Missing Values

Raymond KimPink PreechavanichwongAndrew Wendel

October 27, 2015

I. Intro Missing Values and BiasII. Simulations and ImputationIII. Deletion MethodologyIV. 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?

,

https://www.utexas.edu/cola/prc/_files/cs/Missing-Data.pdf

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

https://www.utexas.edu/cola/prc/_files/cs/Missing-Data.pdf

Missing Values and Bias

Are missing values moving us away or closer to the true DGP?

𝐵𝑖𝑎𝑠�̂�=𝐸𝜃 [ �̂� ]−𝜃=𝐸𝜃[ �̂�−𝜃 ]

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

Example: Sea Level =

Normal Data MCAR

NMAR MAR

Statistical Models- A.C. Davison- Cambridge University Press

Bias Matrix – Does Bias Exist?

Deletion Mean Imputation

None (but reduced power)

None (but reduced power)

None < 0

ConditionalNone

UnconditionalYes

ConditionalNone

UnconditionalYes

ConditionalNone

UnconditionalYes

ConditionalYes < 0

UnconditionalYes

Yes Yes 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 ImputationMAR

• Sensitivity Analysis• Pattern Mixture Models• Selection Model• Maximum Entropy

NMAR

https://www.utexas.edu/cola/prc/_files/cs/Missing-Data.pdf

Listwise and Pairwise Deletion

Missing values are MCAR

MAR

BIASEDNMAR

Conditonal

UNBIASEDMCAR

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

https://www.utexas.edu/cola/prc/_files/cs/Missing-Data.pdf

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

http://missingdata.lshtm.ac.uk/

Multiple Imputations

• A more complex way to impute missing values.

• Imputes and analyzes data to replace missing values within the data set.

http://www.stefvanbuuren.nl/mi/MI.html

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.

http://gking.harvard.edu/files/gking/files/amelia_jss.pdf?m=1360040717

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

https://cran.r-project.org/web/packages/mi/mi.pdf

Additional Resources

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

http://www.stefvanbuuren.nl/mi/Software.html

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: email 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: http://www.stat.columbia.edu/~gelman/arm/missing.pdf]

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: https://www.statisticssolutions.com/missing-data-listwise-vs-pairwise/Credit: http://files.eric.ed.gov/fulltext/ED281854.pdf

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 dataMissing values are NMAR

Our sample

Meboot Package

Our NMAR missing values introduce the most unsolvable estimator bias We don’t know the true distribution. But we can infer a similar distribution

for imputation. Maximum Entropy is for time series statistical inference when traditional

assumptions are unreliable For the worst case scenario:

• Missing values are NMAR

• Missing values follow a different distribution

• Extraction of this distribution is not available from historical data– i.e. company stock enters bankruptcy

– Company stock trading is halted

– Your client is calling and wants to know whether they should sell or hold

– This is a methodology for a “best guess” in the worst possible case

Our sample

https://cran.r-project.org/web/packages/meboot/index.html

Evaluation of a Fund Manager

Bond Fund Equity Fund2007 8.54% 7.58%2008 9.58% NA2009 -1.87% 23.14%2010 5.46% 13.44%

Yearly Returns10 Year Treasury S&P 500

2007 10.21% 5.48%2008 20.10% -36.55%2009 -11.12% 25.94%2010 8.46% 14.82%

Yearly Returns

• 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

Equity Fund '07,'09, '10 US MarketsIT 20.7% 20.4%Financials 17.5% 16.5%Health Care 15.8% 14.7%Cons. D 15.3% 13.1%Industrials 10.8% 10.1%Cons. S 10.1% 9.9%Energy 8.2% 6.9%Utilities 3.5% 3.1%Materials 3.0% 2.8%Telecom 2.8% 2.4%

Sector Breakdown

• 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

• Data dependent nonstandard bootstrap

• Creates a population of time series that is non-stationary (i.e. mean changes over time)

• Creates a large number of replicates based on your provided ensemble 1. Sorts provided data in increasing order2. Compute intermediate points of sorted data3. Compute min/max4. Compute mean preserving constraints5. Generate random U[0,1] interval iterations6. Repeat

https://cran.r-project.org/web/packages/meboot/index.html

Meboot Maximum Entropy

Bond Fund Equity Fund2007 8.54% 7.58%2008 9.58% -35.47%2009 -1.87% 23.14%2010 5.46% 13.44%m 5.43% 14.72%s 5.17% 7.86%

mME 2.17% s ME 25.90%

Yearly Returns

• 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?

Questions?

THANK YOU!