Bayesian Model Averaging (BMA) for settings with a high number of regressors and a limited number of observations. The package provides standard BMA tools, including the estimation of BMA statistics, Extreme Bounds Analysis, Bayesian model selection, and jointness analysis. It also offers graphical functions for exploring the model space and coefficient distributions. Moreover, it introduces a non-empirical dilution prior to address multicollinearity in the presence of model uncertainty. When the model space is too large to enumerate, it can be explored by MC^3 sampling instead.
You can install the released version of rmsBMA from CRAN with:
install.packages("rmsBMA")
You can install the development version of rmsBMA from GitHub with:
# install.packages("pak")
# pak::pak("becku182/rmsBMA")
install.packages("rmsBMA", repos = c('https://becku182.r-universe.dev', 'https://cloud.r-project.org'))
Example of finding robust determinants of the dependent variable in the context of linear regression using Bayesian model averaging with reduced model space. The example uses data on the determinants of international trade for 25 European Union countries.
# loading the package
library(rmsBMA)
# inspecting the trade data set
round(head(Trade_data)[,1:16],2)
LNTRADE B LNDGEO L LNRGDPPROD RGDPpcDIFF GOV HUMAN CPW INFVAR ARABLE ARABLEpw LAND LANDpc EPCpc FDI
22.45 0 7.01 1 25.16 0.09 0.03 0.23 9955.85 0.19 11.01 2.15 52292 7154 587.15 13.74
20.68 0 6.92 0 23.76 1.21 0.14 0.07 189220.51 2.10 13.28 4.10 26375 4016 3632.99 8.97
18.37 0 7.61 0 22.18 0.44 0.02 0.09 19423.21 0.40 5.89 32.91 73332 2265 3797.95 5.49
22.93 1 5.68 0 24.74 0.50 0.11 0.63 89762.36 0.71 24.81 4.04 5318 2560 1668.44 3.49
21.20 0 7.00 0 24.62 0.02 0.06 0.12 24264.09 0.24 38.74 15.43 40142 2248 1309.72 4.12
18.93 0 7.42 0 22.41 0.83 0.12 0.47 150494.26 1.61 2.53 20.01 40182 21311 2288.56 5.86
# First we build model space using model space function. At this point we need to choose maximum model size (M)
# and g prior. We start with M = 7 considering only models with 7 or less regressors and Unit Informatio Piror.
modelSpace <- model_space(Trade_data, M = 7, g = "UIP")
# Using the modelSpace object, BMA statistics can be computed under the selected model prior specifications.
bma_list <- bma(modelSpace, round = 3)
# Now we can see the results obtained using uniform model prior.
bma_list[[1]]
PIP PM PSD PMcon PSDcon P(+) PSC
CONST NA 8.093 0.759 8.093 0.759 1.000 1.000
B 0.817 0.347 0.208 0.424 0.142 0.816 0.907
LNDGEO 1.000 -1.175 0.081 -1.175 0.081 0.000 1.000
L 0.185 0.063 0.161 0.343 0.210 0.175 0.583
LNRGDPPROD 1.000 0.887 0.021 0.887 0.021 1.000 1.000
RGDPpcDIFF 0.074 -0.007 0.064 -0.094 0.220 0.026 0.511
GOV 0.984 -4.011 1.197 -4.074 1.093 0.000 0.992
HUMAN 0.056 -0.004 0.050 -0.068 0.199 0.021 0.508
CPW 0.147 0.000 0.000 0.000 0.000 0.136 0.562
INFVAR 0.931 -0.038 0.015 -0.040 0.012 0.000 0.965
ARABLE 0.210 0.001 0.003 0.006 0.003 0.202 0.597
ARABLEpw 0.702 -0.006 0.005 -0.009 0.003 0.003 0.847
LAND 0.060 0.000 0.000 0.000 0.000 0.035 0.505
LANDpc 0.749 0.000 0.000 0.000 0.000 0.747 0.872
EPCpc 0.290 0.000 0.000 0.000 0.000 0.279 0.634
FDI 0.829 0.023 0.014 0.028 0.010 0.827 0.913
KSI 0.401 -0.414 0.590 -1.033 0.479 0.006 0.694
BCIDIFF 0.062 0.000 0.001 0.000 0.003 0.024 0.507
# We can compare them with the results obtained using binomial-beta model prior.
bma_list[[2]]
PIP PM PSD PMcon PSDcon P(+) PSC
CONST NA 8.106 0.760 8.106 0.760 1.000 1.000
B 0.797 0.338 0.212 0.424 0.142 0.795 0.897
LNDGEO 1.000 -1.178 0.083 -1.178 0.083 0.000 1.000
L 0.201 0.068 0.165 0.338 0.210 0.190 0.589
LNRGDPPROD 1.000 0.887 0.021 0.887 0.021 1.000 1.000
RGDPpcDIFF 0.083 -0.007 0.068 -0.087 0.220 0.030 0.512
GOV 0.984 -4.021 1.201 -4.085 1.098 0.000 0.992
HUMAN 0.066 -0.005 0.054 -0.071 0.199 0.024 0.509
CPW 0.167 0.000 0.000 0.000 0.000 0.155 0.572
INFVAR 0.934 -0.038 0.015 -0.041 0.012 0.000 0.967
ARABLE 0.232 0.001 0.003 0.006 0.003 0.223 0.607
ARABLEpw 0.663 -0.006 0.005 -0.009 0.003 0.003 0.828
LAND 0.070 0.000 0.000 0.000 0.000 0.042 0.507
LANDpc 0.745 0.000 0.000 0.000 0.000 0.742 0.869
EPCpc 0.298 0.000 0.000 0.000 0.000 0.286 0.637
FDI 0.782 0.022 0.015 0.028 0.010 0.780 0.889
KSI 0.398 -0.407 0.584 -1.020 0.480 0.007 0.693
BCIDIFF 0.072 0.000 0.001 0.000 0.003 0.028 0.508
# We might compare those results with the one obtained using Extreme Bounds Analysis.
bma_list[[3]]
Variable Lower bound Minimum Mean Maximum Upper bound EBA test %(+)
CONST -4.966 -3.240 15.249 33.075 35.147 FAIL 87.098
B -0.073 0.235 1.217 2.720 3.440 FAIL 100.000
LNDGEO -1.874 -1.563 -1.182 -0.743 -0.357 PASS 0.000
L -0.990 0.066 1.039 3.275 4.374 FAIL 100.000
LNRGDPPROD 0.792 0.844 0.894 0.992 1.062 PASS 100.000
RGDPpcDIFF -4.621 -3.605 -0.707 0.586 1.488 FAIL 11.766
GOV -26.535 -20.032 -6.657 0.223 3.948 FAIL 0.029
HUMAN -2.368 -1.133 -0.208 0.885 1.996 FAIL 18.137
CPW 0.000 0.000 0.000 0.000 0.000 FAIL 71.011
INFVAR -0.225 -0.158 -0.045 0.033 0.105 FAIL 3.571
ARABLE -0.050 -0.030 -0.003 0.015 0.025 FAIL 50.570
ARABLEpw -0.093 -0.076 -0.027 0.014 0.022 FAIL 11.230
LAND 0.000 0.000 0.000 0.000 0.000 FAIL 60.468
LANDpc 0.000 0.000 0.000 0.000 0.000 FAIL 31.876
EPCpc 0.000 0.000 0.000 0.000 0.000 FAIL 81.988
FDI -0.159 -0.120 0.023 0.134 0.180 FAIL 76.193
KSI -11.977 -9.997 -3.459 0.358 2.608 FAIL 0.105
BCIDIFF -0.062 -0.037 0.013 0.061 0.085 FAIL 69.551
# Using bma_list objest we can calculate jointness measures for the regressors.
jointness(bma_list)[1:10,1:10]
B LNDGEO L LNRGDPPROD RGDPpcDIFF GOV HUMAN CPW INFVAR ARABLE
B NA 0.635 -0.561 0.635 -0.529 0.622 -0.545 -0.363 0.556 -0.275
LNDGEO 0.593 NA -0.631 1.000 -0.853 0.969 -0.888 -0.705 0.861 -0.580
L -0.487 -0.599 NA -0.631 0.520 -0.613 0.543 0.409 -0.507 0.331
LNRGDPPROD 0.593 1.000 -0.599 NA -0.853 0.969 -0.888 -0.705 0.861 -0.580
RGDPpcDIFF -0.461 -0.833 0.491 -0.833 NA -0.855 0.747 0.588 -0.747 0.470
GOV 0.584 0.969 -0.581 0.969 -0.835 NA -0.857 -0.673 0.847 -0.549
HUMAN -0.478 -0.869 0.513 -0.869 0.719 -0.838 NA 0.609 -0.753 0.498
CPW -0.270 -0.665 0.389 -0.665 0.561 -0.632 0.577 NA -0.561 0.451
INFVAR 0.533 0.868 -0.479 0.868 -0.732 0.856 -0.739 -0.526 NA -0.458
ARABLE -0.174 -0.537 0.328 -0.537 0.451 -0.506 0.475 0.466 -0.417 NA
# Next we can examine which models are the best according to PMP calculated
# using unform model prior.
Best <- best_models(bma_list, best = 5, round = 3)
# We can look at which variables are present in the best models
# (PMPs and R^2 are in the two last rows).
Best[[3]]
#> | | 'No. 1' | 'No. 2' | 'No. 3' | 'No. 4' | 'No. 5' |
#> |:----------|:-------:|:-------:|:-------:|:-------:|:-------:|
#> |Const | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
#> |B | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
#> |LNDGEO | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
#> |L | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
#> |LNRGDPPROD | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
#> |RGDPpcDIFF | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
#> |GOV | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
#> |HUMAN | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
#> |CPW | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
#> |INFVAR | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
#> |ARABLE | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 |
#> |ARABLEpw | 1.000 | 1.000 | 1.000 | 1.000 | 0.000 |
#> |LAND | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
#> |LANDpc | 1.000 | 1.000 | 0.000 | 1.000 | 1.000 |
#> |EPCpc | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 |
#> |FDI | 1.000 | 1.000 | 1.000 | 1.000 | 0.000 |
#> |KSI | 0.000 | 1.000 | 1.000 | 0.000 | 0.000 |
#> |BCIDIFF | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
#> |PMP | 0.107 | 0.051 | 0.037 | 0.031 | 0.031 |
#> |R^2 | 0.927 | 0.928 | 0.927 | 0.927 | 0.923 |
# We can also see at the best models themselves.
Best[[4]]
#> | | 'No. 1' | 'No. 2' | 'No. 3' | 'No. 4' | 'No. 5' |
#> |:----------|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|
#> |Const | 8.075 (0.68)*** | 8.093 (0.675)*** | 8.169 (0.675)*** | 7.793 (0.694)*** | 8.109 (0.643)*** |
#> |B | 0.44 (0.136)*** | 0.407 (0.136)*** | 0.405 (0.136)*** | 0.461 (0.136)*** | 0.417 (0.139)*** |
#> |LNDGEO | -1.188 (0.064)*** | -1.144 (0.067)*** | -1.1 (0.066)*** | -1.175 (0.064)*** | -1.205 (0.065)*** |
#> |L | NA | NA | NA | NA | NA |
#> |LNRGDPPROD | 0.886 (0.019)*** | 0.883 (0.019)*** | 0.873 (0.019)*** | 0.89 (0.019)*** | 0.892 (0.018)*** |
#> |RGDPpcDIFF | NA | NA | NA | NA | NA |
#> |GOV | -4.313 (0.864)*** | -3.583 (0.927)*** | -3.461 (0.93)*** | -4.341 (0.86)*** | -3.906 (0.867)*** |
#> |HUMAN | NA | NA | NA | NA | NA |
#> |CPW | NA | NA | NA | NA | NA |
#> |INFVAR | -0.041 (0.01)*** | -0.033 (0.011)*** | -0.039 (0.011)*** | -0.041 (0.01)*** | -0.043 (0.01)*** |
#> |ARABLE | NA | NA | NA | 0.006 (0.003)** | NA |
#> |ARABLEpw | -0.009 (0.003)*** | -0.008 (0.003)*** | -0.01 (0.003)*** | -0.009 (0.003)*** | NA |
#> |LAND | NA | NA | NA | NA | NA |
#> |LANDpc | NA | NA | NA | NA | NA |
#> |EPCpc | NA | NA | NA | NA | NA |
#> |FDI | 0.028 (0.007)*** | 0.034 (0.008)*** | 0.031 (0.008)*** | 0.03 (0.008)*** | NA |
#> |KSI | NA | -0.867 (0.417)** | -1.116 (0.418)*** | NA | NA |
#> |BCIDIFF | NA | NA | NA | NA | NA |
#> |PMP | 0.107 | 0.051 | 0.037 | 0.031 | 0.031 |
#> |R^2 | 0.927 | 0.928 | 0.927 | 0.927 | 0.923 |
# rmsBMA package allows drawing graphs of prior and posterior model probabilities.
# Firsly, it can be done over the models sizes.
model_sizes(bma_list)
# Secondly, it can be done for the 20 best models.
# We can also use rmsBMA package to depict distribiution of coeffciients over
# the entire models space (on the condition of the inclusion
# of the variable in a model).
# First, we can use the histogram of the coeffcients on the distance between two countries. Let's
# first specify the number of bins to be 40 for each variable.
bin_sizes <- matrix(40, nrow = 18, ncol = 1)
Coef <- coef_hist(bma_list, BN = 1, num = bin_sizes)
Coef$LNDGEO
# Secondly, we can use the kernel density of the coeffcients on
# the GDP product of two countries.
Coef2<-coefHist(Post,kernel=1)
Coef2$LNRGDPPRO
The rmsBMA package can be used to perform standard Bayesian Model Averaging over the entire model space. To do so, the user can either omit the M argument in the model_space() function or set M equal to the total number of regressors.
# We use model_space and bma functions again.
modelSpace <- model_space(Trade_data, g = "UIP")
bma_list <- bma(modelSpace, round = 3)
# Again we can have a look at posterior statistics for uniform model prior.
bma_list[[1]]
PIP PM PSD PMcon PSDcon P(+) PSC
CONST NA 8.093 0.759 8.093 0.759 1.000 1.000
B 0.817 0.347 0.208 0.424 0.142 0.816 0.907
LNDGEO 1.000 -1.175 0.081 -1.175 0.081 0.000 1.000
L 0.185 0.063 0.161 0.343 0.210 0.175 0.583
LNRGDPPROD 1.000 0.887 0.021 0.887 0.021 1.000 1.000
RGDPpcDIFF 0.074 -0.007 0.064 -0.094 0.220 0.026 0.511
GOV 0.984 -4.011 1.197 -4.074 1.093 0.000 0.992
HUMAN 0.056 -0.004 0.050 -0.068 0.199 0.021 0.508
CPW 0.147 0.000 0.000 0.000 0.000 0.136 0.562
INFVAR 0.931 -0.038 0.015 -0.040 0.012 0.000 0.965
ARABLE 0.210 0.001 0.003 0.006 0.003 0.202 0.597
ARABLEpw 0.702 -0.006 0.005 -0.009 0.003 0.003 0.847
LAND 0.060 0.000 0.000 0.000 0.000 0.035 0.505
LANDpc 0.749 0.000 0.000 0.000 0.000 0.747 0.872
EPCpc 0.290 0.000 0.000 0.000 0.000 0.279 0.634
FDI 0.829 0.023 0.014 0.028 0.010 0.827 0.913
KSI 0.401 -0.414 0.590 -1.033 0.479 0.006 0.694
BCIDIFF 0.062 0.000 0.001 0.000 0.003 0.024 0.507
# We might compare those results with the one obtained using Extreme Bounds Analysis.
bma_list[[3]]
Variable Lower bound Minimum Mean Maximum Upper bound EBA test %(+)
CONST -4.966 -3.240 15.249 33.075 35.147 FAIL 87.100
B -0.073 0.235 1.217 2.720 3.440 FAIL 100.000
LNDGEO -1.874 -1.563 -1.182 -0.743 -0.357 PASS 0.000
L -0.990 0.066 1.039 3.275 4.374 FAIL 100.000
LNRGDPPROD 0.792 0.844 0.894 0.992 1.062 PASS 100.000
RGDPpcDIFF -4.621 -3.605 -0.707 0.586 1.488 FAIL 11.768
GOV -26.535 -20.032 -6.656 0.223 3.948 FAIL 0.029
HUMAN -2.368 -1.133 -0.208 0.885 1.996 FAIL 18.134
CPW 0.000 0.000 0.000 0.000 0.000 FAIL 71.019
INFVAR -0.225 -0.158 -0.045 0.033 0.105 FAIL 3.571
ARABLE -0.050 -0.030 -0.003 0.015 0.025 FAIL 50.581
ARABLEpw -0.093 -0.076 -0.027 0.014 0.022 FAIL 11.227
LAND 0.000 0.000 0.000 0.000 0.000 FAIL 60.474
LANDpc 0.000 0.000 0.000 0.000 0.000 FAIL 31.891
EPCpc 0.000 0.000 0.000 0.000 0.000 FAIL 81.993
FDI -0.159 -0.120 0.023 0.134 0.180 FAIL 76.199
KSI -11.977 -9.997 -3.458 0.358 2.608 FAIL 0.105
BCIDIFF -0.062 -0.037 0.013 0.061 0.085 FAIL 69.539
# In the case of M = K, we have really informative graphs depicting prior and
# posterior model probabilities over the model sizes.
model_sizes(bma_list)
# In this case it is also very instructive to have a look a prior and posterior
# probabilities of the best 200 models.
model_pmp(bma_list, top = 200)
# Lets now have a look a the histogram of the coefficients of the product of GDP of
# the trading countries (LNRGDPPROD).
bin_sizes <- matrix(80, nrow = 18, ncol = 1)
Coef <- coef_hist(bma_list, BN = 1, num = bin_sizes)
Coef$LNDGEO
# Now we use the kernel density of the coeffcients on the distance
# between two countries (LNDGEO).
Coef2 <- coef_hist(bma_list, kernel = 1)
Coef2$LNDGEO
# Finally, lets have a look at the 5 best models according to PMPs based
# on uniform model prior.
Best <- best_models(bma_list, best = 5, round = 3)
Best[[4]]
#> | | 'No. 1' | 'No. 2' | 'No. 3' | 'No. 4' | 'No. 5' |
#> |:----------|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|
#> |Const | 8.075 (0.68)*** | 8.093 (0.675)*** | 8.169 (0.675)*** | 7.793 (0.694)*** | 8.109 (0.643)*** |
#> |B | 0.44 (0.136)*** | 0.407 (0.136)*** | 0.405 (0.136)*** | 0.461 (0.136)*** | 0.417 (0.139)*** |
#> |LNDGEO | -1.188 (0.064)*** | -1.144 (0.067)*** | -1.1 (0.066)*** | -1.175 (0.064)*** | -1.205 (0.065)*** |
#> |L | NA | NA | NA | NA | NA |
#> |LNRGDPPROD | 0.886 (0.019)*** | 0.883 (0.019)*** | 0.873 (0.019)*** | 0.89 (0.019)*** | 0.892 (0.018)*** |
#> |RGDPpcDIFF | NA | NA | NA | NA | NA |
#> |GOV | -4.313 (0.864)*** | -3.583 (0.927)*** | -3.461 (0.93)*** | -4.341 (0.86)*** | -3.906 (0.867)*** |
#> |HUMAN | NA | NA | NA | NA | NA |
#> |CPW | NA | NA | NA | NA | NA |
#> |INFVAR | -0.041 (0.01)*** | -0.033 (0.011)*** | -0.039 (0.011)*** | -0.041 (0.01)*** | -0.043 (0.01)*** |
#> |ARABLE | NA | NA | NA | 0.006 (0.003)** | NA |
#> |ARABLEpw | -0.009 (0.003)*** | -0.008 (0.003)*** | -0.01 (0.003)*** | -0.009 (0.003)*** | NA |
#> |LAND | NA | NA | NA | NA | NA |
#> |LANDpc | NA | NA | NA | NA | NA |
#> |EPCpc | NA | NA | NA | NA | NA |
#> |FDI | 0.028 (0.007)*** | 0.034 (0.008)*** | 0.031 (0.008)*** | 0.03 (0.008)*** | NA |
#> |KSI | NA | -0.867 (0.417)** | -1.116 (0.418)*** | NA | NA |
#> |BCIDIFF | NA | NA | NA | NA | NA |
#> |PMP | 0.107 | 0.051 | 0.037 | 0.031 | 0.031 |
#> |R^2 | 0.927 | 0.928 | 0.927 | 0.927 | 0.923 |
Enumerating the model space means fitting $2^K$ models, which is 131,072
for the seventeen regressors in Trade_data and quickly becomes
impossible as $K$ grows. Setting mc3 = TRUE explores the space by MC^3
sampling instead, which visits models in proportion to their posterior
probability and so never spends time on the ones that carry no mass.
set.seed(1)
mc3Space <- model_space(Trade_data, mc3 = TRUE, draws = 50000, burn = 25000, g = "UIP")
mc3Space
#> model space
#> regressors : 17
#> maximum size : 17 (full model space)
#> models : 1,512 visited by MC3, of 131,072
#> g prior : UIP (g = 0.003077)
#> covariance : conventional
#> chain : 50000 draws after 25000 burn-in
#> acceptance rate : 0.213
#> convergence : cor(analytic PMP, visit frequency) = 0.9942
The chain found the 1,512 models that matter out of 131,072, in about four seconds against roughly thirty for enumeration. The last line is the convergence check: it compares the posterior mass each visited model earns analytically with how often the chain actually visited it, and the two agree only once the chain has settled.
The result is passed to bma() in the usual way. Note that Extreme
Bounds Analysis is not available for a sampled model space, because a
sampler does not visit the extremes of a distribution it is exploring by
posterior mass; element 3 of the returned object is NULL.
bma_mc3 <- bma(mc3Space, EMS = 5, round = 3)
summary(bma_mc3)
#> Bayesian model averaging -- binomial model prior
#> regressors: 17 models: 1,512 (MC3)
#> prior model size: 5 posterior model size: 6.94
#> chain: acceptance 0.213 cor(analytic PMP, visit frequency) 0.9942
#> extreme bounds analysis is not available for an MC3 model space
#>
#> regressors ordered by posterior inclusion probability
#> PIP PM PSD PMcon PSDcon P(+) PSC
#> LNDGEO 1.000 -1.211 0.084 -1.211 0.084 0.000 1.000
#> LNRGDPPROD 1.000 0.892 0.020 0.892 0.020 1.000 1.000
#> GOV 0.989 -4.019 1.078 -4.064 0.996 0.000 0.994
#> INFVAR 0.919 -0.039 0.016 -0.042 0.012 0.000 0.959
#> LANDpc 0.774 0.000 0.000 0.000 0.000 0.773 0.886
#> B 0.647 0.272 0.231 0.421 0.141 0.646 0.822
#> FDI 0.493 0.012 0.014 0.025 0.011 0.491 0.744
#> ...
Reduced model spaces can be sampled as well: M behaves as it does
under enumeration, and the sampler applies the correction that the size
constraint requires.
model_space and bma objects have print, summary and coef
methods. coef() returns the posterior means as a named vector, and
takes prior = "random" for the binomial-beta model prior or
conditional = TRUE for means conditional on inclusion.
round(coef(bma_mc3), 3)
#> CONST B LNDGEO L LNRGDPPROD RGDPpcDIFF GOV
#> 8.216 0.272 -1.211 0.060 0.892 -0.003 -4.019
#> HUMAN CPW INFVAR ARABLE ARABLEpw LAND LANDpc
#> -0.001 0.000 -0.039 0.000 -0.003 0.000 0.000
#> EPCpc FDI KSI BCIDIFF
#> 0.000 0.012 -0.231 0.000
model_sizes() and model_pmp() take type = "histogram" to draw the
prior and posterior as side-by-side bars rather than lines.