Standardised Effects
Calculate fully standardised effects (model coefficients) in standard deviation units, adjusted for multicollinearity by default.
stdEff( mod, weights = NULL, data = NULL, term.names = NULL, unique.eff = TRUE, unique.x = TRUE, cen.x = TRUE, cen.y = TRUE, std.x = TRUE, std.y = TRUE, refit.x = TRUE, r.squared = FALSE, incl.raw = FALSE, env = parent.frame(), ... )
mod |
A fitted model object, or a list or nested list of such objects. |
weights |
An optional numeric vector of weights to use for model
averaging, or a named list of such vectors. The former should be supplied
when |
data |
An optional dataset, used to first refit the model(s). |
term.names |
An optional vector of names used to extract and/or sort effects from the output. |
unique.eff |
Logical, whether unique effects should be calculated (adjusted for multicollinearity among predictors). |
unique.x |
Deprecated but allowed temporarily. Use |
cen.x, cen.y |
Logical, whether effects should be calculated as if from mean-centred variables. |
std.x, std.y |
Logical, whether effects should be scaled by the standard deviations of variables. |
refit.x |
Logical, whether the model should be refit with mean-centred predictor variables (see Details). |
r.squared |
Logical, whether R-squared values should also be calculated. |
incl.raw |
Logical, whether to append the raw (unstandardised) effects to the output. |
env |
Environment in which to look for model data (if none supplied). |
... |
Arguments to |
stdEff will calculate fully standardised effects
(coefficients) in standard deviation units for a fitted model or list of
models. It achieves this via adjusting the 'raw' model coefficients, so no
standardisation of input variables is required beforehand. Users can simply
specify the model with all variables in their original units and the
function will do the rest. However, the user is free to scale and/or centre
any input variables should they choose, which should not affect the outcome
of standardisation (provided any scaling is by standard deviations). This
may be desirable in some cases, such as to increase numerical stability
during model fitting when variables are on widely different scales.
If arguments cen.x or cen.y are TRUE, effects will be
calculated as if all predictors (x) and/or the response variable (y) were
mean-centred prior to model-fitting (including any dummy variables arising
from categorical predictors). Thus, for an ordinary linear model where
centring of x and y is specified, the intercept will be zero - the mean (or
weighted mean) of y. In addition, if cen.x = TRUE and there are
interacting terms in the model, all effects for lower order terms of the
interaction are adjusted using an expression which ensures that each main
effect or lower order term is estimated at the mean values of the terms
they interact with (zero in a 'centred' model) - typically improving the
interpretation of effects. The expression used comprises a weighted sum of
all the effects that contain the lower order term, with the weight for the
term itself being zero and those for 'containing' terms being the product
of the means of the other variables involved in that term (i.e. those not
in the lower order term itself). For example, for a three-way interaction
(x1 * x2 * x3), the expression for main effect β1 would be:
β1 + (β12 * Mx2) + (β13 * Mx3) + (β123 * Mx2 * Mx3)
(adapted from here)
In addition, if std.x = TRUE or unique.eff = TRUE (see
below), product terms for interactive effects will be recalculated using
mean-centred variables, to ensure that standard deviations and variance
inflation factors (VIF) for predictors are calculated correctly (the model
must be refit for this latter purpose, to recalculate the
variance-covariance matrix).
If std.x = TRUE, effects are scaled by multiplying by the standard
deviations of predictor variables (or terms), while if std.y = TRUE
they are divided by the standard deviation of the response variable (minus
any offsets). If the model is a GLM, this latter is calculated using the
link-transformed response (or an estimate of same) generated using the
function glt. If both arguments are true, the effects
are regarded as 'fully' standardised in the traditional sense, often
referred to as 'betas'.
If unique.eff = TRUE (default), effects are adjusted for
multicollinearity among predictors by dividing by the square root of the
VIFs (Dudgeon 2016, Thompson et al. 2017;
RVIF). If they have also been scaled by the standard
deviations of x and y, this converts them to semipartial correlations, i.e.
the correlation between the unique components of predictors (residualised
on other predictors) and the response variable. This measure of effect size
is arguably much more interpretable and useful than the traditional
standardised coefficient, as it always represents the unique effects of
predictors and so can more readily be compared both within and across
models. Values range from zero to +/- one rather than +/- infinity (as in
the case of betas) - putting them on the same scale as the bivariate
correlation between predictor and response. In the case of GLMs however,
the measure is analogous but not exactly equal to the semipartial
correlation, so its values may not always be bound between +/- one (such
cases are likely rare). Importantly, for ordinary linear models, the square
of the semipartial correlation equals the increase in R-squared when that
variable is included last in the model - directly linking the measure to
unique variance explained. See
here
for additional arguments in favour of the use of semipartial correlations.
If refit.x, cen.x, and unique.eff are TRUE and
there are interaction terms in the model, the model will be refit with any
(newly-)centred continuous predictors, in order to calculate correct VIFs
from the variance-covariance matrix. However, refitting may not be
necessary in some circumstances, for example where predictors have already
been mean-centred, and whose values will not subsequently be resampled
(e.g. parametric bootstrap). Setting refit.x = FALSE in such cases
will save time, especially with larger/more complex models and/or bootstrap
runs.
If r.squared = TRUE, model R-squared values are appended to effects
via the R2 function, with any additional arguments
being passed via ....
If incl.raw = TRUE, raw (unstandardised) effects can also be
appended, i.e. those with all centring and scaling options set to
FALSE (though still adjusted for multicollinearity, where
applicable). These may be of interest in some cases, for example to compare
their bootstrapped distributions with those of standardised effects.
Finally, if weights are specified, the function calculates a
weighted average of the standardised effects across models (Burnham &
Anderson 2002) via avgEst.
A numeric vector of the standardised effects, or a list or nested list of such vectors.
Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.). New York: Springer-Verlag. Retrieved from https://www.springer.com/gb/book/9780387953649
Dudgeon, P. (2016). A Comparative Investigation of Confidence Intervals for Independent Variables in Linear Regression. Multivariate Behavioral Research, 51(2-3), 139-153. https://doi.org/gfww3f
Thompson, C. G., Kim, R. S., Aloe, A. M., & Becker, B. J. (2017). Extracting the Variance Inflation Factor and Other Multicollinearity Diagnostics from Typical Regression Results. Basic and Applied Social Psychology, 39(2), 81-90. https://doi.org/gfww2w
library(lme4)
# Standardised (direct) effects for SEM
m <- Shipley.SEM
stdEff(m)
stdEff(m, cen.y = FALSE, std.y = FALSE) # x-only
stdEff(m, std.x = FALSE, std.y = FALSE) # centred only
stdEff(m, cen.x = FALSE, cen.y = FALSE) # scaled only
stdEff(m, unique.eff = FALSE) # include multicollinearity
stdEff(m, r.squared = TRUE) # add R-squared
stdEff(m, incl.raw = TRUE) # add unstandardised
# Demonstrate equality with effects from manually-standardised variables
# (gaussian models only)
m <- Shipley.Growth[[3]]
d <- data.frame(scale(na.omit(Shipley)))
e1 <- stdEff(m, unique.eff = FALSE)
e2 <- coef(summary(update(m, data = d)))[, 1]
stopifnot(all.equal(e1, e2))
# Demonstrate equality with square root of increment in R-squared
# (ordinary linear models only)
m <- lm(Growth ~ Date + DD + lat, data = Shipley)
r2 <- summary(m)$r.squared
e1 <- stdEff(m)[-1]
en <- names(e1)
e2 <- sapply(en, function(i) {
f <- reformulate(en[!en %in% i])
r2i <- summary(update(m, f))$r.squared
sqrt(r2 - r2i)
})
stopifnot(all.equal(e1, e2))
# Model-averaged standardised effects
m <- Shipley.Growth # candidate models
w <- runif(length(m), 0, 1) # weights
stdEff(m, w)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.