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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. [email protected] Sessions in Four Parts. (1) Introduction: Missing Data Theory

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missing data in randomized control trials

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

[email protected]

sessions in four parts
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
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.
problem with missing data
Problem with Missing Data
  • Analysis procedures were designed for complete data. . .
solution 1
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
Solution 2
  • Data based procedures
    • 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
slide8
FAQ
  • Aren\'t you somehow helping yourself with imputation?. . .
no missing data imputation
NO. Missing data imputation . . .
  • does NOT give you something for nothing
  • DOES let you make use of all data you have

. . .

slide10
FAQ
  • Is the imputed value what the person would have given?
no when we impute a value
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
We want . . .
  • unbiased parameter estimation
    • e.g., b-weights
  • Good estimate of variability
    • e.g., standard errors
  • best statistical power
causes of missingness
Causes of Missingness
  • Ignorable
    • MCAR: Missing Completely At Random
    • MAR: Missing At Random
  • Non-Ignorable
    • MNAR: Missing Not At Random
mcar missing completely 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
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
MNAR (Missing Not At Random)
  • Even after controlling for measured variables ...
  • Residual relationship with unmeasured variables
  • Example: drug use reason for absence
mnar causes
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
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
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
old procedures analyze complete cases listwise deletion
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
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
Other "Old" Procedures
  • Pairwise deletion
    • 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
Recommended Model-Based Procedures
  • Multiple Group SEM (Structural Equation Modeling)
  • LatentTransitionAnalysis (Collins et al.)
    • A latent class procedure
recommended model based procedures25
Recommended Model-Based Procedures
  • Raw Data Maximum Likelihood SEMaka 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
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
Limitation with Model-Based Procedures
  • That particular model must be what you want
recommended data based procedures
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
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
Multiple Imputation
  • Problem with Single Imputation:Too Little Variability
    • Because of Error Variance
    • Because covariance matrix is only one estimate
too little error variance
Too Little Error Variance
  • Imputed value lies on regression line
restore error
Restore Error . . .
  • Add random normal residual
covariance matrix 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
Data Augmentation
  • stochastic version of EM
  • EM
    • 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 augmentation37
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
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
slide39

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.

relevant papers
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
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
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 mnar44
MAR vs MNAR
  • Better to think of MAR and MNAR asforming a continuum
  • MAR vs MNAR NOT even the dimension of interest
mar vs mnar what is 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
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 mnar47
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
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
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
slide50
rYZ
  • Correlation between
    • cause of missingness (Z)
      • e.g., rebelliousness (or smoking itself)
    • and the variable of interest (Y)
      • e.g., Cigarette Smoking
r z r
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
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
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
But Even CSK ResultsToo 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
r z r 24 very plausible
rZR < .24 Very Plausible

(estimated)

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
csk and follow up simulations
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
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 attrition58
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
Taxonomy of Attrition
  • Cases 1-4
    • often little or no problem
  • Cases 5-8
    • 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
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
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
Empirical Examples
  • Hedeker & Gibbons (1997)
    • 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
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
From Hedeker & Gibbons (1997)

Placebo Control

IMPS

low = better outcomes

Drug Treatment

Weeks of Treatment

longitudinal diagnostics hedeker gibbons example
Longitudinal DiagnosticsHedeker & 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
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
slide67
AAPT

Control

Program

Control

Cigarette Smoking

(high = more smoking; arbitrary scale)

Program

th

th

th

th

th

longitudinal diagnostics aapt example
Longitudinal DiagnosticsAAPT Example
  • Treatment
    • droppers do WORSE than stayers
      • little steeper increase
  • Control
    • droppers do WORSE than stayers
      • little steeper increase
  • Little evidence for Prog X DV interaction
  • Very likely MAR methods allow good conclusions (CSK scenario holds)
use of auxiliary variables
Use of Auxiliary Variables
  • Reduces attrition bias
  • Restores some power lost due to attrition
what is an auxiliary variable
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
benefit of auxiliary variables
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 variables73
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 variables74
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 r yz
Effective Sample Size by rYZ

Effective

Sample

Size

rYZ

conclusions
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
multiple imputation basic steps
Multiple Imputation:Basic Steps
  • Impute
  • Analyze
  • Combine results
imputation and analysis
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
Combine the ResultsParameter Estimates to Report
  • Average of estimate (b-weight) over 40 imputed datasets
combine the results standard errors to report
Combine the ResultsStandard 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
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

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