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

stdEff(
mod,
weights = NULL,
data = NULL,
term.names = NULL,
unique.eff = TRUE,
cen.x = TRUE,
cen.y = TRUE,
std.x = TRUE,
std.y = TRUE,
refit.x = TRUE,
incl.raw = FALSE,
R.squared = FALSE,
R2.arg = NULL,
env = NULL
)

## Arguments

mod A fitted model object, or a list or nested list of such objects. 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. An optional dataset, used to first refit the model(s). An optional vector of names used to extract and/or sort effects from the output. Logical, whether unique effects should be calculated (adjusted for multicollinearity among predictors). Logical, whether effects should be calculated as if from mean-centred variables. Logical, whether effects should be scaled by the standard deviations of variables. Logical, whether the model should be refit with mean-centred predictor variables (see Details). Logical, whether to append the raw (unstandardised) effects to the output. Logical, whether R-squared values should also be calculated (via R2()). A named list of additional arguments to R2() (where applicable), excepting argument env. Ignored if R.squared = FALSE. Environment in which to look for model data (if none supplied). Defaults to the formula() environment.

## 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 $$\beta1$$ would be:

$$\beta_{1} + \beta_{12}\bar{x}_{2} + \beta_{13}\bar{x}_{3} + \beta_{123}\bar{x}_{2}\bar{x}_{3}$$ (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 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.

If R.squared = TRUE, model R-squared values are appended to effects via the R2() function, with any additional arguments passed via R2.arg.

Finally, if weights are specified, the function calculates a weighted average of standardised effects across a set (or sets) of different candidate models for a particular response variable(s) (Burnham & Anderson, 2002), via the avgEst() function.

## References

Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.). Springer-Verlag. 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. doi: 10/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. doi: 10/gfww2w

## Examples

library(lme4)
#> Attaching package: ‘lme4’#> The following object is masked from ‘package:semEff’:
#>
#>     getData
# Standardised (direct) effects for SEM
m <- shipley.sem
stdEff(m)
#> $DD #> (Intercept) lat #> -0.05600661 -0.68772025 #> #>$Date
#> (Intercept)          DD
#> -0.01493651 -0.62813666
#>
#> $Growth #> (Intercept) Date #> -0.2917507 0.3824224 #> #>$Live
#> (Intercept)      Growth
#>   0.3105220   0.3681961
#> stdEff(m, cen.y = FALSE, std.y = FALSE)  # x-only
#> $DD #> (Intercept) lat #> 143.183138 -7.066909 #> #>$Date
#> (Intercept)          DD
#>   126.79703    -5.11375
#>
#> $Growth #> (Intercept) Date #> 48.955514 2.448161 #> #>$Live
#> (Intercept)      Growth
#>    5.452982    2.226862
#> stdEff(m, std.x = FALSE, std.y = FALSE)  # centred only
#> $DD #> (Intercept) lat #> -0.5755154 -0.8354729 #> #>$Date
#> (Intercept)          DD
#>  -0.1216002  -0.4976475
#>
#> $Growth #> (Intercept) Date #> -1.8677067 0.3007147 #> #>$Live
#> (Intercept)      Growth
#>   1.8780467   0.3478536
#> stdEff(m, cen.x = FALSE, cen.y = FALSE)  # scaled only
#> $DD #> (Intercept) lat #> 19.1373370 -0.6877202 #> #>$Date
#> (Intercept)          DD
#>  24.3624481  -0.6281367
#>
#> $Growth #> (Intercept) Date #> 1.6853620 0.3824224 #> #>$Live
#> (Intercept)      Growth
#>  -2.0214940   0.3681961
#> stdEff(m, unique.eff = FALSE)  # include multicollinearity
#> $DD #> (Intercept) lat #> -0.05600661 -0.68772025 #> #>$Date
#> (Intercept)          DD
#> -0.01493651 -0.62813666
#>
#> $Growth #> (Intercept) Date #> -0.2917507 0.3824224 #> #>$Live
#> (Intercept)      Growth
#>   0.3105220   0.3681961
#> stdEff(m, R.squared = TRUE)  # add R-squared
#> $DD #> (Intercept) lat (R.squared) (R.squared.adj) #> -0.05600661 -0.68772025 0.70800826 0.70747906 #> (R.squared.pred) #> 0.68356361 #> #>$Date
#>      -0.01493651      -0.62813666       0.98555565       0.98553510
#> (R.squared.pred)
#>       0.98202414
#>
#> $Growth #> (Intercept) Date (R.squared) (R.squared.adj) #> -0.2917507 0.3824224 0.7938367 0.7934879 #> (R.squared.pred) #> 0.7552854 #> #>$Live
#>        0.3105220        0.3681961        0.2668652        0.2655971
#> (R.squared.pred)
#>               NA
#> stdEff(m, incl.raw = TRUE)  # add unstandardised
#> $DD #> (Intercept) lat (raw)_(Intercept) (raw)_lat #> -0.05600661 -0.68772025 196.65237838 -0.83547294 #> #>$Date
#>       (Intercept)                DD (raw)_(Intercept)          (raw)_DD
#>       -0.01493651       -0.62813666      198.33816379       -0.49764747
#>
#> $Growth #> (Intercept) Date (raw)_(Intercept) (raw)_Date #> -0.2917507 0.3824224 10.7892162 0.3007147 #> #>$Live
#>       (Intercept)            Growth (raw)_(Intercept)      (raw)_Growth
#>         0.3105220         0.3681961       -12.2260588         0.3478536
#>
# 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)
#>  (Intercept)         Date           DD          lat
#> -0.294231350  0.248803366 -0.003638724 -0.039021926