![[Experimental]](figures/lifecycle-experimental.svg)
Use an analysis of variance or an analysis of deviance to estimate the
contribution of each variable in reducing variance or deviance. This
contribution could be explained as a proportion of the total deviance
(total contribution), of the residual deviance (partial contribution)
or of the explained deviance (relative contribution), see details for
more information. contributions() computes the different contributions.
tbl_contributions() displays the results as a formatted gt
table, taking into account variable labels.
Arguments
- mod
a statistical model
- type
type of Anova, Roman numerals being equivalent to the corresponding Arabic numerals:
stats::anova()will be used for type-I, Anova,car::Anova()for types II and III; an alternative method,"drop1"usesstats::drop1()to calculate the reduction of deviance associated with each predictor (in the absence of interactions, is equivalent to a type-II Anova);"add1"usesstats::add1()to calculate the reduction of deviance associated with each predictor in an univariable model (model with just this predictor compared to the null model), in this scenario, partial and relative contribution are not defined.- ...
additional parameters passed to
stats::anova(),car::Anova(),stats::drop1()orstats::add1()- show
list of contributions to display
- notes
should table notes be added?
Details
In linear regression, the squared multiple correlation, R2 is used
to assess goodness of fit as it represents the proportion of variance in the
criterion that is explained by the predictors. It could be expressed as
1 - RSS / TSS where TSS represents the total sum of squares (the overall
observed variance) and RSS represents the residual sum of squares (i.e.
the variance not explained by the full model).
For generalized linear models (GLM), model fitting does not rely on ordinary
least squares (OLS). It is achieved by maximum likelihood. Therefore, sum
of squares is not available and R-squared cannot be computed. An alternative
goodness of fit measure used in such case is pseudo R2. The
McFadden pseudo R2 (sometimes called likelihood ratio index) could
be expressed using deviance rather than likelihood. In such case, it is equal
to 1 - Dr / Dt where Dt represents the full deviance
(i.e. the deviance of the null model, a model with no
predictor) and Dr the residual deviance (i.e. the deviance of
the full model). The McFadden pseudo R2 corresponds the
proportion of deviance reduced by the model. Deviance is a generalization of
the idea of using the sum of squares of residuals in the cases where
model-fitting is achieved by maximum likelihood.
In R, the residual deviance (Dr) of a GLM could be obtain with
stats::deviance(). For a linear model, stats::deviance() reports RSS.
The function total_deviance() could be used to get the total deviance
(Dt), i.e. the deviance of the null model. For a linear model, it provides
TSS.
Analysis of variance (Anova) is common to identify of the different
predictors have a significant effect on a model. Anova approaches have been
extended to also covers analysis of deviance. In the context of Anova-like
tests, it is common to report effect sizes indicators representing the amount
of variance or deviance explained by each variable included in the model.
These indicators are based on Dpred or the delta of deviance/variance
reduced by the inclusion of a specific predictor in the model.
A first measure, known as Eta-squared (η2) in the context of
linear models, expresses this delta of deviance/variance as a
proportion of the total deviance (Dpred / Dt). This indicator represents
the total contribution of a predictor in the reduction of deviance. In
the context of a linear model, it represents the proportion of variance
explained by this predictor.
An alternative, known as partial Eta-squared
(ηp2), could be expressed as
Dpred / (Dr + Dpred), where Dr represents the residual deviance of the
full model. This partial contribution is the proportion of partial
variance/deviance uniquely explained by the associated effect.
That is, the variance/deviance uniquely explained by the effect expressed
as the proportion of variance/deviance not explained by the other effects.
Here the variance/deviance explained by the other effects in the model is
completely partialed out.
Finally, it is possible to express a relative contribution as the
proportion of the variance/deviance explained by the model, that could be
be expressed as Dt - Dr. Therefore, relative contribution is equal to
Dpred / (Dt - Dr) and represents the relative reduction of deviance of the
predictor compared to the total reduction of the deviance by the full model.
It is crucial to understand the different types of Anova.
In a type-I Anova,
as performed by stats::anova(), the different predictors are included
sequentially and in-order into the model. Such analysis is therefore
order-dependant. The effect of a predictor is computed once taken into
account the previous predictors (but not the other one introduced later).
The first factor is tested without adjustment. The second factor is tested
after removing the effect of the first. The third is tested after removing
the effect of the first and the second.
Type II-Anova (default of car::Anova()) tests each main effect adjusted
for all other effects of the same order or lower, but not for interactions
involving that factor. Each main effect is tested as if it were the last
main effect entered. Type II are order-independent for main effects and are
generally preferred when there is no interaction. They have higher power
than Type III for testing main effects because they do not adjust for the
interaction term.
Type III-Anova (also done with car::Anova()) tests each effect adjusted
for all other effects in the model, including higher-order interactions.
Each effect is tested as if it were the last one entered into a model
containing all other effects. Type III are order-independent.
They require a specific contrast coding (typically sum-to-zero or Helmert)
to be interpretable.
car::Anova() calculates type-II or type-III Anova tables
indicating for each predictor the variance or deviance explained
by adding this predictor in the model compared to a model without this
predictor (but keeping all other predictors in the model).
contributions() also includes two alternatives: "drop1" and "add1".
type = "drop1" uses stats::drop1() which is equivalent, in the absence,
of interaction terms to a type-II Anova. For some models (such as
survey::svyglm() models), it provides better estimates than car::Anova().
type = "add1" uses stats::add1() to calculate the reduction of deviance
associated with each predictor in an univariable model (a model with just
this predictor compared to the null model). In this scenario, partial and
relative contribution are not defined. It provides an estimate of the
contribution of a variable in the absence of all other predictors.
Regarding η2 indicators, more details are provided in a
dedicated vignette
of the effectsize package. This vignette also presents variations of these
indicators. Some explanations are also available in the
documentation of the GAMLj
package for Jamovi.
To be noted, GAMLj highlights an potential issue regarding the computation
of total variance in effectsize::eta_squared(partial = FALSE). The
effectsize package sum the values displayed in the Anova object instead
of performing a null model. Here, we rely on total_deviance() and therefore
estimates for η2 are not equal to those performed by
effectsize::eta_squared(partial = FALSE). To be noted,
effectsize::eta_squared(partial = TRUE) (partial η2), is
not impacted by this difference in approaches.
In a type-II or type-III Anova, the sum of relative contributions would equal 100% only if all predictors are perfectly independent. In practice, the sum is never equal to 100% due to some correlation between predictors.
For survey::svyglm() object, only type II and III are supported. The
type = "drop1" is recommended, in the absence of interactions, for
such models.
contributions() and tbl_contributions() have been tested so far with
stats::lm(), stats::glm(), MASS::glm.nb(), survey::svyglm() and
survival::coxph() models.
In the field of linear models, several authors have been working on decomposing the coefficient of determination (R2) into individual contributions of predictors in regression models, accounting for correlations between predictors. It addresses a key challenge in relative importance analysis: how to allocate shared variance when predictors are collinear. It includes Shapley value-based approaches, Genizi method, Proportional Marginal Variance Decomposition (PMVD), relative weight analysis (RWA) or dominance analysis.
Examples
# Linear model
i <- iris |>
labelled::set_variable_labels(
Sepal.Width = "Sepal's width",
Petal.Length = "Petal's length"
)
m <- lm(Sepal.Length ~ Sepal.Width + Species + Petal.Length, data = i)
m |> contributions()
#> Error in eval(mf, parent.frame()): object 'i' not found
m |> tbl_contributions()
#> Error in eval(mf, parent.frame()): object 'i' not found
# \donttest{
m |> tbl_contributions(type = 1)
#> Error in eval(mf, parent.frame()): object 'i' not found
# GLM
m2 <- glm(Survived == "Yes" ~ ., data = titanic, family = binomial)
m2 |> contributions()
#> Analysis of Deviance Table (Type II tests)
#>
#> Response: Survived == "Yes"
#> LR Chisq Total Partial Relative Df Pr(>Chisq)
#> Class 119.03 0.042981 0.051107 0.21279 3 < 2.2e-16 ***
#> Sex 352.91 0.127430 0.137696 0.63088 1 < 2.2e-16 ***
#> Age 18.85 0.006807 0.008458 0.03370 1 1.413e-05 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
m2 |> tbl_contributions()
Total deviance (null model): 2 769.5
Residual deviance (full model): 2 210.1
McFadden pseudo R2: 20.2%
m2 |> tbl_contributions(show = "Relative", notes = FALSE)
# Survey-weighted GLM
library(survey)
#> Loading required package: grid
#> Loading required package: Matrix
#> Loading required package: survival
#>
#> Attaching package: ‘survey’
#> The following object is masked from ‘package:graphics’:
#>
#> dotchart
m3 <- survey::svyglm(
Survived == "Yes" ~ Class + Sex + Age,
design = srvyr::as_survey(titanic),
family = quasibinomial
)
m3 |> tbl_contributions(type = "drop1")
Total deviance (null model): 2 769.5
Residual deviance (full model): 2 210.1
McFadden pseudo R2: 20.2%
# }