Multivariate Cox PH model with log-skew-normal frailties

Transcription

1 Multivariate Cox PH model with log-skew-normal frailties Department of Statistical Sciences, University of Padua, Padua (IT)

2 Multivariate Cox PH model A standard statistical approach to model clustered failure time data is given by the Cox PH frailty models λ(t ij x ij, b ij ) = λ 0 (t ij ) exp(x T ij β + b i ), where λ 0 is an unspecified baseline hazard function, β is a vector of the regression parameters, and b i, i = 1,..., G are independent and identically distributed random effects (frailties) with density function f (b).

3 Frailty distribution Different distributions of the frailty exp(b i ) have been proposed. The most common are gamma: (Clayton, 1978; Vaupel et al., 1979; Klein,1992; Yashin et al.,1995) log-normal: (McGilchrist & Aisbett, 1991; Xue & Brookmeyer, 1996; Yau & McGilchrist, 1997; Vaida & Xu, 2000; Ripatti & Palmgren, 2000) positive stable: (Hougaard, 1985; Hougaard, 1986; Lam & Kuk, 1997; Oaks, 1994).

4 Frailty distribution Gamma and the positive stable distributions have simple Laplace transforms and are thus mathematically convenient to use. log-normal frailty model allows a relatively simple extension to the multivariate case with general variance-covariance matrix (Ripatti & Palmgren, 2000) which is far more complex to pursue with other distributions.

5 Choice of the frailty distribution The choice of the frailty distribution is crucial to obtain correct estimates of the dependence structure (Duchateau & Jannsen, 2008), but the researcher often has no prior information with which to choose among the distributions in many situations the distribution of the frailty is neither gamma, nor log-normal distributed.

6 Standard Frailty distributions Probability density function b Densities of (i) log-normal distribution with (µ, σ) = (0, 0.8) (continuous line), (ii) gamma distribution with mean 1 and variance θ = 0.5 (dashed line) and (iii) positive stable distribution with dependence parameter θ = 0.84 (dotted line).

7 Flexible (Skew normal) Frailty distribution Probability density function b Densities of log-skew-normal distribution with (µ, σ) = (0, 0.8) shape parameter (i) α = 0 (continuous line), (ii) α = 4 (dashed line), and (iii) α = 4 (dotted line)

8 Skew normal distribution In its basic form, a continuous random variable Z is said to have a skew-normal distribution (Azzalini, 1985), if it has density function φ(z; α) = 2 φ(z) Φ(α z), < z <, where φ and Φ denote the standard normal density and distribution function, respectively, and α ( < α < ) is a parameter such that α > 0 produces a distribution with positive skewness, and α < 0 corresponds to negative skewness; if α = 0, we are back to the usual standard normal density. The mean and variance of Z are E(Z) = 2 π α 1 + α 2 = µ z, var(z) = 1 µ 2 z = σ 2 z.

9 Skew normal distribution Now assume that the log-transformed frailty b i is skew-normally distributed with mean 0, standard deviation σ, and shape parameter α. The logarithm of its density function can then be written as ln f (b i ; σ, α) = 1 2 ln(2π) + ln σ z σ z2 i 2 + ζ 0(αz i ), where z i = µ z + σ z b i /σ, and ζ 0 (x) = ln(2φ(x)).

10 Dependence structure induced by the frailty model The cross-ratio function can be expressed as the ratio of two hazards, i.e. the ratio of the hazard of subject j given that subject l died at t l and the hazard of j given that T l > t l. CR(t 1, t 2 ) = λ(t 1 T 2 = t 2 ) λ(t 1 T 2 > t 2 ) = S(t 1, t 2 )D 1 D 2 S(t 1, t 2 ) D 1 S(t 1, t 2 )D 2 S(t 1, t 2 ) where D j = t j and, in an obvious notation, S(t 1, t 2 ) = denotes the bivariate survival function. S(t 1 b i ) S(t 2 b i )f (b i ) db i

11 Dependence structure induced by the frailty model (a) (b) Marginal survival function t time t 1 (c) (d) t t t 1 t 1 (a) Marginal survival functions; (b) CR function of positive stable frailty distribution; (c) CR function of log-normal frailty distribution; (d) Cross-ratio function of log-skew-normal frailty distribution with (µ, σ, α) = (0, 0.8, 4).

12 Marginal likelihood The contribution to the conditional log-likelihood for the i-th cluster is given by n i l C i (β, λ 0, b i ) = [δ ij {ln λ 0 (t ij )+xij T β+b i } Λ 0 (t ij ) exp(xij T β+b i )], j=1 where Λ 0 denotes the cumulative baseline hazard function, and the augmented log-likelihood is given by l A i (β, λ 0, b i ) = l C i (β, λ 0, b i ) + log f (b i ), for i = 1,..., G. The marginal likelihood for cluster i is L M i (β, λ 0, θ) = { } exp l A i (β, λ 0, b i ) db i, where θ denotes the vector parameters of the distribution f (b).

13 EM algorithm: E-step The EM algorithm has been applied by several authors to estimate the parameters of Cox random effects models by treating the random effects as missing data. The expectation of the logarithm of the augmentated likelihood can be expressed as the sum of two terms, namely G n i Q 1 ( β, λ 0 ) = [δ ij {ln λ 0 (t ij ) + xij T β + E[b i ]}+ i=1 j=1 Λ 0 (t ij ) exp(x T ij β + ln E[e b i ]], and Q 2 (θ) = G E[ln f (b i )]. i=1

14 EM algorithm: E-step These two quantities involve expectations of the form E[g(b i )] = g(b i)e la i ( β, λ 0,b i ) db i ela i ( β, λ 0,b i ) db i. which are not amenable to explicit expression. Therefore we compute them using the Gaussian quadrature E[g(b i )] k g(u k)w k e la i ( β, λ 0,u k )+uk 2 k w ke la i ( β, λ, say, 0,u k )+uk 2 where w k and u k are the weights and the nodes of the Gaussian quadrature, respectively.

15 EM algorithm: M-step In the M-step new estimates β and θ are found by maximizing the functions Q 1 and Q 2, respectively. Parameters β can be estimated by maximizing the partial likelihood of the Cox model with offset given by ln E(e b i ); see Vaida & Xu (2000). Using the current estimates of β it is possible to derive the baseline hazard using the Nelson-Aalen estimator λ 0 (t ij ) = δ ij t kl t ij exp(x T kl β + ln E[exp(b k )]). The new estimate of θ is obtained by maximizing Q 2. To prevent bias in the estimation of the shape parameter α, we maximize a penalized Q 2 function.

16 EM algorithm: M-step In the spirit of the work of Sartori (2006), here we adopted the method for bias reduction due to Firth (1993), as the occurrence of ˆα = ± with non-0 probability generates infinite bias. Simulation results showed that maximum likelihood estimates of σ are robust with respect to the bias on the shape parameter. The nice behavior of ˆσ suggests a two step estimation procedure. In the first step we maximize Q 2 in order to estimate ˆα = (ˆσ, ˆα). In the second step, with fixed σ = ˆσ, we maximize the expectation of the Jeffrey-like penalized expected likelihood only with respect to the shape parameter α Q 2 (α) = Q 2 (α) log{ Q 2 (α)}.

17 Variance matrix of the estimates The variance-covariance matrix of the solution ( ˆβ, ˆλ 0, ˆθ) obtained from the EM algorithm can be estimated using the inverse of the observed information matrix (Louis, 1982) I (β, λ 0, θ) = E[ l A ] E[l A l A T ] where l A and l A are the first and the second derivatives with respect to (β, λ 0, θ) of the augmented log-likelihood.

18 Simulation study The data were generated using the following hazard model λ(t ij X 1,ij, X 2,ij, X 3,ij, b i ) = λ 0 (t ij ) exp(β 1 X 1,ij +β 2 X 2,ij +β 3 X 3,ij +b i ). G = 100 or 150 clusters containing n i = 10 or 20, (i = 1,..., G) subjects. b i was SN distributed with µ = 0, σ = 0.7, and shape parameter α = 4, 0 or 4. X k N(0, 1), k = 1, 2, 3 independent and normally distributed with β = (β 1, β 2, β 3 ) = (0.5, 1, 0.3). λ 0 (t) = ϱψt ψ 1, ϱ = 1/80 and ψ = 5. Right-censoring around 20 %. We used 100 nodes to compute the Gaussian quadrature.

19 Simulation study Table: Estimated parameters and their estimated and empirical standard errors in 250 simulations based on a shared frailty model with G clusters and n i repetitions per cluster with variance σ 2 = and skew parameter α = 4, 0, or 4. G=250, n i = 2 G=100, n i = 5 Empirical Mean Empirical Mean Mean SE SE Mean SE SE β 1 = β 2 = β 3 = σ = α = β 1 = β 2 = β 3 = σ = α = β 1 = β 2 = β 3 = σ = α =

20 Case Study: Multiple myeloma patients with autologous transplantation As an illustration, we applied the proposed method to a real data set from the European Group for Blood and Marrow Transplantation (EBMT) registry. We analyzed a subset of the Multiple Myeloma (MM) patients with autologous transplantation. The original MM autologous registry includes about cases. we considered a subset of N=3081 patients with common myelomas (IgA, IgG), transplanted after 1998, without inconsistent records or missing values in a list of relevant prognostic factors. we studied the Progression Free Survival (PFS). The number of observed events is 1548 (51%) and the median PFS time is months (minimum=0.03, maximum=119.18).

21 Case Study: Multiple myeloma patients with autologous transplantation Patients are clustered into G =190 centers with a mean of 16 individuals per cluster (median 7, minimum=1, maximum=134). We fitted three different Cox models to the data: a model without random effect (Model 1), a model with log-normal distributed frailties (Model 2), and a model with log-skew-normal distributed frailties (Model 3).

22 Case Study: Multiple myeloma patients with autologous transplantation Table: Model fits to the multiple myeloma data ˆβ (SE) Variable Model 1 Model 2 Model 3 Stage 0.15(0.03) 0.16(0.03) 0.16(0.04) Status at cond. (no CR vs CR) 0.35(0.09) 0.37(0.09) 0.37(0.11) Serum beta(2)-mic. > 4 vs < (0.05) 0.28(0.05) 0.28(0.06) Age at transplant (+1 yr) 0.08(0.04) 0.09(0.04) 0.09(0.04) Year (+1) 0.02(0.01) 0.02(0.01) 0.02(0.01) Gender (F vs M) -0.15(0.05) -0.15(0.05) -0.15(0.06) Interval diag.-auto. 0.03(0.02) 0.03(0.02) 0.03(0.02) Frailty SD (ˆσ) 0.17(-) 0.17(0.06) Frailty Shape parameter ( ˆα) 4.21(2.07)

23 Case Study: Multiple myeloma patients with autologous transplantation (a) (b) Probability of remaining progression free (%) profile log likelihood Time since transplant (months) α

24 Conclusions: skewness of the frailty When sample size is large it is possible to estimate the skewness of the distribution estimates of the parameters are robust with respect to the skewness parameter BUT the structure of the dependence (e.g. the cross-ratio function) depends on the skewness of the frailty. It follows that the log-skew-normal frailty model is useful to better describe the dependence structure in the data. the shared frailty model can be extended to multivariate frailties using the multivariate skew-normal distribution (Azzalini & Capitanio, 1999).

25 Acknowledgements We acknowledge the Chronic Leukemia Working Party of the EBMT for the availability of the data used in the application. Prof. A. Azzalini: Dept Statistical Sciences University of Padua (Italy) S. Iacobelli: University Tor Vergata, Rome (Italy), on the behalf of the EBMT Chronic Leukemia Working Party