say hi to the Bell regression model

A few days ago I made a post about the Bell distribution, and showed some of its properties. Well, at the end of that post I said I would post about adjusting the mortality model using this, and here I am doing this post about adjusting the mortality model with the Bell distribution.

As I mentioned, the Bell distribution is part of the exponential Family, and that makes it possible to work with it as a regression model (linear or not). So here we go:

Let’s consider \(D_t\) as the number of deaths in a given interval \([t, t+1)\), with \(t = 0, 1, \dots, m\). If the number of deaths and the number of person-years exposed to the risk of dying can be observed, then we can assume that \(D_t\) follows a Poisson distribution (as well a Bell distribution), with \(\mathbb{E}(D_t) = \mu(t; \mathbf{\theta})E_t\), as proposed by Brillinger (1986). As you already know, \(\mu(t; \mathbf{\theta})\) represent the mortality rate at age \(t\), \(E_t\) denotes the number of person-years with age \(t\) exposed to the risk of dying, and \(\mathbf{\theta} = (\theta_1, \dots, \theta_d) '\) the \(d\)-dimentional parameter vector that characterizes the mortality rate.

For the likelihood function, let’s consider \(\mathbf{D} = (D_0, D_1, \dots, D_m)'\) and \(\mathbf{E} = (E_0, E_1, \dots, E_m)'\). So, considering the Poisson distribution for the death count, the likelihood function for the parameter vector \(\mathbf{\theta}\) is given by

\[L(\mathbf{\theta}|\mathbf{D},\mathbf{E}) = \prod_{t=0}^m\frac{e^{-\mu(t; \mathbf{\theta}) E_t} [\mu(t; \mathbf{\theta}) E_t]^{D_t}}{D_t!},\]

and the log-likelihood function, eliminating the additive constants, is given by

\[\ell(\mathbf{\theta}|\mathbf{D},\mathbf{E})= \ln(L(\mathbf{\theta})) = \sum_{t=0}^m\left[D_t\ln(\mu(t; \mathbf{\theta})E_t) -\mu(t; \mathbf{\theta})E_t\right].\]

Therefore, if you consider that the number of deaths has a Poisson distribution, to estimate the model what you need to do is just maximize the log-likelihood and obtain the estimates (see, for example, Castellares (2020)).

But the Poisson regression model has been explored for many years, so come with me to explore the new Bell regression model. As I had commented, from the Poisson model we can easily derive the Bell model, because both are one-parameter models. Thus, considering the Bell distribution for the death count, the likelihood function for the parameter vector $\mathbf{\theta}$ assumes the form

\[L(\mathbf{\theta}|\mathbf{D},\mathbf{E}) = \prod_{t=0}^m \exp\left(1-e^{W_0(\mu(t; \mathbf{\theta}) E_t)}\right)\frac{W_0(\mu(t; \mathbf{\theta}) E_t)^{D_t}B_{D_t}}{D_t!}\]

and the log-likelihood function, unless constant terms, can be expressed as

\[\ell(\mathbf{\theta}|\mathbf{D},\mathbf{E})= \ln(L(\mathbf{\theta})) = \sum_{t=0}^m\left[D_t\ln\left(W_0(\mu(t; \mathbf{\theta})E_t)\right) -e^{W_0(\mu(t; \mathbf{\theta})E_t)}\right].\]

Again, to obtain estimates of the parameters that characterize the mortality rate considering that the number of deaths has a Bell distribution, it is enough to maximize the likelihood function with respect to \(\mathbf\theta\). Simpler than you imagined, isn’t it?

In the next post I will apply and compare these two regression models to estimate mortality.

I see ya in the next post, or on twitter.

References

Brillinger, David R. A biometrics invited paper with discussion: the natural variability of vital rates and associated statistics. Biometrics, p. 693-734, 1986.

Castellares, Fredy, Silvio C. Patrício, and Artur J. Lemonte. “On gamma-Gompertz life expectancy.” Statistics & Probability Letters (2020): 108832.

Castellares, Fredy., Patrício, Silvio C., Lemonte, Artur J., & Queiroz, Bernardo L. On closed-form expressions to Gompertz–Makeham life expectancy. Theoretical Population Biology. (2020)

Castellares, Fredy, Silvia LP Ferrari, and Artur J. Lemonte. “On the Bell distribution and its associated regression model for count data.” Applied Mathematical Modelling 56 (2018): 172-185.