A Transition Model for the RSV data

Chapter 1 Introduction

6.5 A Transition Model for the RSV data

Diggle et al (2002) state that transition models are considered as extensions of generalized linear models (GLMs) for describing the conditional distribution of each responsey_ij as an explicit function of past responsesyij−1, . . . , y_i1 and covariates x_ij. Hence the past outcomes are treated as predictor variables.

If we consider the generalized linear transition model with respect to the Kil- ifi data set, we can model the conditional distribution of Y_ij given the past as an explicit function of the qpreceding responses. We can assume that the probability of RSV for child i at visit j has a direct dependence on whether or not the child had RSV at visit j−1 as well as on explanantory variables, x_ij. This is the first case of a first order transition model. If we take the logit link then a first order transition model is given by

logit[P(Y_ij = 1|Yij−1, . . . , Y_i1)]=x⁰_ijβ+αYij−1.

Therefore the probability of RSV at time t_ij depends on the measured co- variates or explanatory variables but also on whether or not the child had RSV at the previous visit. The parameter exp(α) is the ratio of the odds of infection among the children who did and did not have RSV at the prior visit. The β coefficient is the change per unit change in x in the log odds of infection among children who were free of RSV at the previous visit. The transition model stated above is a first order Markov chain according to Feller (1968, vol 1, p. 132). At equally spaced time intervals the 2×2 transition matrix whose elements are P(Y_ij =y_ij|Yij−1 = yij−1) where each of Y_ij and Yij−1 may take values of 0 and 1 is given by inverting the logistic regression equation for every pair (y_ij, yij−1) as

Yij

0 1

Y_ij−1 0 ¹

1+exp(x⁰_ijβ

exp(x⁰_ijβ) 1+exp(x⁰_ijβ)

1 ¹

1+exp(x⁰_ijβ+α)

exp(x⁰_ijβ+α) 1+exp(x⁰_ijβ+α)

However in the general transition model we let H_ij = {Y_i1, . . . , Y_ij−1} rep- resent the past responses for the i-th subject, µ^c_ij = E(Y_ij|H_ij) and let v^c_ij = var(Y_ij|H_ij) be the conditional mean and variance of Y_ij given past responses and the explanatory variables. We can specify the model analo- gous to the GLM for independent data, where we assume:

g(µ^c_ij) =x⁰_ijβ+

s

X

r=1

f_r(H_ij;α) = x⁰_ijβ+h⁰_ijα (6.10) and

v_ij^c =v(µ⁰_ij)φ

We model the transition from the prior state by the functions f_r to the present response. The past outcomes after transformation by the known

functions f_r are treated as explanatory variables. Interactions among the prior responses may be considered. We can then fit the transition model using GLM techniques and treat the repeated transitions for a child/subject as independent events.

General

Diggle et al (2002) focus on the case where the observation times t_ij are equally spaced. The history for subject i at visit j is denoted as H_ij = {y_ik, k = 1, . . . , j−1}. The most useful transition models are Markov chain for which the conditional distribution of Y_ij given H_ij depends only on the q prior responses Yij−1, . . . , Y_ij−q. The integer q represents the order of the model. Writing the conditional p.d.f of Y_ij as an exponential family type of distribution gives

f(y_ij|H_ij) = exp{[y_ijθ_ij −ψ(θ_ij)]/φ+c(y_ij, φ)} (6.11)

for known functionsψ(θ_ij) andc(y_ij, φ). The conditional mean and variance:

µ^c_ij =E(Y_ij|H_ij) =ψ⁰(θ_ij) and v⁰_ij =var(Y_ij|H_ij) = ψ⁰⁰(θ_ij)φ

Diggle et al. (2002) consider models where the conditional mean and vari- ance satisfy the equations

g(µ^c_ij) =x⁰_ijβ+

s

X

r=1

f_r(H_ij;α) for suitable functions f_r and

v_ij^c =v(µ⁰_ij)φ

where h and v are known link and variance functions determined from the

density function. Hence the transition model expresses the conditional mean as a function of both covariates x_ij and of the past responsesYij−1, . . . , Yij−q

in a much more general setting. We assume that the past affects the present through the sum of sterms each of which may depend on theq prior values.

As an example: A logistic regression model for binary responses assuming a first order Markov chain (Cox, 1970, Korn and Whittemore, 1979, Zeger et al., 1985) specified as:

g(µ^c_ij) = x⁰_ijβ+αyij−1 (6.12) where g(µ^c_ij) = logit(µ^c_ij), v(µ^c_ij) = µ^c_ij(1−µ^c_ij), f_r(H_ij, α) =α_ryij−r,

s =q = 1 and µ^c_ij = Prob(Y_ij = 1|H_ij).

A first order Markov model can be fitted by making use of the likelihood function. The contribution to the likelihood for thei^thsubject can be written as:

L_i(y_i1, . . . , y_ini =f(y_i1)Qni

j=2f(y_ij|H_ij) whereH_ij is the history measure- ment at occasion j given by H_ij ={y_ij−1}

In a Markov model of order q, the conditional distribution of Y_ij is

f(y_ij|H_ij) =f(y_ij|yij−1, . . . , yij−q) so that the likelihood is:

f(y_i1, . . . , y_iq)Qni

j=q+1f(y_ij|yij−1, . . . , yij−q)

The GLM in Eq.(6.8) specifies only the conditional distribution f(y_ij|H_ij) whilst the likelihood of the first q observations f(y_i1, . . . , y_iq) is not specified directly. In the logistic and log-linear modelsf(y_i1, . . . , y_iq) is not determined from the GLM assumption about the conditional model and the full likeli- hood is unavailable. An alternative is to estimate β and α by maximizing

the conditional likelihood given by:

QN

i=1f(y_iq+1, . . . , y_ini|y_i1, . . . , y_iq) = QN i=1

Qni

j=q+1f(y_ij|H_ij) where N is the number of subjects or clusters in the study.

There are 2 distinct cases to consider in the maximization process of the likelihood

CASE 1

f_r(H_ij,α, β)=α_rf_r(H_ij) so that

g(µ^c_ij)=x⁰_ijβ+Ps

r=1α_rf_r(H_ij)

Clearly g(µ^c_ij) is a linear function of both β and α=(α₁, . . . , α_s)⁰ so that estimation is the same as for GLMs for independent data. We regress Y_ij on the (p+s) dimensional vector of extended explanatory variables (x⁰_ij, f₁(H_ij), . . . , f_s(H_ij))⁰.

CASE 2

Case 2 occurs when functions of past responses include both β and α. Ex- amples are linear and log-linear models. The Iterative weighted least squares method is used to estimate β and α. This exposition is given in Diggle et al (2002, pg. 193-194). As a summary; the derivative of the log conditional likelihood or conditional score function has the form

S⁰(δ) =

m

X

i=1 ni

X

j=q+1

∂µ^c_ij

∂δ v_ij^c−1(y_ij −µ^c_ij) = 0 (6.13) where δ = (β,α). The above equation is analogous to the GLM score equation. The derivative ^∂µ

c ij

∂δ is analogous to x_ij but it can depend on both α and β. The iterative weighted least squares procedure is formulated as

follows. Let Y_i be the (n_i−q) vector of responses forj =q+ 1, . . . , n_i and µ^c_ij its expectation given byH_ij.

Let X_i^∗ be an (n_i−q)x (p+s) matrix with the k^th row given by

∂µ_iq+k

∂δ and W_i = diag

1 v⁰_ik+q

, k = 1, . . . , ni−q be an (n_i −q)x (n_i−q) diagonal weighting matrix.

Finally let Z_i=X_i^∗δ+ (Yˆ _i − µˆ^c_i) then an updated δˆ can be obtained by iteratively regressing Z onX^∗using weights in W. When the correct model is assumed for the conditional mean and variance, the solution ˆδ of Eq.(6.13) asymptotically follows a Gaussian distribution, as N goes to infinity, with mean equal to the true value δ and (p+s)×(p+s) variance matrix:

V_δ =

N

X

i=1

X_i^0∗W_iX_i^∗

!⁻¹

The variance V_δ depends on both α and β and a consistent estimate ˆV_δ is obtained by replacing α and β by their estimates ˆα and ˆβ. However when the conditional mean is correctly specified and variance is not, consistent inferences about δ can be still obtained using the robust variance:

V_R=

m

X

i=1

X_i^0∗W_iX_i^∗

!−1 m

X

i=1

X_i^0∗W_iV_iW_iX_i^∗

! _m X

i=1

X_i^0∗W_iX_i^∗

!−1

.

A consistent estimate of V_R can be obtained by replacing V_i = var(Y_ij|H_i) by its estimate (Y_i−µc_i^c)(Y_i−µc_i^c)⁰. Interestingly, even when the Markov assumtption is violated, the robust variance will give more consistent confi- dence intervals for ˆδ. This concludes the estimation process for the transition model.

6.6 Software for fitting Conditional Models