Introduction An approximated EM algorithm Simulation studies Discussion

Transcription

1 1 / 33 An Approximated Expectation-Maximization Algorithm for Analysis of Data with Missing Values Gong Tang Department of Biostatistics, GSPH University of Pittsburgh NISS Workshop on Nonignorable Nonresponse November 12-13, 2015

2 2 / 33 1 Introduction Background & Current Approaches Expectation-Maximization (EM) Algorithm for Regression Analysis of Data with Nonresponse 2 An approximated EM algorithm The algorithm Application in contingency table analysis Application in regression analyses with a continuous outcome 3 Simulation studies 4 Discussion

3 3 / 33 Regression Analysis of Data with Nonresponse Consider bivariate data {x i, y i, i = 1, 2,..., n} where x i s are fully observed, y i s are only observed for i = 1,..., m. Denote R i to be the missing-data indicator: R i = 1 if y i is observed and R i = 0 otherwise. Assume that [y i x i ] g(y i x i ; θ) exp{θs(x i, y i ) + a(θ)} pr[r i = 1 x i, y i ] = w(x i, y i ; ψ) and the parameter of interest is θ. Goal: to avoid modeling w(x i, y i ; ψ).

4 4 / 33 Likelihood-based Inference (I) The observed data are D obs = {x i, R i, R i y i ; i = 1,..., n}. When w(x i, y i ; ψ) is parametrically modeled, the likelihood function is L(θ, ψ; D obs ) = = n p(r i, R i y i x i ; θ, ψ) m g(y i x i ; θ)w(x i, y i ; ψ) n i=m+1 g(y i x i ; θ){1 w(x i, y i ; ψ)} dy i When w(x i, y i ; ψ) = w(x i ; ψ), data are called missing at random (MAR) and L(θ, ψ; D obs ) L(θ; D obs )L(ψ; D obs ) (Rubin, 1976).

5 5 / 33 Likelihood-based Inference (II) When data are MAR plus θ and ψ are distinct, the modeling of w(x, y; ψ) is not necessary under the likelihood-based inference. When data are not MAR, the missingness has to be modeled and the inference on θ and ψ are made together. Misspecification of the missing-data model w(x, y; ψ) often leads to biased estimate of θ.

6 6 / 33 A conditional likelihood Assume that [y i x i ] g(y i x i ; θ) (A parametric regression) pr[r i = 1 x i, y i ] = w(x i, y i ; ψ) = w(y i ; ψ). Then [X Y, R = 1] = [X Y, R = 0] = [X Y ]. Consider the following conditional likelihood: CL(θ) = R i =1 p(x i y i ; θ, F X ) (F X is the CDF of X) = R i =1 g(y i x i ; θ)p(x i ) p(y i ; θ, F X ) (Bayes formula) R i =1 g(y i x i ; θ) g(yi x; θ) df X (x) Requires knowing F X ( )!

7 A pseudolikelihood method (Tang, Little & Raghunathan, 2003) In alternative, we may either model X f (x; α), obtain α = arg max α n f (x i; α), then consider a pseudolikelihood function PL 1 (θ) = R i =1 g(y i x i ; θ) g(yi x; θ) df X (x; α) or substitute F X ( ) by the empirical distribution F n ( ): PL 2 (θ) = R i =1 p(y i x i ; θ) p(yi x; θ) df n (x) = p(y i x i ; θ) 1 n R i =1 n j=1 p(y i x j ; θ) 7 / 33

8 8 / 33 Exponential tilting (Kim & Yu, 2011) Consider the following semiparametric logistic regression model: pr[r i = 1 x i, y i ] = logit 1 {h(x i ) ψy i }, where h( ) is unspecified and ψ is either known or estimated from an external dataset with the missing values recovered. Subsequently we would have: exp(ψy) p(y x, r = 0) = p(y x, r = 1) E[exp(ψy) x, r = 1]. With both p(y x, r = 1) and E[exp(ψy) x, r = 1] empirically estimated, one will obtain a density estimator for p(y x, r = 0) and estimate θ = E[H(X, Y )] via: θ = 1 n { ri =1 H(x i, y i ) + n Ê[H(x i, y i ) x i, r i = 0]} r i =0

9 9 / 33 An instrumental variable approach (Wang, Shao & Kim, 2014) Assume that X = (Z, U) and E(Y X) = β 0 + β 1 u + β 2 z pr(r = 1 Z, U, Y ) = pr(r = 1 U, Y ) = π(ψ; U, Y ) one may use the following estimating equations to estimate ψ: n r i { π(ψ; u i, y i ) 1}(1, u i, z i ) = 0. Then estimate β s via n r i π( ψ; u i, y i ) (y i β 0 β 1 u i + β 2 z i )(1, u i, z i ) = 0.

10 10 / 33 The likelihood function From the following representation: L(θ, ψ; D obs ) = = = = n p(r i, R i y i x i ; θ, ψ) n n n p(r i, R i y i, (1 R i )y i x i ; θ, ψ) p((1 R i )y i R i, R i y i, x i ; θ, ψ) p(r i, y i x i ; θ, ψ) p((1 R i )y i R i, R i y i, x i ; θ, ψ) p(y i x i ; θ)p(r i x i, y i ; ψ) p((1 R i )y i R i, R i y i, x i ; θ, ψ),

11 11 / 33 The log-likelihood function We have Let Q(θ, ψ; θ, ψ) = l(θ, ψ; y obs ) = log L(θ, ψ; y obs ) nx = {log p(y i x i ; θ) + log p(r i x i, y i ; ψ) nx log p((1 R i )y i R i, R i y i, x i ; θ, ψ)}. E[log p(y i x i ; θ) + log p(r i x i, y i ; ψ) R i, R i y i, x i ; θ, ψ], H(θ, ψ; θ, nx ψ) = E[log p((1 R i )y i R i, R i y i, x i ; θ, ψ) R i, R i y i, x i ; θ, ψ]. Then we have l(θ, ψ; y obs ) = Q(θ, ψ; θ, ψ) H(θ, ψ; θ, ψ), for any ( θ, ψ).

12 12 / 33 The Expectation-Maximization (EM) Algorithm (Dempster, Laird and Rubin, 1977) From the Jansen s inequality, we have H(θ, ψ; θ, ψ) H( θ, ψ; θ, ψ). The EM algorithm is an iterative algorithm to find a sequence {(θ (t), ψ (t) ), t = 0, 1, 2,...} such that (θ (t+1), ψ (t+1) ) = arg max (θ,ψ) Q(θ, ψ; θ (t), ψ (t) ). This assures that l(θ (t), ψ (t) ; y obs ) l(θ (t+1), ψ (t+1) ; y obs ). Typically logistic or probit regression models are used for w(x, y; ψ). Numerical integrations or the Monte Carlo method are often necessary in the implementation.

13 13 / 33 When ψ = ψ 0 is known Now consider when the true value of ψ, ψ 0, is known: l(θ, ψ 0 ; y obs ) = log L(θ, ψ 0 ; y obs ) nx = {log p(y i x i ; θ) + log p(r i x i, y i ; ψ 0 ) log p((1 R i )y i D i,obs ; θ, ψ 0 )}. With current estimate θ (t), the Q-function becomes Q(θ; θ (t) ) = nx E[log p(y i x i ; θ) + log p(r i x i, y i ; ψ 0 ) R i, R i y i, x i ; θ (t), ψ 0 ] nx E[log p(y i x i ; θ) R i, R i y i, x i ; θ (t), ψ 0 ], because the second term does not involve θ.

14 14 / 33 Another view of the likelihood Without loss of generality, assume that the regression model follows a canonical exponential family. Then Ω(θ; θ (t) ) = mx nx logg(y i x i ; θ) + E[log g(y i x i ; θ) x i, R i = 0; θ (t), ψ 0 ] mx logg(y i x i ; θ) + i=m+1 i=m+1 nx {θe[s(x i, y i ) x i, R i = 0; θ (t), ψ 0 ] + a(θ)}, we need to obtain E[S(x i, y i ) x i, R i = 0; θ (t), ψ 0 ] in order to carry out the EM algorithm. With w(x i, y i ; ψ 0 ) known, R E[S(x i, y i ) x i, R i = 0; θ (t) s(xi, y i )g(y i x i ; θ (t) ){1 w(x i, y i ; ψ 0 )} dy i, ψ 0 ] = R g(yi x i ; θ (t) ){1 w(x i, y i ; ψ 0 )} dy i

15 15 / 33 An alternative look of the E-step On the other hand, E[S(x i, y i ) x i, θ (t) ] = E[E[S(x i, y i ) R i, x i ] x i ] We would have E[S(x i, y i ) x i, R i = 0; θ (t), ψ 0 ] = E[S(x i, y i ) x i, R i = 1; θ (t) ]pr[r i = 1 x i ; θ (t), ψ 0 ] +E[S(x i, y i ) x i, R i = 0; θ (t), ψ 0 ]pr[r i = 0 x i ; θ (t), ψ 0 ], = E[S(x i, y i ) x i, θ (t) ] E[S(x i, y i ) x i, R i = 1; θ (t), ψ 0 ]pr[r i = 1 x i ; θ (t), ψ 0 ] pr[r i = 0 x i ; θ (t), ψ 0 ]

16 16 / 33 Empirical replacements It is noted that if we use some empirical estimates of E[S(x i, y i ) x i, R i = 1; θ (t), ψ 0 ] and pr[r i = 0 x i ; θ (t), ψ 0 ] to replace them, we would be able to carry out an iterative algorithm as the following: At the E-step be[s(x i, y i ) x i, R i = 0; θ (t), ψ 0 ] = E[S(x i, y i ) x i, θ (t) ] b E[S(x i, y i ) x i, R i = 1] bpr[r i = 1 x i ] 1 bpr[r i = 1 x i ] At the M-step, find θ = θ (t+1) that solves: θ (t+1) = arg max θ Q (θ; θ (t) ) mx := arg max θ [θ{ S(x i, y i ) + nx i=m+1 be[s(x i, Y ) x i, R i = 0; θ (t) ]} + na(θ)]

17 17 / 33 An important observation In usual an EM algorithm, θ (t) may be far away from the truth θ 0. However, in order for the empirical estimates pr[r i = 1 x i ] and Ê[S(x i, y i ) x i, R i = 1] to be self-consistent with the corresponding terms under (θ (t), ψ 0 ), it is required that θ (t) is a consistent estimate of θ. Therefore we should start with a consistent initial estimate of θ and carry out this iterative algorithm to obtain another consistent estimate of θ at convergence.

18 A short summary of the modified EM 1 Choose initial value of θ (0) from either a complete dataset from subjects with similar characteristics or a completed subset recovered from recalls. 2 Compute the Nadaraya-Watson estimates or local polynomial estimates for pr[r = 1 x i ] and E[S(x i, Y ) x i, R = 1], for each i = m + 1, m + 2,..., n. 3 At the E-step of the tth iteration, calculate be[s(x i, y i ) x i, R i = 0; θ (t), ψ 0 ]. 4 At the M-step of the tth iteration, find θ = θ (t+1) that solves: nx mx nx E[S(x i, y i ) x i ; θ] = S(x i, y i ) + be[s(x i, Y ) x i, R i = 0; θ (t) ] i=m+1 With a complete external dataset {x i, y i ; i = n + 1,..., n + n E }, we would solve θ = θ (t+1) from: n+n nx X E E[S(x i, y i ) x i ; θ] + E[S(x i, y i ) x i ; θ] i=n+1 = mx S(xi, y i ) + nx be[s(xi, Y ) x i, R i = 0; θ (t) ] + n+n X E S(xi, y i ) 18 / 33

19 19 / 33 Here we design an iterative algorithm with a sequence {θ (t), t = 0, 1, 2,...} such that θ (t+1) = arg max θ Q (θ; θ (t) ) = arg max θ {Q(θ; θ (t) ) + o(1)} This algorithm will yield l(θ (t+1), ψ 0 ; y obs ) l(θ (t), ψ 0 ; y obs ) + o(1).

20 An example in contingency table analysis Consider discrete data {x i, y i, z i, i = 1,..., n}: x i s and y i s are fully observed. z i is observed for i = 1,..., c; missing for i = c + 1,..., n. Let m = n c. Essentially the observed include the fully classified table {c jkl } and the partially classified table {m jk+ }. Complete external data D E = {x i, y i, z i, i = n + 1,..., n + n E } are fully observed. A log-linear model is assumed with parameter θ, which leads to π jkl = pr[x = i, Y = j, Z = l], j = 1,..., J; k = 1,..., K ; l = 1,..., L. Initial estimates {π (0) jkl } are derived from the external complete data D E. 20 / 33

21 21 / 33 Implementation under the discrete setting Initial estimations: bpr[z = l x = j, y = k, R = 1] = bpr[r = 1 x = j, y = k] = At the tth step, for (j, k, l), we update S jkl = c jkl + P c I{x i = j, y i = k, z i = l} P c I{x i = j, y i = k} P c I{x i = j, y i = k} P n I{x i = j, y i = k} = c jk+ n jk+. = c jkl c jk+ nx I{x i = j, y i = k} bpr[z = l x = j, y = k, R = 0] i=c+1 = c jkl + m jk+ pr[z = l x = j, y = k; θ (t) ] cjkl c jk+ c jk+ n jk+ 1 c jk+ n jk+ = c jkl + m jk+ π (t) jkl c jkl π (t) n jk+ jk+ m jk+ n jk+ = n jk+ π (t) jkl π (t) jk+

22 22 / 33 Impression on the discrete setting In the E-step, we ended up as if imputing all Z i s based on (x i, y i ), including those observed ones. If we include the external data D E in the algorithm with updating the sufficient statistics through π (t) Sjkl = njkl E + S jkl = njkl E jkl + n jk+ π (t), jk+ the modified EM algorithm is equivalent to running a regular EM algorithm on the fully classified table {n E jkl } and the partial classified table {n jk+ }, or, removing the observed z i s from the complete cases (not part of the external complete data).

23 23 / 33 Implementation under the continuous setting Consider the regression analysis of [Y X] where Y is continuous and subject to nonresponse If X is discrete, the empirical approximations are: r j =1,x j =x i =k Ê[S(x i, Y ) x i = k, r i = 1] = s(x i, y j ) #{j : r j = 1, x j = x i = k} pr[r i = 1 x i = k] = #{j : r j = 1, x j = x i = k} #{j : x j = x i = k} If X is continuous, we use either the Nadaraya-Watson estimate or local polynomial estimate as Ê[S(x i, Y ) x i = k, r i = 1]; a kernel estimator for pr[r i = 1 x i = k].

24 24 / 33 Simulation settings when data are NMAR We simulated bivariate data {x i, y i, i = 1, 2,..., N} following: (i) x i N(0, 1) or x i Bin(5, 0.3). (ii) [y i x i ] N(β 0 + β 1 x i, σ 2 ), where θ = (β 0, β 1, σ 2 ) = (1, 1, 1). Keep n E = 200 of the above observations as the external datasets for obtaining initial values for θ. Simulate missing y i s from the rest n = 1000 subjects with: pr[r i = 1 x i, y i ] = Φ(ψ 0 + ψ 1 x i + ψ 2 y i ), where Φ() is the CDF of the Gaussian distribution. Compare the complete-case estimate ( θ cc ) based on the 1000 observations with missing data, MLE from the external data ( θ E ) and the proposed approximated EM estimates ( θ AEM ).

25 25 / 33 Simulation results with X N(0, 1) Methods β 0 β 1 σ 2 Complete-case analysis Empirical Bias* Empirical SD* Coverage of 95% CI 0% 8.8% 79.5% MLEs from external subset Empirical Bias* Empirical SD* Coverage of 95% CI 96.1% 93.8% 92.6% Approximated EM** Empirical Bias* Empirical SD* Coverage of 95% CI 95.0% 95.2% 93.4% The proportion of nonresponse was about 40%. * Empirical biases and SDs were the actual numbers times ** The Epanechnikov kernel was used in the approximated EM. Bootstrap was used to derive the standard errors of the b θ AEM

26 26 / 33 Simulation results with X Bin(5, 0.3) Methods β 0 β 1 σ 2 Complete-case analysis Empirical Bias* Empirical SD* Coverage of 95% CI 94.6% 11.6% 86.5% MLEs from external subset Empirical Bias* Empirical SD* Coverage of 95% CI 94.7% 94.2% 92.5% Approximated EM** Empirical Bias* Empirical SD* Coverage of 95% CI 94.7% 93.9% 93.6% The proportion of nonresponse was about 40% * Empirical biases and SDs were the actual numbers times ** The empirical averages were used in the approximated EM. Bootstrap was used to derive the standard errors of the b θ AEM.

27 27 / 33 Simulation settings when data MAR We simulated bivariate data {x i, y i, i = 1, 2,..., N} following: (i) x i N(0, 1). (ii) [y i x i ] N(β 0 + β 1 x i, σ 2 ), where θ = (β 0, β 1, σ 2 ) = (1, 1, 1). Keep n E = 200 of the above observations as the external datasets for obtaining initial values for θ. Simulate missing y i s from the rest n = 1000 subjects with: pr[r i = 1 x i, y i ] = Φ(ψ 0 + ψ 1 x i ), where Φ() is the CDF of the Gaussian distribution. Compare the complete-case estimate ( θ cc ) based on the 1000 observations with missing data, MLE from the external data ( θ E ) and the proposed approximated EM estimates ( θ AEM ).

28 28 / 33 Simulation results with X N(0, 1) Methods β 0 β 1 σ 2 Complete-case analysis Empirical Bias* Empirical SD* Coverage of 95% CI 94% 94.5% 93.4% MLEs from external subset Empirical Bias* Empirical SD* Coverage of 95% CI 96.1% 93.8% 92.6% Approximated EM** Empirical Bias* Empirical SD* Coverage of 95% CI 93.9% 94.5% 93.7% The proportion of nonresponse was about 40%. * Empirical biases and SDs were the actual numbers times ** The Epanechnikov kernel was used in the modified EM. Bootstrap was used to derive the standard errors of the b θ AEM

29 29 / 33 Without an external complete dataset... If one could obtain a reliable initial estimate θ (0) and the associated variance-covariance matrix estimate V, then we can use it as the initial and perform the M-steps via θ (t+1) = arg max θ {(θ θ (0) ) T V 1 (θ θ (0) ) + 1 n Q (θ; θ (t) )}

30 30 / 33 Without external complete dataset... Consider missing-data mechanisms such as pr[r i = 1 x i, y i ] = w(x i, y i ; ψ) = w(y i ; ψ), There are two approaches for implementing the approximated EM algorithm for data with outcome-dependent nonresponses: 1 Use the pseudolikelihood estimate as the initial estimate and run the approximated EM. 2 Use the pseudolikelihood estimate as the initial estimate and incorporate the pseudolikehood function as a component in each M-step: θ (t+1) = arg max θ {l pl (θ) + Q (θ; θ (t) )} The second approach is more computationally intensive.

31 31 / 33 Future works Need to monitor {l(θ (t) ; ψ 0 ), Q (θ (t+1) ; θ (t) ) Q(θ (t+1) ; θ (t)) ; t = 0, 1, 2...}. (X,Y,R) Search for a target function l(θ; P n (X,Y,R) a stationary point of l(θ; P n, θ (0) )., θ (0) ) so that θ AEM is Variance estimation. Link to integrative data analysis.

32 32 / 33 Acknowledgment Megan Hunt Olson, University of Wisconsin, Green Bay. Yang Zhang, Amgen Inc.

33 33 / 33 Reference 1. Dempster, A. P., N. M. Laird and D. B. Rubin (1977). Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society: Series B 39, Tang, G., Little, R.J.A. and Raghunathan, T. (2003). Analysis of Multivariate Missing Data with Nonignorable Nonresponse. Biometrika, 90, Kim, J. K. and C. L. Yu (2011). A semi-parametric estimation of mean functionals with non-ignorable missing data, Journal of the American Statistical Association 106, Wang, S., J. Shao and J. K. Kim (2014). Identifiability and estimation in problems with nonignorable nonresponse, Statistica Sinica 24,