Analysis
We will use g-computation to estimate the ADRF of logBLL
on Math adjusting for the covariates. It is also possible
to use weighted g-computation or propensity score weighting alone to
adjust for the covariates; we will demonstrate those in the section
“Additional Scenarios”. The procedure for these is essentially the
same.
We will use adrftools to answer the following research questions:
- What does the ADRF look like?
- Is the ADRF flat?
- Is the ADRF linear?
- What is the linear projection of the ADRF? Is its slope zero?
- What is the expected difference in Math scores under a low and high level of blood lead exposure?
Fitting the outcome model
The first step is to fit the outcome model. This model is not to be
interpreted and should flexible enough to capture nonlinear
relationships, especially between the treatment and the outcome.
adrftools supports any model also supported by marginaleffects,
but performance is improved with models fit with lm() or
glm().
Below, we use lm() to fit a linear regression of
Math on logBLL, the covariates, and their
interaction. We will supply logBLL as a natural cubic
spline with 5 degrees of freedom to capture potential nonlinearities
using splines::ns(). We will also allow the full ADRF to
vary across levels of Male to allow us to examine sex
differences later.
fit <- lm(Math ~ splines::ns(logBLL, df = 5) * Male *
(Age + Race + PIR + Enough_Food +
Smoke_in_Home + Smoke_Pregnant + NICU),
data = nhanes3lead)We won’t even look at this model; none of its coefficients are interpretable. Instead, we’ll use functions in adrftools to characterize the ADRF and perform inference on it.
Computing and plotting the ADRF
To compute the ADRF, we supply the outcome model to
adrf(), which also requires the name of the treatment to be
supplied to treat.
The ADRF is only evaluated for a range of treatment values. That
range can be controlled by the range argument to
adrf(). By default, the range is the middle 95% of
treatment values, which in the case corresponds to values from -0.3567
to 2.4248. It is important to know this range because all inferences
made generalize only to treatment values in this range. Typically,
values on the edges of the observed treatment range are estimated with
less certainty, and values beyond the observed treatment range involve
extrapolation.
adrf() also provides options for uncertainty estimation
using the vcov argument. The default,
vcov = "unconditional", uses M-estimation to account for
uncertainty in estimation and sampling analytically as described by
Hansen and Overgaard (2024); this
requires the model object to have estfun() and
bread() methods. Setting vcov = "boot" or
vcov = "fwb" use the traditional or fractional weighted
bootstrap (Xu et al.
2020), respectively, as implemented in fwb; these both also
account for sampling and estimation uncertainty. Finally, any argument
to the vcov argument of
marginaleffects::get_vcov() can be supplied to perform
conditional inference (i.e., treating the sample as fixed); the
“default” variance can be requested by setting
vcov = "conditional" (which typically just uses the
vcov() method for the model object).
The output of adrf() is an
<effect_curve> object. Printing the object displays
some information about the effect curve.
adrf_bll
## An <effect_curve> object
##
## - curve type: ADRF
## - response: Math
## - treatment: logBLL
## + range: -0.3567 to 2.4248
## - inference: unconditionalAn <effect_curve> object is a function that takes
in values of the treatment and returns estimates of the ADRF at those
points. For example, to display the ADRF estimates at values of
logBLL equal to 0, 1, and 2, we can run the following:
adrf_bll(logBLL = c(0, 1, 2))
## ADRF Estimates
## ────────────────
## logBLL Estimate
## 0 8.421
## 1 8.025
## 2 7.146
## ────────────────Perhaps the most useful summary of an effect curve is its plot, which
we can produce simply by running plot() on the
<effect_curve> object:
The x-axis is the treatment and the y-axis is the value of the ADRF at the treatment level (labeled \(\text{E[Math]}\)). The solid red line is the ADRF and the pink ribbon is the 95% uniform confidence band.
By default, confidence bands are computed providing uniform coverage,
meaning the nominal rate refers to simultaneous coverage of all points
on the line rather for a single given point (Montiel Olea and Plagborg-Møller
2019). Pointwise confidence bands can instead be requested by
setting simultaneous = FALSE in the call to
plot(). This is generally not recommended as it overstates
the precision of the estimated ADRF. The confidence level can be set
with conf_level, and if bootstrapping is used in
adrf(), the method of computing the confidence interval can
be controlled by the ci.type argument, which is passed to
fwb::summary.fwb(). Simultaneous coverage is only possible
when ci.type is "perc" (percentile confidence
intervals, the default for bootstrapping) or "wald".
To compute confidence intervals around specific points of the ADRF,
one can use summary() on the output of the
<effect_curve> object:
adrf_bll(logBLL = c(0, 1, 2)) |>
summary()
## ADRF Estimates
## ──────────────────────────────────────────
## logBLL Estimate Std. Error CI Low CI High
## 0 8.421 0.1678 8.021 8.822
## 1 8.025 0.1124 7.757 8.293
## 2 7.146 0.2162 6.630 7.661
## ──────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.386, df = 2401)summary() accepts the same arguments as
plot() to control the confidence interval coverage and
whether inference is simultaneous. Unlike with plot(), with
summary(), simultaneous inference only accounts for the
estimates requested, not the full effect curve.
Points along the ADRF can be contrasted pairwise using
point_contrast() on the output of the
<effect_curve> object (and then using
summary() on that output):
adrf_bll(logBLL = c(0, 1, 2)) |>
point_contrast() |>
summary()
## ADRF Point Contrasts
## ────────────────────────────────────────────────────────────────────────────────
## Term Estimate Std. Error t P-value CI Low CI High
## [logBLL = 1] - [logBLL = 0] -0.3964 0.2127 -1.863 0.1476 -0.8938 0.1010
## [logBLL = 2] - [logBLL = 0] -1.2757 0.2800 -4.556 < 0.0001 -1.9305 -0.6210
## [logBLL = 2] - [logBLL = 1] -0.8793 0.2642 -3.328 0.0025 -1.4972 -0.2615
## ────────────────────────────────────────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.338, df = 2401)
## Null value: 0In addition to the contrast estimates and confidence intervals, test
statistics and p-values for the tests that the contrasts are equal to 0
are displayed. The Null value note below the table
indicates the null value used for the tests.
Testing the ADRF
To test omnibus hypotheses about the ADRF, such as whether it is flat
or linear, we can use summary() on the
<effect_curve> object itself, rather than on its
output. In particular, summary() tests whether a given
projection of the ADRF is sufficient to characterize it. This projection
is determined by the argument to hypothesis. By default,
this is "flat", which tests that null hypothesis that the
ADRF is flat. If this hypothesis is rejected, this implies there is some
effect of the treatment.
summary(adrf_bll)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF is flat for values of logBLL between -0.3567
## and 2.4248
##
## P-value
## < 0.0001
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximationThis test works in two steps: first, the projection \(\hat{P}(a)\) is computed, and second, the residuals of the projection are used to compute the test statistic. The test statistic \(T\) is defined as
\[ T=\int_{a_\text{low}}^{a_\text{high}} {\left(\hat{\theta}(a) - \hat{P}(a) \right)^2 da} \]
where \(\hat{\theta}(a)\) is the ADRF estimate at treatment value \(a\) and \(a_\text{low}\) and \(a_\text{high}\) are the lower and upper bounds of the treatment domain, respectively. If the ADRF is perfectly flat, \(T=0\) because the difference between the ADRF and its flat projection will be 0. It is possible to compute the cumulative distribution function (CDF) of \(T\) under the null hypothesis that the true ADRF is equal to the specified projection, and that CDF is used to compute the p-value along with the test statistic \(T\) computed in the sample.
There are several methods of approximating the CDF, and this choice
is controlled by the method argument to
summary():
- Monte Carlo simulation of the distribution of \(T\) under the null hypothesis
(
method = "sim") - Analytic approximation using functionality in the CompQuadForm
package (
method = "imhof","davies", or"liu") - Satterthwaite’s approximation based on moment matching to a \(\chi^2\) distribution
(
method = "satterthwaite") - Kuonen’s saddlepoint approximation using functionality in the survey
package (
method = "saddlepoint")
method = "imhof" and method = "sim" are the
most reliable. The former is fast and deterministic, and so is used by
default when the CompQuadForm package is installed. The latter
is slower and subject to Monte Carlo error, but does not require any
additional packages. method = "saddlepoint" performs well,
too, and is used by default if the CompQuadForm package is not
installed but the survey package is.
hypothesis can also be "linear",
"quadratic", "cubic", or a formula
representing the projection to test. Rejecting the null hypothesis
indicates that the ADRF is more complicated than the supplied model; for
example, rejecting the null hypothesis for
hypothesis = "linear" means the ADRF is nonlinear, and
failing to reject the null hypothesis for for
hypothesis = "quadratic" means there is not enough evidence
to claim the ADRF is more complicated than a quadratic model. Rejecting
any of these hypotheses also rejects the null hypothesis for a
lower-order model, including "flat".
Finally, hypothesis can be a single number, indicating
the null hypothesis that the ADRF is flat and equal to that value.
Typically, this is more useful when testing other effect curves that
have a natural null value (e.g., 0).
In the example above, the p-value for the test that the ADRF is flat
in the range listed is less than .0001, so we can reject the null
hypothesis that the line is flat and therefore claim that there is an
effect of logBLL on Math. We can also test
whether the ADRF is approximately linear:
summary(adrf_bll, hypothesis = "linear")
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF is linear for values of logBLL between
## -0.3567 and 2.4248
##
## P-value
## 0.4782
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximationIn this case, the high p-value indicates that we cannot reject the null hypothesis that the ADRF is linear, even though there appears to be a curve in the plot. We can interpret this as indicating that a linear ADRF is consistent with the data, or, more precisely, that there isn’t evidence that the ADRF is nonlinear.
It’s important to only interpret these tests within the range listed in the test output. It is possible for the ADRF to have a different shape beyond this range, and it is possible that including a different range could change the results of the test. For example, including parts of the ADRF that are estimated with less precision can make the whole test less precise.
Estimating a linear projection
Though we have characterized, visualized, and tested the ADRF in a
flexible way, there may be use in finding a simpler representation of it
that can be used as a more interpretable summary. We were unable to
reject the null hypothesis that the ADRF is linear, so we can try
projecting the ADRF onto a linear model to more easily interpret its
form as a slope and intercept. This will also allow us to make a firmer
statement about the magnitude and direction of the effect of
logBLL on Math. We don’t need to re-fit the
outcome model; the outcome model does not correspond to the ADRF because
it is a conditional model, whereas the ADRF marginalizes over the
distribution of covariates. Instead, we can use
curve_projection() to perform the projection.
curve_projection() takes in an
<effect_curve> object and a projection model that
includes only the treatment. This model can be in any form, but in this
case, it makes sense to supply a simple linear model. Any value that can
be supplied to hypothesis in summary() can be
supplied to the model argument of
curve_projection(), which essentially fits a linear
regression of the effect curve estimates on the treatment values and
correctly accounts for the uncertainty in the effect curve estimates for
estimating the uncertainty of the projection model parameters. The
projection is formed by minimizing the loss function
\[ \int_{a_\text{low}}^{a_\text{high}}{\left(\hat{\theta}(a)-P(a; \mathbf{\hat{b}})\right)^2 \,da} \]
where \(P(a; \mathbf{\hat{b}})\) is the projection model with parameters \(\mathbf{\hat{b}}=\{\hat{b}_0,\ \dots,\ \hat{b}_k \}\) evaluated at treatment value \(a\). When requesting a linear projection,
\[ P(a; \mathbf{\hat{b}})=\hat b_0 + \hat b_1 a \]
Below, we fit the linear projection using
curve_projection():
proj <- curve_projection(adrf_bll, "linear")
summary(proj)
## ADRF Projection Coefficients
## ──────────────────────────────────────────────────────────────
## Term Estimate Std. Error t P-value CI Low CI High
## (Intercept) 8.485 0.1322 64.19 < 0.0001 8.226 8.745
## logBLL -0.612 0.1267 -4.83 < 0.0001 -0.861 -0.364
## ──────────────────────────────────────────────────────────────
## Inference: unconditional
## Confidence level: 95% (t* = 1.961, df = 2401)
## Null value: 0The output of summary() is similar to that when applied
to an <lm> object; we can see the coefficient
estimates, standard errors, tests, and confidence intervals. As
expected, we find a negative slope that is significant at the .001
level, indicating that an increase in logBLL corresponds to
a decrease in average Math scores.
We can plot the projection either alone or on top of the ADRF plot.
To plot it alone, we can simply use plot() on the
curve_projection() output:
This plot is similar in interpretation to the ADRF plot.
To add the projection to the original estimated ADRF plot, we can
supply the curve_projection() output to the
proj argument of plot() applied to the
original <effect_curve> object:
The dotted line is the projection, and the solid line and shaded region correspond to the original ADRF estimate and the original ADRF estimate’s confidence bands.
Two nested projection models can be compared with a Wald test using
anova(), just as for typical linear models.
Other effect curves
There is more to learn about the effect of the treatment on the outcome than the ADRF alone can provide. Other effect curves help us answer substantive questions about this effect. Below, we’ll describe computing reference effect curves, the average marginal effect function (AMEF), subgroup ADRFs, and contrasts between subgroup ADRFs. All these steps use the ADRF as a building block.
Reference effect curves
A reference effect curve is a function representing the contrast between the ADRF and the ADRF at a single level of the treatment. For example, one may be interested in characterizing the effect of treatment as the difference between the ADRF at nonzero treatment values and the ADRF with treatment set to 0.
To compute a reference effect curve for the ADRF, we can supply the
ADRF to reference_curve(), supplying an argument to
reference corresponding to the level of the treatment to
use as the reference. Below, we set the reference to 0 (which is
arbitrary in this example because the treatment is on a log scale).
This produces a <reference_curve> object, which,
when printed, displays the reference level.
adrf_bll_ref
## An <effect_curve> object
##
## - curve type: ADRF reference
## - response: Math
## - treatment: logBLL
## + range: -0.3567 to 2.4248
## - reference level: 0
## - inference: unconditionalWe can plot the reference curve, which will essentially be the original ADRF shifted so that the curve height at the reference level is equal to 0.
Notice that the confidence bands narrow close for treatment levels close to the reference level; we have greater certainty about ADRF deviations from the its value at the reference level in that region.
We can use summary() to test whether the reference curve
is equal to 0 everywhere, which tests whether the ADRF differs from its
value at the reference level:
summary(adrf_bll_ref)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF difference from reference (logBLL = 0) is 0
## for values of logBLL between -0.3567 and 2.4248
##
## P-value
## < 0.0001
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximationTheoretically, this is equivalent to testing whether the ADRF is flat (no matter what the reference level is set to), though there may be slight differences between the results of this test and those of the test for flatness applied directly to the ADRF.
We can test whether individual values of the ADRF differ from the
value of the ADRF at the reference level by calling
summary() on the output of the
<reference_curve> object applied to the desired
treatment levels:
adrf_bll_ref(logBLL = c(.5, 1, 1.5)) |>
summary()
## ADRF Reference Estimates
## ───────────────────────────────────────────────────────────
## logBLL Estimate Std. Error t P-value CI Low CI High
## 0.5 -0.1113 0.1466 -0.760 0.7642 -0.4528 0.2301
## 1.0 -0.3964 0.2127 -1.863 0.1452 -0.8918 0.0990
## 1.5 -0.9598 0.2205 -4.354 < 0.0001 -1.4732 -0.4464
## ───────────────────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.329, df = 2401)
## Reference: logBLL = 0 | Null value: 0From this output, we can see that the value of the ADRF at
logBLL = 1.5 differs from that at logBLL = 0
with a tiny p-value. The note below the table indicates the value of the
reference and the null value for the hypothesis tests.
The AMEF
The AMEF is the derivative of the ADRF. Because it is a level of
abstraction away from the ADRF, it can be more challenging to interpret.
Each point on the AMEF represents the instantaneous effect of the
treatment on the outcome. Where the AMEF is 0, the ADRF is flat. In
adrftools, amef() can be used to compute the AMEF
by supplying to it an <effect_curve> object.
amef_bll <- amef(adrf_bll)
amef_bll
## An <effect_curve> object
##
## - curve type: AMEF
## - response: Math
## - treatment: logBLL
## + range: -0.3567 to 2.4248
## - inference: unconditionalThe result is an <amef_curve> object, which, like
all <effect_curve> objects, is a function. We can
plot the AMEF with plot().
Unlike when plotting the ADRF, plotting the AMEF automatically
includes a horizontal line with a y-intercept of 0. This can be
controlled with the null argument to
plot().
We can compute estimates of the AMEF as specific values by supplying
those values to the <amef_curve> object and using
summary() to examine its output:
amef_bll(c(0, 1, 2)) |>
summary()
## AMEF Estimates
## ───────────────────────────────────────────────────────────
## logBLL Estimate Std. Error t P-value CI Low CI High
## 0 -0.0982 0.5166 -0.1902 0.9965 -1.3313 1.1348
## 1 -0.8166 0.6599 -1.2375 0.5136 -2.3914 0.7583
## 2 -0.1605 0.6479 -0.2478 0.9923 -1.7068 1.3858
## ───────────────────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.387, df = 2401)
## Null value: 0Here, a p-value for the test that the AMEF value is equal to 0 is
displayed along with confidence intervals. The null value for this test
can be controlled with the null argument to
summary().
Testing whether the AMEF is uniformly equal to 0 is theoretically
equivalent testing whether the ADRF is flat. We can perform this test
using summary() on the <amef_curve>
object itself. The default value for hypothesis is 0 to
test whether the AMEF is 0 everywhere.
summary(amef_bll)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: AMEF is 0 for values of logBLL between -0.3567 and
## 2.4248
##
## P-value
## 0.3529
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximationConclusions from the test that the ADRF is flat and that the AMEF is 0 may differ due to differing sensitivities to local alternatives between the tests. More research is needed to understand these differences. For now, we recommend not using the test for the AMEF and instead testing the flatness of the ADRF directly.
Subgroup ADRFs
The ADRF can be computed within subgroups to further probe the effect
of treatment for different unit profiles. To compute the ADRF in a
subgroup, one can simply use the subset argument to
adrf() as one would in a call to lm(). The
supplied argument to subset should evaluate to a logical
vector using the data fit to the original model.
Below, we compute the ADRF of logBLL for the group
Male == 0.
All functionality for full-sample ADRFs can be used with a subgroup ADRF.
It’s also possible to compute the ADRF in multiple subgroups
simultaneously. To do so, one can use the by argument to
adrf(), which accepts a one-sided formula with the
subgrouping variables on the right side.
Printing the stratified <effect_curve> object
displays which variables were supplied to by:
adrf_bll_by_male
## An <effect_curve> object
##
## - curve type: ADRF
## - response: Math
## - treatment: logBLL
## + range: -0.3567 to 2.4248
## - by: Male
## - inference: unconditionalWhen we plot the ADRF, we can see curves for both groups:
Indeed, all functions that operate on the ADRF or its estimates produce results for all subgroups. Below, we test the null hypotheses that the ADRF for each subgroup is flat:
summary(adrf_bll_by_male)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF is flat for values of logBLL between -0.3567
## and 2.4248
##
## Male P-value
## 0 0.0670
## 1 < 0.0001
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximationWe can reject the null hypothesis for the Male = 1
group, but not for the Male = 0 group at the .05 level.
We can display ADRF estimates for each subgroup by supplying values
of the treatment to the <effect_curve> object and
calling summary() on its output:
adrf_bll_by_male(logBLL = c(0, 1, 2)) |>
summary()
## ADRF Estimates
## ───────────────────────────────────────────────
## Male logBLL Estimate Std. Error CI Low CI High
## 0 0 8.469 0.2351 7.850 9.088
## 0 1 8.460 0.1523 8.059 8.861
## 0 2 7.619 0.3274 6.757 8.480
## 1 0 8.375 0.2392 7.746 9.005
## 1 1 7.605 0.1641 7.173 8.036
## 1 2 6.689 0.2833 5.943 7.434
## ───────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.631, df = 2401)Columns for the subgrouping variables are displayed on the left.
When we project the ADRFs onto a linear model, we also get separate projection coefficients for each subgroup:
proj_by_male <- curve_projection(adrf_bll_by_male, "linear")
# Coefficient estimates
summary(proj_by_male)
## ADRF Projection Coefficients
## ───────────────────────────────────────────────────────────────────
## Male Term Estimate Std. Error t P-value CI Low CI High
## 0 (Intercept) 8.623 0.1867 46.18 < 0.0001 8.257 8.989
## 0 logBLL -0.441 0.1863 -2.37 0.0180 -0.806 -0.076
## 1 (Intercept) 8.353 0.1871 44.65 < 0.0001 7.986 8.720
## 1 logBLL -0.778 0.1720 -4.52 < 0.0001 -1.115 -0.441
## ───────────────────────────────────────────────────────────────────
## Inference: unconditional
## Confidence level: 95% (t* = 1.961, df = 2401)
## Null value: 0
Note that any corrections for multiple comparisons when
simultaneous = TRUE in plot() and
summary() will be made accounting for all subgroups, which
can quickly escalate with the additional estimates within subgroups.
To display results for a subset of the subgroups,
plot(), the <effect_curve> object, and
summary() all accept a subset argument that
works like that supplied to adrf(), but it operates only on
the subgroup estimates included in the <effect_curve>
object by the by argument. For example, to plot each of the
subgroups separately, we can use the following:
This produces the same output as plotting the ADRF computed by
supplying subset to adrf() with the same
arguments. We can do the same with other functions as well:
Male = 0
# Testing whether the ADRF is flat
summary(adrf_bll_by_male, subset = Male == 0)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF is flat for values of logBLL between -0.3567
## and 2.4248
##
## Male P-value
## 0 0.067
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximation
# Estimates along the ADRF
adrf_bll_by_male(logBLL = c(0, 1, 2), subset = Male == 0) |>
summary()
## ADRF Estimates
## ───────────────────────────────────────────────
## Male logBLL Estimate Std. Error CI Low CI High
## 0 0 8.469 0.2351 7.908 9.030
## 0 1 8.460 0.1523 8.096 8.823
## 0 2 7.619 0.3274 6.837 8.400
## ───────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.386, df = 2401)Male = 1
# Testing whether the ADRF is flat
summary(adrf_bll_by_male, subset = Male == 1)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF is flat for values of logBLL between -0.3567
## and 2.4248
##
## Male P-value
## 1 < 0.0001
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximation
# Estimates along the ADRF
adrf_bll_by_male(logBLL = c(0, 1, 2), subset = Male == 1) |>
summary()
## ADRF Estimates
## ───────────────────────────────────────────────
## Male logBLL Estimate Std. Error CI Low CI High
## 1 0 8.375 0.2392 7.804 8.946
## 1 1 7.605 0.1640 7.213 7.996
## 1 2 6.689 0.2833 6.013 7.365
## ───────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.386, df = 2401)Notice the critical \(t\) value
(displayed as t* in the output) differs when examining
estimates within a subgroup and including both subgroups because the
adjustment to the value for simultaneous inference is smaller with fewer
estimates.
Subgroup ADRF contrasts
When by is used to compute the ADRF within subgroups, we
compute a new effect curve corresponding to the contrast between
subgroup ADRFs. Using curve_contrast() produces a
<contrast_curve> object, which is an
<effect_curve> object.
Printing the object displays the contrast(s) computed. With more than two subgroups, all pairs of subgroups are compared.
adrf_bll_male_contrast
## An <effect_curve> object
##
## - curve type: ADRF contrast
## - response: Math
## - treatment: logBLL
## + range: -0.3567 to 2.4248
## - contrast: [Male = 1] - [Male = 0]
## - inference: unconditionalWe can plot the difference between the two ADRFs using
plot():
Here, the y-axis corresponds to the difference between the subgroup
ADRF estimates, and a horizontal line at 0 is displayed. We can use
summary() to test whether the ADRF contrast curve is equal
to 0 everywhere, which would indicate that the ADRF does not differ
across subgroups:
summary(adrf_bll_male_contrast)
## Omnibus Curve Test
## ───────────────────────────────────────────────────────
## H₀: ADRF contrast is 0 for values of logBLL between
## -0.3567 and 2.4248
##
## Contrast P-value
## [Male = 1] - [Male = 0] 0.01
## ───────────────────────────────────────────────────────
## Computed using the Imhof approximationGiven the small p-value, we can reject the null hypothesis that the ADRF contrast is 0, implying the subgroup ADRFs differ from each other.
We can use summary() on the output of the
<contrast_curve> object to test for differences
between the subgroup ADRFs at specific points:
adrf_bll_male_contrast(logBLL = c(0, 1, 2)) |>
summary()
## ADRF Contrast Estimates
## ──────────────────────────────────────────────────────────────────────────────────
## Contrast logBLL Estimate Std. Error t P-value CI Low CI High
## [Male = 1] - [Male = 0] 0 -0.0938 0.3354 -0.280 0.9890 -0.8942 0.7066
## [Male = 1] - [Male = 0] 1 -0.8551 0.2239 -3.820 0.0004 -1.3892 -0.3209
## [Male = 1] - [Male = 0] 2 -0.9300 0.4329 -2.148 0.0914 -1.9631 0.1030
## ──────────────────────────────────────────────────────────────────────────────────
## Inference: unconditional, simultaneous
## Confidence level: 95% (t* = 2.386, df = 2401)
## Null value: 0Here we see evidence that the ADRF values at logBLL = 1
differ between the Male = 1 and Male = 0
subgroups.
