Missing Data in Randomized Control Trials John W. Graham The Prevention Research Center and Department of Biobehavioral Health Penn State University IES/NCER Summer Research Training Institute, August 2, 2010 jgraham@psu.edu

Sessions in Three Parts (1) Introduction: Missing Data Theory (2) Attrition: Bias and Lost Power After the break ... (3) Hands-on with Multiple Imputation Multiple Imputation with NORM SPSS Automation Utility (New!) SPSS Regression HLM Automation Utility (New!) 2-Level Regression with HLM 6

Recent Papers Graham, J. W., (2009). Missing data analysis: making it work in the real world. Annual Review of Psychology, 60, 549-576. Graham, J. W., Cumsille, P. E., & Elek-Fisk, E. (2003). Methods for handling missing data. In J. A. Schinka & W. F. Velicer (Eds.). Research Methods in Psychology (pp. 87_114). Volume 2 of Handbook of Psychology (I. B. Weiner, Editor-in-Chief). New York: John Wiley & Sons. Graham, J. W. (2010, forthcoming). Missing Data: Analysis and Design. New York: Springer. Chapter 4: Multiple Imputation with Norm 2.03 Chapter 6: Multiple Imputation and Analysis with SPSS 17/18 Chapter 7: Multiple Imputation and Analysis with Multilevel (Cluster) Data

Recent Papers Collins, L. M., Schafer, J. L., & Kam, C. M. (2001). A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychological Methods, 6, 330-351. Schafer, J. L., & Graham, J. W. (2002). Missing data: our view of the state of the art. Psychological Methods, 7, 147-177. Graham, J. W., Taylor, B. J., Olchowski, A. E., & Cumsille, P. E. (2006). Planned missing data designs in psychological research. Psychological Methods, 11, 323-343.

Part 1: A Brief Introduction to Analysis with Missing Data

Problem with Missing Data Analysis procedures were designed for complete data . . .

Solution 1 Design new model-based procedures Missing Data + Parameter Estimation in One Step Full Information Maximum Likelihood (FIML) SEM and Other Latent Variable Programs (Amos, LISREL, Mplus, Mx, LTA)

Solution 2 Data based procedures Two Steps e.g., Multiple Imputation (MI) Two Steps Step 1: Deal with the missing data (e.g., replace missing values with plausible values Produce a product Step 2: Analyze the product as if there were no missing data

FAQ Aren't you somehow helping yourself with imputation? . . .

NO. Missing data imputation . . . does NOT give you something for nothing DOES let you make use of all data you have . . .

FAQ Is the imputed value what the person would have given?

NO. When we impute a value . . We do not impute for the sake of the value itself We impute to preserve important characteristics of the whole data set . . .

We want . . . unbiased parameter estimation e.g., b-weights Good estimate of variability e.g., standard errors best statistical power

Causes of Missingness Ignorable Non-Ignorable MCAR: Missing Completely At Random MAR: Missing At Random Non-Ignorable MNAR: Missing Not At Random

MCAR (Missing Completely At Random) MCAR 1: Cause of missingness completely random process (like coin flip) MCAR 2: (essentially MCAR) Cause uncorrelated with variables of interest Example: parents move No bias if cause omitted

MAR (Missing At Random) Missingness may be related to measured variables But no residual relationship with unmeasured variables Example: reading speed No bias if you control for measured variables

MNAR (Missing Not At Random) Even after controlling for measured variables ... Residual relationship with unmeasured variables Example: drug use reason for absence

MNAR Causes The recommended methods assume missingness is MAR But what if the cause of missingness is not MAR? Should these methods be used when MAR assumptions not met? . . .

YES! These Methods Work! Suggested methods work better than “old” methods Multiple causes of missingness Only small part of missingness may be MNAR Suggested methods usually work very well

Methods: "Old" vs MAR vs MNAR MAR methods (MI and ML) are ALWAYS at least as good as, usually better than "old" methods (e.g., listwise deletion) Methods designed to handle MNAR missingness are NOT always better than MAR methods

Analysis: Old and New

Old Procedures: Analyze Complete Cases (listwise deletion) may produce bias you always lose some power (because you are throwing away data) reasonable if you lose only 5% of cases often lose substantial power

Analyze Complete Cases (listwise deletion) 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 very common situation only 20% (4 of 20) data points missing but discard 80% of the cases

Other "Old" Procedures Pairwise deletion Mean substitution May be of occasional use for preliminary analyses Mean substitution Never use it Regression-based single imputation generally not recommended ... except ...

Recommended Model-Based Procedures Multiple Group SEM (Structural Equation Modeling) Latent Transition Analysis (Collins et al.) A latent class procedure

Recommended Model-Based Procedures Raw Data Maximum Likelihood SEM aka Full Information Maximum Likelihood (FIML) Amos (James Arbuckle) LISREL 8.5+ (Jöreskog & Sörbom) Mplus (Bengt Muthén) Mx (Michael Neale)

Amos, Mx, Mplus, LISREL 8.8 Structural Equation Modeling (SEM) Programs In Single Analysis ... Good Estimation Reasonable standard errors Windows Graphical Interface

Limitation with Model-Based Procedures That particular model must be what you want

Recommended Data-Based Procedures EM Algorithm (ML parameter estimation) Norm-Cat-Mix, EMcov, SAS, SPSS Multiple Imputation NORM, Cat, Mix, Pan (Joe Schafer) SAS Proc MI SPSS 17/18 (not quite yet) LISREL 8.5+ Amos

EM Algorithm Expectation - Maximization Alternate between E-step: predict missing data M-step: estimate parameters Excellent (ML) parameter estimates But no standard errors must use bootstrap or multiple imputation

Multiple Imputation Problem with Single Imputation: Too Little Variability Because of Error Variance Because covariance matrix is only one estimate

Too Little Error Variance Imputed value lies on regression line

Imputed Values on Regression Line

Restore Error . . . Add random normal residual

Regression Line only One Estimate

Covariance Matrix (Regression Line) only One Estimate Obtain multiple plausible estimates of the covariance matrix ideally draw multiple covariance matrices from population Approximate this with Bootstrap Data Augmentation (Norm) MCMC (SAS)

Data Augmentation stochastic version of EM EM Data Augmentation E (expectation) step: predict missing data M (maximization) step: estimate parameters Data Augmentation I (imputation) step: simulate missing data P (posterior) step: simulate parameters

Data Augmentation Parameters from consecutive steps ... too related i.e., not enough variability after 50 or 100 steps of DA ... covariance matrices are like random draws from the population

Multiple Imputation Allows: Unbiased Estimation Good standard errors provided number of imputations (m) is large enough too few imputations  reduced power with small effect sizes

ρ From Graham, J.W., Olchowski, A.E., & Gilreath, T.D. (2007). How many imputations are really needed? Some practical clarifications of multiple imputation theory. Prevention Science, 8, 206-213.