Handling Data with Three Types of Missing Values

7/31/2019 Handling Data with Three Types of Missing Values

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Missing DataMultiple Imputation

Proposed Research

Handling Data with Three Types of Missing Values

Jennifer Boyko

Department of StatisticsUniversity of Connecticut

Storrs, CT

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Proposed Research

Outline

1 Missing DataProblemCharacterizationMethods for Handling

2 Multiple ImputationStandard MITwo Stage MI

3 Proposed ResearchProcedureCombining RulesIgnorability and Rates of Missing InformationApplication

4 Conclusion

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Proposed Research

ProblemCharacterizationMethods for Handling

The Missing Data Problem

Present in many areas of research

Small amounts can cause issues (Belin, 2009)Most statistical package defaults use complete case analysis

Problems include

biasinefficiency

unrealistic standard errors

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Proposed Research


Pattern of Missingness

Maps which values are missing in a data set

Figure: Schafer & Graham (2002)Jennifer Boyko Handling Data with Three Types of Missing Values 4 / 3 3
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Mi i D t P bl


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Proposed Research


Mechanisms of Missingness

Missing At Random (MAR)

P(R|Y, ) = P(R|Yobs, )Missingness depends on observed values of Y only

Missing Completely At Random (MCAR)P(R|Y, ) = P(R, )Missingness not dependent on observed or unobserved valuesof YSpecial case of MAR

Missing Not At Random (MNAR)Occurs when condition of MAR is violatedMissingness is dependent on Ymis or some unobserved covariate


Missing Data Problem
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Proposed Research


Ignorability

A missing data mechanism is classified as ignorable if twoconditions are met:

1 The data must be MAR or MCAR2 and must be distinct

P(, ) = P()P()Joint parameter space is the Cartesian cross-product of theindividual parameter spaces

Ignorability representes the weakest set of conditions under whichthe distribution of R does not need to be considered in Bayesian orlikelihood-based inference of (Rubin, 1976)


Missing Data Problem
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Proposed Research


Older Methods

Complete Case Analysis (CCA)

Can produce biased resultsDefault in many statistical packages

Loss of information

Single Imputation

Fills in missing values with plausible valuesImputing unconditional meansHot deck imputationConditional mean imputationLast Observation Carried Forward (LOCF)



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Missing DataS d d MI


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Proposed Research

Standard MITwo Stage MI

Standard Multiple Imputation

Multiple imputation (Rubin, 1987) uses a three step process to

analyze incomplete data sets:1 Imputation

2 Analysis

3 Combination

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Missing DataSt d d MI
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ss g tMultiple Imputation

Proposed Research


Imputation Stage

Idea: fill inm

> 1 plausible values for the missing data toaccount for model uncertainty

Create m complete data sets by drawing from the posteriorpredictive distribution of the missing values


Missing DataStandard MI


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gMultiple Imputation

Proposed Research


Analysis Stage

Analyze each of the m data sets using complete data methods

Let Q denote the parameter of interestLet Q be the complete data estimate

Let U be the variance of Q

Assumption: (Q

Q)/

U

N(0, 1)


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gMultiple Imputation

Proposed Research


Combination Stage

Q =1

m

m

j=1Q(j)

U =1

m

mj=1

U(j)

B =1

m 1

m

j=1

Q(j) Q2

T = U + (1 + m1)B


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Multiple ImputationProposed Research


Combination Stage

(Q

Q)

T t

= (m 1)1 +U

(1 + m1)B2


Missing DataM l i l I i

Standard MI
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Benefits of Multiple Imputation

Adds variability to the imputed values

Uses standard data analysis procedures after imputation

Can be very efficient

Can use the same set of imputations for several analyses


Missing DataM lti l I t ti

Standard MI


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STwo Stage MI

Two Stage Multiple Imputation

Two stage multiple imputation (Harel, 2009) considers a situationwhere we can have data missing for two different reasons

Dropout in a longitudinal study vs. intermittent missingfollow-up

Refusal to answer a question vs. a dont know response

Latent variable vs. missing planned observed values

Death vs. dropout for other reasonsUnit nonresponse vs. item nonresponse



Standard MI
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Two Stage MI

Computational Efficiency

Originally developed by Shen (2000) with the intention ofimproving computational efficiency.

Y1 Y2 Y3 Y4 Y5?

?

?

? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?



Standard MI
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Two Stage MI

Procedure

Imputation step is broken into two stages:

1 First draw m imputations of YAmis2 Conditioned on YAmis, draw n imputations of Y

Bmis

Yields a total of mn completed data sets



Standard MI
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Two Stage MI

Two Stage MI Combining Rules

Q =1

mn

mj=1

nk=1

Q(j,k)

U = 1mn

mj=1

nk=1

U(j,k)

B =1

m 1m

j=1 Qj. Q..

2

W =1

m(n 1)m

j=1

nk=1

Q(j,k) Qj.

2T = U + (1 + m1)B + (1 n1)W



Standard MIT S MI
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Two Stage MI

Two Stage MI Combining Rules

Q QT

t

1

=1

m(n

1)

(1 1/n)WT

2

+1

m

1

(1 + 1/m)B

T 2



Standard MIT St MI


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p pProposed Research

Two Stage MI

Benefits

Can simplify imputation computationally

Able to quantify how much missing information is due to eachtype of missing value which can help in planning future studies

Allows for different mechanisms of missingness for each typeof missing value (one ignorable and one nonignorable type ofmissing data)



ProcedureCombining Rules
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p pProposed Research

gIgnorability and Rates of Missing Information

Proposed Research

1 Multiple imputation in three stages including derivation of

combining rules

2 Ignorability and rates of missing information

3 Application of methodology to cognitive functioning data





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Proposed Research Ignorability and Rates of Missing Information

Benefits

Extend the benefits of two stage MI to allow for greaterspecificity regarding the data analysis

Allows for missing data to be of three different types

Allows for three different assumptions of the mechanisms ofmissingness

Can quantify the variability and missing information due to

each type of missing value



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Example 1

Example of missing data due to dropout, intermittent missingness,and a missing covariate

Y1 Y2 Y3 Y4 Y5?

?

??

? ?? ? ?

? ?? ? ?

Y1 Y2 Y3 Y4 Y5A

B

CB

A BC C C

B CC C C



P d R h

ProcedureCombining RulesI bili d R f Mi i I f i
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Example 2

Example with missing values due to item nonresponse, unitnonresponse, and latent class

Y1 Y2 Y3 Y4 Y5? ??? ???? ?

? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?

Y1 Y2 Y3 Y4 Y5A BAA BAAA B

A BA C C C CA C C C CA C C C CA C C C C



P d R h

ProcedureCombining RulesI bilit d R t f Mi i I f ti
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Process

Same as standard and two stage MI but with three stages in theimputation step and different combining rules

1 Impute L values of YAmis2 Conditioned on YAmis, impute M values of Y

Bmis

3 Conditioned on YAmis and YBmis, impute N values of Y

Cmis

Yields a total of LMN completed data sets

A second, but equivalent, method draws simultaneously from thejoint distribution of YAmis, Y

Bmis, and Y

Cmis



Proposed Research

ProcedureCombining RulesIgnorability and Rates of Missing Information
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Three Stage MI Combining Rules

Q =1

LMN

Ll=1

Mm=1

Nn=1

Q(l,m,n)

U =

1

LMN

Ll=1

Mm=1

Nn=1

U(l,m,n)

B =1

L 1L

l=1

Ql.. Q...

2

W1 =1

L(M 1)L

l=1

Mm=1

Qlm. Ql..

2

W2 =1

LM(N

1)

L

l=1

M

m=1

N

n=1

Q(l,m,n) Qlm.2



Proposed Research



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Three Stage MI Combining Rules

T = U + (1 + L1)B + (1 M1)W1 + (1 N1)W2

1 =

1 + 1

L

B

T

2(L 1)1 +

1 1

M

W1

T

2(L(M 1))1

+1

1

NW2

T2

(LM(N 1))1



Proposed Research



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Ignorability

Extension of Rubins theory of MAR and ignorability aspresented in Rubin (1976)

Harel & Schafer (2009) present an extension to two types ofmissing values

Conditional ignorability; possible to define weaker conditionsunder which M+ can be ignored in one or more stages



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Rates of Missing Information

Helps with determination of number of imputations requiredat each stage

Small numbers of imputations are required when the main

concern is relative efficiency of point estimatesEstimates for rates of missing information can be noisy forsmall numbers of imputations

Derivation of the asymptotic distribution of rates of missing

informationI will derive the estimates and asymptotic distribution for therates of missing information for three types of missing values



Proposed Research

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p g y g

Application

Cognitive functioning data

Three types of missing values will be dropout due todementia, dropout due to death unrelated to dementia, andan intermittently missing covariate

Large amounts of missing data are common in studies ofcognitive functioning (Coley et al., 2011)



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p

Conclusion

Applicable in analysis of many types of data sets

Allows researchers to quantify amount of variance attributableto each type of missing value

Informative in analysis of data and planning of future studies



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Belin, T. (2009). Missing data: what a little can do and whatresearchers can do in response. American Journal of Opthalmology

148, 820822.Coley, N. et al. (2011). How should we deal with missing data in

clinical trials involving alzheimers disease patients? CurrentAlzheimers Research 8, 421433.

Harel, O. (2009). Strategies for Data Analysis with Two Types of

Missing Values: From Theory to Application. Saarbrucken, Germany:Lambert Academic Publishing.

Harel, O. & Schafer, J. L. (2009). Partial and latent ignorability inmissing-data problems. Biometrika 96, 3750.

Rubin, D. B. (1976). Inference and missing data. Biometrika 64,

581592.Rubin, D. B. (1987). Multiple Imputation for Nonresponse in Surveys.

Hoboken, New Jersey: John Wiley & Sons, Ltd, 1st ed.

Shen, Z. (2000). Nested Multiple Imputation. Ph.D. thesis, Departmentof Statistics, Harvard University, Cambridge, MA.

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Handling Data with Three Types of Missing Values

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