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, July 2008 jgraham@psu.edu

Sessions in Four Parts (1) Introduction: Missing Data Theory (2) Attrition: Bias and Lost Power (3) A brief analysis demonstration Multiple Imputation with NORM and Proc MI (4) Hands-on Intro to Multiple Imputation

Recent Papers 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., (2009, in press). Missing data analysis: making it work in the real world. Annual Review of Psychology, 60. 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.

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: 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 7, 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 LISREL 8.5+ Amos 7

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 8.2, 9)

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.

Part 2 Attrition: Bias and Loss of Power

Relevant Papers Graham, J.W., (in press). Missing data analysis: making it work in the real world. Annual Review of Psychology, 60. 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. Hedeker, D., & Gibbons, R.D. (1997). Application of random-effects pattern-mixture models for missing data in longitudinal studies, Psychological Methods, 2, 64-78. Graham, J.W., & Collins, L.M. (2008). Using Modern Missing Data Methods with Auxiliary Variables to Mitigate the Effects of Attrition on Statistical Power. Annual Meetings of the Society for Prevention Research, San Francisco, CA. (available upon request) Graham, J.W., Palen, L.A., et al. (2008). Attrition: MAR & MNAR missingness, and estimation bias. Annual Meetings of the Society for Prevention Research, San Francisco, CA. (available upon request)

What if the cause of missingness is MNAR? Problems with this statement MAR & MNAR are widely misunderstood concepts I argue that the cause of missingness is never purely MNAR The cause of missingness is virtually never purely MAR either.

MAR vs MNAR "Pure" MCAR, MAR, MNAR never occur in field research Each requires untenable assumptions e.g., that all possible correlations and partial correlations are r = 0

MAR vs MNAR Better to think of MAR and MNAR as forming a continuum MAR vs MNAR NOT even the dimension of interest

MAR vs MNAR: What IS the Dimension of Interest? How much estimation bias? when cause of missingness cannot be included in the model

Bottom Line ... All missing data situations are partly MAR and partly MNAR Sometimes it matters ... bias affects statistical conclusions Often it does not matter bias has tolerably little effect on statistical conclusions (Collins, Schafer, & Kam, Psych Methods, 2001)

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

Yardstick for Measuring Bias Standardized Bias = (average parameter est) – (population value) -------------------------------------------------------- X 100 Standard Error (SE) |bias| < 40 considered small enough to be tolerable

A little background for Collins, Schafer, & Kam (2001; CSK) Example model of interest: X  Y X = Program (prog vs control) Y = Cigarette Smoking Z = Cause of missingness: say, Rebelliousness (or smoking itself) Factors to be considered: % Missing (e.g., % attrition) rYZ rZR

rYZ Correlation between cause of missingness (Z) e.g., rebelliousness (or smoking itself) and the variable of interest (Y) e.g., Cigarette Smoking

rZR Correlation between cause of missingness (Z) e.g., rebelliousness (or smoking itself) and missingness on variable of interest e.g., Missingness on the Smoking variable Missingness on Smoking (often designated: R or RY) Dichotomous variable: R = 1: Smoking variable not missing R = 0: Smoking variable missing

CSK Study Design (partial) Simulations manipulated amount of missingness (25% vs 50%) rZY (r = .40, r = .90) rZR held constant r = .45 with 50% missing (applies to "MNAR-Linear" missingness)

CSK Results (partial) (MNAR Missingness) 25% missing, rYZ = .40 ... no problem 25% missing, rYZ = .90 ... no problem 50% missing, rYZ = .40 ... no problem 50% missing, rYZ = .90 ... problem * "no problem" = bias does not interfere with inference These Results apply to the regression coefficient for X  Y with "MNAR-Linear" missingness (see CSK, 2001, Table 2)

But Even CSK Results Too Conservative Not considered by CSK: rZR In their simulation rZR = .45 Even with 50% missing and rYZ = .90 bias can be acceptably small Graham et al. (2008): Bias acceptably small (standardized bias < 40) as long as rZR < .24

rZR < .24 Very Plausible Study rZR _________ _____ HealthWise (Caldwell, Smith, et al., 2004) .106 AAPT (Hansen & Graham, 1991) .093 Botvin1 .044 Botvin2 .078 Botvin3 .104 All of these yield standardized bias < 10 (estimated)

Attrition in HealthWise Best (pretest) predictors of Attrition in HW Gender Lifetime Sex Lifetime Alcohol Use Lifetime Smoking Lifetime Dagga Use

CSK and Follow-up Simulations Results very promising Suggest that even MNAR biases are often tolerably small But these simulations still too narrow

Beginnings of a Taxonomy of Attrition Causes of Attrition on Y (main DV) Case 1: not Program (P), not Y, not PY interaction Case 2: P only Case 3: Y only . . . (CSK scenario) Case 4: P and Y only

Beginnings of a Taxonomy of Attrition Causes of Attrition on Y (main DV) Case 5: PY interaction only Case 6: P + PY interaction Case 7: Y + PY interaction Case 8: P, Y, and PY interaction

Taxonomy of Attrition Cases 1-4 Cases 5-8 often little or no problem Jury still out (more research needed) Very likely not as much of a problem as previously though Use diagnostics to shed light

Use of Missing Data Diagnostics Diagnostics based on pretest data not much help Hard to predict missing distal outcomes from differences on pretest scores Longitudinal Diagnostics can be much more helpful

Hedeker & Gibbons (1997) Plot main DV over time for four groups: for Program and Control for those with and without last wave of data Much can be learned

Empirical Examples Hedeker & Gibbons (1997) Hansen & Graham (1991) Drug treatment of psychiatric patients Hansen & Graham (1991) Adolescent Alcohol Prevention Trial (AAPT) Alcohol, smoking, other drug prevention among normal adolescents (7th – 11th grade)

Empirical Example Used by Hedeker & Gibbons (1997) IV: Drug Treatment vs. Placebo Control DV: Inpatient Multidimensional Psychiatric Scale (IMPS) 1 = normal 2 = borderline mentally ill 3 = mildly ill 4 = moderately ill 5 = markedly ill 6 = severely ill 7 = among the most extremely ill

From Hedeker & Gibbons (1997) Placebo Control IMPS low = better outcomes Drug Treatment Weeks of Treatment

Longitudinal Diagnostics Hedeker & Gibbons Example Treatment droppers do BETTER than stayers Control droppers do WORSE than stayers Example of Program X DV interaction But in this case, pattern would lead to suppression bias Not as bad for internal validity in presence of significant program effect

AAPT (Hansen & Graham, 1991) IV: Normative Education Program vs Information Only Control DV: Cigarette Smoking (3-item scale) Measured at one-year intervals 7th grade – 11th grade

AAPT Control Program Control Cigarette Smoking (high = more smoking; arbitrary scale) Program th th th th th

Longitudinal Diagnostics AAPT Example Treatment droppers do WORSE than stayers little steeper increase Control Little evidence for Prog X DV interaction Very likely MAR methods allow good conclusions (CSK scenario holds)

Use of Auxiliary Variables Reduces attrition bias Restores some power lost due to attrition

What Is an Auxiliary Variable? A variable correlated with the variables in your model but not part of the model not necessarily related to missingness used to "help" with missing data estimation Best auxiliary variables: same variable as main DV, but measured at waves not used in analysis model

Model of Interest

Benefit of Auxiliary Variables Example from Graham & Collins (2008) X Y Z 1 1 1 500 complete cases 1 0 1 500 cases missing Y X, Y variables in the model (Y sometimes missing) Z is auxiliary variable

Benefit of Auxiliary Variables Effective sample size (N') Analysis involving N cases, with auxiliary variable(s) gives statistical power equivalent to N' complete cases without auxiliary variables

Benefit of Auxiliary Variables It matters how highly Y and Z (the auxiliary variable) are correlated For example increase rYZ = .40 N = 500 gives power of N' = 542 ( 8%) rYZ = .60 N = 500 gives power of N' = 608 (22%) rYZ = .80 N = 500 gives power of N' = 733 (47%) rYZ = .90 N = 500 gives power of N' = 839 (68%)

Effective Sample Size by rYZ

Conclusions Attrition CAN be bad for internal validity But often it's NOT nearly as bad as often feared Don't rush to conclusions, even with rather substantial attrition Examine evidence (especially longitudinal diagnostics) before drawing conclusions Use MI and ML missing data procedures! Use good auxiliary variables to minimize impact of attrition

Part 3: Illustration of Missing Data Analysis: Multiple Imputation with NORM and Proc MI

Multiple Imputation: Basic Steps Impute Analyze Combine results

Imputation and Analysis Impute 40 datasets a missing value gets a different imputed value in each dataset Analyze each data set with USUAL procedures e.g., SAS, SPSS, LISREL, EQS, STATA Save parameter estimates and SE’s

Combine the Results Parameter Estimates to Report Average of estimate (b-weight) over 40 imputed datasets

Combine the Results Standard Errors to Report Weighted sum of: “within imputation” variance average squared standard error usual kind of variability “between imputation” variance sample variance of parameter estimates over 40 datasets variability due to missing data

Materials for SPSS Regression Starting place http://methodology.psu.edu downloads (you will need to get a free user ID to download all our free software) missing data software Joe Schafer's Missing Data Programs John Graham's Additional NORM Utilities http://mcgee.hhdev.psu.edu/missing/index.html