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.
Arguments
- object
A supported model object (not a reference grid)
- specs
Specifications for what marginal trends are desired – as in
emmeans. Ifspecsis missing orNULL,emmeansis not run and the reference grid for specified trends is returned.- var
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,
varshould be a numeric predictor that interacts with at least one factor inspecs. Then instead of computing EMMs, we compute and compare the slopes of thevartrend over levels of the specified other predictor(s). As in EMMs, marginal averages are computed for the predictors inspecsandby. See also the “Generalizations” section below.- delta.var
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
varover the dataset.- max.degree
Integer value. The maximum degree of trends to compute (this is capped at 5). If greater than 1, an additional factor
degreeis added to the grid, with corresponding numerical derivatives of orders1, 2, ..., max.degreeas the estimates.- ...
Additional arguments passed to
ref_gridoremmeansas appropriate. See Details.
Value
An emmGrid or emm_list object, according to specs.
See emmeans for more details on when a list is returned.
Details
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.
Note
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.
Generalizations
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.
Examples
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
#> disp cyl am disp.trend SE df
#> 105 4 0 -0.0949 0.0829 20
#> 183 6 0 -0.0024 0.0496 20
#> 353 8 0 -0.0106 0.0105 20
#> 105 4 1 -0.1212 0.0338 20
#> 183 6 1 -0.0217 0.0573 20
#> 353 8 1 -0.0147 0.0645 20
#>
emmeans(mtcTrends.rg, "am", weights = "prop")
#> am disp.trend SE df lower.CL upper.CL
#> 0 -0.0378 0.0312 20 -0.103 0.02733
#> 1 -0.0529 0.0260 20 -0.107 0.00145
#>
#> 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