Missing Values Raymond Kim Pink Preechavanichwong Andrew Wendel October 27, 2015

I.Intro Missing Values and Bias II.Simulations and Imputation III.Deletion Methodology IV.Not Missing at Random

Initial Steps Why is our data missing? What is the characteristic of our missing data? How will that affect the bias? Mean? Std?

OLS Unbiased Estimator

Initial Steps 1.Identify the reason for missing data  Marriage, graduation, death, etc. 2.Understand the distribution of missing data  Certain groups more likely to have missing values 3.Decide on the best method of analysis  Deletion methods – Listwise, pairwise deletion  Single Imputation Methods – Mean substitution, dummy variable, single regression  Model based methods – Maximum likelihood and multiple imputation 4.Power and Bias  Too many missing variables reduces power  Introduction of bias in your estimator

Missing Values and Bias Are missing values moving us away or closer to the true DGP?

Conditional Distribution MCAR (missing completely at random) Probability ( Y = Missing | X,Y) = Probability (Y=Missing) Probability that Y is missing does not depend on X or Y MAR (missing at random) Probability ( Y = Missing | X,Y) = Probability (Y=Missing | X) Probability that Y is missing depends on X but not Y NMAR (not missing at random) Probability ( Y = Missing | X,Y) = Probability (Y=Missing | X,Y) Probability that Y is missing depends on Y and possibly on X Statistical Models- A.C. Davison- Cambridge University Press

Normal Data MCAR NMAR MAR Statistical Models- A.C. Davison- Cambridge University Press

Bias Matrix – Does Bias Exist? DeletionMean Imputation None (but reduced power) None< 0 Conditional None Unconditional Yes Conditional Yes < 0 Unconditional Yes Yes Statistical Models- A.C. Davison- Cambridge University Press

Working with Missing Data Deletion Maximum Likelihood Multiple Imputation Single Imputation MCAR Maximum Likelihood Multiple Imputation Single Imputation MAR Sensitivity Analysis Pattern Mixture Models Selection Model Maximum Entropy NMAR

Listwise and Pairwise Deletion Missing values are MCAR MAR BIASED NMAR Conditonal UNBIASED MCAR MAR

Single Imputation Replace missing data with mean or mode Introduces bias in estimated variance Mean Mode Substition Create indicator (1=missing, 0=not missing) Impute missing values to a constant Dummy Variable Control Replace missing values with predicted score from a regression Overestimates model fit Conditional Mean Substitution

PRESENTATION TITLE HERE Simulations and Imputation

Imputing Values Deal with missing data by generating values for those that are missing. Use a variety of methods to impute these values varying in accuracy and complexity. We will focus on single imputation methods and a few multiple imputation methods.

Mean Imputation We can use the mean in place of the missing values This will retain the mean from the dataset This will also cause a negative bias in the variance

Regression Mean Imputation Instead of using the mean, we can use regression to give us predicted values for those missing. This may allow us to achieve better estimates

Multiple Imputations A more complex way to impute missing values. Imputes and analyzes data to replace missing values within the data set.

A Few R Methods How can we do this in R?  Amelia  mi  There are many others, and some can be used to treat specific conditions for certain data sets.

Amelia Amelia is an algorithm that bootstraps data and uses that data in a multiple imputation process.

mi “mi” imputes missing values using Bayesian regression methods, which are run a number of times and analyzed for convergence. This method is very customizable, but is also very costly

Additional Resources Additional packages that can be used in R can be found here:

Imputation Summary  In order to use imputation based methods we need to first understand the data and the reason for the “missingness” of the data.  By knowing this we can fit the method that we feel is most appropriate to our data set.  Single imputation methods can give us quick and easy answers to our missing values, but they also bias statistics like the variance.  Multiple imputation methods can handle the bias better but are complex and require more specialized R packages or software

PRESENTATION TITLE HERE Deletion Methodology

Bias 0 means no bias there is a systematic tendency for the estimate to be larger than the parameter it is estimating. there is a systematic tendency for the estimate to be smaller than the parameter it is estimating. Credit: from Dr.Westfall

Listwise Vs Pairwise Deletion What are they? They are methods that discard data. How do they work? Listwise (Complete-case analysis): Excluding all units for which the outcome or any of the inputs are missing. Pairwise (Available-case analysis): Excluding a pair which contains one ore two missing values from data set. What is the difference? Pairwise attempts to minimize the loss that occurs in listwise deletion. Credit:

Listwise Vs Pairwise Deletion (Cont’) Listwise deletion Pairwise deletion

Listwise Vs Pairwise Deletion (Cont’) Pros and Cons of Listwise and Pairwise deletions: Listwise : The sample after deletion may not be representative of the full sample. Reducing power and type II error rates increase. Tendency to get bias results. Pairwise: Preserved or increase statistical power in the analyses. The result will be the same if the data has two variables (columns) Bias (over or underestimated) Credit: Credit:

PRESENTATION TITLE HERE Not Missing at Random

Case of NMAR  Why are our values missing? High income individuals don’t report income  What is the characteristic of the missing data Missing values are NMAR

Meboot Package

Evaluation of a Fund Manager While evaluating a fund manager for investment you notice that the fund did not include 2008 returns for its equity fund You highly suspect it is NMAR – It was left out because returns were bad

Evaluation of a Fund Manager You find out that the equity fund normally held stocks representative of the entire stock market Distribution of the missing data may follow the overall US equity market

Meboot Maximum Entropy

Meboot Maximum Entropy NMAR missing values requires the most assumptions Minimizing bias for NMAR depends heavily on your model setup There is no “right” answer, we do not know the true DGP All we can do is minimize bias with well grounded assumptions

Questions? THANK YOU!