stdEff: Standardised Effects

Description

Calculate fully standardised effects (model coefficients) in standard deviation units, adjusted for multicollinearity among predictors.

Usage

stdEff(
  mod,
  weights = NULL,
  data = NULL,
  term.names = NULL,
  cen.x = TRUE,
  cen.y = TRUE,
  std.x = TRUE,
  std.y = TRUE,
  unique.x = TRUE,
  refit.x = TRUE,
  r.squared = FALSE,
  incl.raw = FALSE,
  env = parent.frame(),
  ...
)

Arguments

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 mod is a list, and the latter when it is a nested list (with matching list names). If set to "equal", a simple average is calculated instead.

data

An optional dataset, used to first re-fit the model(s).

term.names

An optional vector of names used to extract and/or sort effects from the output.

cen.x, cen.y

Logical, whether effects should be calculated using mean-centred variables.

std.x, std.y

Logical, whether effects should be scaled by the standard deviations of variables.

unique.x

Logical, whether effects should be adjusted for multicollinearity among predictors.

refit.x

Logical, whether the model should be re-fit with mean-centred predictors.

r.squared

Logical, whether R-squared values should also be returned.

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 R2.

Value

A numeric vector of the standardised effects, or a list or nested list of such vectors.

Details

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 $β 1$ would be:

$β_{1} + β_{12} {\bar{x}}_{2} + β_{13} {\bar{x}}_{3} + β_{123} {\bar{x}}_{2} {\bar{x}}_{3}$ (adapted from here)

In addition, if std.x = TRUE or unique.x = 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 re-fit 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.x = TRUE (default), effects are adjusted for multicollinearity among predictors by dividing by the square root of the VIFs (Dudgeon 2016, Thompson et al. 2017). 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 is always estimated independent of other 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 added last in the model - directly linking the measure to model fit and 'variance explained'. See here for additional arguments in favour of the use of semipartial correlations.

If refit.x, cen.x, and unique.x are TRUE and there are interaction terms in the model, the model will be re-fit with any (newly-)centred continuous predictors, in order to calculate correct VIFs from the variance-covariance matrix. However, re-fitting 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).

References

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

Examples

Run this code

# NOT RUN {
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.x = 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.x = 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)
# }

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