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 roadmap 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.

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-homogenous 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("qgcomp")
library("knitr")
library("ggplot2")
data("metals", package="qgcomp")
head(metals)
##       arsenic      barium    cadmium    calcium    chloride    chromium
## 1  0.09100165  0.08166362 15.0738845 -0.7746662 -0.15408335 -0.05589104
## 2  0.17018302 -0.03598828 -0.7126486 -0.6857886 -0.19605499 -0.03268488
## 3  0.13336869  0.09934014  0.6441992 -0.1525231 -0.17511844 -0.01161098
## 4 -0.52570747 -0.76616263 -0.8610256  1.4472733  0.02552401 -0.05173287
## 5  0.43420529  0.40629920  0.0570890  0.4103682 -0.24187403 -0.08931824
## 6  0.71832662  0.19559582 -0.6823437 -0.8931696 -0.03919936 -0.07389407
##        copper       iron         lead  magnesium   manganese     mercury
## 1  1.99438050 19.1153352 21.072630908 -0.5109546  2.07630966 -1.20826726
## 2 -0.02490169 -0.2039425 -0.010378362 -0.1030542 -0.36095395 -0.68729723
## 3  0.25700811 -0.1964581 -0.063375935  0.9166969 -0.31075240  0.44852503
## 4  0.75477075 -0.2317787 -0.002847991  2.5482987 -0.23350205  0.20428158
## 5 -0.09919923 -0.1698619 -0.035276281 -0.5109546  0.08825996  1.19283834
## 6 -0.05622285 -0.2129300 -0.118460981 -1.0059145 -0.30219838  0.02875033
##      nitrate    nitrite         ph    selenium     silver      sodium
## 1  1.3649492 -1.0500539 -0.7125482  0.23467592 -0.8648653 -0.41840695
## 2 -0.1478382  0.4645119  0.9443009  0.65827253 -0.8019173 -0.09112969
## 3 -0.3001660 -1.4969868  0.4924330  0.07205576 -0.3600140 -0.11828963
## 4  0.3431814 -0.6992263 -0.4113029  0.23810705  1.3595205 -0.11828963
## 5  0.0431269 -0.5041390  0.3418103 -0.02359910 -1.6078044 -0.40075299
## 6 -0.3986575  0.1166249  1.2455462 -0.61186017  1.3769466  1.83722597
##      sulfate total_alkalinity total_hardness       zinc mage35          y
## 1 -0.1757544      -1.31353389    -0.85822417  1.0186058      1 -0.6007989
## 2 -0.1161359      -0.12699789    -0.67749970 -0.1509129      0 -0.2022296
## 3 -0.1616806       0.42671890     0.07928399 -0.1542524      0 -1.2164116
## 4  0.8272415       0.99173604     1.99948142  0.1843372      0  0.1826311
## 5 -0.1726845      -0.04789549     0.30518957 -0.1529379      0  1.1760472
## 6 -0.1385631       1.98616621    -1.07283447 -0.1290391      0 -0.4100912
##   disease_time disease_state
## 1 6.168764e-07             1
## 2 4.000000e+00             0
## 3 4.000000e+00             0
## 4 4.000000e+00             0
## 5 1.813458e+00             1
## 6 2.373849e+00             1

Example 1: linear model

# we save the names of the mixture variables in the variable "Xnm"
Xnm <- c(
    'arsenic','barium','cadmium','calcium','chromium','copper',
    'iron','lead','magnesium','manganese','mercury','selenium','silver',
    '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()))
## Including all model terms as exposures of interest
##    user  system elapsed 
##   0.020   0.001   0.027
#   user  system elapsed 
#  0.011   0.002   0.018 

# contrasting other methods with computational speed
# WQS regression
#system.time(wqs.fit <- gwqs(y~NULL,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 is very fast

# View results: scaled coefficients/weights and statistical inference about
# mixture effect
qc.fit
## Scaled effect size (positive direction, sum of positive coefficients = 0.39)
##  calcium     iron   barium   silver  arsenic  mercury   sodium chromium 
##  0.72216  0.06187  0.05947  0.03508  0.03447  0.02451  0.02162  0.02057 
##  cadmium     zinc 
##  0.01328  0.00696 
## 
## Scaled effect size (negative direction, sum of negative coefficients = -0.124)
## magnesium    copper      lead manganese  selenium 
##  0.475999  0.385299  0.074019  0.063828  0.000857 
## 
## Mixture slope parameters (Delta method CI):
## 
##              Estimate Std. Error Lower CI Upper CI t value
## (Intercept) -0.356670   0.107878 -0.56811 -0.14523   1e-03
## psi1         0.266394   0.071025  0.12719  0.40560   2e-04

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.

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.

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
## Scaled effect size (positive direction, sum of positive coefficients = 0.392)
##    barium      zinc  chromium magnesium    silver    sodium 
##    0.3520    0.2002    0.1603    0.1292    0.0937    0.0645 
## 
## Scaled effect size (negative direction, sum of negative coefficients = -0.696)
##  selenium    copper   arsenic   calcium manganese   cadmium   mercury      lead 
##    0.2969    0.1627    0.1272    0.1233    0.1033    0.0643    0.0485    0.0430 
##      iron 
##    0.0309 
## 
## Mixture log(OR) (Delta method CI):
## 
##             Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## (Intercept)  0.26362    0.51615 -0.74802  1.27526  0.5107   0.6095
## psi1        -0.30416    0.34018 -0.97090  0.36258 -0.8941   0.3713
# and a qgcomp.boot fit:
qcboot.fit2
## Mixture log(OR) (bootstrap CI):
## 
##             Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## (Intercept)  0.26362    0.38110 -0.48332  1.01056  0.6917   0.4891
## psi1        -0.30416    0.27478 -0.84272  0.23439 -1.1070   0.2683
# and a qgcomp.boot fit, where the risk/prevalence ratio is estimated, 
#  rather than the odds ratio:
qcboot.fit2b
## Mixture log(RR) (bootstrap CI):
## 
##             Estimate Std. Error Lower CI Upper CI Z value  Pr(>|z|)
## (Intercept) -0.56237    0.15716 -0.87040 -0.25434 -3.5783 0.0003458
## psi1        -0.16373    0.13611 -0.43049  0.10304 -1.2029 0.2290108

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 bounds is 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 estimate 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. 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 cutpoint, 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
## Scaled effect size (positive direction, sum of positive coefficients = 0.381)
##  calcium   barium     iron   silver  arsenic  mercury chromium     zinc 
##  0.74466  0.06636  0.04839  0.03765  0.02823  0.02705  0.02344  0.01103 
##   sodium  cadmium 
##  0.00775  0.00543 
## 
## Scaled effect size (negative direction, sum of negative coefficients = -0.124)
## magnesium    copper      lead manganese  selenium 
##   0.49578   0.35446   0.08511   0.06094   0.00372 
## 
## Mixture slope parameters (Delta method CI):
## 
##              Estimate Std. Error Lower CI Upper CI t value
## (Intercept) -0.348084   0.108037 -0.55983 -0.13634  0.0014
## psi1         0.256969   0.071459  0.11691  0.39703  0.0004
plot(qc.fit3)
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
## Mixture slope parameters (bootstrap CI):
## 
##              Estimate Std. Error Lower CI Upper CI t value
## (Intercept) -0.342787   0.114983 -0.56815 -0.11742   3e-03
## psi1         0.256969   0.075029  0.10991  0.40402   7e-04
p = plot(qcboot.fit3)
plot(qcboot.fit3, pointwiseref = 3)

pointwisebound.boot(qcboot.fit3, pointwiseref=3)
##   quantile quantile.midpoint  y.expected  mean.diff    se.diff       ul.pw
## 0        0             0.125 -0.34278746 -0.5139387 0.15005846 -0.04867828
## 1        1             0.375 -0.08581809 -0.2569694 0.07502923  0.06123650
## 2        2             0.625  0.17115127  0.0000000 0.00000000  0.17115127
## 3        3             0.875  0.42812064  0.2569694 0.07502923  0.57517523
##        ll.pw
## 0 -0.6368966
## 1 -0.2328727
## 2  0.1711513
## 3  0.2810660
qgcomp:::modelbound.boot(qcboot.fit3)
##   quantile quantile.midpoint  y.expected      se.pw       ll.pw        ul.pw
## 0        0             0.125 -0.34278746 0.11498301 -0.56815001 -0.117424905
## 1        1             0.375 -0.08581809 0.04251388 -0.16914376 -0.002492425
## 2        2             0.625  0.17115127 0.04065143  0.09147593  0.250826606
## 3        3             0.875  0.42812064 0.11294432  0.20675384  0.649487428
##     ll.simul     ul.simul
## 0 -0.4622572 -0.161947578
## 1 -0.1191843 -0.005233921
## 2  0.1386622  0.223888629
## 3  0.3081934  0.566961520

plot of chunk adjusting for covariatesplot of chunk adjusting for covariatesplot of chunk adjusting for covariates

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.

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. Generally, we cannot overlay the data over this plot since the regression line corresponds to a change in potentially many exposures at once. Hence, it is useful to explore non-linearity by fitting models that allow for non-linear effects.

Example 4: non-linearity (and non-homogeneity)

Let's close with one more feature of qgcomp (and qgcomp.boot): handling non-linearity. 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)

plot of chunk non-linear non-hom intro

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)
qgcomp::pointwisebound.boot(qcboot.fit5)
##   quantile quantile.midpoint  y.expected mean.diff   se.diff      ul.pw
## 0        0             0.125 -0.89239044 0.0000000 0.0000000 -0.8923904
## 1        1             0.375 -0.18559680 0.7067936 0.6560598  1.1002568
## 2        2             0.625  0.12180659 1.0141970 0.6480848  1.3920295
## 3        3             0.875  0.02981974 0.9222102 0.2803460  0.5792877
##        ll.pw
## 0 -0.8923904
## 1 -1.4714504
## 2 -1.1484163
## 3 -0.5196482
qgcomp:::modelbound.boot(qcboot.fit5)
##   quantile quantile.midpoint  y.expected      se.pw      ll.pw       ul.pw
## 0        0             0.125 -0.89239044 0.73835876 -2.3395470 0.554766138
## 1        1             0.375 -0.18559680 0.09639511 -0.3745278 0.003334151
## 2        2             0.625  0.12180659 0.12502104 -0.1232301 0.366843327
## 3        3             0.875  0.02981974 0.82096521 -1.5792425 1.638881990
##      ll.simul    ul.simul
## 0 -2.08917795  0.23867330
## 1 -0.32419926 -0.02762852
## 2 -0.02981037  0.39857469
## 3 -1.10497070  1.47916561
plot(qcboot.fit5)

plot of chunk overall non-linearity

Ideally, the smooth fit will look very similar to the model prediction regression line.

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 couple 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
pl.fit6lin <- plot(qc.fit6lin, suppressprint = TRUE, pointwiseref = 4)
pl.fit6nonlin <- plot(qc.fit6nonlin, suppressprint = TRUE, pointwiseref = 4)
pl.fit6nonhom <- plot(qc.fit6nonhom, suppressprint = TRUE, pointwiseref = 4)

pl.fit6lin + coord_cartesian(ylim=c(-0.75, .75)) + 
  ggtitle("Linear fit: mixture of iron, lead, and cadmium")
pl.fit6nonlin + coord_cartesian(ylim=c(-0.75, .75)) + 
  ggtitle("Non-linear fit: mixture of iron, lead, and cadmium")
pl.fit6nonhom + coord_cartesian(ylim=c(-0.75, .75)) + 
  ggtitle("Non-linear, non-homogeneous fit: mixture of iron, lead, and cadmium")

plot of chunk graphical non-linearityplot of chunk graphical non-linearityplot of chunk graphical non-linearity

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.” 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.

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
## Mixture slope parameters (bootstrap CI):
## 
##              Estimate Std. Error  Lower CI Upper CI t value
## (Intercept) -0.064420   0.123768 -0.307001 0.178162  0.6030
## psi1         0.029869   0.031972 -0.032795 0.092534  0.3507
plot(qc.overfit, pointwiseref = 5)
plot(qc.overfit, flexfit = FALSE, pointwiseref = 5)

plot of chunk graphical cautionplot of chunk graphical caution

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
##                   Estimate Std. Error    t value     Pr(>|t|)
## (Intercept)   -0.052981109 0.08062430 -0.6571357 5.114428e-01
## factor(iron)1  0.046571725 0.09466914  0.4919420 6.230096e-01
## factor(iron)2 -0.056984300 0.09581659 -0.5947227 5.523395e-01
## factor(iron)3  0.143813131 0.09558055  1.5046276 1.331489e-01
## factor(iron)4  0.053319057 0.09642069  0.5529835 5.805601e-01
## factor(iron)5  0.303967859 0.09743261  3.1197753 1.930856e-03
## factor(iron)6  0.246259568 0.09734385  2.5297907 1.176673e-02
## factor(iron)7  0.447045591 0.09786201  4.5681217 6.425565e-06
## lead          -0.009210341 0.01046668 -0.8799679 3.793648e-01
## cadmium       -0.010503041 0.01086440 -0.9667387 3.342143e-01
## mage35         0.081114695 0.07274583  1.1150426 2.654507e-01
## arsenic        0.021755516 0.02605850  0.8348720 4.042502e-01
## magnesium     -0.010758356 0.02469893 -0.4355798 6.633587e-01
## manganese      0.004418266 0.02551449  0.1731670 8.626011e-01
## mercury        0.003913896 0.02448078  0.1598763 8.730531e-01
## selenium      -0.058085344 0.05714805 -1.0164012 3.100059e-01
## silver         0.020971562 0.02407397  0.8711302 3.841658e-01
## sodium        -0.062086322 0.02404447 -2.5821454 1.014626e-02
## zinc           0.017078438 0.02392381  0.7138679 4.756935e-01
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
##                                      Estimate Std. Error      t value
## (Intercept)                     -5.910113e-02 0.05182385 -1.140423351
## I(iron > 4)TRUE                  3.649940e-01 0.06448858  5.659824144
## I(lead > 4)TRUE                 -9.004067e-05 0.06181587 -0.001456595
## cadmium                         -6.874749e-03 0.01078339 -0.637531252
## mage35                           7.613672e-02 0.07255110  1.049422029
## arsenic                          2.042370e-02 0.02578001  0.792230124
## magnesium                       -3.279980e-03 0.02427513 -0.135116878
## manganese                        1.055979e-02 0.02477453  0.426235507
## mercury                          9.396898e-03 0.02435057  0.385900466
## selenium                        -4.337729e-02 0.05670006 -0.765030761
## silver                           1.807248e-02 0.02391112  0.755819125
## sodium                          -5.537968e-02 0.02403808 -2.303831424
## zinc                             2.349906e-02 0.02385762  0.984970996
## I(iron > 4)TRUE:I(lead > 4)TRUE -1.828835e-01 0.10277790 -1.779405131
##                                     Pr(>|t|)
## (Intercept)                     2.547332e-01
## I(iron > 4)TRUE                 2.743032e-08
## I(lead > 4)TRUE                 9.988385e-01
## cadmium                         5.241120e-01
## mage35                          2.945626e-01
## arsenic                         4.286554e-01
## magnesium                       8.925815e-01
## manganese                       6.701456e-01
## mercury                         6.997578e-01
## selenium                        4.446652e-01
## silver                          4.501639e-01
## sodium                          2.169944e-02
## zinc                            3.251821e-01
## I(iron > 4)TRUE:I(lead > 4)TRUE 7.586670e-02
plot(qc.fit7c)

plot of chunk non-linear examplesplot of chunk non-linear examplesplot of chunk non-linear examples

Note one restriction on exploring non-linearity: while we can use flexible function 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-homogenous 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, AICC, or BIC (all various forms of 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$fit)
## [1] 676.0431
AIC(qc.fit6nonlin$fit)
## [1] 682.7442
AIC(qc.fit6nonhom$fit)
## [1] 705.6187
BIC(qc.fit6lin$fit)
## [1] 733.6346
BIC(qc.fit6nonlin$fit)
## [1] 765.0178
BIC(qc.fit6nonhom$fit)
## [1] 898.9617

Example 7: time-to-event analysis and parallel processing

# 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)
## Call:
## survival::coxph(formula = survival::Surv(disease_time, disease_state) ~ 
##     iron + lead + cadmium + arsenic + magnesium + manganese + 
##         mercury + selenium + silver + sodium + zinc + mage35, 
##     data = metals)
## 
##                coef exp(coef)  se(coef)      z      p
## iron      -0.056447  0.945117  0.156178 -0.361 0.7178
## lead       0.440735  1.553849  0.203264  2.168 0.0301
## cadmium    0.023325  1.023599  0.105502  0.221 0.8250
## arsenic   -0.003812  0.996195  0.088897 -0.043 0.9658
## magnesium  0.099399  1.104507  0.064730  1.536 0.1246
## manganese -0.014065  0.986033  0.064197 -0.219 0.8266
## mercury   -0.060830  0.940983  0.072918 -0.834 0.4042
## selenium  -0.231626  0.793243  0.173655 -1.334 0.1823
## silver     0.043169  1.044114  0.070291  0.614 0.5391
## sodium     0.057928  1.059638  0.063883  0.907 0.3645
## zinc       0.057169  1.058835  0.047875  1.194 0.2324
## mage35    -0.458696  0.632107  0.238370 -1.924 0.0543
## 
## Likelihood ratio test=23.52  on 12 df, p=0.02364
## n= 452, number of events= 205
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
## Scaled effect size (positive direction, sum of positive coefficients = 0.32)
##    barium      zinc magnesium  chromium    silver    sodium      iron 
##    0.3432    0.1946    0.1917    0.1119    0.0924    0.0511    0.0151 
## 
## Scaled effect size (negative direction, sum of negative coefficients = -0.554)
##  selenium    copper   calcium   arsenic manganese   cadmium      lead   mercury 
##    0.2705    0.1826    0.1666    0.1085    0.0974    0.0794    0.0483    0.0466 
## 
## Mixture log(hazard ratio) (Delta method CI):
## 
##      Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## psi1 -0.23356    0.24535 -0.71444  0.24732 -0.9519   0.3411
plot(qc.survfit1)

# 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
## Mixture log(hazard ratio) (bootstrap CI):
## 
##      Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## psi1 -0.23360    0.46321  -1.1415  0.67427 -0.5043    0.614
p2 = plot(qc.survfit2, suppressprint = TRUE)  
p2 + labs(title="Linear log(hazard ratio), overall and exposure specific")

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
## Mixture log(hazard ratio) (bootstrap CI):
## 
##      Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## psi1 -0.21122    0.35664 -0.91023  0.48779 -0.5922   0.5537
p3 = plot(qc.survfit3, suppressprint = TRUE) 
p3 + labs(title="Non-linear log(hazard ratio) overall, linear exposure specific ln-HR")

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)
## Warning in fitter(X, Y, istrat, offset, init, control, weights = weights, :
## Loglik converged before variable 1,2 ; coefficient may be infinite.
qc.survfit4
## Mixture log(hazard ratio) (bootstrap CI):
## 
##      Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## psi1 -2.86220   13.71090 -29.7351  24.0107 -0.2088   0.8346
## psi2  0.87417    4.64894  -8.2376   9.9859  0.1880   0.8508
# 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)")
## [1] "Hazard ratios by quartiles (min-25%,25-50%, 50-75%, 75%-max)"
hrs_q
##      Hazard ratio
## [1,]    1.0000000
## [2,]    0.1369648
## [3,]    0.1077738
## [4,]    0.4872055
p4 = plot(qc.survfit4, suppressprint = TRUE) 
p4 + labs(title="Non-linear log(hazard ratio), overall and exposure specific") 

plot of chunk time-to-eventplot of chunk time-to-eventplot of chunk time-to-eventplot of chunk time-to-event

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://arxiv.org/abs/1902.04200

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.