Standardized effect sizes are typically calculated using pairwise differences of estimates,
divided by the SD of the population providing the context for those effects.
This function calculates effect sizes from an emmGrid object,
and confidence intervals for them, accounting for uncertainty in both the estimated
effects and the population SD.
eff_size(object, sigma, edf, method = "pairwise", ...)an emmGrid object containing the effect sizes
an emmGrid object,
typically one defining the EMMs to
be contrasted. If instead, class(object) == "emm_list",
such as is produced by emmeans(model, pairwise ~ treatment),
a message is displayed; the contrasts already therein are used; and
method is replaced by "identity".
numeric scalar, value of the population SD.
numeric scalar that specifies the equivalent degrees of freedom
for the sigma. This is a way of specifying the uncertainty in sigma,
in that we regard our estimate of sigma^2 as being proportional to
a chi-square random variable with edf degrees of freedom. (edf should
not be confused with the df argument that may be passed via ...
to specify the degrees of freedom to use in
the contrast method to use to define the effects.
This is passed to contrast after the elements of object
are scaled.
Additional arguments passed to contrast
This function uses calls to regrid to put the estimated
marginal means (EMMs) on the log scale. Then an extra element is added to
this grid for the log of sigma and its standard error (where we assume
that sigma is uncorrelated with the log EMMs). Then a call to
contrast subtracts log{sigma} from each of the log EMMs,
yielding values of log(EMM/sigma).
Finally, the results are re-gridded back to the original scale and the
desired contrasts are computed using method. In the log-scaling
part, we actually rescale the absolute values and keep track of the signs.
Any by variables specified in object will remain in force in the returned
effects, unless overridden in the optional arguments.
For models having a single random effect, such as those fitted using
lm; in that case, the stats::sigma and
stats::df.residual functions may be useful for specifying sigma
and edf. For models with more than one random effect, sigma may
be based on some combination of the random-effect variances.
Specifying edf can be rather unintuitive but is also relatively
uncritical; but the smaller the value, the wider the confidence intervals for
effect size. The value of sqrt(2/edf) can be interpreted as the
relative accuracy of sigma; for example, with edf = 50,
sigma is accurate to plus or
minus 20 percent. Note in an example below, we tried two different edf
values as kind of a bracketing/sensitivity-analysis strategy. A value of
Inf is allowable, in which case you are assuming that sigma is
known exactly. Obviously, this narrows the confidence intervals for the
effect sizes -- unrealistically if in fact sigma is unknown.
fiber.lm <- lm(strength ~ diameter + machine, data = fiber)
emm <- emmeans(fiber.lm, "machine")
eff_size(emm, sigma = sigma(fiber.lm), edf = df.residual(fiber.lm))
# or equivalently:
eff_size(pairs(emm), sigma(fiber.lm), df.residual(fiber.lm), method = "identity")
### Mixed model example:
if (require(nlme)) withAutoprint({
Oats.lme <- lme(yield ~ Variety + factor(nitro),
random = ~ 1 | Block / Variety,
data = Oats)
# Combine variance estimates
VarCorr(Oats.lme)
(totSD <- sqrt(214.4724 + 109.6931 + 162.5590))
# I figure edf is somewhere between 5 (Blocks df) and 51 (Resid df)
emmV <- emmeans(Oats.lme, ~ Variety)
eff_size(emmV, sigma = totSD, edf = 5)
eff_size(emmV, sigma = totSD, edf = 51)
}, spaced = TRUE)
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