Super C resin

Suppose that an underlying continuous time biological stochastic process of interest is characterized by a latent variable, η_it, where i denotes an individual who is a part of a population being monitored for progression of a specific disease and t is the time since disease acquisition or some other relevant origin. Suppose also that we can measure η_it using a gold standard biomarker measurement methodology, such that
where Y_1i(t) denotes the gold standard value of the biomarker for the ith individual when measured at time t, and ε_1it(η_it) is the within-subject error which we assume to have mean zero and variance
${\sigma }_{{\epsilon }_1}^2({\eta }_{it})$ that may depend on the value of the underlying process.
Next, suppose there exists an established threshold value for gold standard biomarker measurements, c, below which clinical action is taken (the ideas extend easily if clinical action occurs when the biomarker is above c). The rationale for the choice of threshold, though application-specific, will likely be based on some assessment of the balance of risks (and perhaps financial costs) and benefits of clinical intervention at the threshold. Let the times of biomarker measurement for the ith individual be t_ij for j = 1, 2, …. Then, for example, clinical action might be initiated when Y₁ is first observed to be below c, and the time of such action, T_1i, can be defined as
Motivated by our case study, we will henceforth refer to T_1i as the treatment start time (based on the gold standard technology). The probability of crossing the threshold by time t and, hence, of starting treatment, P(T_1i ≤ t), is impacted not only by the latent value of the biomarker and the error, but also by the frequency of measurement. This is because more frequent measurements yield more realizations of the error, and therefore more opportunities for the threshold to be crossed. More generally, T_1i could be defined in terms of multiple values of Y_1i, e.g. the time at which the mean of two successive measurements is first less than c or the time at which both an initial measurement and a confirmatory measurement are less than c.
When a novel measurement method is introduced, we can specify a model for each subject's measurements from this second technology,
where B_2it(η_it) allows for the possibility that Y_2i(t) is a biased assessment of η_it with magnitude that could depend on the value of the underlying process, and ε_2it(η_it) is the within-subject error which we again assume to have mean zero and variance
${\sigma }_{{\epsilon }_2}^2({\eta }_{it})$ that may also depend on the value of the underlying process. Correlation between ε_1it(η_it) and ε_2it(η_it) can be accommodated through specification of a bivariate model. We defer discussion of this to the case study. As for the gold standard, we can define the treatment start time based on the novel method, T_2i as
where we allow for the possibility that the sequence of measurement occasions for the novel method (j′ = 1, 2, …) differs from that of the gold standard (which might arise if, for example, the novel method can be administered more frequently at a rural clinic whereas the gold standard requires a visit to a hospital-based clinic). A comparison of the performance of the two methods might then proceed by comparing the marginal distributions of T₁ and T₂ in the population or by evaluating some aspect of the distribution of gap times for individuals given by T_2i − T_1i. We illustrate how this might be approached in the case study.
As defined here, T₁ and T₂ represent first passage times (FPT) of two stochastic processes to a threshold of interest. Despite the large literature in this area, closed-form expressions for the probability density function (pdf) of the FPT for a discretely-sampled stochastic process are only available for limited special cases. These special cases do not include non-homogeneous stochastic processes (those whose probability distributions change when shifted in time) or dynamic measurement schedules which we study here [10 (link)]. Because of this, we proceed by conducting a simulation study which generates marker trajectories for ndividual subjects based on a longitudinal model for Y₁ and Y₂, and study the corresponding treatment start times based on the gold standard and novel methods.