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)
head(qdat)
## x1 x2 x3 y
## 1 2 2 3 0.5539253
## 2 2 3 1 -0.1005701
## 3 2 2 3 0.3782267
## 4 1 2 2 -0.4557419
## 5 0 0 1 0.6124436
## 6 1 1 2 0.7569574
cor(qdat[,c("x1", "x2", "x3")])
## x1 x2 x3
## x1 1.00000 0.95552 -0.30368
## x2 0.95552 1.00000 -0.29840
## x3 -0.30368 -0.29840 1.00000
qgcomp(y~x1+x2+x3, expnms=c("x1", "x2", "x3"), data=qdat)
## Scaled effect size (positive direction, sum of positive coefficients = 0.3)
## x1 x3
## 0.775 0.225
##
## Scaled effect size (negative direction, sum of negative coefficients = -0.0966)
## x2
## 1
##
## Mixture slope parameters (delta method CI):
##
## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)
## (Intercept) -0.0016583 0.0359609 -0.07214 0.068824 -0.0461 0.9632
## psi1 0.2035810 0.0219677 0.16053 0.246637 9.2673 <2e-16
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:
- Fast estimation of exposure effects under a linear model for quantized exposures for continuous (normal) outcomes
- Estimating conditional and marginal odds/risk ratios of a mixture effect for binary outcomes
- Adjusting for non-exposure covariates when estimating effects of the mixture
- Allowing non-linear and non-homogeneous effects of individual exposures and the mixture as a whole by including product terms
- 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")
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.glm.noboot(y~.,dat=metals[,c(Xnm, 'y')], family=gaussian()))
## Including all model terms as exposures of interest
## user system elapsed
## 0.004 0.000 0.004
# 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.glm.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
## 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 Pr(>|t|)
## (Intercept) -0.356670 0.107878 -0.56811 -0.14523 -3.3062 0.0010238
## psi1 0.266394 0.071025 0.12719 0.40560 3.7507 0.0002001
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)
## arsenic_q barium_q cadmium_q calcium_q chromium_q copper_q iron_q lead_q
## 1 2 2 3 0 1 3 3 3
## 2 2 2 0 0 2 2 1 2
## 3 2 2 3 1 3 3 1 1
## 4 1 0 0 3 1 3 0 2
## 5 2 3 2 3 0 1 2 2
## 6 3 2 0 0 0 2 1 1
## magnesium_q manganese_q mercury_q selenium_q silver_q sodium_q zinc_q
## 1 1 3 0 2 0 0 3
## 2 2 0 1 3 1 3 0
## 3 3 1 2 2 1 3 0
## 4 3 1 2 2 3 3 3
## 5 1 3 3 1 0 0 0
## 6 0 1 2 0 3 3 2
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
##
## Call:
## lm(formula = 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$qx)
##
## Coefficients:
## (Intercept) arsenic_q barium_q cadmium_q calcium_q chromium_q
## -0.3566699 0.0134440 0.0231929 0.0051773 0.2816176 0.0080231
## copper_q iron_q lead_q magnesium_q manganese_q mercury_q
## -0.0476113 0.0241278 -0.0091464 -0.0588190 -0.0078872 0.0095572
## selenium_q silver_q sodium_q zinc_q
## -0.0001059 0.0136815 0.0084302 0.0027125
Here you can see that, for a GLM in which all quantized exposures enter linearly and additively into the underlying model the overall effect from qgcomp is simply the sum of the adjusted coefficients from the underlying model.
sum(newfit$coefficients[-1]) # sum of all coefficients excluding intercept and confounders, if any
## [1] 0.2663942
coef(qc.fit) # overall effect and intercept from qgcomp fit
## (intercept) psi1
## -0.3566699 0.2663942
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.glm.noboot function, which yields a conditional odds ratio or the
qgcomp.glm.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.glm.boot
will estimate the risk ratio by default for binary data (set rr=FALSE to
allow estimation of ORs when using qgcomp.glm.boot).
qc.fit2 <- qgcomp.glm.noboot(disease_state~., expnms=Xnm,
data = metals[,c(Xnm, 'disease_state')], family=binomial(),
q=4)
qcboot.fit2 <- qgcomp.glm.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.glm.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.glm.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
with a qgcomp.glm.boot fit:
qcboot.fit2
## Mixture log(OR) (bootstrap CI):
##
## Estimate Std. Error Lower CI Upper CI Z value Pr(>|z|)
## (Intercept) 0.26362 0.39038 -0.50151 1.02875 0.6753 0.4995
## psi1 -0.30416 0.28009 -0.85314 0.24481 -1.0859 0.2775
with a qgcomp.glm.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 0.16106 -0.87804 -0.24670 -3.4917 0.00048
## psi1 -0.16373 0.13839 -0.43497 0.10752 -1.1830 0.23679
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.glm.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.glm.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 Pr(>|t|)
## (Intercept) -0.348084 0.108037 -0.55983 -0.13634 -3.2219 0.0013688
## psi1 0.256969 0.071459 0.11691 0.39703 3.5960 0.0003601
plot(qc.fit3)
From the first plot we see weights from qgcomp.glm.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.glm.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 Pr(>|t|)
## (Intercept) -0.342787 0.114983 -0.56815 -0.11742 -2.9812 0.0030331
## psi1 0.256969 0.075029 0.10991 0.40402 3.4249 0.0006736
qcee.fit3 <- qgcomp.glm.ee(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)
qcee.fit3
## Mixture slope parameters (robust CI):
##
## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)
## (Intercept) -0.34279 0.10140 -0.54153 -0.14405 -3.3805 0.0007890
## psi1 0.25697 0.06771 0.12426 0.38968 3.7952 0.0001686
We can change the referent category for pointwise comparisons via the pointwiseref parameter:
plot(qcee.fit3, pointwiseref = 3, flexfit = FALSE)
plot(qcboot.fit3, pointwiseref = 3, flexfit = FALSE)
Using qgcomp.glm.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.glm.boot and qgcomp.glm.ee).
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)
## quantile quantile.midpoint linpred mean.diff se.diff ll.diff
## 0 0 0.125 -0.34278746 -0.5139387 0.15005846 -0.8080479
## 1 1 0.375 -0.08581809 -0.2569694 0.07502923 -0.4040240
## 2 2 0.625 0.17115127 0.0000000 0.00000000 0.0000000
## 3 3 0.875 0.42812064 0.2569694 0.07502923 0.1099148
## ul.diff ll.linpred ul.linpred
## 0 -0.2198295 -0.6368966 -0.04867828
## 1 -0.1099148 -0.2328727 0.06123650
## 2 0.0000000 0.1711513 0.17115127
## 3 0.4040240 0.2810660 0.57517523
qgcomp:::modelbound.boot(qcboot.fit3)
## quantile quantile.midpoint linpred m se.pw ll.pw
## 0 0 0.125 -0.34278746 -0.34278746 0.11498301 -0.56815001
## 1 1 0.375 -0.08581809 -0.08581809 0.04251388 -0.16914376
## 2 2 0.625 0.17115127 0.17115127 0.04065143 0.09147593
## 3 3 0.875 0.42812064 0.42812064 0.11294432 0.20675384
## ul.pw ll.simul ul.simul
## 0 -0.117424905 -0.4622572 -0.161947578
## 1 -0.002492425 -0.1191843 -0.005233921
## 2 0.250826606 0.1386622 0.223888629
## 3 0.649487428 0.3081934 0.566961520
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 (specifically qgcomp.*.boot and qgcomp.*.ee methods) addresses non-linearity in a way similar to standard parametric regression models, which lends itself to being able to leverage R language features for n-lin 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.
Aside: some details on qgcomp methods for non-linearity
Note that both qgcomp.*.boot (bootstrap) and qgcomp.*.ee (estimating equations) use standard methods for g-computation, whereas the qgcomp.*.noboot methods use a fast algorithm that works under the assumption of linearity and additivity of exposures (as described in the original paper on quantile-based g-computation). The “standard” method of g-computation with time-fixed exposures involves first fitting conditional models for the outcome, making predictions from those models under set exposure values, and then summarizing the predicted outcome distribution, possibly by fitting a second (marginal structural) model. qgcomp.*.boot follows this three-step process, while qgcomp.*.ee leverages estimating equations (sometimes: M-estimation) to estimate the parameters of the conditional and marginal structural model simultaneously. qgcomp.*.ee uses a sandwich variance estimator, which is similar to GEE (generalized estimating equation) approaches, and thus, when used correctly, can yield inference for longitudinal data in the same way that GEE does. The bootstrapping approach can also do this, but it takes longer. The extension to longitudinal data is representative of the broader concept that qgcomp.*.boot and qgcomp.*.ee can be used in a broader number of settings than qgcomp.*.noboot algorithms, but if one assumes linearity and additivity with no clustering of observations, and conditional parameters are of interest, then they are just a slower way to get equivalent results to qgcomp.*.noboot.
Below, we demonstrate a non-linear conditional fit (with a linear MSM) using the bootstrap approach. Similar approaches could be used to include interaction terms between exposures, as well as between exposures and covariates. Note this example is purposefully done incorrectly, as explained below.
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.glm.boot or qgcomp.glm.ee, 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)
qcee.fit5b <- qgcomp.glm.ee(y~. + .^2,
expnms=Xnm,
metals[,c(Xnm, 'y')], family=gaussian(), q=4, degree=2,
rr=FALSE)
plot(qcee.fit5b)
Note that some features are not availble to qgcomp.*.ee methods, which use estimating equations, rather than maximum likelihood methods. Briefly, these allow assessment of uncertainty under n-lin (and other) scenarios where the qgcomp.*.noboot functions cannot, since they rely on the additivity and linearity assumptions to achieve speed. The qgcomp.*.ee methods will generally be faster than a bootstrapped version, but they are not used extensively here because they are the newest additions to the qgcomp package, and the bootstrapped versions can be made fast (but not accurate) by reducing the number of bootstraps. Where available, the qgcomp.*.ee will be preferred to the qgcomp.*.boot versions for more stable and faster analyses when bootstrapping would otherwise be necessary.
Once again, we can access numerical estimates of uncertainty (answers differ between the qgcomp.*.boot and qgcomp.*.ee fits due to the small number of bootstrap samples):
modelbound.boot(qcboot.fit5)
## quantile quantile.midpoint linpred m se.pw ll.pw
## 0 0 0.125 -0.89239044 -0.89239044 0.70335801 -2.2709468
## 1 1 0.375 -0.18559680 -0.18559680 0.09874085 -0.3791253
## 2 2 0.625 0.12180659 0.12180659 0.14624914 -0.1648365
## 3 3 0.875 0.02981974 0.02981974 0.83609757 -1.6089014
## ul.pw ll.simul ul.simul
## 0 0.486165922 -2.08813110 0.18308760
## 1 0.007931712 -0.32577422 -0.02886665
## 2 0.408449640 -0.06371183 0.43431431
## 3 1.668540859 -1.19007238 1.57263047
pointwisebound.boot(qcboot.fit5)
## quantile quantile.midpoint linpred mean.diff se.diff ll.diff
## 0 0 0.125 -0.89239044 0.0000000 0.0000000 0.0000000
## 1 1 0.375 -0.18559680 0.7067936 0.6165306 -0.5015841
## 2 2 0.625 0.12180659 1.0141970 0.6044460 -0.1704954
## 3 3 0.875 0.02981974 0.9222102 0.3537861 0.2288022
## ul.diff ll.linpred ul.linpred
## 0 0.000000 -0.8923904 -0.8923904
## 1 1.915171 -1.3939746 1.0227810
## 2 2.198889 -1.0628859 1.3064991
## 3 1.615618 -0.6635882 0.7232277
pointwisebound.noboot(qcee.fit5b)
## quantile quantile.midpoint linpred mean.diff se.diff ll.diff
## 1 0 0.125 -0.89239044 0.0000000 0.0000000 0.0000000
## 2 1 0.375 -0.18559680 0.7067936 0.4922129 -0.2579259
## 3 2 0.625 0.12180659 1.0141970 0.9844258 -0.9152420
## 4 3 0.875 0.02981974 0.9222102 1.4766386 -1.9719484
## ul.diff ll.linpred ul.linpred
## 1 0.000000 -0.8923904 -0.8923904
## 2 1.671513 -1.1503163 0.7791227
## 3 2.943636 -1.8076325 2.0512456
## 4 3.816369 -2.8643388 2.9239783
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
## Mixture slope parameters (bootstrap CI):
##
## Estimate Std. Error Lower CI Upper CI t value Pr(>|t|)
## (Intercept) -0.89239 0.70336 -2.27095 0.48617 -1.2688 0.2055
## psi1 0.90649 0.93820 -0.93235 2.74533 0.9662 0.3347
## psi2 -0.19970 0.32507 -0.83682 0.43743 -0.6143 0.5395
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.glm.boot must be used in order to produce the graphics below, as qgcomp.glm.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.glm.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.glm.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.glm.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.glm.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 Pr(>|t|)
## (Intercept) -0.064420 0.129559 -0.318350 0.189511 -0.4972 0.6193
## psi1 0.029869 0.032997 -0.034804 0.094543 0.9052 0.3659
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.glm.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.glm.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.glm.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-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$fit)
## [1] 676.0431
AIC(qc.fit6nonlin$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
More examples on advanced topics can be viewed in the other package vignette.
FAQ
Why don’t I get weights from the boot or ee functions?
Users often use the qgcomp.*.boot or qgcomp.*.ee functions because they want to marginalize over confounders or fit a non-linear joint exposure function. In both cases, the overall exposure response will no longer correspond to a simple weighted average of model coefficients, so none of the qgcomp.*.boot or qgcomp.*.ee functions will calculate weights. In most use cases, the weights would vary according to which level of joint exposure you’re at, so it is not a straightforward proposition to calculate them (and you may not wish to report 4 sets of weights if you use the default q=4). If you fit an otherwise linear model, you can get weights from a qgcomp.*.noboot which will be very close to the weights you might get from a linear model fit via qgcomp.*.boot functions, but be explicit that the weights come from a different model than the inference about joint exposure effects.
Do I need to model non-linearity and non-additivity of exposures?
Maybe. The inferential object of qgcomp is the set of \(\psi\) parameters that correspond to a joint exposure response. As it turns out, with correlated exposures non-linearity can disguise itself as non-additivity (Belzak and Bauer [2019] Addictive Behaviors). If we were inferring independent effects, this distinction would be crucial, but for joint effects it may turn out that it doesn’t matter much if you model non-linearity in the joint response function through non-additivity or non-linearity of individual exposures in a given study. Models fit in qgcomp still make the crucial assumption that you are able to model the joint exposure response via parametric models, so that assumption should not be forgotten in an effort to try to disentagle non-linearity (e.g. quadratic terms of exposures) from non-additivity (e.g. product terms between exposures). The important part to note about parametric modeling is that we have to explicitly tell the model to be non-linear, and no adaptation to non-linear settings will happen automatically. Exploring non-linearity is not a trivial endeavor.
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. Frequently, q=NULL is used in concert with standardizing exposure variables by dividing them by their interquartile ranges (IQR). The joint exposure response can then be interpreted as the effect of an IQR change in all exposures. Using IQR/2 (with or without a log transformation before hand) will yield results that are most (roughly) compatible with the package defaults (q=4) but that does not require quantization. Quantized variables have nice properties: they prevent extrapolation and reduce influence of outliers, but the choice of how to include exposures in the model should be a deliberate and well-informed one.
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 alex.keil@nih.gov to ask if a paper is already in development.)
Where else can I get help?
The vignettes of the package and the help files of the functions give many, many examples of usage. Additionally, some edge case or interesting applications in github gists available at https://gist.github.com/alexpkeil1.
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 has been performed by Nicole Niehoff, Michiel van den Dries, Emily Werder, Jessie Buckley, Barrett Welch, Che-Jung (Rong) Chang, various github users, and Katie O’Brien.
# return to standard processing
future::plan(future::sequential) # return to standard evaluation