# The qgcomp package: g-computation on exposure quantiles In qgcomp: Quantile G-Computation

library("knitr")
#library("gWQS")


## Introduction

qgcomp is a package to implement g-computation for analyzing the effects of exposure mixtures. Quantile g-computation yields estimates of the effect of increasing all exposures by one quantile, simultaneously. This, it estimates a "mixture effect" useful in the study of exposure mixtures such as air pollution, diet, and water contamination.

Using terminology from methods developed for causal effect estimation, quantile g-computation estimates the parameters of a marginal structural model that characterizes the change in the expected potential outcome given a joint intervention on all exposures, possibly conditional on confounders. Under the assumptions of exchangeability, causal consistency, positivity, no interference, and correct model specification, this model yields a causal effect for an intervention on the mixture as a whole. While these assumptions may not be met exactly, they provide a useful road map for how to interpret the results of a qgcomp fit, and where efforts should be spent in terms of ensuring accurate model specification and selection of exposures that are sufficient to control co-pollutant confounding.

### The model

Say we have an outcome $Y$, some exposures $\mathbb{X}$ and possibly some other covariates (e.g. potential confounders) denoted by $\mathbb{Z}$.

The basic model of quantile g-computation is a joint marginal structural model given by

[ \mathbb{E}(Y^{\mathbf{X}_q} | \mathbf{Z,\psi,\eta}) = g(\psi_0 + \psi_1 S_q + \mathbf{\eta Z}) ]

where $g(\cdot)$ is a link function in a generalized linear model (e.g. the inverse logit function in the case of a logistic model for the probability that $Y=1$), $\psi_0$ is the model intercept, $\mathbf{\eta}$ is a set of model coefficients for the covariates and $S_q$ is an "index" that represents a joint value of exposures. Quantile g-computation (by default) transforms all exposures $\mathbf{X}$ into $\mathbf{X}_q$, which are "scores" taking on discrete values 0,1,2,etc. representing a categorical "bin" of exposure. By default, there are four bins with evenly spaced quantile cutpoints for each exposure, so ${X}_q=0$ means that $X$ was below the observed 25th percentile for that exposure. The index $S_q$ represents all exposures being set to the same value (again, by default, discrete values 0,1,2,3). Thus, the parameter $\psi_1$ quantifies the expected change in the outcome, given a one quantile increase in all exposures simultaneously, possibly adjusted for $\mathbf{Z}$.

There are nuances to this particular model form that are available in the qgcomp package which will be explored below. There exists one special case of quantile g-computation that leads to fast fitting: linear/additive exposure effects. Here we simulate "pre-quantized" data where the exposures $X_1, X_2, X_3$ can only take on values of 0,1,2,3 in equal proportions. The model underlying the outcomes is given by the linear regression:

[ \mathbb{E}(Y | \mathbf{X,\beta}) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 ]

with the true values of $\beta_0=0, \beta_1 =0.25, \beta_2 =-0.1, \beta_3=0.05$, and $X_1$ is strongly positively correlated with $X_2$ ($\rho=0.95$) and negatively correlated with $X_3$ ($\rho=-0.3$). In this simple setting, the parameter $\psi_1$ will equal the sum of the $\beta$ coefficients (0.2). Here we see that qgcomp estimates a value very close to 0.2 (as we increase sample size, the estimated value will be expected to become increasingly close to 0.2).

library("qgcomp")
set.seed(543210)
qdat = simdata_quantized(n=5000, outcomtype="continuous", cor=c(.95, -0.3), b0=0, coef=c(0.25, -0.1, 0.05), q=4)
cor(qdat[,c("x1", "x2", "x3")])
qgcomp(y~x1+x2+x3, expnms=c("x1", "x2", "x3"), data=qdat)


## How to use the qgcomp package

Here we use a running example from the metals dataset from the from the package qgcomp to demonstrate some features of the package and method.

Namely, the examples below demonstrate use of the package for: 1. Fast estimation of exposure effects under a linear model for quantized exposures for continuous (normal) outcomes 2. Estimating conditional and marginal odds/risk ratios of a mixture effect for binary outcomes 3. Adjusting for non-exposure covariates when estimating effects of the mixture 4. Allowing non-linear and non-homogeneous effects of individual exposures and the mixture as a whole by including product terms 5. Using qgcomp to fit a time-to-event model to estimate conditional and marginal hazard ratios for the exposure mixture

For analogous approaches to estimating exposure mixture effects, illustrative examples can be seen in the gQWS package help files, which implements weighted quantile sum (WQS) regression, and at https://jenfb.github.io/bkmr/overview.html, which describes Bayesian kernel machine regression.

The metals dataset from the from the package qgcomp, comprises a set of simulated well water exposures and two health outcomes (one continuous, one binary/time-to-event). The exposures are transformed to have mean = 0.0, standard deviation = 1.0. The data are used throughout to demonstrate usage and features of the qgcomp package.

library("ggplot2")
data("metals", package="qgcomp")


### Example 1: linear model

# we save the names of the mixture variables in the variable "Xnm"
Xnm <- c(
'sodium','zinc'
)
covars = c('nitrate','nitrite','sulfate','ph', 'total_alkalinity','total_hardness')

# Example 1: linear model
# Run the model and save the results "qc.fit"
system.time(qc.fit <- qgcomp.noboot(y~.,dat=metals[,c(Xnm, 'y')], family=gaussian()))
#   user  system elapsed
#  0.011   0.002   0.018

# contrasting other methods with computational speed
# WQS regression (v3.0.1 of gWQS package)
#system.time(wqs.fit <- gWQS::gwqs(y~wqs,mix_name=Xnm, data=metals[,c(Xnm, 'y')], family="gaussian"))
#   user  system elapsed
# 35.775   0.124  36.114

# Bayesian kernel machine regression (note that the number of iterations here would
#  need to be >5,000, at minimum, so this underestimates the run time by a factor
#  of 50+
#system.time(bkmr.fit <- kmbayes(y=metals$y, Z=metals[,Xnm], family="gaussian", iter=100)) # user system elapsed # 81.644 4.194 86.520  First note that qgcomp can be very fast relative to competing methods (with their example times given from single runs from a laptop). One advantage of quantile g-computation over other methods that estimate "mixture effects" (the effect of changing all exposures at once), is that it is very computationally efficient. Contrasting methods such as WQS (gWQS package) and Bayesian Kernel Machine regression (bkmr package), quantile g-computation can provide results many orders of magnitude faster. For example, the example above ran 3000X faster for quantile g-computation versus WQS regression, and we estimate the speedup would be several hundred thousand times versus Bayesian kernel machine regression. The speed relies on an efficient method to fit qgcomp when exposures are added additively to the model. When exposures are added using non-linear terms or non-additive terms (see below for examples), then qgcomp will be slower but often still faster than competetive approaches. Quantile g-computation yields fixed weights in the estimation procedure, similar to WQS regression. However, note that the weights from qgcomp.noboot can be negative or positive. When all effects are linear and in the same direction ("directional homogeneity"), quantile g-computation is equivalent to weighted quantile sum regression in large samples. The overall mixture effect from quantile g-computation (psi1) is interpreted as the effect on the outcome of increasing every exposure by one quantile, possibly conditional on covariates. Given the overall exposure effect, the weights are considered fixed and so do not have confidence intervals or p-values. # View results: scaled coefficients/weights and statistical inference about # mixture effect qc.fit  Now let's take a brief look under the hood. qgcomp works in steps. First, the exposure variables are "quantized" or turned into score variables based on the total number of quantiles from the parameter q. You can access these via the qx object from the qgcomp fit object. # quantized data head(qc.fit$qx)


You can re-fit a linear model using these quantized exposures. This is the "underlying model" of a qgcomp fit.

# regression with quantized data
qc.fit$qx$y = qc.fit$fit$data$y # first bring outcome back into the quantized data newfit <- lm(y ~ arsenic_q + barium_q + cadmium_q + calcium_q + chromium_q + copper_q + iron_q + lead_q + magnesium_q + manganese_q + mercury_q + selenium_q + silver_q + sodium_q + zinc_q, data=qc.fit$qx)
newfit


Here you can see that, for a GLM in which all quantized exposures enter linearly and additively into the underlying model the overall effect from qgcomp is simply the sum of the adjusted coefficients from the underlying model.

sum(newfit$coefficients[-1]) # sum of all coefficients excluding intercept and confounders, if any coef(qc.fit) # overall effect and intercept from qgcomp fit  This equality is why we can fit qgcomp so efficiently under such a model, but qgcomp is a much more general method that can allow for non-linearity and non-additivity in the underlying model, as well as non-linearity in the overall model. These extensions are described in some of the following examples. ### Example 2: conditional odds ratio, marginal odds ratio in a logistic model This example introduces the use of a binary outcome in qgcomp via the qgcomp.noboot function, which yields a conditional odds ratio or the qgcomp.boot, which yields a marginal odds ratio or risk/prevalence ratio. These will not equal each other when there are non-exposure covariates (e.g. confounders) included in the model because the odds ratio is not collapsible (both are still valid). Marginal parameters will yield estimates of the population average exposure effect, which is often of more interest due to better interpretability over conditional odds ratios. Further, odds ratios are not generally of interest when risk ratios can be validly estimated, so qgcomp.boot will estimate the risk ratio by default for binary data (set rr=FALSE to allow estimation of ORs when using qgcomp.boot). qc.fit2 <- qgcomp.noboot(disease_state~., expnms=Xnm, data = metals[,c(Xnm, 'disease_state')], family=binomial(), q=4) qcboot.fit2 <- qgcomp.boot(disease_state~., expnms=Xnm, data = metals[,c(Xnm, 'disease_state')], family=binomial(), q=4, B=10,# B should be 200-500+ in practice seed=125, rr=FALSE) qcboot.fit2b <- qgcomp.boot(disease_state~., expnms=Xnm, data = metals[,c(Xnm, 'disease_state')], family=binomial(), q=4, B=10,# B should be 200-500+ in practice seed=125, rr=TRUE)  Compare a qgcomp.noboot fit: qc.fit2  with a qgcomp.boot fit: qcboot.fit2  with a qgcomp.boot fit, where the risk/prevalence ratio is estimated, rather than the odds ratio: qcboot.fit2b  ### Example 3: adjusting for covariates, plotting estimates In the following code we run a maternal age-adjusted linear model with qgcomp (family = "gaussian"). Further, we plot both the weights, as well as the mixture slope which yields overall model confidence bounds, representing the bounds that, for each value of the joint exposure are expected to contain the true regression line over 95% of trials (so-called 95% 'pointwise' bounds for the regression line). The pointwise comparison bounds, denoted by error bars on the plot, represent comparisons of the expected difference in outcomes at each quantile, with reference to a specific quantile (which can be specified by the user, as below). These pointwise bounds are similar to the bounds created in the bkmr package when plotting the overall effect of all exposures. The pointwise bounds can be obtained via the pointwisebound.boot function. To avoid confusion between "pointwise regression" and "pointwise comparison" bounds, the pointwise regression bounds are denoted as the "model confidence band" in the plots, since they yield estimates of the same type of bounds as the predict function in R when applied to linear model fits. Note that the underlying regression model is on the quantile 'score', which takes on values integer values 0, 1, ..., q-1. For plotting purposes (when plotting regression line results from qgcomp.boot), the quantile score is translated into a quantile (range = [0-1]). This is not a perfect correspondence, because the quantile g-computation model treats the quantile score as a continuous variable, but the quantile category comprises a range of quantiles. For visualization, we fix the ends of the plot at the mid-points of the first and last quantile cut-point, so the range of the plot will change slightly if 'q' is changed. qc.fit3 <- qgcomp.noboot(y ~ mage35 + arsenic + barium + cadmium + calcium + chloride + chromium + copper + iron + lead + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=Xnm, metals, family=gaussian(), q=4) qc.fit3 plot(qc.fit3)  From the first plot we see weights from qgcomp.noboot function, which include both positive and negative effect directions. When the weights are all on a single side of the null, these plots are easy to in interpret since the weight corresponds to the proportion of the overall effect from each exposure. WQS uses a constraint in the model to force all of the weights to be in the same direction - unfortunately such constraints lead to biased effect estimates. The qgcomp package takes a different approach and allows that "weights" might go in either direction, indicating that some exposures may beneficial, and some harmful, or there may be sampling variation due to using small or moderate sample sizes (or, more often, systematic bias such as unmeasured confounding). The "weights" in qgcomp correspond to the proportion of the overall effect when all of the exposures have effects in the same direction, but otherwise they correspond to the proportion of the effect in a particular direction, which may be small (or large) compared to the overall "mixture" effect. NOTE: the left and right sides of the plot should not be compared with each other because the length of the bars corresponds to the effect size only relative to other effects in the same direction. The darkness of the bars corresponds to the overall effect size - in this case the bars on the right (positive) side of the plot are darker because the overall "mixture" effect is positive. Thus, the shading allows one to make informal comparisons across the left and right sides: a large, darkly shaded bar indicates a larger independent effect than a large, lightly shaded bar. qcboot.fit3 <- qgcomp.boot(y ~ mage35 + arsenic + barium + cadmium + calcium + chloride + chromium + copper + iron + lead + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=Xnm, metals, family=gaussian(), q=4, B=10,# B should be 200-500+ in practice seed=125) qcboot.fit3 p = plot(qcboot.fit3)  We can change the referent category for pointwise comparisons via the pointwiseref parameter: plot(qcboot.fit3, pointwiseref = 3)  Using qgcomp.boot also allows us to assess linearity of the total exposure effect (the second plot). Similar output is available for WQS (gWQS package), though WQS results will generally be less interpretable when exposure effects are non-linear (see below how to do this with qgcomp.boot). The plot for the qcboot.fit3 object (using g-computation with bootstrap variance) gives predictions at the joint intervention levels of exposure. It also displays a smoothed (graphical) fit. Note that the uncertainty intervals given in the plot are directly accessible via the pointwisebound (pointwise comparison confidence intervals) and modelbound functions (confidence interval for the regression line): pointwisebound.boot(qcboot.fit3, pointwiseref=3) qgcomp:::modelbound.boot(qcboot.fit3)  Because qgcomp estimates a joint effect of multiple exposures, we cannot, in general, assess model fit by overlaying predictions from the plots above with the data. Hence, it is useful to explore non-linearity by fitting models that allow for non-linear effects, as in the next example. ### Example 4: non-linearity (and non-homogeneity) qgcomp (and qgcomp.boot) addresses non-linearity in a way similar to standard parametric regression models, which lends itself to being able to leverage R language features for non-linear parametric models (or, more precisely, parametric models that deviate from a purely additive, linear function on the link function basis via the use of basis function representation of non-linear functions). Here is an example where we use a feature of the R language for fitting models with interaction terms. We use y~. + .^2 as the model formula, which fits a model that allows for quadratic term for every predictor in the model. Similar approaches could be used to include interaction terms between exposures, as well as between exposures and covariates. qcboot.fit4 <- qgcomp(y~. + .^2, expnms=Xnm, metals[,c(Xnm, 'y')], family=gaussian(), q=4, B=10, seed=125) plot(qcboot.fit4)  Note that allowing for a non-linear effect of all exposures induces an apparent non-linear trend in the overall exposure effect. The smoothed regression line is still well within the confidence bands of the marginal linear model (by default, the overall effect of joint exposure is assumed linear, though this assumption can be relaxed via the 'degree' parameter in qgcomp.boot, as follows: qcboot.fit5 <- qgcomp(y~. + .^2, expnms=Xnm, metals[,c(Xnm, 'y')], family=gaussian(), q=4, degree=2, B=10, rr=FALSE, seed=125) plot(qcboot.fit5)  Once again, we can access numerical estimates of uncertainty: qgcomp::pointwisebound.boot(qcboot.fit5) qgcomp:::modelbound.boot(qcboot.fit5)  Ideally, the smooth fit will look very similar to the model prediction regression line. #### Interpretation of model parameters As the output below shows, setting "degree=2" yields a second parameter in the model fit ($\psi_2$). The output of qgcomp now corresponds to estimates of the marginal structural model given by [ \mathbb{E}(Y^{\mathbf{X}_q}) = g(\psi_0 + \psi_1 S_q + \psi_2 S_q^2) ] qcboot.fit5  so that$\psi_2$can be interpreted similar to quadratic terms that might appear in a generalized linear model.$\psi_2$estimates the change in the outcome for an additional unit of squared joint exposure, over-and-above the linear effect given by$\psi_1$. Informally, this is a way of formally assessing specific types of non-linearity in the joint exposure-response curves, and there are many other (slightly incorrect but intuitively useful) ways of interpreting parameters for squared terms in regressions (beyond the scope of this document). Intuition from generalized linear models applies directly to the models fit by quantile g-computation. ### Example 5: Comparing model fits and further exploring non-linearity Exploring a non-linear fit in settings with multiple exposures is challenging. One way to explore non-linearity, as demonstrated above, is to to include all 2-way interaction terms (including quadratic terms, or "self-interactions"). Sometimes this approach is not desired, either because the number of terms in the model can become very large, or because some sort of model selection procedure is required, which risks inducing over-fit (biased estimates and standard errors that are too small). Short of having a set of a priori non-linear terms to include, we find it best to take a default approach (e.g. taking all second order terms) that doesn't rely on statistical significance, or to simply be honest that the search for a non-linear model is exploratory and shouldn't be relied upon for robust inference. Methods such as kernel machine regression may be good alternatives, or supplementary approaches to exploring non-linearity. NOTE: qgcomp necessarily fits a regression model with exposures that have a small number of possible values, based on the quantile chosen. By package default, this is q=4, but it is difficult to fully examine non-linear fits using only four points, so we recommend exploring larger values of q, which will change effect estimates (i.e. the model coefficient implies a smaller change in exposures, so the expected change in the outcome will also decrease). Here, we examine a one strategy for default and exploratory approaches to mixtures that can be implemented in qgcomp using a smaller subset of exposures (iron, lead, cadmium), which we choose via the correlation matrix. High correlations between exposures may result from a common source, so small subsets of the mixture may be useful for examining hypotheses that relate to interventions on a common environmental source or set of behaviors. Note that we can still adjust for the measured exposures, even though only 3 our exposures of interest are considered as the mixture of interest. Note that we will require a new R package to help in exploring non-linearity: splines. Note that qgcomp.boot must be used in order to produce the graphics below, as qgcomp.noboot does not calculate the necessary quantities. #### Graphical approach to explore non-linearity in a correlated subset of exposures using splines library(splines) # find all correlations > 0.6 (this is an arbitrary choice) cormat = cor(metals[,Xnm]) idx = which(cormat>0.6 & cormat <1.0, arr.ind = TRUE) newXnm = unique(rownames(idx)) # iron, lead, and cadmium qc.fit6lin <- qgcomp.boot(y ~ iron + lead + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10) qc.fit6nonlin <- qgcomp.boot(y ~ bs(iron) + bs(cadmium) + bs(lead) + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, degree=2) qc.fit6nonhom <- qgcomp.boot(y ~ bs(iron)*bs(lead) + bs(iron)*bs(cadmium) + bs(lead)*bs(cadmium) + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, degree=3)  It helps to place the plots on a common y-axis, which is easy due to dependence of the qgcomp plotting functions on ggplot. Here's the linear fit : pl.fit6lin <- plot(qc.fit6lin, suppressprint = TRUE, pointwiseref = 4) pl.fit6lin + coord_cartesian(ylim=c(-0.75, .75)) + ggtitle("Linear fit: mixture of iron, lead, and cadmium")  Here's the non-linear fit : pl.fit6nonlin <- plot(qc.fit6nonlin, suppressprint = TRUE, pointwiseref = 4) pl.fit6nonlin + coord_cartesian(ylim=c(-0.75, .75)) + ggtitle("Non-linear fit: mixture of iron, lead, and cadmium")  And here's the non-linear fit with statistical interaction between exposures (recalling that this will lead to non-linearity in the overall effect): pl.fit6nonhom <- plot(qc.fit6nonhom, suppressprint = TRUE, pointwiseref = 4) pl.fit6nonhom + coord_cartesian(ylim=c(-0.75, .75)) + ggtitle("Non-linear, non-homogeneous fit: mixture of iron, lead, and cadmium")  #### Caution about graphical approaches The underlying conditional model fit can be made extremely flexible, and the graphical representation of this (via the smooth conditional fit) can look extremely flexible. Simply matching the overall (MSM) fit to this line is not a viable strategy for identifying parsimonious models because that would ignore potential for overfit. Thus, caution should be used when judging the accuracy of a fit when comparing the "smooth conditional fit" to the "MSM fit." qc.overfit <- qgcomp.boot(y ~ bs(iron) + bs(cadmium) + bs(lead) + mage35 + bs(arsenic) + bs(magnesium) + bs(manganese) + bs(mercury) + bs(selenium) + bs(silver) + bs(sodium) + bs(zinc), expnms=Xnm, metals, family=gaussian(), q=8, B=10, degree=1) qc.overfit plot(qc.overfit, pointwiseref = 5)  Here, there is little statistical evidence for even a linear trend, which makes the smoothed conditional fit appear to be overfit. The smooth conditional fit can be turned off, as below. plot(qc.overfit, flexfit = FALSE, pointwiseref = 5)  ### Example 6: Miscellaneous other ways to allow non-linearity. Note that these are included as examples of how to include non-linearities, and are not intended as a demonstration of appropriate model selection. In fact, qc.fit7b is generally a bad idea in small to moderate sample sizes due to large numbers of parameters. #### using indicator terms for each quantile qc.fit7a <- qgcomp.boot(y ~ factor(iron) + lead + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=20, deg=2) # underlying fit summary(qc.fit7a$fit)$coefficients plot(qc.fit7a)  #### interactions between indicator terms qc.fit7b <- qgcomp.boot(y ~ factor(iron)*factor(lead) + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, deg=3) # underlying fit #summary(qc.fit7b$fit)$coefficients plot(qc.fit7b)  #### breaks at specific quantiles (these breaks act on the quantized basis) qc.fit7c <- qgcomp.boot(y ~ I(iron>4)*I(lead>4) + cadmium + mage35 + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc, expnms=newXnm, metals, family=gaussian(), q=8, B=10, deg=2) # underlying fit summary(qc.fit7c$fit)$coefficients plot(qc.fit7c)  Note one restriction on exploring non-linearity: while we can use flexible functions such as splines for individual exposures, the overall fit is limited via the degree parameter to polynomial functions (here a quadratic polynomial fits the non-linear model well, and a cubic polynomial fits the non-linear/non-homogeneous model well - though this is an informal argument and does not account for the wide confidence intervals). We note here that only 10 bootstrap iterations are used to calculate confidence intervals (to increase computational speed for the example), which is far too low. #### Statistical approach explore non-linearity in a correlated subset of exposures using splines The graphical approaches don't give a clear picture of which model might be preferred, but we can compare the model fits using AIC, or BIC (information criterion that weigh model fit with over-parameterization). Both of these criterion suggest the linear model fits best (lowest AIC and BIC), which suggests that the apparently non-linear fits observed in the graphical approaches don't improve prediction of the health outcome, relative to the linear fit, due to the increase in variance associated with including more parameters. AIC(qc.fit6lin$fit)
AIC(qc.fit6nonlin$fit) AIC(qc.fit6nonhom$fit)

BIC(qc.fit6lin$fit) BIC(qc.fit6nonlin$fit)
BIC(qc.fit6nonhom$fit)  ### Example 7: time-to-event analysis and parallel processing • The qgcomp package utilizes Cox proportional hazards models as the underlying model for time-to-event analysis. The interpretation of a qgcomp.noboot fit parameter is a conditional (on confounders) hazard ratio for increasing all exposures at once. The qc.survfit1 object demonstrates a time-to- event analysis with qgcompcox.noboot. The default plot is similar to that of qgcompcox.noboot, in that it yields weights and an overall mixture effect # non-bootstrapped version estimates a marginal structural model for the # confounder-conditional effect survival::coxph(survival::Surv(disease_time, disease_state) ~ iron + lead + cadmium + arsenic + magnesium + manganese + mercury + selenium + silver + sodium + zinc + mage35, data=metals) qc.survfit1 <- qgcomp.cox.noboot(survival::Surv(disease_time, disease_state) ~ .,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4) qc.survfit1 plot(qc.survfit1)  Marginal hazards ratios (and bootstrapped quantile g-computation in general) uses a slightly different approach to effect estimation that makes it more computationally demanding than other qcomp functions. To estimate a marginal hazards ratio, the underlying model is fit, and then new outcomes are simulated under the underlying model with a baseline hazard estimator (Efron's) - this simulation requires a large sample (controlled by MCsize) for accuracy. This approach is similar to other g-computation approaches to survival analysis, but this approach uses the exact survival times, rather than discretized survival times as are common in most g-computation analysis. Plotting a qgcompcox.bootobject yields a set of survival curves (e.g.qc.survfit2) which comprise estimated survival curves (assuming censoring and late entry at random, conditional on covariates in the model) that characterize conditional survival functions (i.e. censoring competing risks) at various levels of joint-exposure (including the overall average - which may be slightly different from the observed survival curve, but should more or less agree). # bootstrapped version estimates a marginal structural model for the population average effect #library(survival) qc.survfit2 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ .,expnms=Xnm, data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4, B=5, MCsize=1000, parallel=TRUE) qc.survfit2 # testing proportional hazards (note that x=TRUE is not needed (and will cause an error if used)) survival::cox.zph(qc.survfit2$fit)
p2 = plot(qc.survfit2, suppressprint = TRUE)
p2 + labs(title="Linear log(hazard ratio), overall and exposure specific")


All bootstrapped functions in qgcomp allow parellelization via the parallel=TRUE parameter (demonstrated with the non-liner fit in qc.survfit3). Only 5 bootstrap iterations are used here, which is not nearly enough for inference, and will actually be slower for parallel processing due to some overhead when setting up the parallel processes.

qc.survfit3 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ . + .^2,expnms=Xnm,
data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4,
B=5, MCsize=1000, parallel=TRUE)
qc.survfit3
p3 = plot(qc.survfit3, suppressprint = TRUE)
p3 + labs(title="Non-linear log(hazard ratio) overall, linear exposure specific ln-HR")


While qgcompcox.boot fits a smooth hazard ratio function, the hazard ratios contrasting specific quantiles with a referent quantile can be obtained, as demonstrated with qc.survfit4. As in qgcomp.boot plots, the conditional model fit and the MSM fit are overlaid as a way to judge how well the MSM fits the conditional fit (and whether, for example non-linear terms should be added or removed from the overall fit via the degree parameter - we note here that we know of no statistical test for quantifying the difference between these lines, so this is up to user discretion and the plots are provided as visuals to aid in exploratory data analysis).

qc.survfit4 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ . + .^2,expnms=Xnm,
data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4,
B=5, MCsize=1000, parallel=TRUE, degree=2)
qc.survfit4
# examining the overall hazard ratio as a function of overall exposure
hrs_q = exp(matrix(c(0,0,1,1,2,4,3,9), ncol=2, byrow=TRUE)%*%qc.survfit4$msmfit$coefficients)
colnames(hrs_q) = "Hazard ratio"
print("Hazard ratios by quartiles (min-25%,25-50%, 50-75%, 75%-max)")
hrs_q

p4 = plot(qc.survfit4, suppressprint = TRUE)
p4 + labs(title="Non-linear log(hazard ratio), overall and exposure specific")


Testing proportional hazards is somewhat complicated with respect to interpretation. Consider first a linear fit from qgcomp.cox.noboot. Because the underlying model of a linear qgcomp fit is equivalent to the sum of multiple parameters, it is not clear how proportional hazards might be best tested for the mixture. One could examine test statistics for each exposure, but there may be some exposures for which the test indicates non-proportional hazards and some for which the test does not.

The "GLOBAL" test in the cox.zph function from the survival package comes closest to what we might want, and gives an overall assessment of non-proportional hazards for all model terms simultaneously (including non-mixture covariates). While this seems somewhat undesirable due to non-specificity, it is not necessarily important that only the mixture have proportional hazards, so it is useful and easily interpretable to have a global test of fit via GLOBAL.

# testing proportional hazards (must set x=TRUE in function call)
qc.survfit1ph <- qgcomp.cox.noboot(survival::Surv(disease_time, disease_state) ~ .,expnms=Xnm,
data=metals[,c(Xnm, 'disease_time', 'disease_state', "mage35")], q=4,
x=TRUE)
survival::cox.zph(qc.survfit1ph$fit)  For a potentially non-linear/ non-additive fit from qgcomp.cox.boot, the issue is slightly more complicated by the fact that the algorithm will fit both the underlying model and a marginal structural model using the predictions from the underlying model. In order for the predictions to yield valid causal inference, the underlying model must be correct (which implies that proportional hazards hold). The marginal structural model proceeds assuming the underlying model is correct. Currently there is no simple way to allow for non-proportional hazards in the marginal structural model, but non-proportional hazards can be implemented in the conditional model via standard approaches to non-proportional hazards such as time-exposure-interaction terms. This is a rich field and discussion is beyond the scope of this document. # testing global proportional hazards for model (note that x=TRUE is not needed (and will cause an error if used)) phtest3 = survival::cox.zph(qc.survfit3$fit)
phtest3$table[dim(phtest3$table)[1],, drop=FALSE]


Late entry and counting-process style data will currently yield results in qgcomp.cox functions. There has been some testing of this in limited settings, but we note that this is still an experimental feature at this point that may not be valid in all cases and so it is not documented here. As much effort as possible to validate results through other means is needed when using qgcomp in data subject to late-entry or when using counting-process style data.

### Example 8: clustering

Clustering on the individual or group level means that there are individual or group level characteristics which result in covariance between observations (e.g. within individual variance of an outcome may be much lower than the between individual variance). For linear models, the error term is assumed to be independent between observations, and clustering breaks this assumption. Ways to relax this assumption include empirical variance estimation and cluster-appropriate robust variance estimation (e.g. through the sandwich package in R). Another way is to use cluster-based bootstrapping, which samples clusters, rather than individual observations. qgcomp.boot can be leveraged to produce clustering consistent estimates of standard errors for independent effects of exposure as well as the effect of the exposure as a whole. This is done using the id parameter of qgcomp.boot (which can only handle a single variable and so may not efficient for nested clustering, for example).

Below is a simple example with one simulated exposure. First the exposure data are 'pre-quantized' (so that one can verify that standard errors are appropriate using other means - this is just to prove that the clustering works and is not intended to show a suggested practice). Next the data are analyzed using a 1-component mixture in qgcomp - again, this is for verification purposes. The qgcomp.noboot result yields a naive standard error of 0.0310 for the mixture effect:

set.seed(2123)
N = 250
t = 4
dat <- data.frame(row.names = 1:(N*t))
dat <- within(dat, {
id = do.call("c", lapply(1:N, function(x) rep(x, t)))
u =  do.call("c", lapply(1:N, function(x) rep(runif(1), t)))
x1 = rnorm(N, u)
y = rnorm(N) + u + x1
})

# pre-quantize
expnms = c("x1")
datl = quantize(dat, expnms = expnms)

qgcomp.noboot(y~ x1, data=datl$dat, family=gaussian(), q = NULL) # neither of these ways yields appropriate clustering #qgcomp.noboot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL)
#qgcomp.boot(y~ x1, data=datl$dat, family=gaussian(), q = NULL, MCsize=1000)  while the qgcomp.boot result (MCsize=5000, B=500) yields a corrected standard error of 0.0398, which is much closer to the sandwich estimate of 0.0409 than the naive estimator (a second qgcomp.boot fit with fewer bootstrap iterations and smaller MCsize is included for display, but the more accurate results are commented out to reduce runtime). Here, the standard errors from the uncorrected fit are too low, but this may not always be the case. # clustering by specifying id parameter on qgcomp.boot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL, MCsize=1000, B = 5)
#qgcomp.boot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL, MCsize=1000, B = 500) # Mixture slope parameters (bootstrap CI): # # Estimate Std. Error Lower CI Upper CI t value # (Intercept) -0.4632 0.0730 -0.606 -0.32 3.3e-10 # psi1 0.9550 0.0398 0.877 1.03 0 # This can be verified using the sandwich package #fitglm = glm(y~x1, data=datl$dat)
#sw.cov = sandwich::vcovCL(fitglm, cluster=~id, type = "HC0")[2,2]
#sqrt(sw.cov)
# [1] 0.0409


### Example 9: partial effects

Returning to our original example (and adjusting for covariates), note that the default output for a qgcomp.*.noboot object includes "sum of positive/negative coefficients." These can be interpreted as "partial mixture effects" or effects of exposures with coefficients in a particular direction. This is displayed graphically via a plot of the qgcomp "weights," where all exposures that contribute to a given partial effect are on the same side of the plot. Unfortunately, this does not yield valid inference for a true partial effect because it is a parameter conditional on the fitted results and thus does not represent the type of a priori hypothesis that is amenable to hypothesis testing and confidence intervals. Another way to think about this is that it is a data adaptive parameter and thus is subject to issues of overfit that are similar to issues with making inference from step-wise variable selection procedures.

(qc.fit.adj <- qgcomp.noboot(y~.,dat=metals[,c(Xnm, covars, 'y')], expnms=Xnm, family=gaussian()))


Fortunately, there is a way towards estimation of "partial effects." One way to do this is sample splitting, where the data are first randomly partitioned into a "training" and a "validation" data set. The assessment of whether a coefficient is positive or not occurs in the "training" set and then effect estimation for positive/negative partial effects occurs in the validation data. Basic simulations can show that such a procedure can yield valid (often called "honest") hypothesis tests and confidence intervals for partial effects, provided that there is no separate data exploration in the combined dataset to select the models. In the qgcomp package, the partitioning of the datasets into "training" and "validation" sets is done by the user, which prevents issues that may arise if this is done naively on a dataset that contains clusters (where we should partition based on clusters, rather than observations) or multiple observations per individual (all observations from an individual should be partitioned together). Here is an example of simple partitioning on a dataset that contains one observation per individual with no clustering. The downside of sample splitting is the loss of precision, because the final "validation" dataset comprises only a fraction of the original sample size. Thus, the estimation of partial effects is most appropriate with large sample sizes. We also note that these partial effect are only well defined when all effects are linear and additive, since whether a variable contributes to a positive or negative partial effect would depend on the value of that variable, so the valid estimation of "partial effects" is limited to settings in which the qgcomp.\*.noboot objects are used for inference.

# 40/60% training/validation split
set.seed(1231124)
trainidx <- sample(1:nrow(metals), round(nrow(metals)*0.4))
valididx <- setdiff(1:nrow(metals),trainidx)
traindata <- metals[trainidx,]
validdata <- metals[valididx,]
dim(traindata) # 181 observations = 40% of total
dim(validdata) # 271 observations = 60% of total


The qgcomp package then facilitates the analysis of these partitioned data to allow valid estimation and hypothesis testing of partial effects. The qgcomp.partials function is used to estimate partial effects. Note that the variables with "negative effect sizes" differs slightly from the overall analysis given in the qc.fit object that represents our first pass analysis on these data. This is to be expected, and is a feature of this approach: different random subsets of the data will be expected to yield different estimated effects. If the true effect is null, then the estimated effects will vary from positive to negative around the null, and sample splitting is an important way to distinguish between estimates that reflect underlying patterns in the entire dataset from estimates that are simply due to natural sampling variation inherent to small and moderate samples.

splitres <- qgcomp.partials(fun="qgcomp.noboot", f=y~., q=4,
traindata=traindata[,c(Xnm, covars, "y")],validdata=validdata[,c(Xnm, covars, "y")], expnms=Xnm)
splitres


### Do I have to use quantiles?

No. You can turn off "quantization" by setting q=NULL or you can supply your own categorization cutpoints via the "breaks" argument. It is up to the user to interpret the results if either of these options is taken.

### Can I cite this document?

Probably not in a scientific manuscript. If you find an idea here that is not published anywhere else and wish to develop it into a full manuscript, feel free! (But probably check with akeil@unc.edu to ask if a paper is already in development.)

## References

Alexander P. Keil, Jessie P. Buckley, Katie M. O’Brien, Kelly K. Ferguson, Shanshan Zhao, Alexandra J. White. A quantile-based g-computation approach to addressing the effects of exposure mixtures. https://doi.org/10.1289/EHP5838

## Acknowledgments

The development of this package was supported by NIH Grant RO1ES02953101. Invaluable code testing was performed by Nicole Niehoff, Michiel van den Dries, Emily Werder, Jessie Buckley, and Katie O'Brien.

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qgcomp documentation built on July 16, 2021, 9:06 a.m.