The emtrends function is useful when a fitted model involves a
numerical predictor \(x\) interacting with another predictor a
(typically a factor). Such models specify that \(x\) has a different trend
depending on \(a\); thus, it may be of interest to estimate and compare
those trends. Analogous to the emmeans setting, we construct a
reference grid of these predicted trends, and then possibly average them over
some of the predictors in the grid.
emtrends(object, specs, var, delta.var = 0.001 * rng, max.degree = 1, ...)A supported model object (not a reference grid)
Specifications for what marginal trends are desired – as in
emmeans. If specs is missing or NULL,
emmeans is not run and the reference grid for specified trends
is returned.
Character value giving the name of a variable with respect to
which a difference quotient of the linear predictors is computed. In order
for this to be useful, var should be a numeric predictor that
interacts with at least one factor in specs. Then instead of
computing EMMs, we compute and compare the slopes of the var trend
over levels of the specified other predictor(s). As in EMMs, marginal
averages are computed for the predictors in specs and by.
See also the “Generalizations” section below.
The value of h to use in forming the difference
quotient \((f(x+h) - f(x))/h\). Changing it (especially changing its
sign) may be necessary to avoid numerical problems such as logs of negative
numbers. The default value is 1/1000 of the range of var over the
dataset.
Integer value. The maximum degree of trends to compute (this
is capped at 5). If greater than 1, an additional factor degree is
added to the grid, with corresponding numerical derivatives of orders
1, 2, ..., max.degree as the estimates.
Additional arguments passed to ref_grid or
emmeans as appropriate. See Details.
An emmGrid or emm_list object, according to specs.
See emmeans for more details on when a list is returned.
The function works by constructing reference grids for object with
various values of var, and then calculating difference quotients of predictions
from those reference grids. Finally, emmeans is called with
the given specs, thus computing marginal averages as needed of
the difference quotients. Any ... arguments are passed to the
ref_grid and emmeans; examples of such optional
arguments include optional arguments (often mode) that apply to
specific models; ref_grid options such as data, at,
cov.reduce, mult.names, nesting, or transform;
and emmeans options such as weights (but please avoid
trend or offset.
In earlier versions of emtrends, the first argument was named
model rather than object. (The name was changed because of
potential mis-matching with a mode argument, which is an option for
several types of models.) For backward compatibility, model still works
provided all arguments are named.
It is important to understand that trends computed by emtrends are
not equivalent to polynomial contrasts in a parallel model where
var is regarded as a factor. That is because the model object
here is assumed to fit a smooth function of var, and the estimated
trends reflect local behavior at particular value(s) of var;
whereas when var is modeled as a factor and polynomial contrasts are
computed, those contrasts represent the global pattern of changes over
all levels of var.
See the pigs.poly and pigs.fact examples below for an
illustration. The linear and quadratic trends depend on the value of
percent, but the cubic trend is constant (because that is true of
a cubic polynomial, which is the underlying model). The cubic contrast
in the factorial model has the same P value as for the cubic trend,
again because the cubic trend is the same everywhere.
Instead of a single predictor, the user may specify some monotone function of
one variable, e.g., var = "log(dose)". If so, the chain rule is
applied. Note that, in this example, if object contains
log(dose) as a predictor, we will be comparing the slopes estimated by
that model, whereas specifying var = "dose" would perform a
transformation of those slopes, making the predicted trends vary depending on
dose.
fiber.lm <- lm(strength ~ diameter*machine, data=fiber)
# Obtain slopes for each machine ...
( fiber.emt <- emtrends(fiber.lm, "machine", var = "diameter") )
#> machine diameter.trend SE df lower.CL upper.CL
#> A 1.104 0.194 9 0.666 1.54
#> B 0.857 0.224 9 0.351 1.36
#> C 0.864 0.208 9 0.394 1.33
#>
#> Confidence level used: 0.95
# ... and pairwise comparisons thereof
pairs(fiber.emt)
#> contrast estimate SE df t.ratio p.value
#> A - B 0.24714 0.296 9 0.835 0.6919
#> A - C 0.24008 0.284 9 0.845 0.6863
#> B - C -0.00705 0.306 9 -0.023 0.9997
#>
#> P value adjustment: tukey method for comparing a family of 3 estimates
# Suppose we want trends relative to sqrt(diameter)...
emtrends(fiber.lm, ~ machine | diameter, var = "sqrt(diameter)",
at = list(diameter = c(20, 30)))
#> diameter = 20:
#> machine sqrt(diameter).trend SE df lower.CL upper.CL
#> A 9.88 1.73 9 5.96 13.8
#> B 7.67 2.00 9 3.14 12.2
#> C 7.73 1.86 9 3.52 11.9
#>
#> diameter = 30:
#> machine sqrt(diameter).trend SE df lower.CL upper.CL
#> A 12.10 2.12 9 7.30 16.9
#> B 9.39 2.45 9 3.84 14.9
#> C 9.47 2.28 9 4.31 14.6
#>
#> Confidence level used: 0.95
# Obtaining a reference grid
mtcars.lm <- lm(mpg ~ poly(disp, degree = 2) * (factor(cyl) + factor(am)), data = mtcars)
# Center trends at mean disp for each no. of cylinders
mtcTrends.rg <- emtrends(mtcars.lm, var = "disp",
cov.reduce = disp ~ factor(cyl))
summary(mtcTrends.rg) # estimated trends at grid nodes
#> Warning: Negative variance estimate obtained!
#> Warning: Negative variance estimate obtained!
#> disp cyl am disp.trend SE df
#> 105 4 0 -0.0949 0.0829 20
#> 183 6 0 -0.0024 NaN 20
#> 353 8 0 -0.0106 0.8690 20
#> 105 4 1 -0.1212 0.0338 20
#> 183 6 1 -0.0217 NaN 20
#> 353 8 1 -0.0147 0.8710 20
#>
emmeans(mtcTrends.rg, "am", weights = "prop")
#> am disp.trend SE df lower.CL upper.CL
#> 0 -0.0378 0.352 20 -0.771 0.696
#> 1 -0.0529 0.351 20 -0.786 0.680
#>
#> Results are averaged over the levels of: cyl
#> Confidence level used: 0.95
### Higher-degree trends ...
pigs.poly <- lm(conc ~ poly(percent, degree = 3), data = pigs)
emt <- emtrends(pigs.poly, ~ degree | percent, "percent", max.degree = 3,
at = list(percent = c(9, 13.5, 18)))
# note: 'degree' is an extra factor created by 'emtrends'
summary(emt, infer = c(TRUE, TRUE))
#> percent = 9.0:
#> degree percent.trend SE df lower.CL upper.CL t.ratio p.value
#> linear 2.39923 3.6500 25 -5.119 9.917 0.657 0.5170
#> quadratic -0.22674 1.1000 25 -2.498 2.044 -0.206 0.8387
#> cubic 0.00548 0.0825 25 -0.164 0.175 0.066 0.9475
#>
#> percent = 13.5:
#> degree percent.trend SE df lower.CL upper.CL t.ratio p.value
#> linear 0.69212 1.5600 25 -2.528 3.912 0.443 0.6618
#> quadratic -0.15277 0.1750 25 -0.513 0.207 -0.874 0.3903
#> cubic 0.00548 0.0825 25 -0.164 0.175 0.066 0.9475
#>
#> percent = 18.0:
#> degree percent.trend SE df lower.CL upper.CL t.ratio p.value
#> linear -0.34928 4.1200 25 -8.830 8.131 -0.085 0.9331
#> quadratic -0.07880 1.1500 25 -2.448 2.291 -0.068 0.9459
#> cubic 0.00548 0.0825 25 -0.164 0.175 0.066 0.9475
#>
#> Confidence level used: 0.95
# Compare above results with poly contrasts when 'percent' is modeled as a factor ...
pigs.fact <- lm(conc ~ factor(percent), data = pigs)
emm <- emmeans(pigs.fact, "percent")
contrast(emm, "poly")
#> contrast estimate SE df t.ratio p.value
#> linear 23.837 14.70 25 1.617 0.1184
#> quadratic -5.500 6.29 25 -0.874 0.3903
#> cubic 0.888 13.40 25 0.066 0.9475
#>
# Some P values are comparable, some aren't! See Note in documentation