed event times to assess the proportional hazards assumption. Under proportional hazards, the residuals should fluctuate randomly around zero with no systematic trend over time. Systematic patterns or smooth trends may indicate time-varying covariate effects and violation of the proportional hazards assumption. Note that the implementation here corresponds to using transform = "identity" in cox.zph, in contrast to the default setting for that function which applies a Kaplan–Meier transformation of time.

The data must either be stored on the object (keep.data=TRUE in ameras, the default) or supplied via the data argument.

Value

The amerasfit object x is returned invisibly.

See Also

residuals.amerasfit for computing residuals, ameras for model fitting, confint for confidence intervals, plot.lm for the equivalent method for linear models.

Examples

data("data", package="ameras")
dosevars <- paste0("V", 1:10)

## Binomial model
fit <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
              data=data, family="binomial", methods="RC")

## Both diagnostic plots for RC
plot(fit)

## Residuals vs fitted only
plot(fit, which="residuals-vs-fitted")

## Deviance residuals
plot(fit, type="deviance")

## Multiple methods

fit2 <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
               data=data, family="binomial", methods=c("RC", "ERC"))
plot(fit2)


## With keep.data=FALSE, supply data explicitly

fit3 <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
               data=data, family="binomial", methods="RC",
               keep.data=FALSE)
plot(fit3, data=data)


## Schoenfeld residual plot

fit4 <- ameras(Surv(time, status) ~ dose(all_of(dosevars), model = "ERR"),
              data = data, family = "prophaz", methods = "RC")
plot(fit4)

---

Simple Summary for an amerasfit Object

Description

Prints a simple summary of a fitted amerasfit object.

Usage

## S3 method for class 'amerasfit'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x	A fitted model object of class amerasfit, as returned by ameras.
digits	Number of significant digits to be printed. Default is 'max(3, getOption("digits") - 3)'
...	Additional arguments, currently unused.

Value

Prints the 'ameras' call, number of rows and dose realizations in the data, runtime, and model coefficients.

See Also

ameras for model fitting, summary for a more detailed summary of the fitted models including confidence intervals if computed.

Examples

data("data", package = "ameras")

## Fit the model
fit <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial", 
              methods = c("RC","ERC"))

## Default print
fit
print(fit)

## More digits
print(fit, digits=5)

---

Residuals for an amerasfit Object

Description

Computes residuals for a fitted amerasfit object.

Usage

## S3 method for class 'amerasfit'
residuals(object,
          method = "RC",
          type = NULL,
          data = NULL,
          dose.col = NULL,
          scaled.schoenfeld = TRUE,
          ...)

Arguments

object	A fitted model object of class amerasfit, as returned by ameras.
method	Character string specifying which estimation method to compute residuals for. One of "RC", "ERC", "MCML", "FMA", or "BMA". Defaults to "RC".
type	The type of residuals to compute. Defaults to "pearson" for all families except "prophaz", for which the default is "schoenfeld". For types other than "schoenfeld", let $Y_i$ and $\mu_i$ denote the observed response and fitted mean response for individual $i$, respectively. Then the residuals are defined as follows: "pearson" Pearson residuals. For "gaussian": $(Y_i - \mu_i) / \sigma$ where $\sigma$ is the estimated residual standard deviation (note: glm returns raw residuals for Gaussian). For "binomial" and "clogit": $(Y_i - \mu_i) / \sqrt{\mu_i(1 - \mu_i)}$. For "poisson": $(Y_i - \mu_i) / \sqrt{\mu_i}$. For "multinomial": per-category Pearson residuals $(Y_{iz} - \mu_{iz}) / \sqrt{\mu_{iz}(1 - \mu_{iz})}$ where $\mu_{iz}$ is the fitted probability of category $z$ for individual $i$. "deviance" Deviance residuals, defined as the signed square root of the individual contribution to the deviance. For "gaussian": the raw residual $Y_i - \mu_i$. For "binomial" and "clogit": $\pm\sqrt{-2\log(\hat{p}_i)}$ where $\hat{p}_i = \mu_i$ if $Y_i = 1$ and $\hat{p}_i = 1 - \mu_i$ if $Y_i = 0$, with sign equal to $\text{sign}(Y_i - \mu_i)$. For "poisson": $\pm\sqrt{2(Y_i \log(Y_i/\mu_i) - (Y_i - \mu_i))}$ for $Y_i > 0$ and $\pm\sqrt{2\mu_i}$ for $Y_i = 0$, with sign equal to $\text{sign}(Y_i - \mu_i)$. For "multinomial": per-category deviance residuals as for "binomial". "response" Raw residuals $Y_i - \mu_i$. For "multinomial": the matrix $\bm{Y} - \hat{\bm{P}}$ where $\bm{Y}$ is the $N \times Z$ indicator matrix of observed categories and $\hat{\bm{P}}$ is the $N \times Z$ matrix of fitted probabilities. "schoenfeld" Schoenfeld residuals for family="prophaz" only. For each event time, the unscaled residual for the individual experiencing the event is the observed covariate vector minus the risk-set weighted mean covariate vector at that event time. See Details.
data	The original data frame used for fitting. Only required when the model was fitted with keep.data=FALSE.
dose.col	Character string specifying the dose column to use when computing fitted values. If NULL (the default), the dose column is selected automatically: the mean dose across realizations for RC and ERC, the realization with the highest likelihood for MCML and FMA, and the most frequently selected realization for BMA. Can be set to any dose column present in the data to override the automatic selection.
scaled.schoenfeld	Logical. If TRUE (the default), scaled Schoenfeld residuals are returned, obtained by multiplying the raw Schoenfeld residuals by the estimated coefficient variance-covariance matrix following Grambsch and Therneau (1994). If FALSE, raw Schoenfeld residuals are returned. Only used when type="schoenfeld".
...	Additional arguments, currently unused.

Details

Fitted values are computed using the dose column specified by dose.col.

For family="clogit", fitted values $\mu_i$ are the conditional probabilities of being a case within each matched set, computed as $R_i / \sum_{j \in \text{set}(i)} R_j$ where $R_i$ is the relative risk for individual $i$ and the sum is over all individuals in the same matched set.

For family="prophaz" and type="schoenfeld", the Schoenfeld residual for the individual failing at time $t$ is:

$\bm{r} = \bm{x} - \bar{\bm{x}}(t)$

where $\bm{x}$ is the observed covariate vector for the failing individual and

$\bar{\bm{x}}(t) = \frac{\sum_{j \in \mathcal{R}(t)} \bm{x}_j R_j} {\sum_{j \in \mathcal{R}(t)} R_j}$

is the weighted mean covariate vector in the risk set $\mathcal{R}(t)$ at time $t$, with weights equal to the relative risks $R_j$. Ties in event times are handled using the Breslow approximation.

Value

For families "gaussian", "binomial", "poisson", and "clogit", a numeric vector of length $N$ containing the residuals.

For family="multinomial", a numeric matrix of dimension $N \times Z$ where $Z$ is the number of outcome categories. Column names correspond to the factor levels. Note that when plotting via plot, the reference category is excluded since its residuals are a linear combination of the other categories.

For family="prophaz" with type="schoenfeld", a data frame with columns id (row index of the event), time (event time), and one additional column per model parameter containing the Schoenfeld residuals. Only rows corresponding to events (status=1) are included.

References

Schoenfeld, D. (1982). Partial residuals for the proportional hazards regression model. Biometrika, 69(1), 239–241. doi:10.1093/biomet/69.1.239

Grambsch, P. M. and Therneau, T. M. (1994). Proportional hazards tests and diagnostics based on weighted residuals. Biometrika, 81(3), 515–526. doi:10.1093/biomet/81.3.515

See Also

plot for diagnostic plots using these residuals, ameras for model fitting, residuals for the generic function.

Examples

data("data", package="ameras")
dosevars <- paste0("V", 1:10)

fit <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
              data=data, family="binomial", methods="RC")

## Pearson residuals (default)
res <- residuals(fit)
summary(res)

## Deviance residuals
res_dev <- residuals(fit, type="deviance")

## Response residuals
res_raw <- residuals(fit, type="response")

## Specific dose column
res_v1 <- residuals(fit, dose.col="V1")

## Multiple methods

fit2 <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
               data=data, family="binomial", methods=c("RC", "ERC"))
res_erc <- residuals(fit2, method="ERC")


## With keep.data=FALSE

fit3 <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
               data=data, family="binomial", methods="RC",
               keep.data=FALSE)
res <- residuals(fit3, data=data)


## Schoenfeld residuals for proportional hazards model

fit4 <- ameras(Surv(time, status) ~ dose(all_of(dosevars), model="ERR"),
               data=data, family="prophaz", methods="RC")
res_sch <- residuals(fit4)
res_raw_sch <- residuals(fit4, scaled.schoenfeld=FALSE)

---

Extract MCMC Convergence Diagnostics from an amerasfit Object

Description

Extracts the Gelman-Rubin convergence diagnostics from the BMA results of a fitted amerasfit object.

Usage

Rhat(x, ...)

## S3 method for class 'amerasfit'
Rhat(x, ...)

Arguments

x	A fitted model object of class amerasfit, as returned by ameras, with BMA results present.
...	Additional arguments, currently unused.

Value

A data frame with columns Rhat and n.eff, containing the Gelman-Rubin statistic and effective sample size for each parameter. Values of Rhat substantially above 1.05 indicate potential convergence problems, in which case longer chains via niter.BMA and nburnin.BMA are recommended.

See Also

ameras, summary.amerasfit, included_realizations.amerasfit

Examples

data("data", package="ameras")
dosevars <- paste0("V", 1:10)
fit <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
              data=data, family="binomial", methods="BMA")
Rhat(fit)

---

Summarize an amerasfit Object

Description

Produces a summary of a fitted amerasfit object, including parameter estimates, standard errors, and confidence intervals if computed via confint.

Usage

## S3 method for class 'amerasfit'
summary(object, ...)

## S3 method for class 'amerasfit'
summary_table(object, ...)

## S3 method for class 'summary.amerasfit'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

object	A fitted model object of class amerasfit, as returned by ameras.
x	An object of class summary.amerasfit, as returned by summary.amerasfit.
digits	The number of significant digits to use. Defaults to max(3, getOption("digits") - 3).
...	Additional arguments, currently unused.

Details

summary.amerasfit collects results from all estimation methods present in the fitted object into a single summary table. Columns for confidence intervals are only printed if they have been computed by confint. When BMA results are present in the fitted object, the summary table includes columns Rhat and n.eff, with NA values for all other methods.

Printing the summary prints the original call to ameras, the runtime (total and by method), and the table described above. This table can also be accessed directly (i.e., to retrieve confidence intervals) using summary_table.

Value

summary.amerasfit returns an object of class summary.amerasfit, which is a list containing the following elements:

call

The matched call from the original call to ameras.

summary_table

A data frame with one row per parameter per method, containing columns:

Method

The estimation method (RC, ERC, MCML, FMA, or BMA).

Term

The parameter name.

Estimate

The parameter estimate.

SE

The standard error.

CI.lower

The lower confidence bound, if confidence intervals have been computed via confint.

CI.upper

The upper confidence bound, if confidence intervals have been computed via confint.

pval.lower

p-value associated with the lower bound of the profile likelihood CI, the assess validity of the obtained bound. Only included if profile likelihood confidence intervals were computed via confint.

pval.upper

p-value associated with the upper bound of the profile likelihood CI, the assess validity of the obtained bound. Only included if profile likelihood confidence intervals were computed via confint.

Rhat

The Gelman-Rubin convergence diagnostic, included only when BMA results are present. Values above 1.05 indicate potential convergence problems.

n.eff

The effective sample size, included only when BMA results are present.

runtime_table

A data frame with columns Method and Runtime, reporting the computation time in seconds for each method.

total_runtime_seconds

The total computation time in seconds across all methods.

CI.computed

Logical. TRUE if confidence intervals have been computed via confint, FALSE otherwise.

See Also

ameras for model fitting, confint for computing confidence intervals, print for a shorter printed summary, coef for extracting coefficients.

Examples

data("data", package = "ameras")

## Fit the model
fit <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial",
              methods = "RC")

## Summary without confidence intervals
summary(fit)

## Summary with confidence intervals
fit <- confint(fit, method = "wald.orig")
summary(fit)

## Access the summary table directly
s <- summary_table(fit)

## Multiple methods

fit2 <- ameras(Y.binomial~dose(V1:V10, model="ERR"), data = data, family = "binomial",
              methods = c("RC", "ERC", "MCML"))
fit2 <- confint(fit2, method = "wald.orig")
summary(fit2)

---

Traceplots for MCMC Samples

Description

Produce MCMC traceplots for amerasfit objects.

Usage

traceplot(object, ...)

## S3 method for class 'amerasfit'
traceplot(object, iter = 5000, Rhat = TRUE, n.eff = TRUE, pdf = FALSE, ...)

Arguments

object	a amerasfit object containing BMA output to be plotted
iter	number of iterations to include in the traceplot (defaults to last 5000)
Rhat	logical; whether to include R-hat diagnostics in the plot (default TRUE)
n.eff	logical; whether to include effective sample size in the plot (default TRUE)
pdf	logical; whether to save the output as a PDF (default FALSE)
...	additional arguments passed to MCMCtrace

Details

Wrapper for MCMCvis::MCMCtrace to produce MCMC diagnostic plots. See ?MCMCtrace for more plotting options that can be provided through ....

Value

Traceplots and posterior density plots.

See Also

MCMCtrace

Examples

data(data, package="ameras")
   fit <- ameras(Y.gaussian~dose(V1:V10), data, methods="BMA")
   traceplot(fit)

---

Exponential Parameter Transformation

Description

Applies exponential transformation $f(\theta_i)=\exp(\theta_i)+L_i$ to one or multiple components of parameter vector $\bm \theta$, where $L_i$ are lower limits that can be different for each component

Usage

transform1(params, index.t=1:length(params), lowlimit=rep(0,length(index.t)), 
  boundcheck=FALSE, boundtol=1e-3, ... )

Arguments

params	full input parameter vector
index.t	indices of parameters to be transformed (default all)
lowlimit	lower limits to be applied (default zero), where the k-th component of lowlimit is applied to the k-th index in index.t
boundcheck	whether to produce a warning when any of the transformed parameters are within boundtol of lowlimit
boundtol	tolerance for producing a warning for reaching the boundary
...	not used

Value

Transformed parameter vector.

Examples

params <- c(.1, .5, 1)
transform1(params, lowlimit=c(0, -1, 1))

---

Inverse of Exponential Parameter Transformation

Description

Inverse of transform1 for the purpose of deriving initial values.

Usage

transform1.inv(params, index.t=1:length(params), lowlimit=rep(0,length(index.t)), ... )

Arguments

params	full input parameter vector
index.t	indices of parameters to be transformed (default all)
lowlimit	lower limits to be applied (default zero), where the k-th component of lowlimit is applied to the k-th index in index.t
...	not used

Value

Transformed parameter vector.

Examples

params <- c(.1, .5, 1) # Desired initial values on original scale
transform1.inv(params, lowlimit=c(0, -1, 1)) # Initial values to use on transformed scale

---

Jacobian of the Exponential Parameter Transformation

Description

Computes the Jacobian matrix of transform1. Note that lower limits do not need to be specified as the Jacobian is independent of those

Usage

transform1.jacobian(params, index.t=1:length(params), ... )

Arguments

params	input parameter vector (before transformation) to evaluate the Jacobian at
index.t	indices of parameters to be transformed (default all)
...	not used

Value

Jacobian matrix.

Examples

params <- c(.1, .5, 1)
transform1.jacobian(params)

---

Extract Variance-Covariance Matrices from an amerasfit Object

Description

Extracts the variance-covariance matrices of the parameter estimates from a fitted amerasfit object.

Usage

## S3 method for class 'amerasfit'
vcov(object, methods = c("RC", "ERC", "MCML", "FMA", "BMA"), ...)

Arguments

object	A fitted model object of class amerasfit, as returned by ameras.
methods	Character vector specifying which estimation methods to extract variance-covariance matrices for. One or more of "RC", "ERC", "MCML", "FMA", and "BMA". Defaults to all. Methods that were not run are silently skipped.
...	Additional arguments, currently unused.

Value

If a single method is requested or only one method was run, a named numeric matrix with rows and columns corresponding to model parameters. If multiple methods are requested and multiple methods were run, a named list of such matrices, one per method.

Note that for FMA and BMA the variance-covariance matrix is based on the generated samples rather than a Hessian-based approximation, and may differ in interpretation from the RC, ERC, and MCML variance-covariance matrices. For BMA, the reliability of the variance-covariance matrix depends on MCMC convergence as indicated by Rhat.

See Also

ameras for model fitting, coef for extracting coefficients, summary.amerasfit for a summary including standard errors, vcov for the generic function.

Examples

data("data", package="ameras")
dosevars <- paste0("V", 1:10)

fit <- ameras(Y.binomial ~ dose(all_of(dosevars), model="ERR"),
              data=data, family="binomial", methods=c("RC", "ERC"))

## Extract vcov for all available methods
vcov(fit)

## Extract vcov for a single method, returns a matrix
vcov(fit, methods="RC")

## Extract vcov for multiple methods, returns a named list
vcov(fit, methods=c("RC", "ERC"))

---