,..., θ(2),..., θ(n)

Transcription

1 Likelihoods for Multivariate Binary Data Log-Linear Model We have 2 n 1 distinct probabilities, but we wish to consider formulations that allow more parsimonious descriptions as a function of covariates. One choice is the log-linear model: 0 1 Pr(Y = y) = c(θ) X θ (1) j y j + X θ (2) j 1 j y j1 y j θ (n) n y 1...y n A, j j 1 <j 2 with 2 n 1 parameters θ = (θ (1) 1,..., θ(1) n, θ (2) and where c(θ) is the normalizing constant. 12,..., θ(2) n 1,n,..., θ(n) 12...n )T, This formulation allows calculation of cell probabilities, but is less useful for describing Pr(Y = y) as a function of x. Note that we have 2 n 1 parameters and we have two aims: reduce this number, and introduce a regression model. 324 Example: n = 2. We have Pr(Y 1 = y 1, Y 2 = y 2 ) = c(θ) exp θ (1) 1 y 1 + θ (1) 2 y 2 + θ (2) 12 y 1y 2, where θ = (θ (1) 1, θ(1) 2, θ(2) 12 )T and c(θ) 1 = 1X 1X y 1 =0 y 2 =0 exp θ (1) 1 y 1 + θ (1) 2 y 2 + θ (2) 12 y 1y 2 y 1 y 2 Pr(Y 1 = y 1, Y 2 = y 2 ) 0 0 c(θ) 1 0 c(θ) exp(θ (1) 1 ) 0 1 c(θ) exp(θ (1) 2 ) 1 1 c(θ) exp(θ (1) 1 + θ (1) 2 + θ (2) 12 ) 325

2 Hence we have interpretations: exp(θ (1) 1 ) = Pr(Y 1 = 1, Y 2 = 0) Pr(Y 1 = 0, Y 2 = 0) = Pr(Y 1 = 1 Y 2 = 0) Pr(Y 1 = 0 Y 2 = 0) the odds of an event at trial 1, given no event at trial 2; exp(θ (1) 2 ) = Pr(Y 1 = 0, Y 2 = 1) Pr(Y 1 = 0, Y 2 = 0) = Pr(Y 2 = 1 Y 1 = 0) Pr(Y 2 = 0 Y 1 = 0) the odds of an event at trial 2, given an event at trial 1; exp(θ (12) 12 ) = Pr(Y 1 = 1, Y 2 = 1)Pr(Y 1 = 0, Y 2 = 0) Pr(Y 1 = 1, Y 2 = 0)Pr(Y 1 = 0, Y 2 = 1) = Pr(Y 2 = 1 Y 1 = 1)/ Pr(Y 2 = 0 Y 1 = 1) Pr(Y 2 = 1 Y 1 = 0)/ Pr(Y 2 = 0 Y 1 = 0) the ratio of odds of an event at trial 2 given an event at trial 1, divided by the odds of an event at trial 2 given a at event at trial 1. Hence if this parameter is larger than 1 we have positive dependence. 326 Quadratic Exponential Log-Linear Model We describe three approaches to modeling binary data: conditional odds ratios, correlations, marginal odds ratios. Zhao and Prentice (1990) consider the log-linear model with third and higher-order terms set to zero, so that 0 1 Pr(Y = y) = c(θ) X j θ (1) j y j + X θ (2) jk y jy k A. j<k For this model Interpretation: Pr(Y j = 1 Y k = y k, Y l = 0, l j, k) Pr(Y j = 0 Y k = y k, Y l = 0, l j, k) = exp(θ(1) j + θ (2) jk y k). exp(θ (1) j ) is the odds of a success, given all other responses are zero. exp(θ (2) jk ) is the odds ratio describing the association between Y j and Y k, given all other responses are fixed (equal to zero). 327

3 Limitations: 1. Suppose we now wish to model θ as a function of x. Example: Y respiratory infection, x mother s smoking (no/yes). Then we could let the parameters θ depend on x, i.e. θ(x). But the difference between θ (1) j (x = 1) and θ (1) j (x = 0) represent the effect of smoking on the conditional probability of respiratory infection at visit j, given that there was no infection at any other visits. Difficult to interpret, and we would rather model the marginal probability. 2. The interpretation of the θ parameters depends on the number of responses n particularly a problem in a longitudinal setting with different n i. 328 Bahadur Representation Another approach to parameterizing a multivariate binary model was proposed by Bahadur (1961) who used marginal means, as well as second-order moments specified in terms of correlations. Let Then we can write 0 R j = Y j µ j [µ j (1 µ j )] 1/2 ρ jk = corr(y j, Y k ) = E[R j R k ] ρ jkl = E[R j R k R l ] ρ 1,...,n = E[R 1...R n ] Pr(Y = y) + X ρ jk r j r k + X j<k j<k<l ny j=1 µ y j j (1 µ j) 1 y j 1 ρ jkl r j r k r l ρ 1,...,n r 1 r 2...r n A Appealing because we have the marginal means µ j and nuisance parameters. 329

4 Limitations: Unfortunately, the correlations are constrained in complicated ways by the marginal means. Example: consider measurements on a single individual, Y 1 and Y 2, with means µ 1 and µ 2. We have and corr(y 1, Y 2 ) = Pr(Y 1 = 1, Y 2 = 1) µ 1 µ 2 {µ 1 (1 µ 1 )µ 2 (1 µ 2 )} 1/2 max(0, µ 1 + µ 2 1) Pr(Y 1 = 1, Y 2 = 1) min(µ 1, µ 2 ), which implies complex constraints on the correlation. For example, if µ 1 = 0.8 and µ 2 = 0.2 then 0 corr(y 1, Y 2 ) Marginal Odds Ratios An alternative is to parameterize in terms of the marginal means and the marginal odds ratios defined by γ jk = Pr(Y j = 1, Y k = 1) Pr(Y j = 0, Y k = 0) Pr(Y j = 1, Y k = 0)Pr(Y j = 0, Y k = 1) = Pr(Y j = 1 Y k = 1)/Pr(Y j = 0 Y k = 1) Pr(Y j = 1 Y k = 0)/Pr(Y j = 0 Y k = 0) which is the odds that the j-th observation is a 1, given the k-th observation is a 1, divided by the odds that the j-th observation is a 1, given the k-th observation is a 0. Hence if γ jk > 1 we have positive dependence between outcomes j and k. It is then possible to obtain the joint distribution in terms of the means µ, where µ j = Pr(Y j = 1) the odds ratios γ = (γ 12,..., γ n 1,n ) and contrasts of odds ratios 331

5 We need to find E[Y j Y k ] = µ jk, so that we can write down the likelihood function, or an estimating function. For the case of n = 2 we have and so γ 12 = Pr(Y 1 = 1, Y 2 = 1) Pr(Y 1 = 0, Y 2 = 0) Pr(Y 1 = 1, Y 2 = 0)Pr(Y 1 = 0, Y 2 = 1) = µ 12(1 µ 1 µ 2 + µ 12 ) (µ 1 µ 12 )(µ 2 µ 12 ), µ 2 12(γ 12 1) + µ 12 b + γ 12 µ 1 µ 2 = 0, where b = (µ 1 + µ 2 )(1 γ 12 ) 1, to give µ 12 = b ± p b 2 4(γ 12 1)µ 1 µ 2. 2(γ 12 1) Y Y µ 1 1 µ 12 µ 1 1 µ 2 µ Limitations In a longitudinal setting (we add an i subscript to denote individuals), finding the µ ijk terms is computationally complex. Large numbers of nuisance odds ratios if n i s are large assumptions such as γ ijk = γ for all i, j, k may be made. Another possibility is to take log γ ijk = α 0 + α 1 t ij t ik 1, so that the degree of association is inversely proportional to the time between observations. 333

6 Modeling Multivariate Binary Data Using GEE For a marginal Bernoulli outcome we have Pr(Y ij = y ij x ij ) = µ y ij ij (1 µ ij) 1 y ij = exp(y ij θ ij log{1 + e θ ij }), where θ ij = log(µ ij /(1 µ ij ), an exponential family representation. For independent responses we therefore have the likelihood mx Xn i mx Xn i mx Xn i Pr(Y x) = exp 4 y ij θ ij log{1 + e θ ij } 5 = exp 4 j=1 j=1 To find the MLEs we consider the score equation: G(β) = l β = mx Xn i j=1 l ij θ ij θ ij β = mx Xn i x ij (y ij µ ij ) = j=1 j=1 l ij 3 5. mx x T i (y i µ i ). So GEE with working independence can be implemented with standard software, though we need to fix-up the standard errors via sandwich estimation. 334 Non-independence GEE Assuming working correlation matrices: R i (α) and estimating equation G(β) = where W i = 1/2 i R i (α) 1/2 i. mx D T i W 1 i (y i µ i ), Here α are parameters that we need a consistent estimator of (Newey 1990, shows that the choice of estimator for α has no effect on the asymptotic efficiency). Define a set of n i (n i 1)/2 empirical correlations R ijk = (Y ij µ ij )(Y ik µ ik ) [µ ij (1 µ ij )µ ik (1 µ ik )] 1/2. We can then define a set of moment-based estimating equations to obtain estimates of α. 335

7 First Extension to GEE Rather than have a method of moments estimator for α, Prentice (1988) proposed using a second set of estimating equations for α. In the context of data with var(y ij ) = v(µ ij ): G 1 (β, α) = G 2 (β, α) = mx mx D T i W 1 i (Y i µ i ) E T i H 1 i (T i Σ i ) where R ij = {Y ij µ ij }/v(µ ij ) 1/2, to give data T T i = (R i1r i2,..., R ini 1R ini, R 2 i1,..., R2 in i ), Σ i (α) = E[T i ] is a model for the correlations and variances of the standardized residuals, E i = Σ i α, and H i = cov(t i ) is the working covariance model. The vector T i has n i (n i 1)/2 + n i elements in general the working covariance model H i is in general complex. 336 In the context of binary data Prentice (1988) the variances are determined by the mean and so the last n i terms of T i are dropped; he also suggests taking a diagonal working covariance model, H i, i.e. ignoring the covariances. The theoetical variances ae given by var(r ij R ik ) = 1+(1 2p ij )(1 2p ik ){p ij (1 p ij )p ik (1 p ik )} 1/2 Σ(α) ijk Σ(α) 2 ijk which depend on the assumed correlation model Σ these may be taken as the diagonal elements of H i. 337

8 Application of GEE Extension to Marginal Odds Model We have the marginal mean model logit E[Y ij X ij ] = βx ij. Suppose we specify a model for the associations in terms of the marginal log odds ratios: j ff Pr(Yij = 1, Y ik = 1) Pr(Y ij = 0, Y ik = 0) α ijk = log. Pr(Y ij = 1, Y ik = 0) Pr(Y ij = 0, Y ik = 1) These are nuisance parameters, but how do we estimate them? Carey et al. (1992) suggest the following approach fo estimating β and α. Let µ ij = Pr(Y ij = 1) µ ik = Pr(Y ik = 1) µ ijk = Pr(Y ij = 1, Y ik = 1) 338 It is easy to show that Pr(Y ij = 1 Y ik = y ik ) Pr(Y ij = 0 Y ik = y ik ) = exp(y ik α ijk ) Pr(Y ij = 1, Y ik = 0) Pr(Y ij = 0, Y ik = 0) «µ ij µ ijk = exp(y ik α ijk ) 1 µ ij µ ik + µ ijk which can be written as a logistic regression model in terms of conditional probabilities: «Pr(Yij = 1 Y ik = y ik ) logit E[Y ij Y ik ] = log Pr(Y ij = 0 Y ik = y ik ) «µ ij µ ijk = y ik α ijk + log 1 µ ij µ ik + µ ijk where the term on the right is a known offset (the µ s are a function of β only). Suppose for simplicity that α ijk = α then given current estimates of β, α, we can fit a logistic regression model by regressing Y ij on Y ik for 1 j < k n i, to estimate α this can then be used within the working correlation model. Carey et al. (1992) refer to this method as alternating logistic regressions. 339

9 Indonesian Children s Health Example > summary(geese(y ~ xero+age,corstr="independence",id=id,family="binomial")) Mean Model: Mean Link: logit Variance to Mean Relation: binomial Coefficients: (Intercept) e+00 age e-07 xero e-02 Scale Model: Scale Link: identity Estimated Scale Parameters: (Intercept) Correlation Model: Correlation Structure: independence Number of clusters: 275 Maximum cluster size: > summary(geese(y ~ age+xero, corstr="exchangeable", id=id,family="binomial")) (Intercept) e+00 age e-06 xero e-01 Estimated Scale Parameters: (Intercept) Estimated Correlation Parameters: alpha > summary(geese(y ~ age+xero, corstr="ar1", id=id,family="binomial")) Coefficients: (Intercept) e+00 age e-07 xero e-01 Estimated Scale Parameters: (Intercept) Estimated Correlation Parameters: alpha

10 Conditional Likelihood: Binary Longitudinal Data Recall that conditional likelihood is a technique for eliminating nuisance parameters, here what we have previously modeled as random effects. Consider individual i with binary observations y i1,..., y ini and assume the random intercepts model Y ij γ i, β Bernoulli(p ij ), where «pij log = x ij β + γ i 1 p ij where γ i = x i β + b i and x ij (a slight change from our usual notation), are those covariates which change within an individual. 342 We have Pr(y i1,..., y ini γ i, β) = = = n i Y exp (γ i y ij + x ij βy ij ) 1 + exp (γ j=1 i + x ij β) P ni exp γ i j=1 y ij + P n i j=1 x ijy ij β Q ni j=1 [1 + exp (γ i + x ij β)] exp (γ i t 2i + t 1i β) Q ni j=1 [1 + exp (γ i + x ij β)] = exp (γ it 2i + t 1i β) k(γ i, β) = p(t 1i, t 2i γ i, β) where t 1i = k(γ i, β) = n i X X x ij y ij, t 2i = j=1 n i Y j=1 n i j=1 y ij [1 + exp (γ i + x ij β)]. 343

11 We have where L c (β) = p(t 2i γ i, β) = my p(t 1i t 2i, β) = P ni y i+ l=1 exp my p(t 1i, t 2i γ i, β) p(t 2i γ i, β) γ i P ni j=1 y ij + P n i k=1 x iky l ik β k(γ i, β), where the summation is over the ` n i y i+ ways of choosing yi+ ones out of n i, and y l i = (yl i1,..., yl in i ), l = 1,..., ` n i y i+ is the collection of these ways. Hence L c (β) = = my my P ni y i+ P ni exp γ i j=1 y ij + P n i j=1 x ijy ij β l=1 exp P ni y i+ P ni γ i j=1 y ij + P n i k=1 x ikyik l β exp Pni j=1 x ijy ij β l=1 exp `P n i k=1 x ikyik l β 344 Notes Can be computationally expensive to evaluate likelihood if n i is large, e.g. if n i = 20 and y i+ = 10, ` n i y i+ = 184,756. There is no contribution to the conditional likelihood from individuals: With n i = 1. With y i+ = 0 or y i+ = n i. For those covariates with x i1 =... = x ini = x i. The conditional likelihood estimates β s that are associated with within-individual covariates. If a covariate only varies between individuals, then it cannot be estimated using conditional likelihood. For covariates that vary both between and within individuals, only the within-individual contrasts are used. The similarity to Cox s partial likelihood may be exploited to carry out computation. We have not made a distributional assumption for the γ i s! 345

12 Examples: If n i = 3 and y i = (0,0, 1) so that y i+ = 1 then y 1 i = (1, 0,0), y2 i = (0, 1,0), y3 i = (0,0, 1), and the contribution to the conditional likelihood is exp(x i3 β) exp(x i1 β) + exp(x i2 β) + exp(x i3 β). If n i = 3 and y i = (1,0, 1) so that y i+ = 2 then y 1 i = (1, 1,0), y2 i = (1, 0,1), y3 i = (0,1, 1), and the contribution to the conditional likelihood is exp(x i1 β + x i3 β) exp(x i1 β + x i2 β) + exp(x i1 β + x i3 β) + exp(x i2 β + x i3 β). 346