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Variable contribution and Dominance analysis

Source: R/contributions.R
contributions.Rd

[Experimental]
Use an analysis of variance or an analysis of deviance to estimate the contribution of each variable in reducing variance or deviance (sort of R2 decomposition). 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. Alternatively, tbl_domimance() could be used to perform a dominance analysis with dominanceanalysis::dominanceAnalysis() and display the results as a formatted gt table.

Usage

contributions(mod, type = c("II", "III", "I", 1, 2, 3, "drop1", "add1"), ...)

tbl_contributions(
  mod,
  type = c("II", "III", "I", 1, 2, 3, "drop1", "add1"),
  ...,
  show = c("Total", "Partial", "Relative"),
  decimals = 1,
  notes = TRUE
)

total_deviance(mod)

tbl_dominance(mod, indice = NULL, decimals = 1, totals = TRUE, notes = TRUE)

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" uses stats::drop1() to calculate the reduction of deviance associated with each predictor (in the absence of interactions, is equivalent to a type-II Anova); "add1" uses stats::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() or stats::add1()

show

list of contributions to display

decimals

number of decimals for deviance and contributions

notes

should table notes be added?

indice

fit indice name, see dominanceanalysis::dominanceAnalysis()

totals

add a summary row with totals

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 R2 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 if 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. Such approach is also known as R2 decomposition. These indicators are based on Dpred (the delta of deviance) or SSpred (the delta of the sum of squares, i.e. the delta of variance) reduced by the inclusion of a specific predictor in the model.

A first measure, known as Eta-squared in the context of linear models, expresses this delta of sum of squares (variance) as a proportion of the total variance (SSpred / TSS). In the context of GLMs, the delta of deviance could similarly expressed as a proportion of the total deviance (Dpred / Dt), also known as the semi-partial pseudo-R2. This indicator represents the total contribution of a predictor in the reduction of deviance.

An alternative, known as partial Eta-squared, in the context of linear models, could be expressed as SSpred / (RSS + SSpred), the variance uniquely explained by the effect expressed as the proportion of variance not explained by the other effects. Here the variance explained by the other effects in the model is completely partialed out. Similarly, for GLMs, partial pseudo-R2 is 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.

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 TSS - RSS or Dt - Dr. Therefore, relative contribution is equal to SSpred / (TSS - RSS) or Dpred / (Dt - Dr) and represents the relative reduction of variance/deviance of the predictor compared to the total reduction of the variance / 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. The advantage of such approach is that the sum of relative contributions is equal to 100%. However, the contribution of each variable changes if the variables are entered in a different order.

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.

For survey::svyglm() object, only type II and III are supported. The type = "drop1" is recommended, in the absence of interactions, for such models.

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.

Some approaches, such as dominance analysis, have been developed for decomposing the coefficient of determination (R2) into individual contributions of predictors in regression models, accounting for correlations between predictors. They address a key challenge in relative importance analysis: how to allocate shared variance when predictors are collinear.

Dominance analysis follows a systematic, computationally intensive approach based on all-subsets regression. Models of every possible combination of predictors are generated, each pair of predictors are compared by examining their incremental contributions within matching subsets. Finally, the dominance relationship is determined based on three levels: complete dominance (variable A always contributes more), conditional dominance (higher average contribution), and general dominance (weighted average across subset sizes).

The last dimension (the average contribution of each variable) provides a metric similar to semi-partial R2, and could be expressed as a proportion of the global R2 (relative contribution). The sum of these relative contributions is equal 100%.

tbl_dominance() is a quick helper to perform a dominance analysis with dominanceanalysis::dominanceAnalysis() and to display the average total contribution variable in a nicely formatted gt::gt() table. The function also displays the relative contribution (i.e. the contribution of each variable expressed as a proportion of the R2).

To be noted, alternative implementations of dominance analysis in R include parameters::dominance_analysis() or domir::domin().

Note

Regarding eta-squared 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 eta-squared are not equal to those performed by effectsize::eta_squared(partial = FALSE). To be noted, effectsize::eta_squared(partial = TRUE) (partial eta-squared), is not impacted by this difference in approaches.

Examples

# Linear model
m <- lm(Sepal.Length ~ Sepal.Width + Species + Petal.Length, data = iris)
m |> contributions()
#> Anova Table (Type II tests)
#> 
#> Response: Sepal.Length
#>               Sum Sq    Total Partial Relative Df F value    Pr(>F)    
#> Sepal.Width   2.7161 0.026585 0.16282 0.030794  1  28.201 4.026e-07 ***
#> Species       2.3632 0.023131 0.14473 0.026793  2  12.268 1.195e-05 ***
#> Petal.Length 14.0382 0.137402 0.50130 0.159158  1 145.754 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
m |> tbl_contributions()



  

      Sum of Squares
      Total contribution (η2)
      Partial contribution  (ηp2)
      Relative contribution
      p-value
    
Sepal.Width
2.7
2.7%
16.3%
3.1%
<0.001
Species
2.4
2.3%
14.5%
2.7%
<0.001
Petal.Length
14.0
13.7%
50.1%
15.9%
<0.001
Total sum of squares (TSS): 102.2
    
Residual sum of squares (RSS): 14.0
    
R2: 86.3%
    
m |> tbl_dominance()



  

      Average total contribution
      Relative contribution
    
Sepal.Width
4.5%
5.2%
Species
34.5%
40.0%
Petal.Length
47.3%
54.8%
Total
86.3%
100.0%
R2: 86.3%
    

# \donttest{
m |> tbl_contributions(type = 1)



  

      Sum of Squares
      Total contribution (η2)
      Partial contribution  (ηp2)
      Relative contribution
      p-value
    
Sepal.Width
1.4
1.4%
9.2%
1.6%
<0.001
Species
72.8
71.2%
83.9%
82.5%
<0.001
Petal.Length
14.0
13.7%
50.1%
15.9%
<0.001
Total sum of squares (TSS): 102.2
    
Residual sum of squares (RSS): 14.0
    
R2: 86.3%
    

# 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()



  

      Deviance
      Total contribution
(semi-partial pseudo-R2)		
      Partial contribution
(partial pseudo-R2)		
      Relative contribution
      p-value
    
Passenger's class
119.0
4.3%
5.1%
21.3%
<0.001
Sex of passenger
352.9
12.7%
13.8%
63.1%
<0.001
Adult or Child
18.9
0.7%
0.8%
3.4%
<0.001
Total deviance (null model): 2 769.5
    
Residual deviance (full model): 2 210.1
    
McFadden pseudo R2: 20.2%
    
m2 |> tbl_contributions(decimals = 2)



  

      Deviance
      Total contribution
(semi-partial pseudo-R2)		
      Partial contribution
(partial pseudo-R2)		
      Relative contribution
      p-value
    
Passenger's class
119.03
4.30%
5.11%
21.28%
<0.001
Sex of passenger
352.91
12.74%
13.77%
63.09%
<0.001
Adult or Child
18.85
0.68%
0.85%
3.37%
<0.001
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)



  

      Deviance
      Relative contribution
      p-value
    
Passenger's class
119.0
21.3%
<0.001
Sex of passenger
352.9
63.1%
<0.001
Adult or Child
18.9
3.4%
<0.001
m2 |> tbl_dominance()



  

      Average total contribution
      Relative contribution
    
Passenger's class
5.4%
26.6%
Sex of passenger
14.2%
70.2%
Adult or Child
0.7%
3.2%
Total
20.2%
100.0%
McFadden pseudo R2: 20.2%
    

# custom column labels
tbl <- m2 |> tbl_contributions()
colnames(tbl$`_data`) # list of columns
#> [1] "Predictor" "Deviance"  "Total"     "Partial"   "Relative"  "p-value"  
tbl |>
  gt::cols_label(
    Predictor = "Variable",
    Total = "Custom label (total)",
    Partial = gt::html("Custom <em>label</em> with <strong>HTML</strong>"),
    Relative = gt::md("Custom _label_ with **markdown**")
  )



  

      Deviance
      Custom label (total)
      Custom label with HTML
      Custom label with markdown
      p-value
    
Passenger's class
119.0
4.3%
5.1%
21.3%
<0.001
Sex of passenger
352.9
12.7%
13.8%
63.1%
<0.001
Adult or Child
18.9
0.7%
0.8%
3.4%
<0.001
Total deviance (null model): 2 769.5
    
Residual deviance (full model): 2 210.1
    
McFadden pseudo R2: 20.2%
    

# interaction terms
m3 <- glm(
  Survived == "Yes" ~ Class * Sex + Age,
  data = titanic,
  family = binomial,
  contrasts = list(Class = contr.sum, Sex = contr.sum, Age = contr.sum)
)
m3 |> tbl_contributions(type = "III")



  

      Deviance
      Total contribution
(semi-partial pseudo-R2)		
      Partial contribution
(partial pseudo-R2)		
      Relative contribution
      p-value
    
Passenger's class
155.5
5.6%
6.8%
24.8%
<0.001
Sex of passenger
290.5
10.5%
11.9%
46.4%
<0.001
Adult or Child
20.3
0.7%
0.9%
3.2%
<0.001
Passenger's class × Sex of passenger
66.7
2.4%
3.0%
10.6%
<0.001
Total deviance (null model): 2 769.5
    
Residual deviance (full model): 2 143.4
    
McFadden pseudo R2: 22.6%
    

# 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
m4 <- svyglm(
  Survived == "Yes" ~ Class + Sex + Age,
  design = srvyr::as_survey(titanic),
  family = quasibinomial
)
m4 |> tbl_contributions(type = "drop1")



  

      Deviance
      Total contribution
(semi-partial pseudo-R2)		
      Partial contribution
(partial pseudo-R2)		
      Relative contribution
      p-value
    
Passenger's class
119.0
4.3%
5.1%
21.3%
<0.001
Sex of passenger
352.9
12.7%
13.8%
63.1%
<0.001
Adult or Child
18.9
0.7%
0.8%
3.4%
<0.001
Total deviance (null model): 2 769.5
    
Residual deviance (full model): 2 210.1
    
McFadden pseudo R2: 20.2%
    
m4 |> tbl_dominance()



  

      Average total contribution
      Relative contribution
    
Passenger's class
5.4%
26.6%
Sex of passenger
14.2%
70.2%
Adult or Child
0.7%
3.2%
Total
20.2%
100.0%
McFadden pseudo R2: 20.2%
    
# }