We define the hazard function for the ithindividual in the jthcluster as:
λij(t|J = j, X = x) = λ0(t) exp(b0j+ (β1+ bij)xij) with
b0j∼N (0, φ2)
b1j∼N (0, τ2)
(6.3.1)
where τ2is a measure of the heterogeneity between clusters at the VE level and φ2 of the heterogeneity between clusters at the baseline risk level. We can also write λ0j= λ0(t) exp(b0j).
6.4
Statistical intervals
Different types of statistical intervals can be built based on the estimations obtained from our proposed analysis model. The type of interval to be computed will strongly depend on the underlying problem or application (Krishnamoorthy and Mathew 2009). Intervals are given different interpretations in the Bayesian and the frequentist set- tings. The frequentist approach regards the model parameters as fixed unknown quan- tities. In this approach, a statistical interval is interpreted as a range of values that are believed to likely contain the true parameter value (Hahn and Meeker 2011). More precisely, it is interpreted as a range in which the parameter of interest would occur 100(1 − α)% of the time with repeated sampling. Bayesians treat model parameters as unknown random quantities and described them probabilistically. They interpret intervals as the region of values, denoted [L, U ], that contains 100(1 − α)% of the posterior probability of the parameter of interest such that
Z U
L
p(θ|y)dθ = 1 − α (6.4.1) Following this methodology, intervals can be similarly derived for a function of the model parameter(s).
For a parameter θ, the highest posterior density (HPD) interval is the shortest interval containing 100(1 − α)% of the posterior sample for θ while the equal-tail is obtained using the empirical percentiles of the posterior distribution of θ. This latter interval definition has been privileged in this chapter.
In this section, we give the definition of each type of interval used in our work. Prac- tical computation of the intervals will later be presented in section 6.5.
Credible intervals
Conventionally, significant VE is statistically showed when the lower bound of the two-sided 95% confidence interval of VE is substantially above zero (CDC 2011) (see Chapter 2 for more details).
A credible interval is the Bayesian analogue of a confidence interval in the frequentist setting.
As shown by Higgins et al. (2009), the presentation of inference only for the mean VE provides an incomplete summary and is highly misleading when there is hetero- geneity. In a context such as seasonal influenza where VE includes a cluster-specific component, a credible interval for the trial mean VE does not address the question of the protection of subjects from new clusters (new season and/or new country). A one-sided interval for the parameter of interest β1 can be defined as the interval
[L(β1), +∞[ (lower bound) or ] − ∞, U (β1)] (upper bound) containing 100(1 − α)%
of the posterior density of β1. Once the posterior distribution of β1 is known, the
problem reduces to the computation of L(β1|y) or U (β1|y) that satisfies
P (β1≥ L(β1|y)) = 1 − α
or
P (β1≤ U (β1|y)) = 1 − α
In the Bayesian setting, a 100(1−α)% credible interval for β1is a subset of B1, where
B1is the set of possible values for β1, such that:
Z +∞ L p(β1|y)dβ1= 1 − α or Z U −∞ p(β1|y)dβ1= 1 − α
The posterior density of β1, denoted p(β1|y) can be derived by averaging the joint
posterior density of the model over the nuisance parameters. When this integral has no analytic form, it can be evaluated numerically.
An equal-tail two-sided interval for β1can be defined as the interval [L(β1), U (β1)]
6.4. Statistical intervals 95
of β1is known, the problem reduces to the computation of L(β1|y) and U (β1|y) that
satisfies P (β1≥ L(β1|y)) = 1 − α 2 and P (β1≤ U (β1|y)) = 1 − α 2
In the Bayesian setting, a 100(1 − α)% credible interval for for β1is a subset of B1
such that:
Z U
L
p(β1|y)dβ1= 1 − α
Prediction intervals
To derive information about a new random effect, we can compute a prediction inter- val for the random effect of interest, denoted β1,new. Heterogeneity around the mean
effect β1is accounted for in the computation of the prediction interval for a new ran-
dom effect. This interval does also consider the uncertainty of the parameter estimates, that is the imprecision in the estimations of the mean effect β1and the variance of the
random effect, τ2. Senn (2004) suggests that such an interval provides a reasonable prediction for a randomly chosen future unit, here a cluster.
We define a new random effect as β1,new = (β1 + b1|y). A one-sided interval
for β1,new can be defined as the interval [L(β1,new), +∞[ (lower bound) or ] −
∞, U (β1,new)] (upper bound) containing 100(1 − α)% of the posterior density of
β1,new. Once the posterior distribution of β1,new is known, the problem reduces to
the computation of L(β1,new|y) or U (β1,new|y) that satisfies
P (β1,new≥ L(β1,new|y)) = 1 − α
or
P (β1,new≤ U (β1,new|y)) = 1 − α
Z +∞
L
p(β1,new|y)dβ1,new= 1 − α
or Z U
−∞
p(β1,new|y)dβ1,new= 1 − α
A two-sided prediction interval for a new random effect, β1,new, can be defined as the
interval [L(β1,new), U (β1,new)] containing 100(1 − α)% of the posterior density of
β1,new. Once the posterior distribution of β1,new is known, the problem reduces to
the computation of L(β1,new|y) and U (β1,new|y) that satisfies
P (β1,new≥ L(β1,new|y)) = 1 −
α 2 and
P (β1,new≤ U (β1,new|y)) = 1 −
α 2
In the Bayesian setting, a 100(1 − α)% prediction interval is a subset of B1such that:
Z U
L
p(β1,new|y)dβ1,new = 1 − α
Tolerance intervals
Prediction and tolerance intervals are closely related (Krishnamoorthy and Mathew 2009). Indeed, 100(1−α)% level p-content tolerance intervals are prediction intervals for a fixed proportion p of new random effects. Statistically, a p-content tolerance interval is a prediction interval for specific quantiles of the posterior distribution of a new cluster-specific VE. Analytically deriving these quantities is very complex and numerical solutions are often preferred.
A one-sided p-content tolerance interval is an interval on the distribution of a quantile of the distribution of the random new random effect, β1,new. We require a 100(1−α)%
upper or lower confidence limit for gp(θ), the p quantile of β1,new, based on the pos-
terior distribution p(gp(β1,new)|y). Once the posterior distribution of gp(β1,new) is
known, the problem reduces to the computation of L(gp(β1,new)|y) or U (gp(β1,new)|y)