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, Dynamic Linear Models and Hierarchical Linear Models.
Features:
lmWithCov() and mlmWithCov() methods, respectively.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 average contribution of each predictor is also presented, that defines defines general 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 0.007The dominance brief and average contribution of each predictor could be retrieved separately using dominanceBriefing() and averageContribution() methods, respectively.
dominanceBriefing(da.attitude, abbrev = TRUE)$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
averageContribution(da.attitude)
#>
#> Average Contribution by predictor
#> complaints privileges learning raises critical advance
#> r2 0.371 0.051 0.156 0.12 0.007 0.028The summary() method shows the complete dominance analysis matrix, that presents all fit differences between levels. Also, provides the average contribution of each variable.
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 complaints
#> 1 0 0 0.681
#> complaints 1 0.681
#> privileges 1 0.182 0.501
#> learning 1 0.389 0.319
#> raises 1 0.348 0.336
#> critical 1 0.024 0.657
#> advance 1 0.024 0.658
#> Average level 1 1 0.494
#> 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 0.307
#> privileges+raises 2 0.382 0.305
#> privileges+critical 2 0.191 0.493
#> privileges+advance 2 0.182 0.502
#> learning+raises 2 0.451 0.258
#> learning+critical 2 0.396 0.312
#> learning+advance 2 0.432 0.293
#> raises+critical 2 0.353 0.331
#> raises+advance 2 0.399 0.291
#> critical+advance 2 0.038 0.645
#> Average level 2 2 0.374
#> 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 0.257
#> privileges+learning+critical 3 0.413 0.302
#> privileges+learning+advance 3 0.458 0.271
#> privileges+raises+critical 3 0.386 0.301
#> privileges+raises+advance 3 0.443 0.247
#> privileges+critical+advance 3 0.191 0.493
#> learning+raises+critical 3 0.451 0.257
#> learning+raises+advance 3 0.552 0.176
#> learning+critical+advance 3 0.453 0.274
#> raises+critical+advance 3 0.401 0.288
#> Average level 3 3 0.287
#> 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 0.256
#> privileges+learning+raises+advance 4 0.563 0.169
#> privileges+learning+critical+advance 4 0.476 0.255
#> privileges+raises+critical+advance 4 0.445 0.246
#> learning+raises+critical+advance 4 0.553 0.176
#> Average level 4 4 0.22
#> 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 0.169
#> Average level 5 5 0.169
#> complaints+privileges+learning+raises+critical+advance 6 0.733
#> privileges learning raises critical advance
#> 0.182 0.389 0.348 0.024 0.024
#> 0.002 0.027 0.003 0 0.001
#> 0.226 0.2 0.009 0
#> 0.019 0.062 0.007 0.043
#> 0.033 0.102 0.005 0.05
#> 0.166 0.372 0.329 0.013
#> 0.158 0.408 0.375 0.014
#> 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
#> 0.051 0.005 0.05
#> 0.077 0.004 0.061
#> 0.222 0.195 0
#> 0.276 0.261 0.009
#> 0.008 0 0.102
#> 0.016 0.055 0.057
#> 0.026 0.12 0.021
#> 0.033 0.098 0.048
#> 0.044 0.154 0.003
#> 0.153 0.416 0.363
#> 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 0.104
#> 0.046 0.063
#> 0.105 0.018
#> 0.073 0.059
#> 0.12 0.002
#> 0.285 0.254
#> 0.008 0.102
#> 0.011 0.001
#> 0.022 0.1
#> 0.044 0.152
#> 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
#> 0.105
#> 0.001
#> 0.088
#> 0.119
#> 0.011
#> 0.005 0.055 0.02 0.001 0.032
#> 0.017
#> 0.001
#> 0.002
#> 0.042
#> 0.003
#>
#> 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 \(R^2\) as a fit index and 100 permutations. For precise results, you need to run at least 1000 replications.
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 0.975 0.1095 0.95 0.00 0.05 0.95
#> complete complaints learning 1.0 0.930 0.1883 0.87 0.01 0.12 0.87
#> 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 0.970 0.1193 0.94 0.00 0.06 0.94
#> complete privileges learning 0.0 0.270 0.2603 0.01 0.47 0.52 0.47
#> complete privileges raises 0.5 0.465 0.1282 0.00 0.07 0.93 0.93
#> complete privileges critical 1.0 0.510 0.1586 0.06 0.04 0.90 0.06
#> complete privileges advance 0.5 0.495 0.0500 0.00 0.01 0.99 0.99
#> complete learning raises 1.0 0.625 0.2876 0.32 0.07 0.61 0.32
#> complete learning critical 1.0 0.700 0.2659 0.42 0.02 0.56 0.42
#> complete learning advance 1.0 0.725 0.2500 0.45 0.00 0.55 0.45
#> complete raises critical 1.0 0.565 0.1833 0.14 0.01 0.85 0.14
#> complete raises advance 0.5 0.555 0.1572 0.11 0.00 0.89 0.89
#> complete critical advance 0.5 0.535 0.1629 0.09 0.02 0.89 0.89
#> conditional complaints privileges 1.0 0.990 0.0704 0.98 0.00 0.02 0.98
#> conditional complaints learning 1.0 0.940 0.1781 0.89 0.01 0.10 0.89
#> conditional complaints raises 1.0 0.995 0.0500 0.99 0.00 0.01 0.99
#> conditional complaints critical 1.0 0.985 0.0857 0.97 0.00 0.03 0.97
#> conditional complaints advance 1.0 0.975 0.1095 0.95 0.00 0.05 0.95
#> conditional privileges learning 0.0 0.170 0.2484 0.01 0.67 0.32 0.67
#> conditional privileges raises 0.5 0.340 0.2449 0.01 0.33 0.66 0.66
#> conditional privileges critical 1.0 0.600 0.3178 0.32 0.12 0.56 0.32
#> conditional privileges advance 0.5 0.575 0.2057 0.17 0.02 0.81 0.81
#> conditional learning raises 1.0 0.685 0.3383 0.48 0.11 0.41 0.48
#> conditional learning critical 1.0 0.830 0.2862 0.71 0.05 0.24 0.71
#> conditional learning advance 1.0 0.805 0.2451 0.61 0.00 0.39 0.61
#> conditional raises critical 1.0 0.660 0.2648 0.35 0.03 0.62 0.35
#> conditional raises advance 0.5 0.610 0.2082 0.22 0.00 0.78 0.78
#> conditional critical advance 0.5 0.475 0.3128 0.17 0.22 0.61 0.61
#> general complaints privileges 1.0 1.000 0.0000 1.00 0.00 0.00 1.00
#> general complaints learning 1.0 0.970 0.1714 0.97 0.03 0.00 0.97
#> general complaints raises 1.0 1.000 0.0000 1.00 0.00 0.00 1.00
#> 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 0.070 0.2564 0.07 0.93 0.00 0.93
#> general privileges raises 0.0 0.070 0.2564 0.07 0.93 0.00 0.93
#> general privileges critical 1.0 0.750 0.4352 0.75 0.25 0.00 0.75
#> general privileges advance 1.0 0.730 0.4462 0.73 0.27 0.00 0.73
#> general learning raises 1.0 0.680 0.4688 0.68 0.32 0.00 0.68
#> general learning critical 1.0 0.910 0.2876 0.91 0.09 0.00 0.91
#> general learning advance 1.0 0.980 0.1407 0.98 0.02 0.00 0.98
#> general raises critical 1.0 0.920 0.2727 0.92 0.08 0.00 0.92
#> general raises advance 1.0 0.970 0.1714 0.97 0.03 0.00 0.97
#> general critical advance 0.0 0.380 0.4878 0.38 0.62 0.00 0.62Another 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( f = general~picture+blocks+maze+reading+vocab,
x = 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 0.043
summary(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 reading
#> 1 0 0 0.217 0.304 0.116 0.332
#> picture 1 0.217 0.121 0.065 0.221
#> blocks 1 0.304 0.034 0.011 0.166
#> maze 1 0.116 0.167 0.2 0.273
#> reading 1 0.332 0.106 0.138 0.057
#> vocab 1 0.265 0.125 0.155 0.054 0.077
#> Average level 1 1 0.108 0.153 0.047 0.184
#> picture+blocks 2 0.338 0.015 0.156
#> picture+maze 2 0.282 0.07 0.193
#> picture+reading 2 0.439 0.055 0.036
#> picture+vocab 2 0.39 0.059 0.033 0.055
#> blocks+maze 2 0.316 0.037 0.164
#> blocks+reading 2 0.47 0.023 0.009
#> blocks+vocab 2 0.42 0.028 0.007 0.053
#> maze+reading 2 0.389 0.086 0.09
#> maze+vocab 2 0.319 0.104 0.108 0.074
#> reading+vocab 2 0.341 0.103 0.131 0.052
#> Average level 2 2 0.064 0.085 0.025 0.116
#> picture+blocks+maze 3 0.353 0.152
#> picture+blocks+reading 3 0.494 0.011
#> picture+blocks+vocab 3 0.448 0.009 0.048
#> picture+maze+reading 3 0.475 0.03
#> picture+maze+vocab 3 0.423 0.035 0.055
#> 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 0.054
#> 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 0.077
#> picture+blocks+maze+reading 4 0.505
#> picture+blocks+maze+vocab 4 0.458 0.049
#> 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 0.049
#> picture+blocks+maze+reading+vocab 5 0.507
#> vocab
#> 0.265
#> 0.172
#> 0.116
#> 0.203
#> 0.009
#>
#> 0.125
#> 0.11
#> 0.141
#> 0.006
#>
#> 0.111
#> 0.002
#>
#> 0.004
#>
#>
#> 0.062
#> 0.105
#> 0.002
#>
#> 0.003
#>
#>
#> 0.002
#>
#>
#>
#> 0.028
#> 0.002
#>
#>
#>
#>
#> 0.002
#> 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: Matrix
lmer.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: n.marg, n.cond, rb.r2.1, rb.r2.2, sb.r2.1, sb.r2.2
#>
#> * Fit index: n.marg
#> complete conditional general
#> N P,K P,K P,K
#> P
#> K P P P
#>
#> Average contribution:
#> N K P
#> 0.199 0.099 0.006
#> * Fit index: n.cond
#> complete conditional general
#> N P,K P,K P,K
#> P
#> K P P P
#>
#> Average contribution:
#> N K P
#> 0.256 0.120 -0.006
#> * 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
#> 0.023 -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 N N N,K
#> K N N N
#>
#> Average contribution:
#> P K N
#> 0 0 0The fit indices used in the analysis were n.marg (Nakagawa’s marginal R²), n.cond (Nakagawa’s conditional R²), rb.r2.1 (R&B \(R^2_1\): Level-1 variance component explained by predictors), rb.r2.2 (R&B \(R^2_2\): Level-2 variance component explained by predictors), sb.r2.1 (S&B \(R^2_1\): Level-1 proportional reduction in error predicting scores at Level-1), and sb.r2.2 (S&B \(R^2_2\): 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: n.marg
#>
#> Average contribution of each variable:
#>
#> N K P
#> 0.199 0.099 0.006
#>
#> Dominance Analysis matrix:
#> model level fit N P K
#> ( 1 | block ) 0 0 0.203 0.009 0.102
#> ( 1 | block )+N 1 0.203 0.004 0.097
#> ( 1 | block )+P 1 0.009 0.198 0.099
#> ( 1 | block )+K 1 0.102 0.198 0.006
#> Average level 1 1 0.198 0.005 0.098
#> ( 1 | block )+N+P 2 0.207 0.097
#> ( 1 | block )+N+K 2 0.3 0.004
#> ( 1 | block )+P+K 2 0.108 0.196
#> Average level 2 2 0.196 0.004 0.097
#> ( 1 | block )+N+P+K 3 0.304
#>
#> * Fit index: n.cond
#>
#> Average contribution of each variable:
#>
#> N K P
#> 0.256 0.120 -0.006
#>
#> Dominance Analysis matrix:
#> model level fit N P K
#> ( 1 | block ) 0 0.248 0.254 -0.007 0.118
#> ( 1 | block )+N 1 0.501 -0.007 0.121
#> ( 1 | block )+P 1 0.241 0.253 0.117
#> ( 1 | block )+K 1 0.366 0.257 -0.008
#> Average level 1 1 0.255 -0.007 0.119
#> ( 1 | block )+N+P 2 0.495 0.123
#> ( 1 | block )+N+K 2 0.623 -0.005
#> ( 1 | block )+P+K 2 0.358 0.26
#> Average level 2 2 0.26 -0.005 0.123
#> ( 1 | block )+N+P+K 3 0.618
#>
#> * 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
#> 0.023 -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:
#>
#> P K N
#> 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 0
sm.rb.r2.1=s.da.lmer$rb.r2.1$summary.matrix
# Nitrogen completely dominates potassium
as.logical(na.omit(sm.rb.r2.1$N > sm.rb.r2.1$K))
#> [1] TRUE TRUE TRUE TRUE
# Nitrogen completely dominates phosphate
as.logical(na.omit(sm.rb.r2.1$N > sm.rb.r2.1$P))
#> [1] TRUE TRUE TRUE TRUE
# Potassium completely dominates phosphate
as.logical(na.omit(sm.rb.r2.1$K > sm.rb.r2.1$P))
#> [1] TRUE TRUE TRUE TRUEDominance 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)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -1.19039 0.20737 -5.740 9.44e-09 ***
#> agegp.L 3.99663 0.69389 5.760 8.42e-09 ***
#> agegp.Q -1.65741 0.62115 -2.668 0.00762 **
#> agegp.C 0.11094 0.46815 0.237 0.81267
#> agegp^4 0.07892 0.32463 0.243 0.80792
#> agegp^5 -0.26219 0.21337 -1.229 0.21915
#> alcgp.L 2.53899 0.26385 9.623 < 2e-16 ***
#> alcgp.Q 0.09376 0.22419 0.418 0.67578
#> alcgp.C 0.43930 0.18347 2.394 0.01665 *
#> tobgp.L 1.11749 0.24014 4.653 3.26e-06 ***
#> tobgp.Q 0.34516 0.22414 1.540 0.12358
#> tobgp.C 0.31692 0.21091 1.503 0.13294
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 367.953 on 87 degrees of freedom
#> Residual deviance: 82.337 on 76 degrees of freedom
#> AIC: 221.39
#>
#> Number of Fisher Scoring iterations: 6We performed dominance analysis on this dataset and the results are shown below. The fit indices were r2.m (\(R^2_M\): McFadden’s measure), r2.cs (\(R^2_{CS}\): Cox and Snell’s measure), r2.n (\(R^2_N\): Nagelkerke’s measure) and r2.e (\(R^2_E\): 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 tbgp
#> alcgp aggp,tbgp aggp,tbgp aggp,tbgp
#> tobgp
#>
#> Average contribution:
#> alcgp agegp tobgp
#> 0.283 0.256 0.052
#> * Fit index: r2.cs
#> complete conditional general
#> agegp tbgp tbgp tbgp
#> alcgp aggp,tbgp aggp,tbgp aggp,tbgp
#> tobgp
#>
#> Average contribution:
#> alcgp agegp tobgp
#> 0.441 0.407 0.113
#> * Fit index: r2.n
#> complete conditional general
#> agegp tbgp tbgp tbgp
#> alcgp aggp,tbgp aggp,tbgp aggp,tbgp
#> tobgp
#>
#> Average contribution:
#> alcgp agegp tobgp
#> 0.442 0.409 0.113
#> * Fit index: r2.e
#> complete conditional general
#> agegp tbgp tbgp tbgp
#> alcgp aggp,tbgp aggp,tbgp aggp,tbgp
#> tobgp
#>
#> Average contribution:
#> alcgp agegp tobgp
#> 0.457 0.422 0.114
summary(da.esoph)
#>
#> * Fit index: r2.m
#>
#> Average contribution of each variable:
#>
#> alcgp agegp tobgp
#> 0.283 0.256 0.052
#>
#> Dominance Analysis matrix:
#> model level fit agegp alcgp tobgp
#> 1 0 0 0.251 0.303 0.058
#> agegp 1 0.251 0.292 0.076
#> alcgp 1 0.303 0.239 0.026
#> tobgp 1 0.058 0.269 0.272
#> Average level 1 1 0.254 0.282 0.051
#> agegp+alcgp 2 0.543 0.049
#> agegp+tobgp 2 0.326 0.265
#> alcgp+tobgp 2 0.329 0.262
#> Average level 2 2 0.262 0.265 0.049
#> agegp+alcgp+tobgp 3 0.591
#>
#> * Fit index: r2.cs
#>
#> Average contribution of each variable:
#>
#> alcgp agegp tobgp
#> 0.441 0.407 0.113
#>
#> Dominance Analysis matrix:
#> model level fit agegp alcgp tobgp
#> 1 0 0 0.747 0.811 0.271
#> agegp 1 0.747 0.202 0.086
#> alcgp 1 0.811 0.138 0.025
#> tobgp 1 0.271 0.562 0.565
#> Average level 1 1 0.35 0.383 0.056
#> agegp+alcgp 2 0.949 0.012
#> agegp+tobgp 2 0.833 0.128
#> alcgp+tobgp 2 0.836 0.125
#> Average level 2 2 0.125 0.128 0.012
#> agegp+alcgp+tobgp 3 0.961
#>
#> * Fit index: r2.n
#>
#> Average contribution of each variable:
#>
#> alcgp agegp tobgp
#> 0.442 0.409 0.113
#>
#> Dominance Analysis matrix:
#> model level fit agegp alcgp tobgp
#> 1 0 0 0.75 0.814 0.272
#> agegp 1 0.75 0.203 0.086
#> alcgp 1 0.814 0.139 0.025
#> tobgp 1 0.272 0.564 0.567
#> Average level 1 1 0.352 0.385 0.056
#> agegp+alcgp 2 0.953 0.012
#> agegp+tobgp 2 0.837 0.128
#> alcgp+tobgp 2 0.84 0.126
#> Average level 2 2 0.126 0.128 0.012
#> agegp+alcgp+tobgp 3 0.965
#>
#> * Fit index: r2.e
#>
#> Average contribution of each variable:
#>
#> alcgp agegp tobgp
#> 0.457 0.422 0.114
#>
#> Dominance Analysis matrix:
#> model level fit agegp alcgp tobgp
#> 1 0 0 0.795 0.862 0.278
#> agegp 1 0.795 0.192 0.091
#> alcgp 1 0.862 0.124 0.026
#> tobgp 1 0.278 0.608 0.61
#> Average level 1 1 0.366 0.401 0.059
#> agegp+alcgp 2 0.986 0.006
#> agegp+tobgp 2 0.886 0.107
#> alcgp+tobgp 2 0.888 0.104
#> Average level 2 2 0.104 0.107 0.006
#> agegp+alcgp+tobgp 3 0.993Then, 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) near 0 and reproducibility (Rep) close to 1, so are fairly robust on all levels.Dominance values of age over alcohol are not easily reproducible and require more research
set.seed(1234)
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 0.438 0.4425 0.335 0.46 0.205 0.460
#> complete agegp tobgp 1 0.988 0.0783 0.975 0.00 0.025 0.975
#> complete alcgp tobgp 1 0.998 0.0354 0.995 0.00 0.005 0.995
#> conditional agegp alcgp 0 0.438 0.4425 0.335 0.46 0.205 0.460
#> conditional agegp tobgp 1 0.988 0.0783 0.975 0.00 0.025 0.975
#> conditional alcgp tobgp 1 0.998 0.0354 0.995 0.00 0.005 0.995
#> general agegp alcgp 0 0.400 0.4911 0.400 0.60 0.000 0.600
#> general agegp tobgp 1 1.000 0.0000 1.000 0.00 0.000 1.000
#> general alcgp tobgp 1 1.000 0.0000 1.000 0.00 0.000 1.000Budescu (1993) shows that dominance analysis can be applied to groups or set of inseparable predictors. The Longley’s economic regression data is know for have a highly collinear set on Employed variable. We can see that GNP.deflator, GNP, Population and Year are highly correlated.
data(longley)
round(cor(longley),2)
#> GNP.deflator GNP Unemployed Armed.Forces Population Year Employed
#> GNP.deflator 1.00 0.99 0.62 0.46 0.98 0.99 0.97
#> GNP 0.99 1.00 0.60 0.45 0.99 1.00 0.98
#> Unemployed 0.62 0.60 1.00 -0.18 0.69 0.67 0.50
#> Armed.Forces 0.46 0.45 -0.18 1.00 0.36 0.42 0.46
#> Population 0.98 0.99 0.69 0.36 1.00 0.99 0.96
#> Year 0.99 1.00 0.67 0.42 0.99 1.00 0.97
#> Employed 0.97 0.98 0.50 0.46 0.96 0.97 1.00We can group GNP and employment related variables, to determine the importance of both groups of variables. The GNP related variables dominates completely population, and we can see that all predictors dominates generally over employment.
terms.r<-c(GNP.rel="GNP.deflator+GNP",
employment="Unemployed+Armed.Forces",
"Population",
"Year")
da.longley<-dominanceAnalysis(lm(Employed~.,longley),terms = terms.r)
print(da.longley)
#>
#> Dominance analysis
#> Predictors: GNP.deflator+GNP, Unemployed+Armed.Forces, Population, Year
#> Terms: GNP.rel = GNP.deflator+GNP ; employment = Unemployed+Armed.Forces ; = Population ; = Year
#> Fit-indices: r2
#>
#> * Fit index: r2
#> complete conditional general
#> GNP.rel Pplt Pplt empl,Pplt,Year
#> employment
#> Population empl
#> Year Pplt empl,Pplt
#>
#> Average contribution:
#> GNP.rel Year Population employment
#> 0.290 0.279 0.267 0.159You can install the stable version from CRAN
Also, you can install the latest version from github with:
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
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
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
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
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
Shou, Y., & Smithson, M. (2015). Evaluating Predictors of Dispersion: A Comparison of Dominance Analysis and Bayesian Model Averaging. Psychometrika, 80(1), 236-256. https://doi.org/10.1007/s11336-013-9375-8