1 Multiple Imputation : Handling Interactions Michael Spratt

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Multiple Imputation : Handling Interactions

Michael Spratt

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Introduction

• Missing data is considerable problem• Complete case analysis will generally lead

to systematic bias• Have to make some assumption

– Most commonly used is Missing at Random (MAR)

– MNAR uses different assumptions

• In this talk we are discussing analysis when MAR assumption is made

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Introduction : MAR

• In MAR, the probability of being missing does not depend on the missing data itself, given the observed data and the model parameters– Unlike MNAR analysis, we do not have to

explicitly the model missingness mechanism

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Introduction : MAR and multiple imputation

• Most common approach : perform imputation of missing data and save multiple imputed datasets– Each imputed dataset differs (slightly) due to

stochastic nature of imputation

• Then carry out substantive analysis on each of the imputed datasets

• Then combine the individual results (using Rubin’s Rules) to obtain combined imputation estimates and standard errors

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MICE/ICE for imputation assuming MAR

• MICE : Multiple Imputation using Chained Equations has been widely used for imputation– Sometimes called FCS (fully conditional specification)– For general missingness patterns (does not not have to assume

monotone missingness)

• Implemented in– MICE package in R (van Buuren et. al.)– ICE command in Stata (Royston)– IVEWARE (Raghunathan et. al.)– Potentially task-specific versions be written in other programs e.g.

WinBUGS

• Ref : Multiple imputation of missing blood pressure covariates in survival analysis. Van Buuren, Boshuizen, Knook. Statistics in Medicine 1999; 18(6): 681–94.

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• X1, X2, X3 …Xn partially observed• Zobs represents set of fully observed variablesChained equations are :• X1 ~ f(X2, X3 , X4 … Xn, Zobs)• X2 ~ f(X1, X3 , X4 … Xn, Zobs)• X3 ~ f(X1, X2 , X4 … Xn, Zobs)etc.

• Comparable to Gibbs Sampler• Much shorter chains which on termination produce an

imputed dataset

MICE/ICE for imputation assuming MAR

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Interactions in the Analysis Model

• A useful practical guide to using imputation to perform analysis in the presence of missing data are

• Multiple imputation: current perspectives (Kenward and Carpenter Statistical Methods in Medical Rearch16: 199–218)

• Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls (Sterne, White, Carpenter et. al. BMJ 2009;338:b2393) also contains useful guidance

• The imputation model should be at least as rich as the substantive model– The imputation model should preserve the structure of

the data

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MICE/ICE for imputation

• For most datasets where distributional assumptions are met, MICE/ICE has been shown in practice to work well for MAR data

• More care is needed when models contain structures such as interactions, multi-level, non-linearity etc. In particular the structure of the substantive model should be reflected in the imputation model

• This talk focuses on interactions

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Why omitting interactions in the imputation may cause problems

• Take as an example 3 binary variables X, Y, Z• We are interested in a substantive analysis in

the presence of missing data of the logistic regression of Y on X and Z with an interaction

• logit(P(Y=1| X, Z)) = 0+ xx + zz + xzx.z

• We initially have a full [X,Y,Z] dataset, but it then becomes subject to missingness (MAR mechanisms)

• We would like the parameter estimates after MAR followed by imputation occurs to be the same as the full data estimates

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• The coefficients are the same as the coefficients of the corresponding log-linear model– logistic :

• logit(Y) = 0+ xx + zz + xzx.z

– log-linear• log(xyz) = 0+ xx + yy + zz +

xyx.y + xzx.z + yzy.z + xyzx.y.z

– Examining the bias of x is equivalent to examining the bias of xy; same for z and yz; and for xz and xyz

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• Omitting interactions terms in the full conditional models will lead to interactions in the log-linear model being underestimated and hence P(X, Y, Z) being incorrectly estimated

• This can also be seen by looking at the number of parameter estimates needed– If just X is subject to missingness, we need to be able to estimate

P(X | Y,Z) P(Y, Z)• 4 parameter estimates needed for P(X | Y, Z)• This cannot be done with chained equation without interaction X = + y Y + z z

as there are only 3 free parameters– If X and Y are subject to missingess, we need to be able to estimate

P(X, Y | Z) P(Z)• 8 parameter estimates in general needed for P(X, Y | Z)• This cannot be done with chained equations without interaction x = x+ xy y + xz z

and y = y + yx x + yz z as there are only 6 free parameters

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Passive Imputation• Imputation interactions are needed. Both the Stata

program ICE (and also the R MICE package) support passive imputation

• The interaction term is recalculated from the main effects after every mice cycle and can then be made use of in the subsequent chained equations in the cycle for the imputation of other variable(s)– Other possible approaches :

– Von Hippel “How to impute interactions, squares and other transformed variables”, Sociological Methodology 39:265-291 2009 is a less established alternative to passive imputation

– It is also worth noting that where a categorical variable is fully observed an alternative method of imputation is to split it by values of the fully observed variable and separately impute subsets of data

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Simulation Structure1. We created a [X,Z] dataset2. We created Y stochastically given X and Z3. We stochastically created missingness (MAR) in 1, 2

or 3 variables4. Using a number of imputation models we did the

imputation and performed the substantive analysis

Steps 2-4 were repeated 100 times and parameter estimates and standard errors were recorded

We tabulated the median of the parameter estimates, the median of the confidence intervals and the coverage of the original data generation parameter within the parameter estimate’s confidence intervals

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Simulations

• We examined the effect of interactions on analysis of imputed data in a series of simulation scenarios involving 3 variables;– Regression with outcome Y and covariates X and Z

• The simulation scenarios ranged through all 3 variables being binary; 2 variables binary and one variable normal; one variable binary and 2 normal variables; to 3 normal variables– In each case varying combinations of outcomes and

covariates complete/incomplete

• We present a subset of the simulation scenarios

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All variables binary; X and Z incomplete

• Dataset : 20,000 observations, Y generated stochasticallylogit(Y) = 0.5 × X + 0.5 × Z + 0.6 × X × Z

• Data divided into 2 sections with Bernoulli distribution (p = 0.5) [splitting allows missingness to be MAR]

• Two stratified MAR patterns :– logit(Z is missing) = -2 + X + Y (In one section of data)– logit(X is missing) = -2 + 1.3 × Z + 0.8 × Y (Other section of data)

• Imputation then substantive analysis performed

• In a second simulation scenario there were 3 stratified MAR patterns :– P(Z missing | X, Y) (In section 1 of data)– P(X missing | Y, Z) (In section 2 of data)– P(X and Z jointly missing | Y) (In section 3 of data)

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All variables binary; X and Z incomplete

Full data

VarMissingness mechanism

% missing values (95% range)

Median OR (median CI)


CI coverage

Median OR (median CI) CI coverage

X 14.0 (13.6, 14.4) 0.49 (0.37,0.62) 0.32 (0.18,0.46) 0.27 0.58 (0.44,0.71) 0.81

Z 23.0 (22.7, 23.4) 0.49 (0.36,0.63) 0.54 (0.39,0.70) 0.90 0.59 (0.44,0.73) 0.82

XZ 0.61 (0.45,0.76) 0.58 (0.40,0.76) 0.91 0.45 (0.28,0.62) 0.62

X 21.8 (21.4, 22.2) 0.49 (0.37,0.62) 0.45 (0.31,0.59) 0.82 0.59 (0.45,0.72) 0.74

Z 23.1 (22.6, 23.5) 0.49 (0.36,0.63) 0.43 (0.28,0.59) 0.86 0.61 (0.46,0.76) 0.69

XZ 0.61 (0.45,0.76) 0.59 (0.41,0.77) 0.90 0.45 (0.28,0.62) 0.58

(Z ~ X + Y; X ~ Y + Z)

Complete case Imputed, no interaction

2 stratified MAR patterns (interaction)

3 stratified MAR patterns



CI coverage

Median OR (median CI) CI coverage Median OR (median CI)

X 0.54 (0.41,0.67) 0.90 0.55 (0.42,0.68) 0.85 0.50 (0.36,0.64)

Z 0.56 (0.41,0.70) 0.87 0.55 (0.40,0.69) 0.89 0.50 (0.35,0.65)

XZ 0.51 (0.33,0.69) 0.89 0.51 (0.34,0.69) 0.88 0.60 (0.42,0.77)

X 0.53 (0.39,0.67) 0.94 0.56 (0.43,0.70) 0.89 0.50 (0.36,0.64)

Z 0.55 (0.39,0.70) 0.86 0.56 (0.41,0.72) 0.85 0.49 (0.33,0.65)

XZ 0.53 (0.35,0.71) 0.87 0.50 (0.32,0.67) 0.81 0.61 (0.42,0.79)

(Z ~ X + Y + XY; X ~ Y + Z + YZ)(Z ~ X + Y + XY; X ~ Y + Z) (Z ~ X + Y; X ~ Y + Z + YZ)

2 stratified MAR patterns (interaction)

0.93

0.95

0.93

CI coverage

Imputed, XY interaction Imputed, YZ interaction Imputed, XY, YZ interaction

3 stratified MAR patterns

0.93

0.94

0.93

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All variables binary; X, Z and Y incomplete

• Data generated stochasticallylogit(Y) = 0.5 × X + 0.5 × Z + 0.6 × X × Z

• Data divided randomly into 3 sections with equal probability• 3 stochastic stratified MAR patterns :

– logit(Z is missing) = -2 + X + Y (In section 1 of data)– logit(X is missing) = -2 + 1.3 × Z + 0.8 × Y (In section 2 of data)– logit(Y is missing) = -1.5 + 1.9 × Z + 0.6 × X (In section 3 of data) data)

• In a second simulation scenario there were 6 stratified MAR patterns :

– P(Z missing | X, Y) (In section 1 of data)– P(X missing | Y, Z) (In section 2 of data)– P(Y missing | X, Z) (In section 3 of data)– P(X and Y jointly missing | Z) (In section 4 of data)– P(X and Z jointly missing | Y) (In section 5 of data)– P(Y and Z jointly missing | X) (In section 6 of data)

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All variables binary; X, Z and Y incompleteFull data


% Missing values (95% range)



CI coverage


CI coverage

X 11.9 (11.6, 12.2) 0.49 (0.37,0.62) 0.44 (0.31,0.58) 0.82 0.56 (0.43,0.70) 0.83

Z 13.0 (12.7, 13.3) 0.49 (0.36,0.63) 0.40 (0.25,0.56) 0.76 0.61 (0.46,0.76) 0.67

XZ 0.61 (0.46,0.77) 0.56 (0.37,0.74) 0.91 0.43 (0.25,0.60) 0.50

Y 17.2 (16.9,17.6)

X 15.2 (14.8, 15.6) 0.49 (0.37,0.62) 0.46 (0.32,0.59) 0.84 0.57 (0.44,0.71) 0.82

Z 20.9 (20.5, 21.2) 0.49 (0.36,0.63) 0.45 (0.30,0.60) 0.85 0.60 (0.45,0.75) 0.73

XZ 0.61 (0.45,0.76) 0.58 (0.40,0.76) 0.92 0.44 (0.26,0.61) 0.57

Y 21.4 (21.0,21.7)


3 stratified MAR patterns for Z, X and Y




CI coverage


CI coverage

CI coverage


CI coverage

X 0.53 (0.39,0.66) 0.96 0.53 (0.40,0.66) 0.96 0.52 (0.39,0.66) 0.49 (0.36,0.63) 0.94

Z 0.55 (0.40,0.70) 0.87 0.53 (0.38,0.68) 0.91 0.54 (0.39,0.70) 0.49 (0.33,0.65) 0.95

XZ 0.52 (0.34,0.69) 0.87 0.55 (0.37,0.73) 0.94 0.54 (0.35,0.71) 0.61 (0.43,0.79) 0.93

X 0.54 (0.40,0.67) 0.93 0.53 (0.40,0.67) 0.94 0.51 (0.38,0.65) 0.50 (0.36,0.63) 0.89

Z 0.55 (0.40,0.71) 0.90 0.53 (0.38,0.69) 0.94 0.52 (0.37,0.67) 0.50 (0.34,0.65) 0.96

XZ 0.51 (0.33,0.68) 0.85 0.55 (0.37,0.72) 0.92 0.56 (0.39,0.74) 0.60 (0.42,0.78) 0.90

Imputed, XY, YZ interactions Imputed, YZ, XZ interactions Imputed, XY, XZ interactionsImputed, XY, YZ, XZ interactions



0.96

0.89

0.90


0.93

0.97

0.92

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Y continuous, X and Z binary; X, Z and Y incomplete

• Data generated stochastically, this time Y is continuousY ~ 0.45 × X + 0.55 × Z + 0.6 × X × Z + N(0, 1)

• Data divided randomly into 3 sections with equal probability

• 3 stochastic stratified MAR patterns :logit(Z is missing) = -2.5 + 1.5 × X + Y (In section 1 of data)logit(X is missing) = -3 + 2.5 × Z + 0.8 × Y (In section 2 of data)logit(Y is missing) = -4 + 2.5 × Z + 2.0 × X (In section 3 of data)

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Y continuous, X and Z binary; X, Z and Y incomplete

VarMedian coef (median CI) CI coverage

Median coef (median CI) CI coverage

CI coverage


X 0.46 (0.40,0.53) 0.87 0.48 (0.42,0.55) 0.83 0.47 (0.40,0.53) 0.46 (0.39,0.52) 0.90

Z 0.57 (0.50,0.64) 0.94 0.58 (0.51,0.65) 0.89 0.59 (0.52,0.67) 0.56 (0.48,0.63) 0.95

XZ 0.56 (0.48,0.65) 0.83 0.55 (0.47,0.63) 0.80 0.55 (0.46,0.63) 0.59 (0.51,0.67) 0.94

Imputed, XY, YZ interaction Imputed, YZ, ZX interaction Imputed, XY, XZ interaction Imputed, XY, YZ, XZ interaction

Median coef (median CI)

0.87

0.81

0.77

Full data

Var% missing values (95% range)



CI coverage


X 13.0 (12.7, 13.3) 0.46 (0.40,0.52) 0.40 (0.34,0.46) 0.58 0.50 (0.43,0.56) 0.74

Z 14.6 (14.2, 14.9) 0.55 (0.49,0.62) 0.47 (0.39,0.54) 0.34 0.62 (0.55,0.69) 0.49

XZ 0.59 (0.52,0.67) 0.47 (0.39,0.55) 0.12 0.48 (0.40,0.56) 0.20

Y 11.2 (10.9, 11.5)


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Further simulations

• In further simulations similar results were obtained, where the distributional assumptions of the imputations models were adhered to

• In each case omitting an interaction in a chained equation produced biased results. All 2-way interactions had to be included

• Starting with a tri-variate normal distribution and introducing a slight interaction (slight non-normality results) also gave imputed estimates closest to the full data estimates when the full interactions were introduced into the imputation model

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Conclusions

• In general the imputation models should reflect the structure of the substantive analysis, and should be at least as rich as the analysis model

• In order to reflect the structure of the substantive model, the imputation model should not exclude its interactions, and should also include any corresponding interactions involving the outcome variable

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Acknowledgements

• This work was done in collaboration with Jonathan Sterne, Kate Tilling and James Carpenter

• Helpful comments and suggestions from Paul Clarke are gratefully acknowledged

Documents

1 Multiple Imputation : Handling Interactions Michael Spratt