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.
qgcomp packageHere 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
# 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 0.028
# 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.
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
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. 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
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.
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)
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)
Ideally, the smooth fit will look very similar to the model prediction regression line.
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.
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")
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)
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)
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-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.
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)
## [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
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 effectqcomp 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.boot
object 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).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.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).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 mixtures. 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 asessment 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. 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.# 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)
# 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)
## chisq df p
## arsenic 1.57e-01 1 0.691
## barium 1.28e-01 1 0.721
## cadmium 5.14e-02 1 0.821
## calcium 9.16e-04 1 0.976
## chromium 1.25e+00 1 0.263
## copper 3.42e-01 1 0.559
## iron 3.51e+00 1 0.061
## lead 1.59e-01 1 0.690
## magnesium 2.08e+00 1 0.149
## manganese 5.78e-01 1 0.447
## mercury 4.87e-06 1 0.998
## selenium 1.32e-01 1 0.716
## silver 1.30e-02 1 0.909
## sodium 1.06e-01 1 0.745
## zinc 1.53e+00 1 0.216
## mage35 1.73e-02 1 0.895
## GLOBAL 9.93e+00 16 0.870
# 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 -0.24020 0.12466 -0.48453 0.0041255 -1.9269 0.054
# testing proportional hazards (note that x=TRUE is not needed (and will cause an error if used))
survival::cox.zph(qc.survfit2$fit)
## chisq df p
## arsenic 0.18440 1 0.668
## barium 0.10819 1 0.742
## cadmium 0.05345 1 0.817
## calcium 0.00206 1 0.964
## chromium 1.23974 1 0.266
## copper 0.28518 1 0.593
## iron 3.46739 1 0.063
## lead 0.17575 1 0.675
## magnesium 2.12900 1 0.145
## manganese 0.58720 1 0.444
## mercury 0.00136 1 0.971
## selenium 0.15247 1 0.696
## silver 0.01040 1 0.919
## sodium 0.09352 1 0.760
## zinc 1.51261 1 0.219
## GLOBAL 9.82045 15 0.831
p2 = plot(qc.survfit2, suppressprint = TRUE)
p2 + labs(title="Linear log(hazard ratio), overall and exposure specific")
qc.survfit3 <- qgcomp.cox.boot(Surv(disease_time, disease_state) ~ . + .^2,expnms=Xnm,
data=metals[,c(Xnm, 'disease_time', 'disease_state')], q=4,
B=5, MCsize=1000, parallel=TRUE)
qc.survfit3
## Mixture log(hazard ratio) (bootstrap CI):
##
## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## psi1 -0.19766 1.05788 -2.2711 1.8758 -0.1868 0.8518
p3 = plot(qc.survfit3, suppressprint = TRUE)
p3 + labs(title="Non-linear log(hazard ratio) overall, linear exposure specific ln-HR")
# testing global proportional hazards for model (note that x=TRUE is not needed (and will cause an error if used))
phtest3 = survival::cox.zph(qc.survfit3$fit)
phtest3$table[dim(phtest3$table)[1],, drop=FALSE]
## chisq df p
## GLOBAL 206.5578 120 1.53907e-06
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
## Mixture log(hazard ratio) (bootstrap CI):
##
## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## psi1 -2.87367 3.34052 -9.4210 3.6736 -0.8602 0.3897
## psi2 0.87788 1.06008 -1.1998 2.9556 0.8281 0.4076
# 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,] 0.1359070
## [3,] 0.1069065
## [4,] 0.4867284
p4 = plot(qc.survfit4, suppressprint = TRUE)
p4 + labs(title="Non-linear log(hazard ratio), overall and exposure specific")
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 naiive standard error of 0.0310 for the mixture effect, 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 naiive 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.
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$dat, family=gaussian(), q = NULL)
## Including all model terms as exposures of interest
## Scaled effect size (positive direction, sum of positive coefficients = 0.955)
## x1
## 1
##
## Scaled effect size (negative direction, sum of negative coefficients = 0)
## None
##
## Mixture slope parameters (Delta method CI):
##
## Estimate Std. Error Lower CI Upper CI t value
## (Intercept) -0.463243 0.057934 -0.57679 -0.34969 3.553e-15
## psi1 0.955015 0.031020 0.89422 1.01581 0
# neither of these ways yields appropriate clustering
#qgcomp.noboot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL)
#qgcomp.boot(y~ x1, data=datl$dat, family=gaussian(), q = NULL, MCsize=1000)
# clustering by specifying id parameter on
qgcomp.boot(y~ x1, data=datl$dat, id="id", family=gaussian(), q = NULL, MCsize=1000, B = 5)
## Including all model terms as exposures of interest
##
## Note: using all possible values of exposure as the
## intervention values
## Mixture slope parameters (bootstrap CI):
##
## Estimate Std. Error Lower CI Upper CI t value
## (Intercept) -0.463243 0.084446 -0.62875 -0.29773 5.223e-08
## psi1 0.955015 0.055037 0.84714 1.06289 0
#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
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
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.