The program deals with a mediation model specified as <jsmath>M = iM + aX + e_1</jsmath> <jsmath>Y = iY + bM + cX + e_2</jsmath> where <jsm>X</jsm>, <jsm>M</jsm>, and <jsm>Y</jsm> are input, mediator, and output variables.
The unique feature of this program is to analyze the mediation effect with missing data. Data can miss on any of the three variables. However, for obvious reasons, we require the data
BMEM implements two types of missing data handling techinques - the pairwise deletion and the Expectation-Maximization (EM) algorithm is used together the MLE estimation method. To assess the mediation effect, either the stratified bootstrap method or the direct bootstrap method can be used to calculate three types of confidence intervals - the percentile interval, the bias-corrected (BC) interval, and the bias corrected and accelerated interval (BCa).
The following files are included in the download.
The program is written in C++ to analyze the meditaion effects with missing data. Two libraries are used: Newmat and newran
There are two ways to use BMEM – the step by step method and the batch method. For both methods, we suggest to put the data file in the same folder the the program file.
After double clicking on the executable file, a DOS window will pop out and require the following 7 parameters to run the program.
1. The output file name: the name of the file in which one wants to save the analysis results.
2. The data file name: the name of the data file. The data file should be a text file with the following sequence of variables, <jsm>X</jsm> , <jsm>M</jsm> , and <jsm>Y</jsm>. The missing data are should be denoted by 99999, five 9s.
3. The <jsm>\alpha</jsm> level for confidence interval. It should be a number between 0 and 1. There is no difference to input .05 or .95. BMEM automatically recognize the level for CI.
4. The random number seed. It should also be between 0 and 1. The random number seeds determine the bootstrap samples. The same analysis can be replicated using the identical random number seed.
5. The bootstrap sample size. We suggest a bootstrap sample size no less than 1000.
6. The missing data handling method. 1 for EM algorithm and 2 for the pairwise deletion method.
7. The bootstrap method. 1 for the stratified bootstrap and 2 for the direct bootstrap.
To use batch method, one can put the seven parameters in the step by step section into a file with each parameter on one line. An example can be to create a file called batch.txt with the following contents
output.txt input.txt .95 .5 1 1
Then, open the DOS windows through start –> Run… –> cmd. In the DOS window, change the directory to where BMEM is located. Then using the command BMEM.ext < batch.txt to run the analysis. The batch file here is saved in the same directory as BMEM. The above batch file conducts the analysis using the EM algorithm and stratified bootstrap and constructs the 95% CIs. All the output is saved in the file output.txt.
The following output is from the analysis of the attached ACTIVE data (active2.txt). There are several important parts of the output.
The first part is the missing data patterns and sample size of each pattern.
The second part is the results from logistic regression on the test of missingness. For any pair of variable AB, it tests whether A can predict the missing of B. If an abosulte number obtained is larger than 2, one may say the missing data are not MCAR. However, one may not be able to conclude MAR. However, even all numbers are less than 2, one still cannot conlude MCAR.
The third part is the estiamted parameters when the mediator is not considered.
The fourth part is the estimated parameters when the mediator is included in the model.
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| Program name: BMEM.exe (V3.0) |
| See manual.pdf for more information |
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The BootMed program is run on Fri Mar 20 15:48:46 2009
The output file is: active2.txt
The data file is: active2.txt
The alpha level is: 0.95
The random number seed is: .5
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| Missing Data Pattern |
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Pattern X M Y Size
1 o o o 63
2 o o x 3
3 o x o 13
4 o x x 1
NOTE: o: observed; x: missing
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| Testing Missing Mechanism |
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X M Y
X -1.06344 -0.933186
M N/A -0.469486
Y N/A 0.911532
NOTE: N/A means no missing data for B in the pair AB
Magnitude larger than 2 can be considered as not MCAR
The bootstrap sample size is: 1000
The missing data are handled by EM algorithm.
The boostrap method is stratified bootstrap.
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| Estimated parameters and confidence interval |
| EM & STRATIFIED BOOTSTRAP |
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| Model without mediator |
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Percentile BC BCa
Parameter Estimate S.E. L U L U L U
iY 14.7674 1.0239 12.9920 16.7140 13.0000 16.7646 13.0882 16.8196
c 5.2592 1.3067 2.7654 7.6528 2.7654 7.6528 2.6529 7.6199
eY2 47.2724 6.8329 33.1208 59.6471 35.1614 61.6163 36.1410 62.4777
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| Model with mediator |
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Percentile BC BCa
Parameter Estimate S.E. L U L U L U
iM 22.3782 1.8012 19.0931 25.7064 19.0488 25.6339 19.1353 25.7568
iY 4.6776 0.8885 2.8068 6.2040 2.8458 6.2411 2.8458 6.2843
a 12.9178 2.5698 8.0375 18.0439 8.3104 18.2173 8.1839 18.0883
b 0.4509 0.0411 0.3769 0.5326 0.3688 0.5214 0.3683 0.5208
c' -0.5651 1.1618 -2.7772 1.7479 -2.7083 1.9486 -2.7395 1.7600
eM2 169.8286 24.9932 120.0316 216.8313 125.6172 222.1035 131.2006 236.0085
eY2 12.7480 2.1469 8.3597 16.4475 9.3054 17.8272 9.5787 17.9865
a*b 5.8243 1.1195 3.8455 8.2087 3.9042 8.3292 3.7854 8.1979
The total running time is 2.1400 seconds.
This is a free program and you can use and distribute it as long as you want. However, we cannot guarantee its performance under any possible circumstances. YOU CAN USE IT FOR FREE BUT AT YOUR OWN RISK. WE ARE NOT RESPONSIBLE FOR ANY LOSS BECAUSE OF THE USE OF BMEM.
Although it is not required, it is appreciated if you can cite the software in the following way if you are willing to use it.
Zhang, Z., & Wang, L. (xxxx) Mediation analysis with missing data using EM
algorithm and bootstrap. xxxx
Please direct questions or comments to ZhiyongZhang@nd.edu.