# Model averaging (and model selection) after multiple imputation

### Description

Performs model selection/averaging on multiply imputed data and combines the resulting estimates. The package also provides access to less frequently used model averaging techniques and offers integrated bootstrap estimation.

### Usage

1 2 3 4 5 | ```
mami(X, method = c("criterion.average", "criterion.selection", "MMA", "LASSO/LAE", "WAML"), criterion = c("AIC", "BIC", "BIC+", "CV"),
B = 20, X.org = NULL, inference = c("standard", "+bootstrapping"), missing.data = c("imputed", "none", "CC"), var.remove = NULL,
user.weights = NULL, candidate.models = c("all", "restricted", "very restricted"), model.family = c("gaussian", "binomial", "poisson",
"coxph"), add.factor = NULL, add.interaction = NULL, add.transformation = NULL, ycol = 1, CI = 0.95, kfold = 5, id = NULL,
use.stratum = NULL, report.exp = FALSE, print.time = FALSE, print.warnings = TRUE, ...)
``` |

### Arguments

`X` |
Either a list of multiply imputed datasets (each of them a dataframe), or an object of class ‘amelia’ created by Amelia II, or a single dataframe. |

`method` |
A character string specifying the model selection or model averaging technique: |

`criterion` |
A character string specifying the model selection criterion used for criterion-based model selection/averaging; currently either |

`B` |
An integer indicating the number of bootstrap replications to use (when |

`X.org` |
A dataframe consisting of the original unimputed data (which needs to be specified when |

`inference` |
A character string, either |

`missing.data` |
A character string, typically |

`var.remove` |
Either a vector of character strings or integers, specifying the variables or columns which are part of the data but not to be considered in the model selection/averaging procedure. |

`user.weights` |
A weight vector that is relevant to the analysis model. |

`candidate.models` |
A character string specifying whether for criterion based model selection/averaging all possible candidate models should be considered ( |

`model.family` |
A character string specifying the model family, either |

`add.factor` |
Either a vector of character strings or integers, specifying the variables which should be treated as categorical/factors in the analysis. Variables which are already defined to be factors in the data are detected automatically and do not necessarily need to be specified with this option. |

`add.interaction` |
A list of vectors of either character strings or integers, specifying the variables which should be added as interactions in the analysis model. |

`add.transformation` |
A vector of character strings, specifying transformations of variables which should be added to the analysis models. |

`ycol` |
A vector or integer specifying the variable(s) or column(s) which should be treated as outcome variable. |

`CI` |
A value greater than 0 and less than 1 specifying the confidence of the confidence interval. |

`kfold` |
An integer specifying |

`id` |
A character vector or integer specifying the variable or column to be used for a random intercept in the analysis model. |

`use.stratum` |
A character vector or integer specifying the variable used as a stratum in Cox regression analysis. |

`report.exp` |
A logical value specifying whether exponentiated coefficients should be reported or not. |

`print.time` |
A logical value specifying whether analysis time and anticipated estimation time for bootstrap estimation should be printed. |

`print.warnings` |
A logical value specifying whether warnings and any other output from the function should be printed or not. |

`...` |
Further arguments to be passed, i.e. for functions |

### Details

Model selection/averaging will be performed on each imputed dataset. The results will be combined according to formulae (7)-(10) in
Schomaker and Heumann (CSDA, 2014), see *References* below for more details. If `inference="+bootstrapping"`

is chosen, then the procedure
described in Table 1 will be performed in addition to standard MI inference. For longitudinal data (specified via `id`

) the bootstrap is
based on the subject/person/id level. To obtain insightful results from bootstrap estimation `B`

should be large, at least *B>200*
and `plot.mami`

may be used.

Note that a variable will be formally selected if it is selected (by means of either model selection or averaging) in at least one imputed set of data, but its overall impact will depend on how often it is chosen. As a result, effects of variables which are not supported throughout imputed datasets and candidate models will simply be less pronounced. Variable importance measures based on model averaging weights are calculated for each imputed dataset and will then be averaged.

If `method="criterion.average"`

is chosen and the number of variables is large, then computation time might be a burden and obtaining
results can even become unfeasible. The reason for this is that for criterion based model *averaging* the implementation of package MuMIn
is used, which considers all possible candidate models, that is *2^p* different candidate models for p parameters to estimate. If it is clear that
only a subset of variables are relevant then the options `candidate.models="restricted/very restricted"`

may be useful which
essentially specifies that only up to a half/fourth of the provided variables can be added to a single candidate model. However, this option
should be used with caution. Alternatively `criterion="BIC+"`

can be used which utilizes efficient Bayesian Model Averaging based on the leaps
algorithm of package `"BMA"`

. Also, one may consider a model selection or averaging strategy not implemented here and combine estimates "by hand"
according to formulae (7)-(10) in the cited reference below.

The function provides access to linear, logistic, Poisson and Cox proportional hazard models; one may add a random intercept to each of these models with the
`id`

option. Other models are not supported yet. Variables used for the imputation model but not needed for the analysis model can be removed with
option `var.remove`

.

### Value

Returns an object of `class`

‘mami’:

`coefficients` |
A matrix of coefficients, standard errors and confidence intervals for model |

`coefficients.boot` |
A matrix of coefficients and bootstrap results (confidence intervals, mean, standard error) for model |

`coefficients.s` |
A matrix of coefficients, standard errors and confidence intervals for model |

`coefficients.boot.s` |
A matrix of coefficients and bootstrap results (confidence intervals, mean, standard error) for model |

`variable.importance` |
A vector containing the variable importance for each variable based on model averaging weights. |

`boot.results` |
A list of detailed estimation results for each bootstrap sample. The first list element refers to the results from model selection the second entry the results from model averaging. |

### Author(s)

Michael Schomaker

### References

Schomaker, M., Heumann, C. (2014) *Model Selection and Model Averaging after Multiple Imputation*,
Computational Statistics & Data Analysis, 71:758-770

### See Also

`plot.mami`

to visualize bootstrap results, `lae`

and `mma`

for model averaging techniques.

### 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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | ```
####################################################
# Example 1: Freetrade example from Amelia package #
# Cross-Section-Time-Series Data #
# Linear and linear mixed model #
####################################################
set.seed(24121980)
library(Amelia)
data(freetrade)
freetrade$pop <- log(freetrade$pop) # in line with original publication
freetrade_imp <- amelia(freetrade, ts = "year", cs = "country", noms="signed", polytime = 2,
intercs = TRUE, empri = 2)
# AIC based model averaging and model selection in a linear model after MI
# (with and without bootstrapping)
mami(freetrade_imp, method="criterion.selection", ycol="tariff",add.factor=c("country"))
mami(freetrade_imp, method="criterion.average", ycol="tariff",add.factor=c("country"))
m1 <- mami(freetrade_imp, method="criterion.selection", ycol="tariff",add.factor=c("country"),
inference="+bootstrapping",B=25,X.org=freetrade,print.time=TRUE)
m1 # be patient with bootstrapping, increase B>=200 for better results
plot(m1, plots.p.page="4")
# For comparison: Mallow's model averaging (MMA) and complete case analysis
mami(freetrade_imp, method="MMA", ycol="tariff",add.factor=c("country"))
mami(freetrade, method="criterion.selection", missing.data="CC", ycol="tariff",
add.factor=c("country")) #Note the difference to imputed analysis (e.g. usheg)
# Use linear mixed model with random intercept for country as alternative, just specify "id" option
# Note: same procedure for random intercepts (frailty) in logistic, Poisson, or Cox Model
mami(freetrade_imp, ycol="tariff", id="country")
####################################################################
# Example 2: HIV treatment data, linear model and Cox model #
####################################################################
# Impute with Amelia
data(HIV)
HIV_imp <- amelia(HIV, m=5, idvars="patient",noms=c("hospital","sex","dead","tb","cm"),
ords=c("period","stage"),logs=c("futime","cd4"),
bounds=matrix(c(3,7,9,11,0,0,0,0,3000,5000,200,150),ncol=3,nrow=4))
# i) Cox PH model
# Model selection (with AIC) to select risk factors for the hazard of death,
# reported as hazard ratios
# Also: add transformations and interaction terms to candidate models
mami(HIV_imp, method="criterion.selection",model.family="coxph", ycol=c("futime","dead"),
add.factor=c("hospital","stage","period"), add.transformation=c("cd4^2","age^2"),
add.interaction=list(c("cd4","age")), report.exp=TRUE, var.remove=c("patient","cd4slope6"))
# Similar as above (= same but no hazard ratios reported, no interaction, hospitals as stratum),
# but with boostrap CI and visualization of bootstrap distribution (be patient...it's worth it)
m2 <- mami(HIV_imp, method="criterion.selection",model.family="coxph", inference="+bootstrapping",
X.org=HIV, ycol=c("futime","dead"), add.factor=c("stage","period"),
add.transformation=c("cd4^2","age^2"), use.stratum="hospital", B=25,
var.remove=c("patient","cd4slope6"),print.time=TRUE,print.warnings=FALSE)
m2
plot(m2)
# ii) Linear model
# Model selection and averaging to identify predictors for immune recovery 6 months
# after starting antiretroviral therapy, presented as CD4 slope which is the average
# change in number of CD4 cells per week (deaths are ignored for this example)
# AIC based model selection (stepAIC) after multiple imputation
mami(HIV_imp, method="criterion.selection", ycol="cd4slope6",
add.factor=c("hospital","stage","period"), var.remove=c("patient","dead","futime"))
# Model averaging (AIC weights) for variables typically captured
mami(HIV_imp,ycol="cd4slope6", add.factor=c("hospital","stage","period"),
var.remove=c("patient","dead","futime","tb","cm","haem"))
# Mallow's model averaging
mami(HIV_imp, method="MMA", ycol="cd4slope6", add.factor=c("hospital","stage","period"),
var.remove=c("patient","dead","futime"))
#########################################################################################
# Example 3: Model selection/averaging with no missing data, using shrinkage #
# Example from Tibshirani, R. (1996) Regression shrinkage and selection via the lasso, #
# Journal of the Royal Statistical Society, Series B 58(1), 267-288. #
# Useful to use mami to obtain Bootstrap CI after model selection/averaging #
#########################################################################################
library(lasso2)
data(Prostate)
mami(Prostate,method="LASSO/LAE",missing.data="none",ycol="lpsa"
,kfold=10) # LASSO (selection/averaging) based on 10-fold CV
mami(Prostate,missing.data="none",ycol="lpsa") # AIC based averaging
m3 <- mami(Prostate,missing.data="none",ycol="lpsa", inference="+bootstrapping",
B=50,print.time=TRUE) # with Boostrap CI
m3
plot(m3) # a few bimodal distributions: effect or not?
``` |

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