Understanding variation in disease risk: the elusive concept of frailty

Abstract

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.

Keywords: Frailty; epigenetics; heterogeneity; random variation.

Figure 1.

Age-standardized rates (ASR) of colorectal cancer, standardized with respect to the world (W) population, in various regions for 2012. Picture constructed by Globocan.

Figure 2.

Various types of possible distributions of the frailty (unexplained risk), $Z$, at an early age. The panels illustrate: (1) small variation in frailty between individuals, (2) large group have moderate frailty, and a smaller group of individuals have a high frailty, (3) very skewed: many individuals have a low frailty and a small group have a high frailty, (4) most individuals have close to zero frailty and a few individuals have a high frailty.

Figure 3.

The familial relative risk, r, associated with a diseased sibling as a function of s according to formula (1) in the text, where s denotes the relative risk associated with a change in a risk factor from mean minus two standard deviations to mean plus two standard deviations. The familial correlation, $\rho$, is set to 0.5. Based on normally distributed variation in risk.

Figure 4.

The modified familial relative risk, $r_F$, associated with a diseased sibling as a function of r according to formula (2) in the text, where the two individuals in the family have a common risk component that is gamma distributed with shape parameter $\delta$. Here r is the familial relative risk from formula (1), that is the familial relative risk without the skewness introduced by the common, gamma distributed component. $r_F$ is plotted for given values of $\delta$. Note that $\delta \to \infty$ implies $r_F=r$.

Figure 5.

Probability density for a random variable X, following either the exponential distribution (solid line) or the gamma distribution with shape parameter 0.5 (dashed line).

Figure 6.

Assume that the hazard rates in two risk groups are $\alpha \left(t\right)$ and $2\alpha \left(t\right)$ respectively. When frailty variables are introduced, the observed relative risk declines over time as shown in the figure. Three frailty distributions are used; one leads to a crossover of the hazard ratio. This case corresponds to a frailty distribution with a positive probability of zero frailty (i.e. a non-susceptible group). See Aalen et al., Chapter 6, for technical details.

Figure 7.

Effect of discontinuing treatment. A control group with hazard rate $2\alpha \left(t\right)$ is compared with a treatment group with hazard rate $\alpha \left(t\right)$. Treatment is discontinued at time point 1.

Figure 8.

Simple illustration of an Armitage-Doll multi-stage model of carcinogenesis. The states represent the stages of the carcinogenic process. State one is the healthy state, state two is an intermediate state, and in state three a malignant cell has developed. State four is a censored state. The $\alpha$s and $\mu$s are transition rates.

Figure 9.

Incidence rates for the model in Figure 8. Assume that 10,000 individuals enter state one per time unit. The transition rates are ${\alpha }_1=0.01$ for time $<20$, and ${\alpha }_1=0.03$ for time $\ge 20$. Also, ${\alpha }_2=0.02$, ${\mu }_1=0.01$ and ${\mu }_2=0.05$ a) 1% of the population is susceptible, i.e. having ${\alpha }_1\ne 0$. b) 90% of the population is susceptible, i.e. having ${\alpha }_1\ne 0$.