Imputation of missing data under missing not at … · Imputation of missing data under missing not...

Introduction

Model fornonignorablenonresponseSelection models

Pattern mixturemodels

application:Leiden 85+

Drawnindicatorimputation

Leiden 85+(re-analysis)

Conclusion

Imputation of missing data under missingnot at random assumption

&sensitivity analysis

S. Jolani

Department of Methodology and Statistics, Utrecht University, theNetherlands

Advanced Multiple Imputation, Utrecht, May 2013

Introduction






Conclusion

Outline

1 Introduction

2 Model for nonignorable nonresponseSelection modelsPattern mixture models

3 application: Leiden 85+

4 Drawn indicator imputation

5 Leiden 85+ (re-analysis)

6 Conclusion

Introduction






Conclusion

Why missing not at random (MNAR)?

There might be a reason to believe that respondersdiffer from non-responders, even after accounting forthe observed informationSome examples:

- Income - some people may not reveal their salaries- Blood pressure - the blood pressure is measured less

frequently for patients with lower blood pressures- Depression - some patients might dropout because

they believe the treatment is not effective

Introduction






Conclusion

Notation

Y : incomplete variableR: response indicator (R = 1 if Y is observed)X : fully observed covariateYobs and Ymis: the observed and missing parts of Y

Introduction






Conclusion

A general strategy

Y and R must be modeled jointly (Rubin, 1976) under anMNAR assumption

soP(Y ,R)

Introduction






Conclusion

Why the classical MI does not work?

Imputation under MAR

P(Y |X ,R = 0) = P(Y |X ,R = 1)

Introduction






Conclusion

Why the classical MI does not work?

Imputation under MAR

P(Y |X ,R = 0) = P(Y |X ,R = 1)

Imputation under MNAR

P(Y |X ,R = 0) 6= P(Y |X ,R = 1)

Introduction






Conclusion

Models for nonignorable nonresponse

Two general approaches (there are some more):

1 Selection models (Heckman, 1976)2 Pattern mixture-models (Rubin, 1977)

Introduction






Conclusion

Selection model

P(Y ,R; ξ, ω) = P(Y ; ξ)P(R|Y ;ω),

where the parameters ξ and ω are a priori independent.

P(Y ; ξ) distribution for the full dataP(R|Y ;ω) response mechanism (selection function)

Introduction






Conclusion

Selection model

Imputation model under MNAR

P(Ymis|X ,Yobs,R)

A simple but possibly inefficient approach (Rubin, 1987):

1 Draw a candidate Y ∗i ∼ P(Yi |Xi ; ξ = ξ∗)

2 Calculate p∗i = P(Ri = 1|Xi ,Yi = Y ∗

i ;ω)

3 Draw R∗i ∼ Ber(1,p∗

i )

4 Impute Y ∗i if R∗

i = 0 otherwise return to (1)

Introduction






Conclusion

Pattern mixture model

P(Y ,R;ψ, θ) = P(R;ψ)P(Y |R; θ),

where the parameters ψ and θ are a priori independent.

P(Y |X ,R = 1; θ1) distribution for the observed dataP(Y |X ,R = 0; θ0) distribution for the missing dataP(R;ψ) marginal response probability

Introduction






Conclusion

Pattern mixture model

The general procedure (Rubin, 1977):

1 Draw θ∗1 from its posterior distribution usingP(Y |X ,R = 1; θ1)

2 Specify the posterior P(θ0|θ1) a priori (e.g., θ0 = θ1 + kwhere k is a fixed constant)

3 Draw θ∗0 ∼ P(θ0|θ∗1)4 Impute Ymis from P(Y |X ,R = 0; θ∗0)

Introduction






Conclusion

An example

Suppose Y is an incomplete variable (continuous)

Yobs ∼ N(µ1, σ21), Ymis ∼ N(µ0, σ

20)

where θ1 = (µ1, σ21) and θ0 = (µ0, σ

20). Now, if we define

µ0 = µ1 + k1, σ20 = k2σ

21

where k1 and k2 are fixed and known values (sensitivityparameters).

Introduction






Conclusion

An example

Suppose Y is an incomplete variable (continuous)

Yobs ∼ N(µ1, σ21), Ymis ∼ N(µ0, σ

20)

where θ1 = (µ1, σ21) and θ0 = (µ0, σ

20). Now, if we define

µ0 = µ1 + k1, σ20 = k2σ

21

where k1 and k2 are fixed and known values (sensitivityparameters).

Sensitivity analysis:repeat the analysis for different choices of k1 and k2

Introduction






Conclusion

Application: Leiden 85+

Leiden 85+ cohort studyN=1236, 85+ on Dec. 1, 1986N=956 were visited (1987-1989)Blood pressure (BP) is missing for 121 patients

* Do anti-hypertensive drugs shorten life in the oldestold?

* Scientific interest: Mortality risk as function of BP andage

Introduction






Conclusion

Survival probability by response groupImputation techniques > Simple non-ignorable models

Survival probability by response group

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

Years since intake

K−

M S

urvi

val p

roba

bilit

y

BP measured

BP missing

Research Master MS08 2013Source: van Buuren et al. (1999)

Introduction






Conclusion

Leiden 85+

From the data we seeThose with no BP measured die earlierThose that die early and that have no hypertensionhistory have fewer BP measurements

Thus, imputations of BP under MAR could be too highvalues.We need to lower the imputed values of BP, and study theinfluence on the outcome

Introduction






Conclusion

A simple model to shift imputations

Y : BPX : age, hypertension, haemoglobin, and etc

Specify P(Y |X ,R)

Model1 Y = Xβ + ε R = 12 Y = Xβ + δ + ε R = 0

Combined formulation:

Y = Xβ + (1− R)δ + ε

δ cannot be estimated (sensitivity parameter)

Introduction






Conclusion

Numerical exampleImputation techniques > Simple non-ignorable models

Both models

Y Mixture model Selection model

Research Master MS08 2013Source: van Buuren et al. (1999)

Introduction






Conclusion

How to impute under MNAR in MICE?

> delta <- c(0,-5,-10,-15,-20)> post <- mice(leiden85,maxit=0)$post> imp.all <- vector("list", length(delta))> for (i in 1:length(delta)) {+ d <- delta[i]+ cmd <- paste("imp[[j]][,i] <- imp[[j]][,i] +",d)+ post["bp"] <- cmd+ imp <- mice(leidan85, post=post, seed=i*22, print=FALSE)+ imp.all[[i]] <- imp+ }

Introduction






Conclusion

Leiden 85+: Sensitivity analysis

Table IV. Numerical example of an NMAR non-responsemechanism, when there are more missing data for lower blood

pressures

Classmidpoint ofSystolic BP(mmHg) p (R"0 DBP) p (BP) p(BP DR"1) p (BP DR"0)

100 0)35 0)02 0)01 0)06110 0)30 0)03 0)02 0)07120 0)25 0)05 0)04 0)10130 0)20 0)10 0)09 0)16140 0)15 0)15 0)15 0)19150 0)10 0)30 0)31 0)25160 0)08 0)15 0)16 0)10170 0)06 0)10 0)11 0)05180 0)04 0)05 0)05 0)02190 0)02 0)03 0)03 0)00200 0)00 0)02 0)02 0)00

Mean(mmHg) 150 151)6 138)6

Table V. Mean and standard deviation of the observed and imputedblood pressures. The statistics of imputed BP are pooled over m"5

multiple imputations

N d SBP DBPMean SD Mean SD

Observed BP 835 152)9 25)7 82)8 13)1Imputed BP 121 0 151)1 26)2 81)5 14)0

121 !5 142)3 24)6 78)4 13)7121 !10 135)9 24)7 78)2 12)8121 !15 128)6 25)0 75)3 12)9121 !20 122)3 25)2 74)0 12)1

to model A of the introduction. It was expected that multiple imputation would raise the riskestimates in comparison with the analysis based on the complete cases, but the results do notconfirm this. Only slight differences in mortality exist, even among non-response models withvery different d’s. It appears that, for this application, risk estimates are insensitive to the missingdata and the various non-response models used to deal with them.

5. CONCLUSION

A critical point in our application is the poor prediction of blood pressure (multiple r2 (SBP)"0)24 and r2 (DBP)"0.17). The generated imputations thus are quite uncertain and containconsiderable residual noise. The increase of precision of risk estimates under the ignorable modelis therefore, at best, remote. This situation is not atypical, as low r2 is common in epidemiological

692 S. VAN BUUREN, H. BOSHUIZEN AND D. KNOOK

Statist. Med. 18, 681—694 (1999)Copyright ( 1999 John Wiley & Sons, Ltd.

Source: van Buuren et al. (1999)

Introduction






Conclusion

Drawn indicator imputation

Combined formulation: Y = Xβ + (1− R)δ + ε

if ε ∼ N(0, σ2), then

Yobs ∼ N(Xβ, σ2) (1)Ymis ∼ N(Xβ + δ, σ2) (2)

logit{P(R = 1|X ,Y )} = log[P(R = 1)P(Y |X ,R = 1)P(R = 0)P(Y |X ,R = 0)

]

= ψ0 + ψ1Y + ψ2X , (3)

where ψ1 = δ/σ2 so that δ = ψ1 ∗ σ2.

Introduction






Conclusion


Assume P(R = 1|X ,Y ) is known (unrealistic!)

Y R 1 - P(R = 1|X, Y) R1

200 1 .00 1195 1 .02 1183 1 .06 1180 1 .09 1176 1 .10 0160 1 .15 0140 1 .20 0

. 0 .25 1

. 0 .30 1

. 0 .38 0

. 0 .42 0

. 0 .45 0

. 0 .50 0

Introduction






Conclusion


Gr Y R R1 E(Y|X,R, R1)1 200 1 1 µ11

195 1 1183 1 1180 1 1

2 176 1 0 µ10160 1 0140 1 0

3 . 0 1 µ01. 0 1

4 . 0 0 µ00. 0 0. 0 0. 0 0

It can be shown that

µ10 = µ01

µ11 − µ10 ' µ01 − µ00

The idea?Impute group 3 fromgroup 2Impute group 4 fromgroups 2 and 1

Introduction






Conclusion


But in reality P(R = 1|X ,Y ) is unknown

XR RY

Figure : The schematicrepresentation of the data

Introduction






Conclusion


But in reality P(R = 1|X ,Y ) is unknown

XR RY

Figure : The schematicrepresentation of the data

Fully Conditional Specification:

Y ∼ P(Y |X ,R,R1)

R1 ∼ P(R1|X ,Y )

Introduction






Conclusion


1 Impute initially missing values (Y ∗)2 Draw R from a Bernouli process (R ∼ Ber(1, π)) whereπ = P(R = 1|X ,Y ∗)

3 Using groups 1 and 2, estimate β and δ fromE(Y |X ,R = 1,R1 = r1) = Xβ + δ(r1 − 1), r1 = 0,1

4 Draw β from its posterior distribution for a given prior forβ

5 Predict the missing data for group 3 using X β − δ6 Predict the missing data for group 4 using X β − 2δ7 Impute the missing data by adding an appropriate

amount of noise to the predicted values8 Return to Step 2

Introduction






Conclusion

How to implement the drawn indicator methodin MICE?

> mice(data, meth = "ri")

The RI function:

> mice.impute.ri(y, ry, x, ri.maxit = 10, ...)

Note:1 only for continuous variables (the current version)2 the same covariates for both models (the current

version)

Introduction






Conclusion

Leiden 85+

SummaryParticipants: 956Observed BP: 835Missing BP: 121

Imputation modelBP ∼ sex, age, hypertension, haemoglobin, etc.

Missingness mechanismlogit{P(R = 1|Y, X)} ∼ BP, type of residence, ADL,previous hypertension, etc.

- Number of iterations: 10- Number of multiple imputations: 50

Introduction






Conclusion

Leiden 85+

Table : Mean and standard error (SE) for the systolic bloodpressure using CC, MI and RI

Total ImputedMethod Mean SE Mean SECC 152.893 0.892 - -MI 152.473 0.924 149.47 2.409RI 151.075 1.109 139.06 2.438

Introduction






Conclusion

Leiden 85+

Imputation techniques > Simple non-ignorable models

Both models

Y Mixture model Selection model

Research Master MS08 2013

An interesting result:

Using the RI method, we are able to estimateδ = 139.1− 152.9 = −13.8. This value is very similar to theamount of the adjustment in van Buuren et al. (1999) basedon a numerical example.

Introduction






Conclusion

Effect of response mechanism on BP

50 100 150 200 250

0.00

00.

005

0.01

00.

015

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50 100 150 200 250

0.00

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005

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50 100 150 200 250

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015

0.02

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Systolic BP (mmHg)

dens

ity

Leiden85+ (the drawn indicator method)

100 120 140 160 180 200

0.00

0.05

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BP

dens

ity

100 120 140 160 180 200

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dens

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100 120 140 160 180 200

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BP

dens

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Numerical example (van Buuren et al. 1999)

Introduction






Conclusion

A summary of the models under MNAR

1 All methods for the incomplete data under MNAR makeunverified assumptions

2 Selection model: the distribution of the full data3 Pattern mixture: the distribution of the missing data4 Drawn indicator: the distribution of the selection

function

Introduction






Conclusion

General advice on MNAR

1 Why is the ignorability assumption is suspected? (whyMNAR assumption)

2 Include as much data as possible in the imputationmodel

3 Limit the possible non-ignorable alternatives

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