Analyze multiple exposure realizations

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(data, family="gaussian", Y, dosevars, M=NULL, X=NULL, offset=NULL, entry=NULL, 
  exit=NULL, setnr=NULL, methods="RC", deg=1, doseRRmod="ERR", transform=NULL,
  transform.jacobian=NULL, inpar=NULL, CI=c("proflik","percentile"),
  params.profCI="dose", maxit.profCI=20, tol.profCI=1e-2, 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.replicates.BMA=1:length(dosevars), 
  optim.method="Nelder-Mead", control=NULL, ... )

Arguments

data: input data frame.
family: outcome model family: "gaussian", "binomial", "poisson", "prophaz", "multinomial" or "clogit" (default "gaussian").
Y: 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: names or column indices of exposure replicate vectors.
M: names or column indices of binary effect modifying variables (optional).
X: names or column indices of other covariates (optional).
offset: name or column index of offset variable for Poisson regression (optional).
entry: name or column index of left truncation time variable for proportional hazards regression (optional).
exit: name or column index of exit time variable, required when family=prophaz.
setnr: name or column index of integer-valued matched set variable, required when family="clogit".
methods: character vector of one or multiple methods to apply. Options: "RC", "ERC", "MCML", "FMA", "BMA" (default "RC").
deg: for doseRRmod="ERR" and doseRRmod="EXP", whether to fit a linear (deg=1) or linear-quadratic (deg=2) dose-response model (default linear).
doseRRmod: the functional form of the dose-response relationship; options are exponential RR ("EXP"), linear ERR ("ERR"), or linear-exponential RR ("LINEXP") (default "ERR").
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).
CI: 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: when CI="proflik", whether to obtain profile-likelihood CIs for all parameters ("all") or only dose-related parameters ("dose", default).
maxit.profCI: maximum iterations for determining profile-likelihood CIs; passed to uniroot (default 20).
tol.profCI: tolerance for determining profile-likelihood CIs; passed to uniroot (default 1e-2).
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 1,000).
niter.BMA: number of MCMC iterations per chain for BMA (default 5,000).
nchains.BMA: number of MCMC chains for BMA (default 2).
thin.BMA: thinning rate for BMA (default 10).
included.replicates.BMA: indices of exposure replicates used in BMA (default \ 1:length(dosevars)).
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)).
...: other arguments, passed to functions such as transform.

Value

The output is an object of class amerasfit with a component call and a component for every method supplied to methods. For each method, the output is a list containing

coefficients: named vector of model coefficients.
sd: named vector of standard deviations.
CI: data frame with columns lower and upper giving 95% confidence bounds or credible interval bounds. When the CI method is "proflik", the data frame also has columns pval.lower and pval.upper (p-values to verify convergence of the root finder) and iter.lower and iter.upper (number of iterations used by uniroot).
runtime: string with the runtime in seconds.

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

vcov: covariance matrix for the full parameter vector.
convergence.optim: convergence code as returned by optim, with 0 indicating convergence and 1 indicating that the maximal number of iterations was reached.
counts.optim: number of function evaluations used in the model fit returned by optim.
loglik: log-likelihood value at the optimum.

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.replicates: indices of replicate 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:

included.samples: the total number of samples included.
included.replicates: indices of replicate 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 coef, summary, and traceplot.

Details

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.

Multiple options for confidence intervals are provided. 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, CI="wald.transformed" yields the CI $h(\bm\eta \pm 1.96 \bm V)$ with $\bm V$ the vector of standard deviations estimated using the inverse Hessian matrix, and CI="wald.orig" uses the delta method to obtain the CI $h(\bm\eta)\pm 1.96 \bm V_*$ 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, CI="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 CI="percentile" which uses 2.5% and 97.5% percentiles, and CI="hpd" which computes highest posterior density intervals using HPDinterval from the coda package, both using the FMA samples or Bayesian posterior samples.

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.

References

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

Examples

# \donttest{
  data(data, package="ameras")
  dosevars <- paste0("V", 1:10)
  ameras(data=data, family="gaussian", Y="Y.gaussian", dosevars=dosevars, 
  M=c("M1", "M2"), X=c("X1","X2")) 
#> Fitting RC
#> Obtaining profile likelihood CI for dose
#> Warning: P-value for dose upper bound more than 0.005 away from 0.05, reducing tol.profCI and/or increasing maxit.profCI is recommended
#> Warning: P-value for dose lower bound more than 0.005 away from 0.05, reducing tol.profCI and/or increasing maxit.profCI is recommended
#> Obtaining profile likelihood CI for dose:M1
#> Warning: P-value for dose:M1 upper bound more than 0.005 away from 0.05, reducing tol.profCI and/or increasing maxit.profCI is recommended
#> Warning: P-value for dose:M1 lower bound more than 0.005 away from 0.05, reducing tol.profCI and/or increasing maxit.profCI is recommended
#> Obtaining profile likelihood CI for dose:M2
#> Warning: P-value for dose:M2 upper bound more than 0.005 away from 0.05, reducing tol.profCI and/or increasing maxit.profCI is recommended
#> Warning: P-value for dose:M2 lower bound more than 0.005 away from 0.05, reducing tol.profCI and/or increasing maxit.profCI is recommended
#> $call
#> ameras(data = data, family = "gaussian", Y = "Y.gaussian", dosevars = dosevars, 
#>     M = c("M1", "M2"), X = c("X1", "X2"))
#> 
#> $RC
#> $RC$coefficients
#> (Intercept)          X1          X2        dose     dose:M1     dose:M2 
#>  -1.3795847   0.4965539  -0.5151334   0.9817904   0.1738918   0.5054444 
#>       sigma 
#>   1.0660556 
#> 
#> $RC$sd
#> (Intercept)          X1          X2        dose     dose:M1     dose:M2 
#>  0.03531066  0.03895627  0.04785715  0.02547356  0.02779861  0.03518118 
#>       sigma 
#>  0.01376271 
#> 
#> $RC$vcov
#>               (Intercept)            X1            X2          dose
#> (Intercept)  1.246842e-03 -7.761770e-04 -4.509516e-04 -3.805202e-04
#> X1          -7.761770e-04  1.517591e-03 -1.236947e-05  1.044708e-05
#> X2          -4.509516e-04 -1.236947e-05  2.290307e-03 -2.348377e-05
#> dose        -3.805202e-04  1.044708e-05 -2.348377e-05  6.489024e-04
#> dose:M1      8.269652e-06 -8.145006e-06 -1.927736e-06 -4.140406e-04
#> dose:M2     -4.742008e-05  4.342286e-05  1.013751e-05 -2.495313e-04
#> sigma        1.222532e-11  6.799541e-12  7.030303e-13  2.357885e-11
#>                   dose:M1       dose:M2         sigma
#> (Intercept)  8.269652e-06 -4.742008e-05  1.222532e-11
#> X1          -8.145006e-06  4.342286e-05  6.799541e-12
#> X2          -1.927736e-06  1.013751e-05  7.030303e-13
#> dose        -4.140406e-04 -2.495313e-04  2.357885e-11
#> dose:M1      7.727630e-04  4.230091e-05  2.615828e-11
#> dose:M2      4.230091e-05  1.237716e-03 -1.319876e-13
#> sigma        2.615828e-11 -1.319876e-13  1.894123e-04
#> 
#> $RC$CI
#>             lower     upper pval.lower pval.upper iter.lower iter.upper
#> dose    0.9333875 1.0301933 0.05749237 0.05749167          3          3
#> dose:M1 0.1210710 0.2267126 0.05749235 0.05749168          3          3
#> dose:M2 0.4385959 0.5722930 0.05749201 0.05749198          3          3
#> 
#> $RC$convergence.optim
#> [1] 0
#> 
#> $RC$counts.optim
#> function gradient 
#>      551       10 
#> 
#> $RC$loglik
#> [1] -4448.713
#> 
#> $RC$runtime
#> [1] "6.7 seconds"
#> 
#> 
#> attr(,"class")
#> [1] "amerasfit"
# }