Morten Valberg

Understanding variation in disease risk: the elusive concept of frailty

The concept of frailty plays a major role in the statistical field of survival analysis. Frailty variation refers to differences in risk between individuals which go beyond known or measured risk factors. In other words, frailty variation is unobserved heterogeneity. Although understanding frailty is of interest in its own right, the literature on survival analysis has demonstrated that existence of frailty variation can lead to surprising artefacts in statistical estimation that are important to examine. We present literature that demonstrates the presence and significance of frailty variation between individuals. We discuss the practical content of frailty variation, and show the link between frailty and biological concepts like (epi)genetics and heterogeneity in disease risk. There are numerous suggestions in the literature that a good deal of this variation may be due to randomness, in addition to genetic and/or environmental factors. Heterogeneity often manifests itself as clustering of cases in families more than would be expected by chance. We emphasize that apparently moderate familial relative risks can only be explained by strong underlying variation in disease risk between families and individuals. Finally, we highlight the potential impact of frailty variation in the interpretation of standard epidemiological measures such as hazard and incidence rates.

A Hierarchical Frailty Model for Familial Testicular Germ-Cell Tumors

Using a 2-level hierarchical frailty model, we analyzed population-wide data on testicular germ-cell tumor (TGCT) status in 1,135,320 two-generational Norwegian families to examine the risk of TGCT in family members of patients. Follow-up extended from 1954 (cases) or 1960 (unaffected persons) to 2008. The first-level frailty variable was compound Poisson-distributed. The underlying Poisson parameter was randomized to model the frailty variation between families and was decomposed additively to characterize the correlation structure within a family. The frailty relative risk (FRR) for a son, given a diseased father, was 4.03 (95% confidence interval (CI): 3.12, 5.19), with a borderline significantly higher FRR for nonseminoma than for seminoma (P = 0.06). Given 1 affected brother, the lifetime FRR was 5.88 (95% CI: 4.70, 7.36), with no difference between subtypes. Given 2 affected brothers, the FRR was 21.71 (95% CI: 8.93, 52.76). These estimates decreased with the number of additional healthy brothers. The estimated FRRs support previous findings. However, the present hierarchical frailty approach allows for a very precise definition of familial risk. These FRRs, estimated according to numbers of affected/nonaffected family members, provide new insight into familial TGCT. Furthermore, new light is shed on the different familial risks of seminoma and nonseminoma.

Exploring Selection Bias by Causal Frailty Models: The Magnitude Matters

Counter-intuitive associations appear frequently in epidemiology, and these results are often debated. In particular, several scenarios are characterized by a general risk factor that appears protective in particular subpopulations, for example, individuals suffering from a specific disease. However, the associations are not necessarily representing causal effects. Selection bias due to conditioning on a collider may often be involved, and causal graphs are widely used to highlight such biases. These graphs, however, are qualitative, and they do not provide information on the real life relevance of a spurious association. Quantitative estimates of such associations can be obtained from simple statistical models. In this study, we present several paradoxical associations that occur in epidemiology, and we explore these associations in a causal, frailty framework. By using frailty models, we are able to put numbers on spurious effects that often are neglected in epidemiology. We discuss several counter-intuitive findings that have been reported in real life analyses, and we present calculations that may expand the understanding of these associations. In particular, we derive novel expressions to explain the magnitude of bias in index-event studies.

Frailty modeling of age–incidence curves of osteosarcoma and Ewing sarcoma among individuals younger than 40 years

The Armitage-Doll model with random frailty can fail to describe incidence rates of rare cancers influenced by an accelerated biological mechanism at some, possibly short, period of life. We propose a new model to account for this influence. Osteosarcoma and Ewing sarcoma are primary bone cancers with characteristic age-incidence patterns that peak in adolescence. We analyze Surveillance, Epidemiology and End Result program incidence data for whites younger than 40 years diagnosed during the period 1975-2005, with an Armitage-Doll model with compound Poisson frailty. A new model treating the adolescent growth spurt as the accelerated mechanism affecting cancer development is a significant improvement over that model. We also model the incidence rate conditioning on the event of having developed the cancers before the age of 40 years and compare the results with those predicted by the Armitage-Doll model. Our results support existing evidence of an underlying susceptibility for the two cancers among a very small proportion of the population. In addition, the modeling results suggest that susceptible individuals with a rapid growth spurt acquire the cancers sooner than they otherwise would have if their growth had been slower. The new model is suitable for modeling incidence rates of rare diseases influenced by an accelerated biological mechanism.

The surprising implications of familial association in disease risk

BackgroundA wide range of diseases show some degree of clustering in families; family history is therefore an important aspect for clinicians when making risk predictions. Familial aggregation is often quantified in terms of a familial relative risk (FRR), and although at first glance this measure may seem simple and intuitive as an average risk prediction, its implications are not straightforward.MethodsWe use two statistical models for the distribution of disease risk in a population: a dichotomous risk model that gives an intuitive understanding of the implication of a given FRR, and a continuous risk model that facilitates a more detailed computation of the inequalities in disease risk. Published estimates of FRRs are used to produce Lorenz curves and Gini indices that quantifies the inequalities in risk for a range of diseases.ResultsWe demonstrate that even a moderate familial association in disease risk implies a very large difference in risk between individuals in the population. We give examples of diseases for which this is likely to be true, and we further demonstrate the relationship between the point estimates of FRRs and the distribution of risk in the population.ConclusionsThe variation in risk for several severe diseases may be larger than the variation in income in many countries. The implications of familial risk estimates should be recognized by epidemiologists and clinicians.

Can chance cause cancer? A causal consideration

The role of randomness, environment and genetics in cancer development is debated. We approach the discussion by using the potential outcomes framework for causal inference. By briefly considering the underlying assumptions, we suggest that the antagonising views arise due to estimation of substantially different causal effects. These effects may be hard to interpret, and the results cannot be immediately compared. Indeed, it is not clear whether it is possible to define a causal effect of chance at all.

Inequality in genetic cancer risk suggests bad genes rather than bad luck

Heritability is often estimated by decomposing the variance of a trait into genetic and other factors. Interpreting such variance decompositions, however, is not straightforward. In particular, there is an ongoing debate on the importance of genetic factors in cancer development, even though heritability estimates exist. Here we show that heritability estimates contain information on the distribution of absolute risk due to genetic differences. The approach relies on the assumptions underlying the conventional heritability of liability model. We also suggest a model unrelated to heritability estimates. By applying these strategies, we describe the distribution of absolute genetic risk for 15 common cancers. We highlight the considerable inequality in genetic risk of cancer using different metrics, e.g., the Gini Index and quantile ratios which are frequently used in economics. For all these cancers, the estimated inequality in genetic risk is larger than the inequality in income in the USA.Cancer heritability estimates can be obtained via decomposing trait variance into genetic and other factors. Here, the authors obtain the distribution of absolute genetic risk for 15 common cancers, and they use a number of metrics to show that the genetic risk varies considerably across individuals.

Can Collider Bias Explain Paradoxical Associations

To the Editor:Limiting the study population to diseased subjects may influence the effect estimates,1–4 because collider bias is introduced. Sperrin et al.5 recently suggested that “the bias is small relative to the causal relationships between the variables.” Furthermore, they stated that collider

Tumor dormancy and frailty models: A novel approach.

Frailty models are here proposed in the tumor dormancy framework, in order to account for possible unobservable dependence mechanisms in cancer studies where a non-negligible proportion of cancer patients relapses years or decades after surgical removal of the primary tumor. Relapses do not seem to follow a memory-less process, since their timing distribution leads to multimodal hazards. From a biomedical perspective, this behavior may be explained by tumor dormancy, i.e., for some patients microscopic tumor foci may remain asymptomatic for a prolonged time interval and, when they escape from dormancy, micrometastatic growth results in a clinical disease appearance. The activation of the growth phase at different metastatic states would explain the occurrence of metastatic recurrences and mortality at different times (multimodal hazard). We propose a new frailty model which includes in the risk function a random source of heterogeneity (frailty variable) affecting the components of the hazard function. Thus, the individual hazard rate results as the product of a random frailty variable and the sum of basic hazard rates. In tumor dormancy, the basic hazard rates correspond to micrometastatic developments starting from different initial states. The frailty variable represents the heterogeneity among patients with respect to relapse, which might be related to unknown mechanisms that regulate tumor dormancy. We use our model to estimate the overall survival in a large breast cancer dataset, showing how this improves the understanding of the underlying biological process.

Authors’ response: Understanding variation in disease risk

First, we thank the authors of the three commentaries1,2,3 for their interesting discussion of our paper.4 Our intention was to shed light on how the concept of frailty can contribute to the understanding of variations in disease risk that are due to unobserved or unknown factors. We discuss how such differences may arise, and what the consequences of such variations are when studying populations. Although Peto’s statement that frailty retreats when biology advances is certainly true to some extent,1 we argue that there will always be some unexplained variation left, due to stochastic elements involved in the development of many diseases. Even if all relevant risk factors, be they environmental or (epi)genetic, could be identified, there would still be considerable variation in risk that is unexplained. In a recent paper, Tomasetti and Vogelstein suggest that ‘only a third of the variation in cancer risk among tissues is attributable to environmental factors or inherited predispositions’.5 They further argue that the majority of cases are ‘bad luck’ due to random mutations. Although it is possible to question the claim that two-thirds of cancer cases occur randomly, which might be an exaggeration, it still gives recognition to the importance of randomness in the development of cancer. Frailty models offer one way of handling this in epidemiological studies. Using frailty models for analysing single time-to-event data may be quite speculative, and they should be based on sound biological knowledge to be used for any kind of mechanistic inferences. Whether the estimate of only 12% of the US population being susceptible to colon cancer, as suggested by Soto-Ortiz and Brody,6 is reasonable or not, is of course a relevant question. Nevertheless, simple frailty models may still be quite useful in a hypothesis-generating way. Mathews and Hopper point out that frailty explanations and biological explanations are not ‘competing’, as we state in the paper, and they further claim that frailty must have a biological explanation.2 The point we wanted to make is that there is a distinction between a mechanistic explanation of, say, a peak in the incidence rate and a selection explanation which is the frailty one. Of course, selection also has a biological basis but might be seen as competing with a mechanistic biological view. For instance, a mechanistic explanation of the peak in testicular cancer incidence at around 30 years of age could be declining testosterone level; however, there is no basis for this view and the competing selection, or frailty, explanation is likely to hold. When times-to-events are related, the amount of speculation is reduced in a frailty model. A major topic in our paper is that of familial clustering of disease. One particular interest lies in what a so-called familial relative risk larger than one implies. Since such estimates compare the risk in individuals who have a certain familial history of the disease in question, with the average risk level in the population (or with the risk in individuals with a different familial history), it does not immediately say anything about how the risk is distributed across the population. An important implication, also appreciated in the commentaries, is that even moderate familial relative risks have to mean that there are potentially very large differences in risk between individuals in the population. Frailty models seem to be very suitable for analysing this kind of data, and are not only able to provide estimates of very detailed familial relative risks (given any kind of familial history), but can also provide information on how the underlying risk is distributed in the population.7,8 Moolgavkar argues that frailty and heterogeneity are two terms that should be kept apart; frailty should be reserved for situations where a subgroup of the population is exclusively at risk, or at vastly increased risk.3 We fail to see the reasoning behind this statement. A population that contains two risk groups is heterogeneous, and a population with a continuous spectrum of underlying risk is also heterogeneous, even if the variation is modest. In our terminology, varying frailty between individuals expresses the heterogeneity in risk in a population, regardless of how it is distributed. That being said, situations where the frailty effects are most striking are perhaps those where the frailty distribution is much skewed. We also provide a brief discussion of ‘frailty and models of carcinogenesis’, which Moolgavkar finds to be ‘unclear.’3 It is true that our wording was not the best when stating that the Armitage-Doll model is a ‘sensible approximation to the carcinogenic process within an individual’, when it actually gives an approximation to the hazard function in an initially homogeneous population. Moolgavkar states that: ‘Inter-individual variations in susceptibility arising from variations in the rates of critical biological processes can be modelled by assuming a distribution on the parameters of the model’. This is exactly what we are suggesting. We randomize a parameter in the Weibull (i.e. Armitage-Doll) model, and let it be distributed over the entire population. The remaining criticism seems to be the use of the ‘poor’ Weibull approximation. However, as long as the probability of the event (cancer) occurring is small, the Weibull approximation will be good. Our point is that we are interested in studying selection effects in a population that is inherently heterogeneous, which can indeed be done by combining a Weibull model with the notion of a varying frailty. The resulting model is intuitively easy to understand, although there are more sophisticated models that provide a more accurate description of carcinogenesis itself.