# calc.relimp: Function to calculate relative importance metrics for linear... In relaimpo: Relative Importance of Regressors in Linear Models

## Description

calc.relimp calculates several relative importance metrics for the linear model. The recommended metrics are `lmg` (R^2 partitioned by averaging over orders, like in Lindemann, Merenda and Gold (1980, p.119ff)) and `pmvd` (a newly proposed metric by Feldman (2005) that is provided in the non-US version of the package only). For completeness and comparison purposes, several other metrics are also on offer (cf. e.g. Darlington (1968)).

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## generic function calc.relimp(object, ...) ## default S3 method ## Default S3 method: calc.relimp(object, x = NULL, ..., type = "lmg", diff = FALSE, rank = TRUE, rela = FALSE, always = NULL, groups = NULL, groupnames = NULL, weights=NULL, design=NULL) ## S3 method for formula object ## S3 method for class 'formula' calc.relimp(formula, data, weights, na.action, ..., subset=NULL) ## S3 method for objects of class lm ## S3 method for class 'lm' calc.relimp(object, type = "lmg", groups = NULL, groupnames=NULL, always = NULL, ...) ```

## Arguments

 `object ` The class of this object determines which of the methods is used: There are special methods for output objects from function `lm` (or linear model objects inheriting from class lm generated by other functions like `glm` and `svyglm`) and for formula objects. For all other types of object, the default method is used. Thus, object can be a formula (e.g. y\~x1+x2+x3+x2:x3) (cf. below for details) OR the output of a linear model call (inheriting from class `lm`, but not `mlm`); output objects from `lm`, `glm`, `svyglm` or `aov` work (if linear with identity link in case of glm's); there may be further functions that output objects inheriting from `lm` which may or may not work reasonably with `calc.relimp`; for `calc.relimp` to be appropriate, the underlying model must at least be linear! The restrictions on usage of interactions listed under item formula below also apply to linear model objects. OR the covariance matrix of a response y and regressors x, (e.g. obtained by cov(cbind(y,x)), if y is a column vector of response values and x a corresponding matrix of regressors) OR a (raw) data matrix or data frame with the response variable in the first column OR a response vector or one-column matrix, if `x` contains the corresponding matrix or data frame of regressors. `formula ` The first object, if a formula is to be given; one response only. Interaction terms are currently limited to second-order. Note: If several interaction terms are given, calculations may be very resource intensive, if these are all connected (e.g. with A:B, B:C, C:D, all A,B,C,D are connected, while with A:B, C:D, D:E there are separate groups A,B and C,D,E). Interaction terms occurring in always do not increase resource usage (but are only permitted if the respective main effects also occur in always). Interactions and groups currently cannot be used simultaneously. `x ` a (raw) data matrix or data frame containing the regressors, if `object` is a response vector or one-column matrix OR NULL, if `object` is anything else `type ` can be a character string, character vector or list of character strings. It is the collection of metrics that are to be calculated. Available metrics: `lmg`, `pmvd` (non-US version only), `last`, `first`, `betasq`, `pratt`, `genizi` and `car`. For brief sketches of their meaning cf. details section. `diff ` logical; if TRUE, pairwise differences between the relative contributions are calculated; default FALSE `rank ` logical; if TRUE, ranks of regressors in terms of relative contributions are calculated; default TRUE `rela ` is a logical requesting relative importances summing to 100% (`rela=TRUE`). If rela is FALSE (default), some of the metrics sum to R^2 (`lmg`, `pmvd`, `pratt`), others do not have a meaningful sum (`last`, `first`, `betasq`). `always ` is a vector of column numbers or names of variables to be always in the model (adjusted for). Valid numbers are 2 to (number of regressors + 1) (1 is reserved for the response), valid character strings are all column names of `object` or `x` respectively that refer to regressor variables. Numbers and names cannot be mixed. Relative importance is only assessed for the variables not selected in `always`. This option currently does not work for metrics `genizi` and `car`. `groups ` is a list of vectors of column numbers or names of variables to be combined into groups. If only one group is needed, a vector can be given. The numbers and character strings needed are of the same form as for `always`. Relative importance is only allocated between groups of regressors, no subdivision within groups is calculated. Regressors that do not occur in any group are included as singletons. A regressor must not occur in `always` and in `groups`. Also, groups cannot be used with a linear model or a formula in case of higher order effects (interactions). Finally, `groups` only works with the four metrics `lmg`, `pmvd`, `last` and `first`. `groupnames ` is a vector of names for the variable groups to be used for annotation of output. `weights ` is a vector of case weights for the observations in the data frame (or matrix). You can EITHER specify `weights` OR a `design`. Note that weights must not be specified for linear model objects (since these should contain their weights as part of the model). `design ` is a design object of class `survey.design` (cf. package survey). You can EITHER specify a `design` OR `weights`. For `calc.relimp`, the design is used for calculating weights only. Note that it is discouraged (though possible) to specify a design for a conventional linear model object (since a survey-specific linear model should be used for survey data, cf. function `svyglm`). Also note that care is needed when using `subset` together with `design`: the `subset`-Option only treats the `data` handed directly to `calc.relimp`, the `design` has to be equivalently treated beforehand. `data` if first object is of class formula: an optional matrix or data frame that the variables in formula and subset come from; if it is omitted, all names must be meaningful in the environment from which calc.relimp is called `subset` if first object is of class formula: an optional expression indicating the subset of the observations of `data` that should be used in the fit. This can be a logical vector, or a numeric vector indicating which observation numbers are to be included, or a character vector of the row names to be included. All (non-missing) observations are included by default. `na.action` if first object is of class formula: an optional function that indicates what should happen when the data contain 'NA's. The default is first, any na.action attribute of data, second the setting given in the call to calc.relimp, third the na.action setting of options. Possible choices are "na.fail", (print an error message and terminate if there are any incomplete observations), "na.omit" or "na.exclude" (equivalent for package `relaimpo`, both analyse complete cases only and print a warning, this is also what is done the default method ). `...` usable for further arguments, particularly most arguments of default method can be given to all other methods (exception: weights and design cannot be given to lm-method)

## Details

lmg

is the R^2 contribution averaged over orderings among regressors, cf. e.g. Lindeman, Merenda and Gold 1980, p.119ff or Chevan and Sutherland (1991).

pmvd

is the proportional marginal variance decomposition as proposed by Feldman (2005) (non-US version only). It can be interpreted as a weighted average over orderings among regressors, with data-dependent weights.

last

is each variables contribution when included last, also sometimes called usefulness.

first

is each variables contribution when included first, which is just the squared covariance between y and the variable.

betasq

is the squared standardized coefficient.

pratt

is the product of the standardized coefficient and the correlation.

genizi

is the R^2 decomposition according to Genizi 1993

car

is the R^2 decomposition according to Zuber and Strimmer 2010, also available from package care (squares of scores produced by function `carscore`

Each metric is calculated using the internal function “metric”`calc`, e.g. `lmgcalc`.

Five of the metrics in `calc.relimp` (`lmg`, `pmvd`, `pratt`, `genizi` and `car`), decompose the model R^2. `calc.relimp` (`lmg`, `pmvd`, `pratt`, `genizi` and `car`) sum to the R^2 that is to be decomposed, if `rela = FALSE` and to 100pct if `rela = TRUE`.

The other metrics also (artificially) sum to 100pct if `rela = TRUE`. If `rela = FALSE`, they are given relative to var(y) (or the conditional variance of y after adjusting out the variables requested in `always`) but do not sum to R^2.

If `always` requests some variables to be always in the model, these are conditioned upon (i.e. included into the model first). Only the remaining R^2 that is not explained by these variables is decomposed among the other regressors. This currently does not work for metrics `genizi` and `car`.

Four of the metrics, `lmg`, `pmvd`, `first` and `last`, are related to the order in which the variables are included into the model. For these it is possible to consider the variables in groups that are always entered into the model together.

Note that relaimpo can only provide metric `lmg` for models with interactions (2-way interactions only). It averages only over those orders, for which the interactions enter the model after both their main effects.

Note that there are different types of weights, weights indicating the variability of the response (observations with a more variable responses receive a lower weight than those with a less variable response, like in the Aitken estimator), frequency weights indicating the number of observations with exactly the observed data pattern of the current observation, or weights indicating the number of population units represented by the current observation (inverse sampling probability, weights typically used in survey situations). All three types of weight alike can be handed to function `calc.relimp` using the `weights=` option. Note, however, that they have to be treated differently for bootstrapping (cf. `boot.relimp`).

Data from complex surveys can be treated by providing a survey design with `design=`-option. For `calc.relimp`, it is also sufficient to provide the weights derived from the design using the `weights=`-option.

`calc.relimp` cannot handle data with missing values directly. It applies complete-case analysis, i.e. drops all units with any missing values by default. While this can be appropriate, if there are only few missing values, data with more severe missingness issues need special treatment. Package relaimpo offers the function `mianalyze.relimp` that handles multiply-imputed datasets (that can be created by several other R-packages). Currently, possibilities in this function are limited due to the fact that it uses complex survey designs and bootstrapping which do not (yet) go together well with factors, interactions and calculated quantities in formulae.

## Value

 `var.y ` the variance of the response `R2 ` the coefficient of determination, R^2 `R2.decomp ` the part of the coefficient of determination that is decomposed among the variables under investigation `lmg ` vector of relative contributions obtained from the `lmg` method, if `lmg` has been requested in `type` `lmg.diff ` vector of pairwise differences between relative contributions obtained from the `lmg` method, if `lmg` has been requested in `type` and `diff=TRUE` `lmg.rank ` rank of the regressors relative contributions obtained from the `lmg` method, if `lmg` has been requested in `type` and `rank=TRUE` `metric, metric.diff, metric.rank ` analogous to `lmg` for other metrics `ave.coeffs` average coefficients for variables not not requested by always only for models of different sizes; note that coefficients refer to modeling residuals after adjusting out variables listed in always (both from response and other explanatory variables) `namen` names of variables, starting with response `type` character vector of metrics available `rela` Have metrics been normalized to sum 100% ? `always` column numbers of variables always in the model; in case of factors, the column numbers given here are not identical to those in the call to `calc.relimp`, but refer to the columns of the model matrix `alwaysnam` names of variables always in the model `call` contains the call that generated the object

## Warning

`lmg` and `pmvd` are computer-intensive. Although they are calculated based on the covariance matrix, which saves substantial computing time in comparison to carrying out actual regressions, these methods still take quite long for problems with many regressors.

`relaimpo` is a package for univariate linear models. Using `relaimpo` on objects that inherit from class `lm` but are not univariate linear model objects may produce nonsensical results without warning. Objects of class `mlm` or `glm` with link functions other than identity or family other than gaussian lead to an error message.

## Note

There are two versions of this package. The version on CRAN is globally licensed under GPL version 2 (or later). There is an extended version with the interesting additional metric `pmvd` that is licensed according to GPL version 2 under the geographical restriction "outside of the US" because of potential issues with US patent 6,640,204. This version can be obtained from Ulrike Groempings website (cf. references section). Whenever you load the package, a display tells you, which version you are loading.

## Author(s)

Ulrike Groemping, BHT Berlin

## References

Chevan, A. and Sutherland, M. (1991) Hierarchical Partitioning. The American Statistician 45, 90–96.

Darlington, R.B. (1968) Multiple regression in psychological research and practice. Psychological Bulletin 69, 161–182.

Feldman, B. (2005) Relative Importance and Value. Manuscript (Version 1.1, March 19 2005), downloadable at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2255827

Genizi, A. (1993) Decomposition of R2 in multiple regression with correlated regressors. Statistica Sinica 3, 407–420. Downloadable at http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=3&num=2&art=10

Groemping, U. (2006) Relative Importance for Linear Regression in R: The Package relaimpo Journal of Statistical Software 17, Issue 1. Downloadable at http://www.jstatsoft.org/v17/i01

Lindeman, R.H., Merenda, P.F. and Gold, R.Z. (1980) Introduction to Bivariate and Multivariate Analysis, Glenview IL: Scott, Foresman.

Zuber, V. and Strimmer, K. (2010) Variable importance and model selection by decorrelation. Preprint, downloadable at http://arxiv.org/abs/1007.5516

Go to http://prof.beuth-hochschule.de/groemping/relaimpo/ for further information and references.

relaimpo, `booteval.relimp`, `mianalyze.relimp`, `classesmethods.relaimpo`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53``` ```##################################################################### ### Example: relative importance of various socioeconomic indicators ### for Fertility in Switzerland ### Fertility is first column of data set swiss ##################################################################### data(swiss) calc.relimp(swiss, type = c("lmg", "last", "first", "betasq", "pratt", "genizi", "car") ) # calculation of all available relative importance metrics # non-US version offers the additional metric "pmvd", # i.e. call would be # calc.relimp(cov(swiss), # type = c("lmg", "pmvd", "last", "first", "betasq, "pratt"), # rela = TRUE ) ## same analysis with formula or lm method and a few modified options crf <- calc.relimp(Fertility~Agriculture+Examination+Education+Catholic+Infant.Mortality,swiss, subset = Catholic>40, type = c("lmg", "last", "first", "betasq", "pratt"), rela = TRUE ) crf linmod <- lm(Fertility~Agriculture+Examination+Education+Catholic+Infant.Mortality,swiss) crlm <- calc.relimp(linmod, type = c("lmg", "last", "first", "betasq", "pratt", "genizi", "car"), rela = TRUE ) plot(crlm) # bar plot of the relative importance metrics #of statistical interest in this context: correlation matrix cor(swiss) #demonstration of conditioning on one regressor using always calc.relimp(swiss, type = c("lmg", "last", "first", "betasq", "pratt"), rela = FALSE, always = "Education" ) # using calc.relimp with grouping of two regressors # and weights (not reasonable here, purely for demo purposes) calc.relimp(swiss, type = c("lmg", "last", "first"), rela = FALSE, groups = c("Education","Examination"), weights = abs(-23:23) ) # using calc.relimp with grouping of two regressors # and a design object (not reasonable here, purely for demo purposes) des <- svydesign(~1, data=swiss, weights=~abs(-23:23)) calc.relimp(swiss, type = c("lmg", "last", "first"), rela = FALSE, groups = c("Education","Examination"), groupnames ="EduExam", design = des) # calc.relimp with factors (betasq and pratt not possible) # (calc.relimp would not be necessary here, # because the experiment is balanced) calc.relimp(1/time~poison+treat,data=poisons, rela = FALSE, type = c("lmg", "last", "first")) # including also the interaction (lmg possible only) calc.relimp(1/time~poison*treat,data=poisons, rela = FALSE) ```

### Example output

```Loading required package: MASS

Attaching package: 'survival'

The following object is masked from 'package:boot':

aml

Attaching package: 'survey'

The following object is masked from 'package:graphics':

dotchart

This is the global version of package relaimpo.

If you are a non-US user, a version with the interesting additional metric pmvd is available

from Ulrike Groempings web site at prof.beuth-hochschule.de/groemping.

Response variable: Fertility
Total response variance: 156.0425
Analysis based on 47 observations

5 Regressors:
Agriculture Examination Education Catholic Infant.Mortality
Proportion of variance explained by model: 70.67%
Metrics are not normalized (rela=FALSE).

Relative importance metrics:

lmg        last     first     betasq      pratt
Agriculture      0.05709122 0.042869607 0.1246649 0.09791973 -0.1104860
Examination      0.17117303 0.007387419 0.4171645 0.02715186  0.1064274
Education        0.26013468 0.161962693 0.4406156 0.44943721  0.4450046
Catholic         0.10557015 0.062372626 0.2150035 0.12082578  0.1611768
Infant.Mortality 0.11276592 0.056945259 0.1735189 0.06306928  0.1046122
genizi          car
Agriculture      0.0484169 0.0005674742
Examination      0.1547082 0.1511339908
Education        0.2719641 0.3232680710
Catholic         0.1146800 0.1150731842
Infant.Mortality 0.1169657 0.1166922814

Average coefficients for different model sizes:

1X         2Xs         3Xs        4Xs        5Xs
Agriculture       0.1942017  0.03949369 -0.06794018 -0.1380370 -0.1721140
Examination      -1.0113173 -0.89693064 -0.72467898 -0.5056072 -0.2580082
Education        -0.8623503 -0.77680153 -0.77395379 -0.8164207 -0.8709401
Catholic          0.1388857  0.09709583  0.08377687  0.0903765  0.1041153
Infant.Mortality  1.7864860  1.59438387  1.46135692  1.2717960  1.0770481
Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Response variable: Fertility
Total response variance: 280.7706
Analysis based on 19 observations

5 Regressors:
Agriculture Examination Education Catholic Infant.Mortality
Proportion of variance explained by model: 80.93%
Metrics are normalized to sum to 100% (rela=TRUE).

Relative importance metrics:

lmg        last      first      betasq       pratt
Agriculture      0.11486583 0.353055869 0.11306600 0.136299551 -0.19101959
Examination      0.17190060 0.001476325 0.20205545 0.001777293  0.02915946
Education        0.30508034 0.074753723 0.31131441 0.301904369  0.47173598
Catholic         0.31494322 0.475491780 0.31290701 0.533989734  0.62898283
Infant.Mortality 0.09321001 0.095222303 0.06065714 0.026029053  0.06114132

Average coefficients for different model sizes:

1X          2Xs         3Xs          4Xs         5Xs
Agriculture       0.3642223 -0.006783255 -0.16636929 -0.229599451 -0.21989767
Examination      -1.4207166 -0.540061683 -0.05947462  0.008361967 -0.07326973
Education        -1.1118076 -1.171641212 -1.14221086 -0.901857999 -0.60205695
Catholic          0.7923300  0.745030377  0.62073008  0.590074848  0.56916445
Infant.Mortality  1.9642435  1.940427202  1.37998276  0.891605693  0.70754883
Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Fertility Agriculture Examination   Education   Catholic
Fertility         1.0000000  0.35307918  -0.6458827 -0.66378886  0.4636847
Agriculture       0.3530792  1.00000000  -0.6865422 -0.63952252  0.4010951
Examination      -0.6458827 -0.68654221   1.0000000  0.69841530 -0.5727418
Education        -0.6637889 -0.63952252   0.6984153  1.00000000 -0.1538589
Catholic          0.4636847  0.40109505  -0.5727418 -0.15385892  1.0000000
Infant.Mortality  0.4165560 -0.06085861  -0.1140216 -0.09932185  0.1754959
Infant.Mortality
Fertility              0.41655603
Agriculture           -0.06085861
Examination           -0.11402160
Education             -0.09932185
Catholic               0.17549591
Infant.Mortality       1.00000000
Response variable: Fertility
Total response variance: 156.0425
Analysis based on 47 observations

5 Regressors:
Proportion of variance explained: 70.67%

One Regressor always included in model:
Education
44.06 % of variance explained by this regressor

Relative importance of 4 regressors assessed:
Agriculture Examination Catholic Infant.Mortality
26.61 % of variance decomposed among these

Metrics are not normalized (rela=FALSE).

Relative importance metrics:

lmg        last       first     betasq      pratt
Agriculture      0.03432758 0.042869607 0.008632775 0.05787163 0.02235157
Examination      0.03884696 0.007387419 0.064868872 0.01390762 0.03003617
Catholic         0.10165854 0.062372626 0.133891476 0.11796552 0.12567648
Infant.Mortality 0.09128627 0.056945259 0.124164362 0.06244711 0.08805513
Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Response variable: Fertility
Total response variance: 215.6661
Analysis based on 47 observations

5 Regressors:
Some regressors combined in groups:
Group  G1 : Examination Education

Relative importance of 4 (groups of) regressors assessed:
G1 Agriculture Catholic Infant.Mortality

Proportion of variance explained by model: 74.01%
Metrics are not normalized (rela=FALSE).

Relative importance metrics:

lmg       last     first
G1               0.49953751 0.39224182 0.6307919
Agriculture      0.07636859 0.02153528 0.1497720
Catholic         0.05614286 0.01776434 0.1363082
Infant.Mortality 0.10801371 0.03827817 0.1690776

Average coefficients for different model sizes:

1group     2groups     3groups     4groups
Agriculture       0.2426390  0.09581090 -0.02102877 -0.13783809
Examination      -0.3580310 -0.32182157 -0.28867678 -0.25727647
Education        -0.8201332 -0.85553230 -0.89525538 -0.93781515
Catholic          0.1333066  0.08962258  0.06341285  0.07193345
Infant.Mortality  2.2787886  1.98814590  1.73745564  1.19503671
Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Response variable: Fertility
Total response variance: 215.6661
Analysis based on 47 observations

5 Regressors:
Some regressors combined in groups:
Group  EduExam : Examination Education

Relative importance of 4 (groups of) regressors assessed:
EduExam Agriculture Catholic Infant.Mortality

Proportion of variance explained by model: 74.01%
Metrics are not normalized (rela=FALSE).

Relative importance metrics:

lmg       last     first
EduExam          0.49953751 0.39224182 0.6307919
Agriculture      0.07636859 0.02153528 0.1497720
Catholic         0.05614286 0.01776434 0.1363082
Infant.Mortality 0.10801371 0.03827817 0.1690776

Average coefficients for different model sizes:

1group     2groups     3groups     4groups
Agriculture       0.2426390  0.09581090 -0.02102877 -0.13783809
Examination      -0.3580310 -0.32182157 -0.28867678 -0.25727647
Education        -0.8201332 -0.85553230 -0.89525538 -0.93781515
Catholic          0.1333066  0.08962258  0.06341285  0.07193345
Infant.Mortality  2.2787886  1.98814590  1.73745564  1.19503671
Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Response variable: 1/time
Total response variance: 1.393729
Analysis based on 48 observations

5 Regressors:
Some regressors combined in groups:
Group  poison : poison2 poison3
Group  treat : treatB treatC treatD

Relative importance of 2 (groups of) regressors assessed:
poison treat

Proportion of variance explained by model: 84.41%
Metrics are not normalized (rela=FALSE).

Relative importance metrics:

lmg      last     first
poison 0.5324323 0.5324323 0.5324323
treat  0.3116435 0.3116435 0.3116435

Average coefficients for different model sizes:

1group    2groups
poison2  0.4686413  0.4686413
poison3  1.9964249  1.9964249
treatB  -1.6574024 -1.6574024
treatC  -0.5721354 -0.5721354
treatD  -1.3583383 -1.3583383
Warning message:
In rev(variances[[p]]) - variances[[p + 1]] :
Recycling array of length 1 in vector-array arithmetic is deprecated.

Response variable: 1/time
Total response variance: 1.393729
Analysis based on 48 observations

11 Regressors:
Some regressors combined in groups:
Group  poison : poison2 poison3
Group  treat : treatB treatC treatD
Group  poison:treat : poison2:treatB poison3:treatB poison2:treatC poison3:treatC poison2:treatD poison3:treatD

Relative importance of 3 (groups of) regressors assessed:
poison treat poison:treat

Proportion of variance explained by model: 86.81%
Metrics are not normalized (rela=FALSE).

Relative importance metrics:

lmg
poison       0.53243232
treat        0.31164349
poison:treat 0.02397933

Average coefficients for different model sizes:

1group    2groups     3groups
poison2         0.4686413  0.4686413  0.78158915
poison3         1.9964249  1.9964249  2.31580446
treatB         -1.6574024 -1.6574024 -1.32341687
treatC         -0.5721354 -0.5721354 -0.62415711
treatD         -1.3583383 -1.3583383 -0.79719886
poison2:treatB        NaN        NaN -0.55166088
poison3:treatB        NaN        NaN -0.45029566
poison2:treatC        NaN        NaN  0.06960632
poison3:treatC        NaN        NaN  0.08645870
poison2:treatD        NaN        NaN -0.76973705
poison3:treatD        NaN        NaN -0.91368123
```

relaimpo documentation built on March 18, 2018, 2:10 p.m.