Calibration Uncertainty and Model-Based Analyses with Applications to Ovarian Cancer Modeling

Abstract

Model-based analyses for comparative analyses within medical decision making require the development of a model of the disease that is being examined. This involves the specification of the model structure and the calibration of model parameters so that model outcomes are consistent with observable disease-related data. There is rarely a unique set of model parameters that are consistent with observable data, and such parameter sets can vary significantly. This phenomenon is known as “calibration uncertainty” and is especially prevalent when models address preclinical phases of the disease. Because model parameters influence comparative analyses, examination of the impact of calibration uncertainty on recommendations derived from the analysis is crucial to developing confidence in the recommendations. In this chapter, we present an approach to the characterization and systematic examination of the set of models that provide plausible representations of the disease. We illustrate our approach within the context of ovarian cancer. In doing so, we illustrate the impact of calibration uncertainty on the potential for early detection of ovarian cancer.

Appendices

Appendix 1: Ovarian Cancer Data

Figures 15.4 illustrates the boxplots of the age and stage at diagnosis from years 2000–2014, where the combined data for all 15 years are included at the last positions on the subplots. Figure 15.5 illustrates survival distributions as a function of the age and stage at diagnosis. These are clearly age-dependent, for all stages.

Fig. 15.4

The boxplots of the age and stage at diagnosis from years 2000–2014. (a) The boxplot of the age at diagnosis across years 2000–2014. (b) The boxplot of the stage at diagnosis across years 2000–2014. A cursory examination of these marginal distributions indicates minimal variation over the 15-year period. Accordingly, the models in this paper were developed using the aggregated data, which appears as “all”

Fig. 15.5

Nonstationarity in post-diagnosis survival

Appendix 2: Validity Conditions

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,2U} &\geq P_{1D,2D} \quad && {} \end{array}\end{aligned} $$

(15.9)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,3U} &\geq P_{1D,3D} \quad && {} \end{array}\end{aligned} $$

(15.10)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,DD} &\geq P_{1D,DD} \quad && {} \end{array}\end{aligned} $$

(15.11)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,3U} &\geq P_{2D,3D} \quad && {} \end{array}\end{aligned} $$

(15.12)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,DD} &\geq P_{2D,DD} \quad && {} \end{array}\end{aligned} $$

(15.13)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3U,DD} &\geq P_{3D,DD} \quad && {} \end{array}\end{aligned} $$

(15.14)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,3U} &\geq P_{1U,2U} \quad && {} \end{array}\end{aligned} $$

(15.15)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3U,DD} &\geq P_{2U,3U} \quad && {} \end{array}\end{aligned} $$

(15.16)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2D,3D} &\geq P_{1D,2D} \quad && {} \end{array}\end{aligned} $$

(15.17)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3D,DD} &\geq P_{2D,3D} \quad && {} \end{array}\end{aligned} $$

(15.18)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,2U} &\geq P_{1U,3U} \quad && {} \end{array}\end{aligned} $$

(15.19)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,3U} &\geq P_{1U,DD} \quad && {} \end{array}\end{aligned} $$

(15.20)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1D,2D} &\geq P_{1D,3D} \quad && {} \end{array}\end{aligned} $$

(15.21)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1D,3D} &\geq P_{1D,DD} \quad && {} \end{array}\end{aligned} $$

(15.22)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,3U} &\geq P_{2U,DD} \quad && {} \end{array}\end{aligned} $$

(15.23)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2D,3D} &\geq P_{2D,DD} \quad && {} \end{array}\end{aligned} $$

(15.24)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,1D} &\geq P_{1U,2D} \quad && {} \end{array}\end{aligned} $$

(15.25)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,2D} &\geq P_{1U,3D} \quad && {} \end{array}\end{aligned} $$

(15.26)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{1U,3D} &\geq P_{1U,DD} \quad && {} \end{array}\end{aligned} $$

(15.27)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,2D} &\geq P_{2U,3D} \quad && {} \end{array}\end{aligned} $$

(15.28)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,3D} &\geq P_{2U,DD} \quad && {} \end{array}\end{aligned} $$

(15.29)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,DD} &\geq P_{1U,DD} \quad && {} \end{array}\end{aligned} $$

(15.30)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3U,DD} &\geq P_{2U,DD} \quad && {} \end{array}\end{aligned} $$

(15.31)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2D,DD} &\geq P_{1D,DD} \quad && {} \end{array}\end{aligned} $$

(15.32)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3D,DD} &\geq P_{2D,DD} \quad && {} \end{array}\end{aligned} $$

(15.33)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,3D} &\geq P_{1U,3D} \quad && {} \end{array}\end{aligned} $$

(15.34)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3U,3D} &\geq P_{2U,3D} \quad && {} \end{array}\end{aligned} $$

(15.35)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,2D} &\geq P_{1U,2D} \quad && {} \end{array}\end{aligned} $$

(15.36)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3U,3D} &\geq P_{3U,DD} \quad && {} \end{array}\end{aligned} $$

(15.37)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{2U,2D} &\geq P_{1U,1D} \quad && {} \end{array}\end{aligned} $$

(15.38)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} P_{3U,3D} &\geq P_{2U,2D} \quad && {} \end{array}\end{aligned} $$

(15.39)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} p_{DO}(a) &= \hat{p}_{DO}(a) \quad && \forall a \in \mathcal{A} {} \end{array}\end{aligned} $$

(15.40)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} \sum_{j \in \mathcal{S}} P_{i,j}(a) &= 1 \quad && \forall i \in \mathcal{S}, a \in \mathcal{A} {} \end{array}\end{aligned} $$

(15.41)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} 0 \leq P_{i,j}(a) &\leq 1 \quad && \forall i,j \in \mathcal{S}, a \in \mathcal{A} {} \end{array}\end{aligned} $$

(15.42)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} 0 \leq \beta &\leq \dfrac{1-p_{DO}(85)}{45} \quad && {} \end{array}\end{aligned} $$

(15.43)

Equations (15.9)–(15.14) state that treatment helps slow down progression. Equations (15.15)–(15.18) impose that progression is more likely when the person is in a more severe health state. Equations (15.19)–(15.39) require that progressing to a more severe health state is less likely than to a less severe health state. Equation (15.40) sets p _DO(a) as the corresponding observed value. Equations (15.41)–(15.42) refer to the basic probability laws, whereas (15.43) ensures that the slope defining the disease activation process is strictly positive and is at most $\dfrac {1-p_{DO}(85)}{45}$. This is derived from the fact that $\hat {p}_{DO}(a)$ is a monotonically decreasing function of a, and that (15.41) must be satisfied.