Parameter identifiability and redundancy: theoretical considerations.

BACKGROUND: Models for complex biological systems may involve a large number of parameters. It may well be that some of these parameters cannot be derived from observed data via regression techniques. Such parameters are said to be unidentifiable, the remaining parameters being identifiable. Closely related to this idea is that of redundancy, that a set of parameters can be expressed in terms of some smaller set. Before data is analysed it is critical to determine which model parameters are identifiable or redundant to avoid ill-defined and poorly convergent regression. METHODOLOGY/PRINCIPAL FINDINGS: In this paper we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 39 (1971) 577-591) and Catchpole and Morgan (Biometrika 84 (1997) 187-196). Within the widely used exponential family, parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be largely equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 183 (2003) 111-134) and Little et al. (J Theoret Biol 254 (2008) 229-238) that generalize a large number of other recently used quasi-biological cancer models. CONCLUSIONS/SIGNIFICANCE: We have shown that the previously developed concepts of parameter local identifiability and redundancy are closely related to the apparently weaker properties of weak local identifiability and gradient weak local identifiability--within the widely used exponential family these concepts largely coincide.

Use of the individual data of the A-bomb survivors for biologically based cancer models.

All recent analyses of the data on solid cancer incidence of the atomic bomb survivors are corrected for migration and random dose errors. In the usual treatment with grouped data and regression calibration, the calibration of doses depends on the used dose response. For solid cancers, it usually is linear. Here, an individual likelihood is presented which works without further adjustment for all dose responses. When the same assumptions are made as in the usual Poisson regression, equivalent results are obtained. But, the individual likelihood has the potential to use more detailed models for dose errors and to estimate non-linear dose responses without recalibration. As an example for the potential of the individual data set for the analysis of risk at low doses, signals for a saturating bystander effect are investigated.

Parameter identifiability and redundancy in a general class of stochastic carcinogenesis models.

BACKGROUND: Heidenreich et al. (Risk Anal 1997 17 391-399) considered parameter identifiability in the context of the two-mutation cancer model and demonstrated that combinations of all but two of the model parameters are identifiable. We consider the problem of identifiability in the recently developed carcinogenesis models of Little and Wright (Math Biosci 2003 183 111-134) and Little et al. (J Theoret Biol 2008 254 229-238). These models, which incorporate genomic instability, generalize a large number of other quasi-biological cancer models, in particular those of Armitage and Doll (Br J Cancer 1954 8 1-12), the two-mutation model (Moolgavkar et al. Math Biosci 1979 47 55-77), the generalized multistage model of Little (Biometrics 1995 51 1278-1291), and a recently developed cancer model of Nowak et al. (PNAS 2002 99 16226-16231). METHODOLOGY/PRINCIPAL FINDINGS: We show that in the simpler model proposed by Little and Wright (Math Biosci 2003 183 111-134) the number of identifiable combinations of parameters is at most two less than the number of biological parameters, thereby generalizing previous results of Heidenreich et al. (Risk Anal 1997 17 391-399) for the two-mutation model. For the more general model of Little et al. (J Theoret Biol 2008 254 229-238) the number of identifiable combinations of parameters is at most less than the number of biological parameters, where is the number of destabilization types, thereby also generalizing all these results. Numerical evaluations suggest that these bounds are sharp. We also identify particular combinations of identifiable parameters. CONCLUSIONS/SIGNIFICANCE: We have shown that the previous results on parameter identifiability can be generalized to much larger classes of quasi-biological carcinogenesis model, and also identify particular combinations of identifiable parameters. These results are of theoretical interest, but also of practical significance to anyone attempting to estimate parameters for this large class of cancer models.

Heterogeneity of cancer risk due to stochastic effects: emphasis on radiation-induced effects.

Persons with exactly the same genetic background, behavior and environment may differ in radiation cancer risk, due to the stochastic nature of cancer development. These differences are estimated quantitatively by means of the two stage clonal expansion model, in which the number of intermediate cells on their way to malignancy varies stochastically between individuals. For liver cancer after injection of Thorotrast, the estimated relative risk for persons without intermediate cells at age 40 is a factor of more than 10 larger than that for persons with a large number of intermediate cells. The population-based estimate of the relative risk represents an underestimation for most persons at most ages, because for persons showing a large number of intermediate cells liver cancer is not a rare disease.

Heterogeneity of cancer risk due to stochastic effects.

Persons with exactly the same genetic background, behavior, environment, etc. may have differences in cancer risk due to a different number of cells on the way to malignancy. These differences are estimated quantitatively by using the two-stage clonal expansion model. For liver cancer the estimated relative risk for persons without intermediate cells at age 40 is less than 10% when compared to the risk of the total population, while the top 0.1% risk group has a more than 100-fold risk compared to the population. The risk of the 1% percentile in risk is more than 100-fold of the risk of the more than 95% persons without intermediate cells. The number of intermediate (premalignant) cells in the risk groups cannot be calculated from incidence data only because they depend strongly on a nonidentifiable parameter. But under plausible assumptions, less than about 1,000 intermediate cells are present at age 40 even in high-risk persons.

Effects of exposure uncertainties in the TSCE model and application to the Colorado miners data.

The simulations in this paper show that exposure measurement error affects the parameter estimates of the biologically motivated two-stage clonal expansion (TSCE) model. For both Berkson and classical error models, we show that likelihood-based techniques of correction work reliably. For classical errors, the distribution of true exposures needs to be known or estimated in addition to the distribution of recorded exposures conditional on true exposures. Usually the exposure uncertainty biases the model parameters toward the null and underestimates the precision. But when several parameters are allowed to be dependent on exposure, e.g. initiation and promotion, then their relative importance is also influenced, and more complicated effects of exposure uncertainty can occur. The application part of this paper shows for two different types of Berkson errors that a recent analysis of the data for the Colorado plateau miners with the TSCE model is not changed substantially when correcting for such errors. Specifically, the conjectured promoting action of radon remains as the dominant radiation effect for explaining these data. The estimated promoting action of radon increases by a factor of up to 1.2 for the largest assumed exposure uncertainties.

Multistage models and the incidence of cancer in the cohort of atomic bomb survivors.

The analyses in this paper show that a number of biologically based models describe cancer incidence among the A-bomb survivors equally well. However, these different models can predict very different temporal patterns of risk after irradiation. No evidence was found to support the previous claim of Pierce and Mendelsohn that excess cancer risks for the solid tumors depend only upon attained age and not on age at exposure or time since exposure. Although the A-bomb survivor cohort is the largest epidemiological data set for the study of radiation and cancer, it is not large enough to discriminate among various possible carcinogenic mechanisms. Unfortunately for hypothesis generation, the data appear to be consistent with a number of different mechanistic interpretations of the role of radiation in carcinogenesis.