1. How to deal with missing data: INTRODUCTION
Li Qi
UNC CHARLOTTE

2. General Steps for Analysis with Missing Data
1. Identify patterns/reasons for missing and recode correctly
2. Decide on best method of analysis
3. Make an inference about some aspect (parameter) of the distribution of the “full” data when some of the data are missing

3. Step 1: Understand your data
Attrition due to social/natural processes
Eg: School graduation, dropout, death…
Skip pattern in survey
Eg: Certain questions only asked to respondents who indicate they are married
Intentional missing as part of data collection process
Respondent refusal/Non-response
Observations are not sampled with the same frequency. 

4. Understand your data (Cont.)
Are certain groups more likely to have missing values?
Example: Respondents in service occupations less likely to report income
Are certain responses more likely to be missing?
Example: Respondents with high income less likely to report income

5. Missing data mechanism
MCAR (missing completely at random): The probability of missingness is independent of the data.
If the data are MCAR, then the complete-case estimator is unbiased and consistent, as our intuition would suggest.
In fact, there is no way that we can distinguish whether the missing data were MCAR or not from the observed data.-- Identifiability problem

6. Missing data mechanism (Cont.)
MAR (missing at random): The probability of missingness depends only on the observed data.
The MAR assumption allows the dependence between missingness δ and the variable Y.
P(δ=1|Y,W)=f(W);
Example: Respondents in service occupations less likely to report income

7. Missing data mechanism (Cont.)
NMAR (nonmissing at random): The probability of missingness may also depend on the unobservable part of the data.
Difficult to deal with

8. Step 2: Deal with missing data
Use what you know about why data is missing. Decide on the best analysis strategy to yield the best estimates.

9. Traditional Approaches
Deletion Methods
Listwise deletion, pairwise deletion
Single imputation methods
Mean/Mode substitution
Regression substitution

10. advanced methods Maximum Likelihood method (ML) Weighing method (IPW)
Multiple imputation method (MI)

11. Model-based Methods: ML
Identify the set of parameter values that produces the highest log-likelihood.
Advantages: Use both complete cases and incomplete cases; Enjoy the optimality properties afforded to an MLE.
Disadvantages: We need correctly specify the two parametric models; Difficult to compute.

12. Inverse probability weighting
Little and Rubin (1987) proposed this method for missing data problems in survey.

13. Inverse probability weighting
Idea: A subject with weight of 4 has a probability of observation of 0.25 (or 1/pi= 0.25). As a result, data from this subject should count once for herself and 3 times for those subjects missing.

14. Inverse probability weighting (Cont.)
Advantages: Full likelihood is not necessary; use GEE. Could be applied widely in different models.
Disadvantages: The selection probability model is not correctly specified, then IPW estimator would be biased. If the true selection probability is very small, then it could be very .

15. Inverse probability weighting (Cont.)
Robins, Rotnitzky and Zhao (1994) discussed the idea of adding an augmentation term to a simple weighted estimation equation.

16. Inverse probability weighting (Cont.)
SAS: The GEE and CAUSALTRT Procedures
R: The ipw and CausalGAM package

17. SINGLE IMPUTATION
Since some of the Y are missing, a natural strategy is to impute or “estimate” a value for such missing data and then estimate the parameter of interest behaving as if the imputed values were the true values.

18. SINGLE IMPUTATION (Cont.)
For monotone missing data patterns
Regression Method
Propensity Score Method
For arbitrary missing data patterns
MCMC method
All these options are available in SAS MI procedure.

19. Multiple Imputation (MI)
Single imputation does not refect the uncertainty about the predictions of the unknown missing values, and the resulting estimated variances of the parameter estimates will be biased toward zero.
Multiple imputation does not attempt to estimate each missing value through simulated values but rather to represent a random sample of the missing values.

20. MI procedure
Multiple imputation inference involves three distinct phases:
The missing data are filled in m times to generate m complete data sets.
The m complete data sets are analyzed by using standard procedures.
The results from the m complete data sets are combined for the inference.

21. Multiple Imputation Process

22. Multiple Imputation (Cont.)
SAS
PROC MI R
mi package

23. Time Series Data Idea: aggregation and interpolation SAS: PROC EXPAND

24. Missing Spatial Data

25. Thank you!