Roberta Harnett
MAR 550
October 30, 2007

 When do we see missing data?
 Types of missing data
 Traditional approaches
 Deletion
 Substitution
 Modern Approaches
 Maximum likelihood and Bayes
 Software

 Medical studies, nonresponse in surveys or censuses, dropouts in clinical trials, censored data
 Loss of information, power
 Bias in results due to differences in missing and observed data
 Complicated analysis with standard software

 MCAR
 MAR
 MNAR

 Missing Completely at Random
 Probability that x i is missing doesn’t depend on its value or on value of other variables

Doesn’t matter if it is associated with other
“missingness”

 Missing at Random

Missingness doesn’t depend on x i controlling for other variable after
 This is not great, but we can deal with it

 Missing Not at Random
 Not MCAR or MAR (anything else)
 BAD!!
 Model missingness

 Deletion
 List-wise

Unbiased, but loses power

Alternatives are really replacements for list-wise
 Pairwise (also called “unwise”) deletion

Leads to different sample sizes for different parts of analysis

Can be a disaster

 Single Imputation
 Hot deck

Census Bureau
 vs. Cold deck
 Mean substitution
 Regression substitution
 Stochastic regression substitution

 Maximum Likelihood
 EM algorithm

Estimate parameters
 Listwise deletion, add some error

Predict missing data

(M): Maximize likelihood. Repeat.
 NORM (http://www.stat.psu.edu/~jls/misoftwa.html)

 Multiple Imputation
 Simple and general – works for any type of analysis
 Validity of method depends on how imputation is carried out

Should reasonably predict missing data, but should also reflect uncertainty in predictions

Using a “sensible” imputation model

 Predict missing values, then add error component drawn randomly from residual distribution of the variable
 Repeat several times to improve error estimates

 Use Bayesian arguments to impute data:
 Parametric model for data

Ignorable missing data

Non-ignorable missing data
 Apply prior for unknown model parameters
 Simulate m independent draws from distribution of Y mis given Y obs
 Calculate values explicitly or through MCMC

 Simulate a random draw of unknown parameters from observed-data posterior
 Simulate a random draw of missing values from conditional predictive distribution
 Repeat, obtaining new parameter estimates from
“complete” data set until stabilizes
 Do 3-5 times total (Rubin)
 MCMC: data augmentation algorithm of Tanner and
Wong (1987)

T = 1 m
 Calculate parameter Q from m data sets
 Estimate of Q is just average of m values of Q
 Variance of Q is T = (1+m -1 ) B + U


Where U is the mean within-imputation variance and B is
B = (1/m) Σ (Q l
-Q ave
) 2
The between-imputation variability.
As m → ∞ , T = B + U and you don’t need to correct B for low numbers of imputations.

 Imputation is computationally distinct from analysis
 Problem if assumptions of imputation are not compatible with analysis assumptions
 Loss of power if imputation makes fewer assumptions than analysis

“Superefficient” if imputation is based on more
(valid) assumptions than analysis

 Inconsistent if imputation makes invalid assumptions that are not included in analysis
 Ex: interaction terms
 Imputation needs to preserve features of data that will be included in analysis

 Approximate Bayesian Bootstrap (Rubin,
1987)
 Fancier version of Hot deck imputation

 Removing entries with missing data vs. MI
 Imputing once vs. MI
 Number of imputations
 Efficiency is (1+
λ/m
) -1
 MI vs. EM

 Ignorable if data are MAR
 MI can be used when there is nonignorable nonresponse
 Missing-data mechanism

 For S-PLUS: www.stat.psu.edu/~jls/misoftwa.html
 For R:
 Amelia (II) (surveys and time-series data)
 Norm (for multivariate normal data)
 SOLAS (tested by Allison, 2000)
 For windows

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Little, R.J.A. and Rubin, D.B. (1987) Statistical Analysis with Missing Data . J. Wiley &
Sons, New York.
Schafer, J.L. (1999) Multiple imputation: a primer. Statistical Methods in Medical
Research , 8, 3-15.
Barnard, J. and X. Meng. (1999) Applications of multiple imputation in medical studies: from AIDS to NHANES. Statistical Methods in Medical Research , 8, 17-36.
http://www.uvm.edu/~dhowell/StatPages/More_Stuff/Missing_Data/Missing.html http://www.stat.psu.edu/~jls/mifaq.html#em
Allison, P.D. (2000) Multiple Imputation for Missing Data: A Cautionary Tale.
Sociological Methods and Research , 28 (3), 301-309.

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AIDS survival time with reporting-delay
(1) Survival-time model
(2) Reporting-lag model using available information
(3) Multiply impute delayed cases using model from step 2
(4) Compute estimates of survival-time model parameters
(5) Combine estimates using repeated-imputation rules

 Effects of school choice on achievement tests (public vs. private schools)
 School vouchers to attend “choice” schools, participating private schools
 Only households with less than 1.75 times poverty line could participate

 Randomized block design
 Outcome variables were scores from ITBS
 Maximum of 4 years observed (1990-1994)
 Higher levels of missingness than in typical medical study
 Pattern in missing data was not monotone