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tips:multiple_imputation_with_mice_and_metafor [2019/10/09 12:46] (current)
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 +===== Multiple Imputation with the mice and metafor Packages =====
 +
 +Meta-analytic data often looks like Swiss cheese -- there are lots of holes in it! For example, due to missing information,​ it may not be possible to compute the effect size estimates (or the corresponding sampling variances) for some of the studies. Similarly, it may not be possible to code certain moderator variables for some studies. Below, I illustrate how to use multiple imputation as a possible way to deal with the latter issue, using the [[https://​cran.r-project.org/​package=mice|mice]] package in combination with metafor.
 +
 +==== Data Preparation ====
 +
 +For the example, I will use data from the meta-analysis by Bangert-Drowns et al. (2004) on the effectiveness of school-based writing-to-learn interventions on academic achievement (''​help(dat.bangertdrowns2004)''​ provides a bit more background). The data can be loaded with:
 +<code rsplus>
 +library(metafor)
 +dat <- dat.bangertdrowns2004
 +</​code>​
 +(I copy the dataset into '​dat',​ which is a bit shorter and therefore easier to type further below). We can look at the first 10 and the last 10 rows of the dataset with:
 +<code rsplus>
 +rbind(head(dat,​ 10), tail(dat, 10))
 +</​code>​
 +<code output>
 +   ​id ​         author year grade length minutes wic feedback info pers imag meta                subject ​ ni    yi    vi
 +1   ​1 ​       Ashworth 1992     ​4 ​    ​15 ​     NA   ​1 ​       1    1    1    0    1                Nursing ​ 60  0.65 0.070
 +2   ​2 ​          Ayers 1993     ​2 ​    ​10 ​     NA   ​1 ​      ​NA ​   1    1    1    0          Earth Science ​ 34 -0.75 0.126
 +3   ​3 ​         Baisch 1990     ​2 ​     2      NA   ​1 ​       0    1    1    0    1                   ​Math ​ 95 -0.21 0.042
 +4   ​4 ​          Baker 1994     ​4 ​     9      10   ​1 ​       1    1    0    0    0                Algebra 209 -0.04 0.019
 +5   ​5 ​         Bauman 1992     ​1 ​    ​14 ​     10   ​1 ​       1    1    1    0    1                   Math 182  0.23 0.022
 +6   ​6 ​         Becker 1996     ​4 ​     1      20   ​1 ​       0    0    1    0    0             ​Literature 462  0.03 0.009
 +7   ​7 ​    Bell & Bell 1985     ​3 ​     4      NA   ​1 ​       1    1    1    0    1                   ​Math ​ 38  0.26 0.106
 +8   ​8 ​        ​Brodney 1994     ​1 ​    ​15 ​     NA   ​1 ​       1    1    1    0    1                   Math 542  0.06 0.007
 +9   ​9 ​         Burton 1986     ​4 ​     4      NA   ​0 ​       1    1    0    0    0                   ​Math ​ 99  0.06 0.040
 +10 10       ​Davis,​ BH 1990     ​1 ​     9      10   ​1 ​       0    1    1    0    0         ​Social Studies ​ 77  0.12 0.052
 +39 39 Ross & Faucette 1994     ​4 ​    ​15 ​     NA   ​0 ​       1    1    0    0    0                Algebra ​ 16  0.70 0.265
 +40 40           Sharp 1987     ​4 ​     2      15   ​1 ​       0    0    1    0    1                Biology 105  0.49 0.039
 +41 41         ​Shepard 1992     ​2 ​     4      NA   ​0 ​       1    1    0    0    0                   Math 195  0.20 0.021
 +42 42         ​Stewart 1992     ​3 ​    ​24 ​      ​5 ​  ​1 ​       1    1    1    0    1                Algebra ​ 62  0.58 0.067
 +43 43         ​Ullrich 1926     ​4 ​    ​11 ​     NA   ​0 ​       1    1    0    0    0 Educational Psychology 289  0.15 0.014
 +44 44 Weiss & Walters 1980     ​4 ​    ​15 ​      ​3 ​  ​1 ​       0    1    0    0    0             ​Statistics ​ 25  0.63 0.168
 +45 45           Wells 1986     ​1 ​     8      15   ​1 ​       0    1    1    0    1                   Math 250  0.04 0.016
 +46 46          Willey 1988     ​3 ​    ​15 ​     NA  NA        0    1    1    0    1                Biology ​ 51  1.46 0.099
 +47 47          Willey 1988     ​2 ​    ​15 ​     NA  NA        0    1    1    0    1         ​Social Studies ​ 46  0.04 0.087
 +48 48       ​Youngberg 1989     ​4 ​    ​15 ​     10   ​1 ​       1    1    0    0    0                Algebra ​ 56  0.25 0.072
 +</​code>​
 +
 +Variable ''​yi''​ contains the effect size estimates (standardized mean differences) and ''​vi''​ the corresponding sampling variances. There are 48 rows of data in this dataset.
 +
 +For illustration purposes, the following variables will be examined as potential moderators of the treatment effect (i.e., the size of the treatment effect may vary in a systematic way as a function of one or more of these variables):
 +
 +  * length: treatment length (in weeks)
 +  * wic: writing tasks were completed in class (0 = no; 1 = yes)
 +  * feedback: feedback on writing was provided (0 = no; 1 = yes)
 +  * info: writing contained informational components (0 = no; 1 = yes)
 +  * pers: writing contained personal components (0 = no; 1 = yes)
 +  * imag: writing contained imaginative components (0 = no; 1 = yes)
 +  * meta: prompts for metacognitive reflection (0 = no; 1 = yes)
 +
 +More details about the meaning of these variables can be found in Bangert-Drowns et al. (2004). For the purposes of this illustration,​ it is sufficient to understand that we have 7 variables that are potential (and a priori plausible) predictors of the size of the treatment effect.
 +
 +To make some of the code below simpler, I will only keep the variables needed for the analyses:
 +<code rsplus>
 +dat <- dat[c("​yi",​ "​vi",​ "​length",​ "​wic",​ "​feedback",​ "​info",​ "​pers",​ "​imag",​ "​meta"​)]
 +</​code>​
 +
 +==== Complete Case Analysis ====
 +
 +Standard model fitting functions in R (including those from the metafor package) will usually apply '​listwise deletion'​ when fitting models with data that contains missing values. In the present case, this implies that any study where the effect size, sampling variance, or any of the moderator variables included in the model is missing will be removed from the dataset before fitting the model. While the effect sizes and sampling variances (i.e., variables ''​yi''​ and ''​vi''​) are complete, there are some missing values in most of the moderator variables:
 +<code rsplus>
 +data.frame(k.NA=colSums(is.na(dat)))
 +</​code>​
 +<code output>
 +         k.NA
 +yi          0
 +vi          0
 +length ​     2
 +wic         2
 +feedback ​   1
 +info        2
 +pers        2
 +imag        0
 +meta        2
 +</​code>​
 +We can also check how many missing values there are for each study:
 +<code rsplus>
 +table(rowSums(is.na(dat)))
 +</​code>​
 +<code output>
 + ​0 ​ 1  3 
 +41  5  2 
 +</​code>​
 +So, for 41 studies, the data are complete, but 5 studies have one missing value, while 2 studies have three missing values. Therefore, when fitting a meta-regression model with all moderators of interest included simultaneously,​ only the 41 studies with complete data will be included in the analysis.
 +
 +Let's try this:
 +<code rsplus>
 +res <- rma(yi, vi, mods = ~ length + wic + feedback + info + pers + imag + meta, data=dat)
 +res
 +</​code>​
 +Note that the ''​rma()''​ function alerts the user of the fact that studies with missing values were omitted from the model fitting:
 +<code output>
 +Warning message:
 +In rma(yi, vi, mods = ~length + wic + feedback + info + pers + imag +  :
 +  Studies with NAs omitted from model fitting.
 +</​code>​
 +The output of ''​res''​ is:
 +<code output>
 +Mixed-Effects Model (k = 41; tau^2 estimator: REML)
 +
 +tau^2 (estimated amount of residual heterogeneity): ​    ​0.0223 (SE = 0.0153)
 +tau (square root of estimated tau^2 value): ​            ​0.1494
 +I^2 (residual heterogeneity / unaccounted variability):​ 36.91%
 +H^2 (unaccounted variability / sampling variability): ​  1.59
 +R^2 (amount of heterogeneity accounted for):            21.01%
 +
 +Test for Residual Heterogeneity:​
 +QE(df = 33) = 51.4963, p-val = 0.0211
 +
 +Test of Moderators (coefficients 2:8):
 +QM(df = 7) = 11.7175, p-val = 0.1102
 +
 +Model Results:
 +
 +          estimate ​     se     ​zval ​   pval    ci.lb   ​ci.ub ​
 +intrcpt ​    ​0.2689 ​ 0.2154 ​  ​1.2484 ​ 0.2119 ​ -0.1533 ​ 0.6910 ​   ​
 +length ​     0.0072 ​ 0.0078 ​  ​0.9240 ​ 0.3555 ​ -0.0081 ​ 0.0225 ​   ​
 +wic        -0.0472 ​ 0.1097 ​ -0.4308 ​ 0.6666 ​ -0.2622 ​ 0.1677 ​   ​
 +feedback ​   0.0677 ​ 0.1080 ​  ​0.6265 ​ 0.5310 ​ -0.1440 ​ 0.2793 ​   ​
 +info       ​-0.2233 ​ 0.2029 ​ -1.1006 ​ 0.2711 ​ -0.6210 ​ 0.1744 ​   ​
 +pers       ​-0.1137 ​ 0.1898 ​ -0.5992 ​ 0.5490 ​ -0.4857 ​ 0.2582 ​   ​
 +imag        0.4106 ​ 0.1847 ​  ​2.2233 ​ 0.0262 ​  ​0.0486 ​ 0.7726 ​ * 
 +meta        0.2010 ​ 0.1742 ​  ​1.1537 ​ 0.2486 ​ -0.1404 ​ 0.5424 ​   ​
 +
 +---
 +Signif. codes: ​ 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 +</​code>​
 +As can be seen from the output, $k = 41$ studies were included in the analysis (i.e., the ones with complete data). So, roughly 15% (i.e., 7 out of 48) of the studies were excluded from the analysis. While this isn't the most horrible example, it illustrates the loss of data (and information) that can occur when conducting a complete case analysis.
 +
 +==== Multiple Imputation ====
 +
 +One way of dealing with missing data is to make use of imputation techniques. The advantage of using [[wp>​Imputation_(statistics)#​Multiple_imputation|multiple imputation]] is that we not only impute once (and then pretend that the imputed values are free of any uncertainty),​ but multiple times from appropriate distributions,​ so that several imputed datasets are generated. The same analysis is then applied to each dataset and the results are pooled, taking into consideration not only the uncertainty in each fitted model, but also across models.
 +
 +The mice package allows us to automate this process and can be used in combination with the metafor package. First, we install and load the mice package and then evaluate some code from the metafor package that generates two helper functions we need so that mice and metafor can interact as necessary:
 +<code rsplus>
 +install.packages("​mice"​)
 +library(mice)
 +eval(metafor:::​.mice)
 +</​code>​
 +
 +Most of the moderators are dummy variables (coded 0 vs 1). Although it isn't typically necessary to encode such variables as "​proper"​ factors, this does become relevant in the present case, because mice (by default) will impute factors with two levels based on logistic regression models (as opposed to numeric variables that are by default imputed using predictive mean matching). We can turn the dummy variables into factors with:
 +<code rsplus>
 +dat$wic ​     <- factor(dat$wic)
 +dat$feedback <- factor(dat$feedback)
 +dat$info ​    <- factor(dat$info)
 +dat$pers ​    <- factor(dat$pers)
 +dat$imag ​    <- factor(dat$imag)
 +dat$meta ​    <- factor(dat$meta)
 +</​code>​
 +
 +Next, we will set up the predictor matrix for the imputations. For this, we run the ''​make.predictorMatrix()''​ function.
 +<code rsplus>
 +predMatrix <- make.predictorMatrix(dat)
 +predMatrix
 +</​code>​
 +<code output>
 +         yi vi length wic feedback info pers imag meta
 +yi        0  1      1   ​1 ​       1    1    1    1    1
 +vi        1  0      1   ​1 ​       1    1    1    1    1
 +length ​   1  1      0   ​1 ​       1    1    1    1    1
 +wic       ​1 ​ 1      1   ​0 ​       1    1    1    1    1
 +feedback ​ 1  1      1   ​1 ​       0    1    1    1    1
 +info      1  1      1   ​1 ​       1    0    1    1    1
 +pers      1  1      1   ​1 ​       1    1    0    1    1
 +imag      1  1      1   ​1 ​       1    1    1    0    1
 +meta      1  1      1   ​1 ​       1    1    1    1    0
 +</​code>​
 +A value of 1 in this matrix indicates that the corresponding column variable is used to predict the corresponding row variable (so each row can be regarded as specifying the predictors for the model used to predict each row variable). Now we will make a few changes to this matrix. First, I set the entire column corresponding to the ''​vi''​ variable to 0, so that the sampling variances are not used for imputing any of the missing values.((In principle, one could also use the sampling variances as a predictor, but this might be considered a bit unusual, so I refrain from doing so in this example.)) Also, I set the rows for variables ''​yi''​ and ''​vi''​ to 0, so that these variables are never imputed (this isn't really relevant here, since these two variables do not actually contain any missing values).
 +<code rsplus>
 +predMatrix[,"​vi"​] <- 0 # don't use vi for imputing
 +predMatrix["​yi",​] <- 0 # don't impute yi (since yi has no NAs, this is actually irrelevant here)
 +predMatrix["​vi",​] <- 0 # don't impute vi (since vi has no NAs, this is actually irrelevant here)
 +predMatrix
 +</​code>​
 +<code output>
 +         yi vi length wic feedback info pers imag meta
 +yi        0  0      0   ​0 ​       0    0    0    0    0
 +vi        0  0      0   ​0 ​       0    0    0    0    0
 +length ​   1  0      0   ​1 ​       1    1    1    1    1
 +wic       ​1 ​ 0      1   ​0 ​       1    1    1    1    1
 +feedback ​ 1  0      1   ​1 ​       0    1    1    1    1
 +info      1  0      1   ​1 ​       1    0    1    1    1
 +pers      1  0      1   ​1 ​       1    1    0    1    1
 +imag      1  0      1   ​1 ​       1    1    1    0    1
 +meta      1  0      1   ​1 ​       1    1    1    1    0
 +</​code>​
 +
 +Next, I create a vector that specifies the method used to predict (and hence impute) each variable:
 +<code rsplus>
 +impMethod <- make.method(dat)
 +impMethod
 +</​code>​
 +<code output>
 +yi       ​vi ​  ​length ​     wic feedback ​    ​info ​    ​pers ​    ​imag ​    ​meta ​
 +"" ​      "" ​   "​pmm"​ "​logreg"​ "​logreg"​ "​logreg"​ "​logreg" ​      ""​ "​logreg" ​
 +
 +</​code>​
 +By default, predictive mean matching (''​pmm''​) is used for numeric variables, while logistic regression (''​logreg''​) is used for two-level factors. One could change these defaults to other methods (see ''​help(mice)''​ for other imputation methods that could be used), but we will stick to the (usually sensible) defaults here. Note that no imputation method is specified for the ''​yi'',​ ''​vi'',​ and ''​imag''​ variables, since these variables do not contain any missing values.
 +
 +Now we are ready to generate the multiple imputations. Often (and by default), only 5 datasets are generated, but I increase this to 20 below. I also set the seed (for the random number generator) to make the following results fully reproducible:​
 +<code rsplus>
 +imp <- mice(dat, print=FALSE,​ m=20, predictorMatrix=predMatrix,​ method=impMethod,​ seed=1234)
 +</​code>​
 +
 +Next, we can fit the model of interest to each of the 20 imputed datasets with:
 +<code rsplus>
 +fit <- with(imp, rma(yi, vi, mods = ~ length + wic + feedback + info + pers + imag + meta))
 +</​code>​
 +
 +And finally, we can pool the results with:
 +<code rsplus>
 +pool <- pool(fit)
 +round(summary(pool),​ 4)
 +</​code>​
 +<code output>
 +          estimate std.error statistic ​     df p.value
 +intrcpt ​    ​0.3970 ​   0.2418 ​   1.6417 37.3121 ​ 0.1089
 +length ​     0.0132 ​   0.0088 ​   1.5052 37.1180 ​ 0.1405
 +wic1       ​-0.0638 ​   0.1303 ​  ​-0.4897 34.5222 ​ 0.6271
 +feedback1 ​ -0.0066 ​   0.1209 ​  ​-0.0543 34.3060 ​ 0.9570
 +info1      -0.3133 ​   0.2307 ​  ​-1.3583 37.5709 ​ 0.1824
 +pers1      -0.3233 ​   0.1937 ​  ​-1.6689 37.5493 ​ 0.1034
 +imag1       ​0.2117 ​   0.2096 ​   1.0099 38.0212 ​ 0.3189
 +meta1       ​0.4623 ​   0.1739 ​   2.6583 37.4993 ​ 0.0114
 +</​code>​
 +
 +For easier comparison, let's look at the coefficient table based on the complete case analysis obtained earlier:
 +<code rsplus>
 +round(coef(summary(res)),​ 4)
 +</​code>​
 +<code output>
 +          estimate ​    ​se ​   zval   ​pval ​  ​ci.lb ​ ci.ub
 +intrcpt ​    ​0.2689 0.2154 ​ 1.2484 0.2119 -0.1533 0.6910
 +length ​     0.0072 0.0078 ​ 0.9240 0.3555 -0.0081 0.0225
 +wic        -0.0472 0.1097 -0.4308 0.6666 -0.2622 0.1677
 +feedback ​   0.0677 0.1080 ​ 0.6265 0.5310 -0.1440 0.2793
 +info       ​-0.2233 0.2029 -1.1006 0.2711 -0.6210 0.1744
 +pers       ​-0.1137 0.1898 -0.5992 0.5490 -0.4857 0.2582
 +imag        0.4106 0.1847 ​ 2.2233 0.0262 ​ 0.0486 0.7726
 +meta        0.2010 0.1742 ​ 1.1537 0.2486 -0.1404 0.5424
 +</​code>​
 +
 +Leaving aside a discussion of the usefulness of p-values, there is an interesting discrepancy in the findings. In particular, while moderator ''​imag''​ was statistically significant in the complete case analysis, the results based on multiple imputation suggest that another moderator (i.e., ''​meta''​) is significant. I obviously cannot tell you which of these findings is the "​correct"​ one (and possibly neither is), but the difference is definitely noteworthy.
 +
 +For more details on multiple imputation and the mice package, I would suggest to take a look at the book by Van Buuren (2018), which you can also read online (https://​stefvanbuuren.name/​fimd/​).
 +
 +==== References ====
 +
 +Van Buuren, S. (2018). //Flexible imputation of missing data// (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC.
  
tips/multiple_imputation_with_mice_and_metafor.txt · Last modified: 2019/10/09 12:46 (external edit)