dominanceanalysis
Dominance Analysis
Dominance analysis is a method that allows to compare the
relative importance of predictors in multiple regression models:
ordinary least squares, generalized linear models,
hierarchical linear models and dynamic linear models. The main principles and methods of
dominance analysis are described in
Budescu, D. V. (1993)
Dominance Analysis (Azen and Budescu, 2003, 2006; Azen and Traxel, 2009; Budescu, 1993; Luo and Azen, 2013), for multiple regression models: Ordinary Least Squares, Generalized Linear Models and Hierarchical Linear Models.
Features:
- Provides complete, conditional and general dominance analysis for lm (univariate and multivariate), lmer and glm (family=binomial) models.
- Covariance / correlation matrixes could be used as input for OLS dominance analysis, using
lmWithCov()andmlmWithCov()methods, respectively. - Multiple criteria can be used as fit indices, which is useful especially for HLM.
Examples
Linear regression
We could apply dominance analysis directly on the data, using lm (see Azen and Budescu, 2003).
The attitude data is composed of six predictors of the overall rating of 35 clerical employees of a large financial organization: complaints, privileges, learning, raises, critical and advancement. The method dominanceAnalysis() can retrieve all necessary information directly from a lm model.
library(dominanceanalysis)lm.attitude<-lm(rating~.,attitude)da.attitude<-dominanceAnalysis(lm.attitude)
Using print() method on the dominanceAnalysis object, we can see that complaints completely dominates all other predictors, followed by learning (lrnn). The remaining 4 variables (prvl,rass,crtc,advn) don't show a consistent pattern for complete and conditional dominance.
The print() method uses abbreviate, to allow complex models to be visualized at a glance.
print(da.attitude)#> Dominance analysis#> Predictors: complaints, privileges, learning, raises, critical, advance#> Fit-indices: r2#>#> * Fit index: r2#> complete conditional#> complaints prvl,lrnn,rass,crtc,advn prvl,lrnn,rass,crtc,advn#> privileges crtc crtc#> learning prvl,rass,crtc,advn prvl,rass,crtc,advn#> raises crtc crtc#> critical#> advance#> general#> complaints prvl,lrnn,rass,crtc,advn#> privileges crtc,advn#> learning prvl,rass,crtc,advn#> raises prvl,crtc,advn#> critical#> advance crtc#>#> Average contribution:#> complaints learning raises privileges advance critical#> 0.371 0.156 0.120 0.051 0.028 0.007
The summary() method provides the average contribution of each variable. This contribution defines general dominance. Also, shows the complete dominance analysis matrix, that presents all fit differences between levels.
summary(da.attitude)#>#> * Fit index: r2#>#> Average contribution of each variable:#>#> complaints learning raises privileges advance critical#> 0.371 0.156 0.120 0.051 0.028 0.007#>#> Dominance Analysis matrix:#> model level fit#> 1 0 0#> complaints 1 0.681#> privileges 1 0.182#> learning 1 0.389#> raises 1 0.348#> critical 1 0.024#> advance 1 0.024#> Average level 1 1#> complaints+privileges 2 0.683#> complaints+learning 2 0.708#> complaints+raises 2 0.684#> complaints+critical 2 0.681#> complaints+advance 2 0.682#> privileges+learning 2 0.408#> privileges+raises 2 0.382#> privileges+critical 2 0.191#> privileges+advance 2 0.182#> learning+raises 2 0.451#> learning+critical 2 0.396#> learning+advance 2 0.432#> raises+critical 2 0.353#> raises+advance 2 0.399#> critical+advance 2 0.038#> Average level 2 2#> complaints+privileges+learning 3 0.715#> complaints+privileges+raises 3 0.686#> complaints+privileges+critical 3 0.683#> complaints+privileges+advance 3 0.683#> complaints+learning+raises 3 0.708#> complaints+learning+critical 3 0.708#> complaints+learning+advance 3 0.726#> complaints+raises+critical 3 0.684#> complaints+raises+advance 3 0.69#> complaints+critical+advance 3 0.682#> privileges+learning+raises 3 0.459#> privileges+learning+critical 3 0.413#> privileges+learning+advance 3 0.458#> privileges+raises+critical 3 0.386#> privileges+raises+advance 3 0.443#> privileges+critical+advance 3 0.191#> learning+raises+critical 3 0.451#> learning+raises+advance 3 0.552#> learning+critical+advance 3 0.453#> raises+critical+advance 3 0.401#> Average level 3 3#> complaints+privileges+learning+raises 4 0.715#> complaints+privileges+learning+critical 4 0.715#> complaints+privileges+learning+advance 4 0.729#> complaints+privileges+raises+critical 4 0.686#> complaints+privileges+raises+advance 4 0.69#> complaints+privileges+critical+advance 4 0.684#> complaints+learning+raises+critical 4 0.708#> complaints+learning+raises+advance 4 0.729#> complaints+learning+critical+advance 4 0.727#> complaints+raises+critical+advance 4 0.69#> privileges+learning+raises+critical 4 0.459#> privileges+learning+raises+advance 4 0.563#> privileges+learning+critical+advance 4 0.476#> privileges+raises+critical+advance 4 0.445#> learning+raises+critical+advance 4 0.553#> Average level 4 4#> complaints+privileges+learning+raises+critical 5 0.715#> complaints+privileges+learning+raises+advance 5 0.732#> complaints+privileges+learning+critical+advance 5 0.731#> complaints+privileges+raises+critical+advance 5 0.691#> complaints+learning+raises+critical+advance 5 0.729#> privileges+learning+raises+critical+advance 5 0.564#> Average level 5 5#> complaints+privileges+learning+raises+critical+advance 6 0.733#> complaints privileges learning raises critical advance#> 0.681 0.182 0.389 0.348 0.024 0.024#> 0.002 0.027 0.003 0 0.001#> 0.501 0.226 0.2 0.009 0#> 0.319 0.019 0.062 0.007 0.043#> 0.336 0.033 0.102 0.005 0.05#> 0.657 0.166 0.372 0.329 0.013#> 0.658 0.158 0.408 0.375 0.014#> 0.494 0.075 0.227 0.194 0.007 0.022#> 0.032 0.003 0 0#> 0.007 0 0 0.018#> 0.002 0.024 0 0.006#> 0.002 0.027 0.003 0.001#> 0.001 0.043 0.007 0#> 0.307 0.051 0.005 0.05#> 0.305 0.077 0.004 0.061#> 0.493 0.222 0.195 0#> 0.502 0.276 0.261 0.009#> 0.258 0.008 0 0.102#> 0.312 0.016 0.055 0.057#> 0.293 0.026 0.12 0.021#> 0.331 0.033 0.098 0.048#> 0.291 0.044 0.154 0.003#> 0.645 0.153 0.416 0.363#> 0.374 0.029 0.137 0.106 0.004 0.034#> 0 0 0.014#> 0.029 0 0.004#> 0.032 0.003 0#> 0.046 0.007 0#> 0.007 0 0.02#> 0.007 0 0.019#> 0.004 0.003 0.002#> 0.002 0.024 0.005#> 0.001 0.039 0#> 0.001 0.045 0.007#> 0.257 0 0.104#> 0.302 0.046 0.063#> 0.271 0.105 0.018#> 0.301 0.073 0.059#> 0.247 0.12 0.002#> 0.493 0.285 0.254#> 0.257 0.008 0.102#> 0.176 0.011 0.001#> 0.274 0.022 0.1#> 0.288 0.044 0.152#> 0.287 0.011 0.084 0.053 0.002 0.039#> 0 0.017#> 0 0.016#> 0.002 0.002#> 0.029 0.004#> 0.041 0#> 0.047 0.007#> 0.007 0.021#> 0.003 0.001#> 0.004 0.002#> 0.001 0.04#> 0.256 0.105#> 0.169 0.001#> 0.255 0.088#> 0.246 0.119#> 0.176 0.011#> 0.22 0.005 0.055 0.02 0.001 0.032#> 0.017#> 0.001#> 0.002#> 0.042#> 0.003#> 0.169#> 0.169 0.003 0.042 0.002 0.001 0.017#>
To evaluate the robustness of our results, we can use bootstrap analysis (Azen and Budescu, 2006).
We applied a bootstrap analysis using bootDominanceAnalysis() method with R2 as a fit index and 100 permutations. For precise results, you need to run at least 1000 replications.
bda.attitude=bootDominanceAnalysis(lm.attitude, R=100)
The summary() method presents the results for the bootstrap analysis. Dij shows the original result, and mDij, the mean for Dij on bootstrap samples and SE.Dij its standard error. Pij is the proportion of bootstrap samples where i dominates j, Pji is the proportion of bootstrap samples where j dominates i and Pnoij is the proportion of samples where no dominance can be asserted. Rep is the proportion of samples where original dominance is replicated.
We can see that the value of complete dominance for complaints is fairly robust over all variables (Dij almost equal to mDij, and small SE), contrarily to learning (Dij differs from mDij, and bigger SE).
summary(bda.attitude)#> Dominance Analysis#> ==================#> Fit index: r2#> dominance i k Dij mDij SE.Dij Pij Pji Pnoij Rep#> complete complaints privileges 1.0 0.980 0.0985 0.96 0.00 0.04 0.96#> complete complaints learning 1.0 0.910 0.2290 0.85 0.03 0.12 0.85#> complete complaints raises 1.0 0.980 0.0985 0.96 0.00 0.04 0.96#> complete complaints critical 1.0 0.975 0.1095 0.95 0.00 0.05 0.95#> complete complaints advance 1.0 0.955 0.1438 0.91 0.00 0.09 0.91#> complete privileges learning 0.0 0.275 0.2694 0.02 0.47 0.51 0.47#> complete privileges raises 0.5 0.460 0.1363 0.00 0.08 0.92 0.92#> complete privileges critical 1.0 0.530 0.1560 0.08 0.02 0.90 0.08#> complete privileges advance 0.5 0.510 0.0704 0.02 0.00 0.98 0.98#> complete learning raises 1.0 0.615 0.2918 0.31 0.08 0.61 0.31#> complete learning critical 1.0 0.715 0.2776 0.46 0.03 0.51 0.46#> complete learning advance 1.0 0.665 0.2363 0.33 0.00 0.67 0.33#> complete raises critical 1.0 0.555 0.1572 0.11 0.00 0.89 0.11#> complete raises advance 0.5 0.525 0.1095 0.05 0.00 0.95 0.95#> complete critical advance 0.5 0.520 0.0985 0.04 0.00 0.96 0.96#> conditional complaints privileges 1.0 0.985 0.0857 0.97 0.00 0.03 0.97#> conditional complaints learning 1.0 0.925 0.2289 0.89 0.04 0.07 0.89#> conditional complaints raises 1.0 0.975 0.1306 0.96 0.01 0.03 0.96#> conditional complaints critical 1.0 0.990 0.0704 0.98 0.00 0.02 0.98#> conditional complaints advance 1.0 0.980 0.0985 0.96 0.00 0.04 0.96#> conditional privileges learning 0.0 0.195 0.3089 0.07 0.68 0.25 0.68#> conditional privileges raises 0.5 0.345 0.2532 0.02 0.33 0.65 0.65#> conditional privileges critical 1.0 0.600 0.2462 0.24 0.04 0.72 0.24#> conditional privileges advance 0.5 0.535 0.1629 0.09 0.02 0.89 0.89#> conditional learning raises 1.0 0.665 0.3697 0.49 0.16 0.35 0.49#> conditional learning critical 1.0 0.790 0.2859 0.62 0.04 0.34 0.62#> conditional learning advance 1.0 0.760 0.2511 0.52 0.00 0.48 0.52#> conditional raises critical 1.0 0.685 0.2626 0.39 0.02 0.59 0.39#> conditional raises advance 0.5 0.595 0.1971 0.19 0.00 0.81 0.81#> conditional critical advance 0.5 0.445 0.2552 0.08 0.19 0.73 0.73#> general complaints privileges 1.0 1.000 0.0000 1.00 0.00 0.00 1.00#> general complaints learning 1.0 0.930 0.2564 0.93 0.07 0.00 0.93#> general complaints raises 1.0 0.980 0.1407 0.98 0.02 0.00 0.98#> general complaints critical 1.0 1.000 0.0000 1.00 0.00 0.00 1.00#> general complaints advance 1.0 1.000 0.0000 1.00 0.00 0.00 1.00#> general privileges learning 0.0 0.100 0.3015 0.10 0.90 0.00 0.90#> general privileges raises 0.0 0.070 0.2564 0.07 0.93 0.00 0.93#> general privileges critical 1.0 0.760 0.4292 0.76 0.24 0.00 0.76#> general privileges advance 1.0 0.770 0.4230 0.77 0.23 0.00 0.77#> general learning raises 1.0 0.600 0.4924 0.60 0.40 0.00 0.60#> general learning critical 1.0 0.930 0.2564 0.93 0.07 0.00 0.93#> general learning advance 1.0 0.970 0.1714 0.97 0.03 0.00 0.97#> general raises critical 1.0 0.940 0.2387 0.94 0.06 0.00 0.94#> general raises advance 1.0 1.000 0.0000 1.00 0.00 0.00 1.00#> general critical advance 0.0 0.440 0.4989 0.44 0.56 0.00 0.56
Another way to perform the dominance analysis is by using a correlation or covariance matrix. As an example, we use the ability.cov matrix which is composed of five specific skills that might explain general intelligence (general). The biggest average contribution is for predictor reading (0.152). Nevertheless, in the output of summary() method on level 1, we can see that picture (0.125) dominates over reading (0.077) on 'vocab' submodel.
lmwithcov<-lmWithCov(general~picture+blocks+maze+reading+vocab, cov2cor(ability.cov$cov))da.cov<-dominanceAnalysis(lmwithcov)print(da.cov)#>#> Dominance analysis#> Predictors: picture, blocks, maze, reading, vocab#> Fit-indices: r2#>#> * Fit index: r2#> complete conditional general#> picture maze maze maze#> blocks pctr,maze pctr,maze,vocb pctr,maze,vocb#> maze#> reading maze,vocb pctr,blck,maze,vocb pctr,blck,maze,vocb#> vocab pctr,maze#>#> Average contribution:#> reading blocks vocab picture maze#> 0.152 0.124 0.096 0.091 0.043summary(da.cov)#>#> * Fit index: r2#>#> Average contribution of each variable:#>#> reading blocks vocab picture maze#> 0.152 0.124 0.096 0.091 0.043#>#> Dominance Analysis matrix:#> model level fit picture blocks maze#> 1 0 0 0.217 0.304 0.116#> picture 1 0.217 0.121 0.065#> blocks 1 0.304 0.034 0.011#> maze 1 0.116 0.167 0.2#> reading 1 0.332 0.106 0.138 0.057#> vocab 1 0.265 0.125 0.155 0.054#> Average level 1 1 0.108 0.153 0.047#> picture+blocks 2 0.338 0.015#> picture+maze 2 0.282 0.07#> picture+reading 2 0.439 0.055 0.036#> picture+vocab 2 0.39 0.059 0.033#> blocks+maze 2 0.316 0.037#> blocks+reading 2 0.47 0.023 0.009#> blocks+vocab 2 0.42 0.028 0.007#> maze+reading 2 0.389 0.086 0.09#> maze+vocab 2 0.319 0.104 0.108#> reading+vocab 2 0.341 0.103 0.131 0.052#> Average level 2 2 0.064 0.085 0.025#> picture+blocks+maze 3 0.353#> picture+blocks+reading 3 0.494 0.011#> picture+blocks+vocab 3 0.448 0.009#> picture+maze+reading 3 0.475 0.03#> picture+maze+vocab 3 0.423 0.035#> picture+reading+vocab 3 0.445 0.051 0.033#> blocks+maze+reading 3 0.479 0.026#> blocks+maze+vocab 3 0.427 0.031#> blocks+reading+vocab 3 0.473 0.023 0.008#> maze+reading+vocab 3 0.394 0.085 0.087#> Average level 3 3 0.041 0.051 0.016#> picture+blocks+maze+reading 4 0.505#> picture+blocks+maze+vocab 4 0.458#> picture+blocks+reading+vocab 4 0.496 0.011#> picture+maze+reading+vocab 4 0.478 0.028#> blocks+maze+reading+vocab 4 0.481 0.026#> Average level 4 4 0.026 0.028 0.011#> picture+blocks+maze+reading+vocab 5 0.507#> reading vocab#> 0.332 0.265#> 0.221 0.172#> 0.166 0.116#> 0.273 0.203#> 0.009#> 0.077#> 0.184 0.125#> 0.156 0.11#> 0.193 0.141#> 0.006#> 0.055#> 0.164 0.111#> 0.002#> 0.053#> 0.004#> 0.074#>#> 0.116 0.062#> 0.152 0.105#> 0.002#> 0.048#> 0.003#> 0.055#>#> 0.002#> 0.054#>#>#> 0.077 0.028#> 0.002#> 0.049#>#>#>#> 0.049 0.002#>
Hierarchical Linear Models
For Hierarchical Linear Models using lme4, you should provide a null model (see Luo and Azen, 2013).
As an example, we use npk dataset, which contains information about a classical N, P, K (nitrogen, phosphate, potassium) factorial experiment on the growth of peas conducted on 6 blocks.
library(lme4)#> Loading required package: Matrixlmer.npk.1<-lmer(yield~N+P+K+(1|block),npk)lmer.npk.0<-lmer(yield~1+(1|block),npk)da.lmer<-dominanceAnalysis(lmer.npk.1,null.model=lmer.npk.0)
Using print() method, we can see that random effects are modeled as a constant (1 | block).
print(da.lmer)#>#> Dominance analysis#> Predictors: N, P, K#> Constants: ( 1 | block )#> Fit-indices: rb.r2.1, rb.r2.2, sb.r2.1, sb.r2.2#>#> * Fit index: rb.r2.1#> complete conditional general#> N P,K P,K P,K#> P#> K P P P#>#> Average contribution:#> N K P#> 0.341 0.148 -0.030#> * Fit index: rb.r2.2#> complete conditional general#> N#> P N,K N,K N,K#> K N N N#>#> Average contribution:#> P K N#> 0.022 -0.112 -0.259#> * Fit index: sb.r2.1#> complete conditional general#> N P,K P,K P,K#> P#> K P P P#>#> Average contribution:#> N K P#> 0.192 0.084 -0.017#> * Fit index: sb.r2.2#> complete conditional general#> N P,K#> P#> K P#>#> Average contribution:#> N K P#> 0 0 0
The fit indices used in the analysis were rb.r2.1 (R&B R12: Level-1 variance component explained by predictors), rb.r2.2 (R&B R22: Level-2 variance component explained by predictors), sb.r2.1 (S&B R12: Level-1 proportional reduction in error predicting scores at Level-1), and sb.r2.2 (S&B R22: Level-2 proportional reduction in error predicting scores at Level-1). We can see that using rb.r2.1 and sb.r2.1 index, that shows influence of predictors on Level-1 variance, clearly nitrogen dominates over potassium and phosphate, and potassium dominates over phosphate.
s.da.lmer=summary(da.lmer)s.da.lmer#>#> * Fit index: rb.r2.1#>#> Average contribution of each variable:#>#> N K P#> 0.341 0.148 -0.030#>#> Dominance Analysis matrix:#> model level fit N P K#> ( 1 | block ) 0 0 0.317 -0.042 0.13#> ( 1 | block )+N 1 0.317 -0.025 0.158#> ( 1 | block )+P 1 -0.042 0.334 0.136#> ( 1 | block )+K 1 0.13 0.345 -0.037#> Average level 1 1 0.34 -0.031 0.147#> ( 1 | block )+N+P 2 0.292 0.167#> ( 1 | block )+N+K 2 0.475 -0.016#> ( 1 | block )+P+K 2 0.094 0.366#> Average level 2 2 0.366 -0.016 0.167#> ( 1 | block )+N+P+K 3 0.459#>#> * Fit index: rb.r2.2#>#> Average contribution of each variable:#>#> P K N#> 0.022 -0.112 -0.259#>#> Dominance Analysis matrix:#> model level fit N P K#> ( 1 | block ) 0 0 -0.241 0.032 -0.099#> ( 1 | block )+N 1 -0.241 0.019 -0.12#> ( 1 | block )+P 1 0.032 -0.254 -0.103#> ( 1 | block )+K 1 -0.099 -0.262 0.028#> Average level 1 1 -0.258 0.023 -0.112#> ( 1 | block )+N+P 2 -0.222 -0.127#> ( 1 | block )+N+K 2 -0.361 0.012#> ( 1 | block )+P+K 2 -0.071 -0.277#> Average level 2 2 -0.277 0.012 -0.127#> ( 1 | block )+N+P+K 3 -0.348#>#> * Fit index: sb.r2.1#>#> Average contribution of each variable:#>#> N K P#> 0.192 0.084 -0.017#>#> Dominance Analysis matrix:#> model level fit N P K#> ( 1 | block ) 0 0 0.179 -0.024 0.073#> ( 1 | block )+N 1 0.179 -0.014 0.089#> ( 1 | block )+P 1 -0.024 0.189 0.077#> ( 1 | block )+K 1 0.073 0.195 -0.021#> Average level 1 1 0.192 -0.017 0.083#> ( 1 | block )+N+P 2 0.165 0.094#> ( 1 | block )+N+K 2 0.268 -0.009#> ( 1 | block )+P+K 2 0.053 0.206#> Average level 2 2 0.206 -0.009 0.094#> ( 1 | block )+N+P+K 3 0.259#>#> * Fit index: sb.r2.2#>#> Average contribution of each variable:#>#> N K P#> 0 0 0#>#> Dominance Analysis matrix:#> model level fit N P K#> ( 1 | block ) 0 0 0 0 0#> ( 1 | block )+N 1 0 0 0#> ( 1 | block )+P 1 0 0 0#> ( 1 | block )+K 1 0 0 0#> Average level 1 1 0 0 0#> ( 1 | block )+N+P 2 0 0#> ( 1 | block )+N+K 2 0 0#> ( 1 | block )+P+K 2 0 0#> Average level 2 2 0 0 0#> ( 1 | block )+N+P+K 3 0sm.rb.r2.1=s.da.lmer$rb.r2.1$summary.matrix# Nitrogen completely dominates potassiumas.logical(na.omit(sm.rb.r2.1$N > sm.rb.r2.1$K))#> [1] TRUE TRUE TRUE TRUE# Nitrogen completely dominates phosphateas.logical(na.omit(sm.rb.r2.1$N > sm.rb.r2.1$P))#> [1] TRUE TRUE TRUE TRUE# Potassium completely dominates phosphateas.logical(na.omit(sm.rb.r2.1$K > sm.rb.r2.1$P))#> [1] TRUE TRUE TRUE TRUE
Logistic regression
Dominance analysis can be used in logistic regression (see Azen and Traxel, 2009).
As an example, we used the esoph dataset, that contains information about a case-control study of (o)esophageal cancer in Ille-et-Vilaine, France.
Looking at the report for standard glm summary method, we can see that the linear effect of each variable was significant (p < 0.05 for agegp.L, alcgp.L and tobgp.L), such as the quadratic effect of predictor age (p < 0.05 for agegp.Q). Even so,it is hard to identify which variable is more important to predict esophageal cancer.
glm.esoph<-glm(cbind(ncases,ncontrols)~agegp+alcgp+tobgp, esoph,family="binomial")summary(glm.esoph)#>#> Call:#> glm(formula = cbind(ncases, ncontrols) ~ agegp + alcgp + tobgp,#> family = "binomial", data = esoph)#>#> Deviance Residuals:#> Min 1Q Median 3Q Max#> -1.6891 -0.5618 -0.2168 0.2314 2.0642#>#> Coefficients:#> Estimate Std. Error z value Pr(>|z|)#> (Intercept) -1.77997 0.19796 -8.992 < 2e-16 ***#> agegp.L 3.00534 0.65215 4.608 4.06e-06 ***#> agegp.Q -1.33787 0.59111 -2.263 0.02362 *#> agegp.C 0.15307 0.44854 0.341 0.73291#> agegp^4 0.06410 0.30881 0.208 0.83556#> agegp^5 -0.19363 0.19537 -0.991 0.32164#> alcgp.L 1.49185 0.19935 7.484 7.23e-14 ***#> alcgp.Q -0.22663 0.17952 -1.262 0.20680#> alcgp.C 0.25463 0.15906 1.601 0.10942#> tobgp.L 0.59448 0.19422 3.061 0.00221 **#> tobgp.Q 0.06537 0.18811 0.347 0.72823#> tobgp.C 0.15679 0.18658 0.840 0.40071#> ---#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#>#> (Dispersion parameter for binomial family taken to be 1)#>#> Null deviance: 227.241 on 87 degrees of freedom#> Residual deviance: 53.973 on 76 degrees of freedom#> AIC: 225.45#>#> Number of Fisher Scoring iterations: 6
We performed dominance analysis on this dataset and the results are shown below. The fit indices were r2.m (RM2: McFadden's measure), r2.cs (RC**S2: Cox and Snell's measure), r2.n (RN2: Nagelkerke's measure) and r2.e (RE2: Estrella's measure). For all fit indices, we can conclude that age and alcohol completely dominate tobacco, while age shows general dominance over both alcohol and tobacco.
da.esoph<-dominanceAnalysis(glm.esoph)print(da.esoph)#>#> Dominance analysis#> Predictors: agegp, alcgp, tobgp#> Fit-indices: r2.m, r2.cs, r2.n, r2.e#>#> * Fit index: r2.m#> complete conditional general#> agegp tbgp tbgp alcg,tbgp#> alcgp tbgp tbgp tbgp#> tobgp#>#> Average contribution:#> agegp alcgp tobgp#> 0.363 0.339 0.061#> * Fit index: r2.cs#> complete conditional general#> agegp tbgp tbgp alcg,tbgp#> alcgp tbgp tbgp tbgp#> tobgp#>#> Average contribution:#> agegp alcgp tobgp#> 2.129 2.023 0.446#> * Fit index: r2.n#> complete conditional general#> agegp tbgp tbgp alcg,tbgp#> alcgp tbgp tbgp tbgp#> tobgp#>#> Average contribution:#> agegp alcgp tobgp#> 2.303 2.188 0.483#> * Fit index: r2.e#> complete conditional general#> agegp tbgp tbgp alcg,tbgp#> alcgp tbgp tbgp tbgp#> tobgp#>#> Average contribution:#> agegp alcgp tobgp#> 1.372 1.288 0.246summary(da.esoph)#>#> * Fit index: r2.m#>#> Average contribution of each variable:#>#> agegp alcgp tobgp#> 0.363 0.339 0.061#>#> Dominance Analysis matrix:#> model level fit agegp alcgp tobgp#> 1 0 -0.649 0.388 0.389 0.078#> agegp 1 -0.261 0.328 0.084#> alcgp 1 -0.26 0.327 0.032#> tobgp 1 -0.571 0.394 0.343#> Average level 1 1 0.36 0.336 0.058#> agegp+alcgp 2 0.067 0.047#> agegp+tobgp 2 -0.177 0.291#> alcgp+tobgp 2 -0.228 0.341#> Average level 2 2 0.341 0.291 0.047#> agegp+alcgp+tobgp 3 0.113#>#> * Fit index: r2.cs#>#> Average contribution of each variable:#>#> agegp alcgp tobgp#> 2.129 2.023 0.446#>#> Dominance Analysis matrix:#> model level fit agegp alcgp tobgp#> 1 0 -4.344 3.381 3.388 0.974#> agegp 1 -0.963 1.121 0.383#> alcgp 1 -0.956 1.114 0.156#> tobgp 1 -3.37 2.789 2.57#> Average level 1 1 1.952 1.846 0.269#> agegp+alcgp 2 0.159 0.095#> agegp+tobgp 2 -0.58 0.834#> alcgp+tobgp 2 -0.8 1.054#> Average level 2 2 1.054 0.834 0.095#> agegp+alcgp+tobgp 3 0.254#>#> * Fit index: r2.n#>#> Average contribution of each variable:#>#> agegp alcgp tobgp#> 2.303 2.188 0.483#>#> Dominance Analysis matrix:#> model level fit agegp alcgp tobgp#> 1 0 -4.699 3.657 3.665 1.054#> agegp 1 -1.042 1.213 0.414#> alcgp 1 -1.034 1.205 0.169#> tobgp 1 -3.645 3.017 2.78#> Average level 1 1 2.111 1.996 0.291#> agegp+alcgp 2 0.171 0.103#> agegp+tobgp 2 -0.628 0.902#> alcgp+tobgp 2 -0.865 1.14#> Average level 2 2 1.14 0.902 0.103#> agegp+alcgp+tobgp 3 0.275#>#> * Fit index: r2.e#>#> Average contribution of each variable:#>#> agegp alcgp tobgp#> 1.372 1.288 0.246#>#> Dominance Analysis matrix:#> model level fit agegp alcgp tobgp#> 1 0 -2.639 1.818 1.823 0.428#> agegp 1 -0.821 0.984 0.297#> alcgp 1 -0.815 0.979 0.117#> tobgp 1 -2.211 1.687 1.513#> Average level 1 1 1.333 1.249 0.207#> agegp+alcgp 2 0.164 0.104#> agegp+tobgp 2 -0.524 0.791#> alcgp+tobgp 2 -0.698 0.965#> Average level 2 2 0.965 0.791 0.104#> agegp+alcgp+tobgp 3 0.267
Then, we performed a bootstrap analysis. Using McFadden's measure (r2.m), we can see that bootstrap dominance of age over tobacco, and of alcohol over tobacco have standard errors (SE.Dij) of 0 and reproducibility (Rep) equal to 1, so are fairly robust on all levels.Dominance values of age over alcohol are not easily reproducible and require more research
da.b.esoph<-bootDominanceAnalysis(glm.esoph,R = 200)print(format(summary(da.b.esoph)$r2.m,digits=3),row.names=F)#> dominance i k Dij mDij SE.Dij Pij Pji Pnoij Rep#> complete agegp alcgp 0.5 0.580 0.4551 0.505 0.345 0.150 0.150#> complete agegp tobgp 1.0 0.998 0.0354 0.995 0.000 0.005 0.995#> complete alcgp tobgp 1.0 1.000 0.0000 1.000 0.000 0.000 1.000#> conditional agegp alcgp 0.5 0.580 0.4551 0.505 0.345 0.150 0.150#> conditional agegp tobgp 1.0 0.998 0.0354 0.995 0.000 0.005 0.995#> conditional alcgp tobgp 1.0 1.000 0.0000 1.000 0.000 0.000 1.000#> general agegp alcgp 1.0 0.590 0.4931 0.590 0.410 0.000 0.590#> general agegp tobgp 1.0 1.000 0.0000 1.000 0.000 0.000 1.000#> general alcgp tobgp 1.0 1.000 0.0000 1.000 0.000 0.000 1.000
Set of predictors
Budescu (1993) shows that dominance analysis can be applied to groups or set of inseparable predictors.
m.budescu.5=matrix(c(1,.30,.41,.33,.30,1,.16,.57,.41,.16,1,.50,.33,.57,.50,1), nrow = 4,ncol = 4,byrow = T,dimnames = list(c('SES','IQ','nAch','GPA'),c('SES','IQ','nAch','GPA')))lmCov.b5<-lmWithCov(GPA~SES+IQ+nAch,m.budescu.5)da.b5<-dominanceAnalysis(lmCov.b5)print(da.b5)#>#> Dominance analysis#> Predictors: SES, IQ, nAch#> Fit-indices: r2#>#> * Fit index: r2#> complete conditional general#> SES#> IQ SES,nAch SES,nAch SES,nAch#> nAch SES SES SES#>#> Average contribution:#> IQ nAch SES#> 0.266 0.186 0.044summary(da.b5)#>#> * Fit index: r2#>#> Average contribution of each variable:#>#> IQ nAch SES#> 0.266 0.186 0.044#>#> Dominance Analysis matrix:#> model level fit SES IQ nAch#> 1 0 0 0.109 0.325 0.25#> SES 1 0.109 0.244 0.16#> IQ 1 0.325 0.028 0.172#> nAch 1 0.25 0.019 0.246#> Average level 1 1 0.023 0.245 0.166#> SES+IQ 2 0.353 0.144#> SES+nAch 2 0.269 0.228#> IQ+nAch 2 0.496 0#> Average level 2 2 0 0.228 0.144#> SES+IQ+nAch 3 0.496da.b5.g<-dominanceAnalysis(lmCov.b5,terms = c("SES","IQ+nAch"))print(da.b5.g)#>#> Dominance analysis#> Predictors: SES, IQ+nAch#> Terms: = SES ; = IQ+nAch#> Fit-indices: r2#>#> * Fit index: r2#> complete conditional general#> SES#> IQ+nAch SES SES SES#>#> Average contribution:#> IQ+nAch SES#> 0.442 0.054summary(da.b5.g)#>#> * Fit index: r2#>#> Average contribution of each variable:#>#> IQ+nAch SES#> 0.442 0.054#>#> Dominance Analysis matrix:#> model level fit SES IQ.nAch#> 1 0 0 0.109 0.496#> SES 1 0.109 0.388#> IQ+nAch 1 0.496 0#> Average level 1 1 0 0.388#> SES+IQ+nAch 2 0.496
Installation
You can install the github version of dominanceanalysis from github with:
install_github("clbustos/dominanceanalysis")
Authors
- Claudio Bustos Navarrete: Creator and maintainer
- Filipa Coutinho Soares: Documentation and testing
References
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Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin, 114(3), 542-551. https://doi.org/10.1037/0033-2909.114.3.542
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Azen, R., & Budescu, D. V. (2003). The dominance analysis approach for comparing predictors in multiple regression. Psychological Methods, 8(2), 129-148. https://doi.org/10.1037/1082-989X.8.2.129
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Azen, R., & Budescu, D. V. (2006). Comparing Predictors in Multivariate Regression Models: An Extension of Dominance Analysis. Journal of Educational and Behavioral Statistics, 31(2), 157-180. https://doi.org/10.3102/10769986031002157
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Azen, R., & Traxel, N. (2009). Using Dominance Analysis to Determine Predictor Importance in Logistic Regression. Journal of Educational and Behavioral Statistics, 34(3), 319-347. https://doi.org/10.3102/1076998609332754
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Luo, W., & Azen, R. (2013). Determining Predictor Importance in Hierarchical Linear Models Using Dominance Analysis. Journal of Educational and Behavioral Statistics, 38(1), 3-31. https://doi.org/10.3102/1076998612458319
News
dominanceanalysis 1.0.0
- First official version. Code coverage of 99% and complete documentation of all methods.
- Added vignette "Exploring predictors' importance in binomial logistic regressions", by Filipa Coutinho Soares.
- Added dominanceMatrix() as a generic for matrix, data.frame and dominanceAnalysis methods
- New retrieval methods for dominanceAnalysis object: getFits(), contributionAverage() and contributionByLevel()
- Removed support for nlme (it never worked well)
dominanceanalysis 0.1.2
- Added parameter
termson dominanceAnalysis, to manually define set of variables - Updated documentation, thanks to Filipa Coutinho Soares
- Test for all relevant functions
dominance analysis 0.1.1
- Allowed multivariate regression with covariance matrix
- Added documentation for multivariate methods
- Added bootstrap analysis for average contribution
- Resolved bug on bootstrap analysis
- Resolved bug on glm.da.fit
dominance analysis 0.1.0
- Complete support for OLS, logistic regression and HLM
- Support for bootstrap analysis