1. Missing Data Mechanisms
MCAR
MAR
MNAR
References:
Schafer, J., Graham, J.W. Missing data: our view of the state of the art. Psychological Methods,7(2), , 2002
Raghunathan, T.E., What do we do with missing data ?
Some options for analysis of incomplete data. Ann. Rev. Public Health 25: , 2004

2. Graphical representation
Y = variable partly missing
X = variable completely observed
Z = cause of missingness (unrelated to Y)
R = represents missingness

3. X Z X Z X Z R Y Y R Y R
MCAR
MAR
MNAR

4. Use of conditional probability
Yc = the complete vector of Y observations
Yc = ( Yo , Ym)
MCAR: P (R | Yc) = P(R)
Prob of missing does not depend on Yo
MAR: P (R | Yc) = P( R | Yo)
Prob of missing depends only on Yo
MNAR: P (R | Yc) = P( R | Ym)
Prob of missing depends on unobserved Ym

5. Methods for analyzing data with missing values in the repeated measures situation
Case deletion: delete subjects with missing components (complete case analysis)
Available case analysis: analysis is based on all observable data (use data from subjects with complete Y vectors as well as incomplete Y vectors)

6. Simulation Study: Parameter: MCAR MAR MNAR
Μean(Y): (7.0) (19.3) (30.7)
Std(Y): (5.3) (5.8) 12.2(13.2)
Rho: (0.2) (0.37) 0.34(0.36)
Beta Y|X: (0.27) (0.51) 0.21(0.43)
Beta X|Y: (0.25) (0.44) (0.52)
Generate: 50 observations from bivariate normal (Y,X)
MCAR: prob Y missing is 0.73 (high !)
MAR: prob Y missing if X < 141
MNAR: prob Y missing if Y < 141

7. Methods for analyzing survey data
Weight responses that are present
Average the available items (social sciences based on standardized scores but not studied in any systematic fashion)

8. Single imputation MS: Mean substitution HD: Hot Deck
CM : conditional mean
PD: predictive distribution

9. Average parameter estimate (RMSE)
MCAR
Average parameter estimate (RMSE) MS HD CM PD
µ = 125.0
125.1
(7.18)
125.2
(7.89)
(6.26)
(6.57)
σ = 25.0
12.3
(13.0)
23.4
(5.40)
18.2
(8.57)
24.7
(5.37)
ρ = .60
.30
(.32)
.16
(.46)
.79
(.27)
.59
(.20)
βy|x = .60
(.45)
(.47)
.61
(.25)
.60
βx|y= .60
(.26)
.17
1.12
(.64)
(.24)

10. Average parameter estimate (RMSE)
MAR
Average parameter estimate (RMSE) MS HD CM PD
µ = 125.0
143.5
(19.4)
(19.5)
124.9
(18.1)
124.8
(18.3)
σ = 25.0
10.6
(14.6)
20.0
(6.68)
20.4
(10.7)
27.0
(8.77)
ρ = .60
.08
(.52)
.04
(.57)
.64
(.48)
.50
(.40)
βy|x = .60
(.56)
.61
.62
βx|y= .60
.20
(.44)
.06
.78
(.75)
.45

11. Average parameter estimate (RMSE)
MNAR
Average parameter estimate (RMSE) MS HD CM PD
µ = 125.0
155.5
(30.7)
(30.73)
151.6
(26.9)
σ = 25.0
6.2
(18.9)
11.7
(13.7)
8.42
(16.9)
12.9
(12.7)
ρ = .60
.08
(.47)
.04
(.53)
.64
(.40)
.50
(.37)
βy|x = .60
(.56)
.61
(.43)
.62
βx|y= .60
.20
(.55)
.06
.78
(1.72)
.45
(.68)

12. ML estimation Widely accepted
Yields unbiased estimators under general regular conditions
Provides a mechanism to do inference: testing hypotheses and confidence intervals
Often relies on the EM algorithm
Newton-Raphson /Fisher scoring used in multilevel modeling

13. Software for ML estimation
SPSS: missing data module
EMCOV
NORM
SAS: Proc Mixed
S-Plus: lme function
STATA
LISREL
Mplus
HLM / MLWin (multi-level models)

14. Simulation Study: ML estimation
Parameter: MCAR MAR MNAR
Μean(Y): (6.5) (16.9) (26.9)
Std(Y): (5.7) (7.4) (13.2)
Rho: (0.2) (0.38) (0.36)
Beta Y|X: (0.27) (0.51) 0.21(0.43)
Beta X|Y: (0.25) (0.38) (0.68)
Generate: 50 observations from bivariate normal (Y,X)
MCAR: prob Y missing is 0.73 (high !)
MAR: prob Y missing if X < 141
MNAR: prob Y missing if Y < 141

15. ML estimation More attractive than ad-hoc methods
Assume a large sample
May or may not be robust to model assumptions
Assume MAR

16. Multiple Imputation
Each missing value replaced by m > 1 values: effectively create m datasets
Efficiency: (1 + λ / m)-1 where λ is the rate of missing information implies m need not be large but certainly larger than 1
Rubin’s rules for combining estimators are now well accepted
Helps to be a Bayesian !
MAR is usually assumed

17. Software NORM Proc MI in SAS: regression, propensity scores, MCMC
This does NORM plus other routines
SAS macro: IVE library
S-Plus: missing data library (NORM)
longitudinal data uses function PAN
LISREL: missing data library like NORM
SOLAS (same as Proc MI ??)

18. Comments on MI methods
Regression based MI methods are really based on Ml estimation: usually require a multivariate normal distribution
Should you transform skewed data to normality (log or power transformation)?
Partial answer: no Graham and Schafer (1999)
Practice of rounding data to create binary/ordinal variables ?
Partial answer: okay even for small samples

19. Comments continued: However: better specialized methods are available
Schaffer (1997) for nominal data
Liu et al (2000) for clustered data
How about propensity scores ?
No: can distort covariance structure in data
(Allison, 2000)

20. Simulation Study: MI (NORM)
Parameter: MCAR MAR MNAR
Μean(Y): (6.5) (17.2) (26.9)
Std(Y): (5.9) (8.2) (12.1)
Rho: (0.2) (0.37) (0.36)
Beta Y|X: (0.27) (0.52) 0.21(0.43)
Beta X|Y: (0.22) (0.38) (0.56)
Generate: 50 observations from bivariate normal (Y,X)
MCAR: prob Y missing is 0.73 (high !)
MAR: prob Y missing if X < 141
MNAR: prob Y missing if Y < 141

21. Methods that do not assume MAR
Selection models
Pattern Mixture models

22. Food for thought
In an longitudinal study on aging many subjects die while on study
Is MAR a reasonable assumption ?
Alternatively: joint modeling of outcome and death may be superior