dominanceanalysis

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:

• Provides complete, conditional and general dominance analysis for lm (univariate and multivariate), dynlm, lmer, betareg and glm (family=binomial) models.
• Covariance / correlation matrixes could be used as input for OLS dominance analysis, using `lmWithCov()` and `mlmWithCov()` 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 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
#> privileges                     crtc                     crtc
#> raises                         crtc                     crtc
#> critical
#>                             general
#> critical
#>
#> Average contribution:
#> complaints   learning     raises privileges    advance   critical
#>      0.371      0.156      0.120      0.051      0.028      0.007``````

The 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
#> privileges                     crtc                     crtc
#> raises                         crtc                     crtc
#> critical
#>                             general
#> critical
averageContribution(da.attitude)
#>
#> Average Contribution by predictor
#>    complaints privileges learning raises critical advance
#> r2      0.371      0.051    0.156   0.12    0.007   0.028``````

The `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.

``````  set.seed(1234)
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 0.975 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 0.62``````

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( 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
#> ``````

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)
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: 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
#>  0.023 -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,K         N,K     N,K
#> K                            N
#>
#> Average contribution:
#> P K N
#> 0 0 0``````

The fit indices used in the analysis were 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:  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 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 ((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 alcg,tbgp
#> alcgp     tbgp        tbgp      tbgp
#> tobgp
#>
#> Average contribution:
#> agegp alcgp tobgp
#> 0.220 0.205 0.037
#> * Fit index:  r2.cs
#>       complete conditional   general
#> agegp     tbgp        tbgp alcg,tbgp
#> alcgp     tbgp        tbgp      tbgp
#> tobgp
#>
#> Average contribution:
#> agegp alcgp tobgp
#> 0.398 0.378 0.084
#> * Fit index:  r2.n
#>       complete conditional   general
#> agegp     tbgp        tbgp alcg,tbgp
#> alcgp     tbgp        tbgp      tbgp
#> tobgp
#>
#> Average contribution:
#> agegp alcgp tobgp
#> 0.404 0.384 0.085
#> * Fit index:  r2.e
#>       complete conditional   general
#> agegp     tbgp        tbgp alcg,tbgp
#> alcgp     tbgp        tbgp      tbgp
#> tobgp
#>
#> Average contribution:
#> agegp alcgp tobgp
#> 0.432 0.410 0.087
summary(da.esoph)
#>
#> * Fit index:  r2.m
#>
#> Average contribution of each variable:
#>
#> agegp alcgp tobgp
#> 0.220 0.205 0.037
#>
#> Dominance Analysis matrix:
#>              model level   fit agegp alcgp tobgp
#>                  1     0     0 0.235 0.236 0.047
#>              agegp     1 0.235       0.199 0.051
#>              alcgp     1 0.236 0.198       0.019
#>              tobgp     1 0.047 0.239 0.208
#>    Average level 1     1       0.218 0.204 0.035
#>        agegp+alcgp     2 0.434             0.028
#>        agegp+tobgp     2 0.286       0.176
#>        alcgp+tobgp     2 0.256 0.207
#>    Average level 2     2       0.207 0.176 0.028
#>  agegp+alcgp+tobgp     3 0.462
#>
#> * Fit index:  r2.cs
#>
#> Average contribution of each variable:
#>
#> agegp alcgp tobgp
#> 0.398 0.378 0.084
#>
#> Dominance Analysis matrix:
#>              model level   fit agegp alcgp tobgp
#>                  1     0     0 0.633 0.634 0.182
#>              agegp     1 0.633        0.21 0.072
#>              alcgp     1 0.634 0.209       0.029
#>              tobgp     1 0.182 0.522 0.481
#>    Average level 1     1       0.365 0.345  0.05
#>        agegp+alcgp     2 0.843             0.018
#>        agegp+tobgp     2 0.704       0.156
#>        alcgp+tobgp     2 0.663 0.197
#>    Average level 2     2       0.197 0.156 0.018
#>  agegp+alcgp+tobgp     3  0.86
#>
#> * Fit index:  r2.n
#>
#> Average contribution of each variable:
#>
#> agegp alcgp tobgp
#> 0.404 0.384 0.085
#>
#> Dominance Analysis matrix:
#>              model level   fit agegp alcgp tobgp
#>                  1     0     0 0.642 0.643 0.185
#>              agegp     1 0.642       0.213 0.073
#>              alcgp     1 0.643 0.212        0.03
#>              tobgp     1 0.185 0.529 0.488
#>    Average level 1     1        0.37  0.35 0.051
#>        agegp+alcgp     2 0.855             0.018
#>        agegp+tobgp     2 0.714       0.158
#>        alcgp+tobgp     2 0.673   0.2
#>    Average level 2     2         0.2 0.158 0.018
#>  agegp+alcgp+tobgp     3 0.873
#>
#> * Fit index:  r2.e
#>
#> Average contribution of each variable:
#>
#> agegp alcgp tobgp
#> 0.432 0.410 0.087
#>
#> Dominance Analysis matrix:
#>              model level   fit agegp alcgp tobgp
#>                  1     0     0 0.681 0.682 0.186
#>              agegp     1 0.681       0.231 0.081
#>              alcgp     1 0.682 0.229       0.033
#>              tobgp     1 0.186 0.576 0.529
#>    Average level 1     1       0.402  0.38 0.057
#>        agegp+alcgp     2 0.911             0.017
#>        agegp+tobgp     2 0.762       0.167
#>        alcgp+tobgp     2 0.715 0.213
#>    Average level 2     2       0.213 0.167 0.017
#>  agegp+alcgp+tobgp     3 0.929``````

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) 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 0.5 0.627 0.4312 0.530 0.275 0.195 0.195
#>     complete agegp tobgp 1.0 0.998 0.0354 0.995 0.000 0.005 0.995
#>     complete alcgp tobgp 1.0 0.998 0.0354 0.995 0.000 0.005 0.995
#>  conditional agegp alcgp 0.5 0.627 0.4312 0.530 0.275 0.195 0.195
#>  conditional agegp tobgp 1.0 0.998 0.0354 0.995 0.000 0.005 0.995
#>  conditional alcgp tobgp 1.0 0.998 0.0354 0.995 0.000 0.005 0.995
#>      general agegp alcgp 1.0 0.600 0.4911 0.600 0.400 0.000 0.600
#>      general agegp tobgp 1.0 1.000 0.0000 1.000 0.000 0.000 1.000
#>      general alcgp tobgp 1.0 0.995 0.0707 0.995 0.005 0.000 0.995``````

Set of predictors

Budescu (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     1.00``````

We 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      0.159``````

Installation

You can install the stable version from CRAN

``install.packages('dominanceanalysis')``

Also, you can install the latest version from github with:

``````library(devtools)
install_github("clbustos/dominanceanalysis")``````

Authors

• Claudio Bustos Navarrete: Creator and maintainer
• Filipa Coutinho Soares: Documentation and testing

Acknowledgments

• Daniel Schlaepfer: Error reporting on logistic regression code.
• Xiong Zhang: Incorporation of dynamic linear models.
• Maartje Hidalgo: Incorporation of beta regression.

References

• 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