An Ensemble Approach to Modelling the Combined Effect of Risk Factors on Age at Parkinson’s Disease Onset

Abstract

Ensemble approaches to statistical modelling combine multiple statistical methods to form a comprehensive analysis. They are of increasing interest for problems that involve diverse data sources, complex systems and subtle outcomes of interest. An example of such an ensemble approach is described in this chapter, in the context of a substantive case study that aimed to tease out factors affecting the age at onset of the neurodegenerative medical condition, Parkinsons Disease (PD), with a particular focus on the role of a particular potential risk factor, pesticide exposure.

Appendix: Supporting Information

Let Y  represent early age at onset and X represent the presence of one or more risk factors, which include smoking, alcohol, head injury or family history. Let X ^c represent the absence of risk factors and define E as,

$$\displaystyle \begin{aligned} E &= Y \cap (X) {} \end{aligned} $$

(11.9)

where E can be a lifestyle or medical history effect conducive to an early age at onset, such that when E represents lifestyle, X = {X ₁ = smoking, X ₂ = alcohol}. When E represents medical history, X = {X ₁ = head injury, X ₂ = family history}.

We wish to quantify the conditional probabilities P(E|X) and the data available is in the form of ORs and marginal probabilities as derived from the literature and from meta-analyses. These ORs take the following form,

(11.10)

Equation 11.10 can be rearranged to express OR(E|X) entirely in terms of P(E|X) and its marginal probabilities,

(11.11)

We solve Eq. 11.11 for P(E|X), which involves solving for the roots of the quadratic on P(E|X), to obtain the following expression for the CPT,

$$\displaystyle \begin{aligned} P(E | X) &= \frac{OR(E | X)P(X) + OR(E | X)P(E) + P(X^c) + P(E) \pm \sqrt{\psi}}{2[OR(E | X)P(X) + P(X)]} \\ \text{where }\, \psi &= [OR(E | X)P(X) + OR(E | X)P(E) + P(X^c) + P(E)]^2 \\ &\qquad - 4[OR(E | X)P(X) + P(X)][OR(E | X)P(E)] {} \end{aligned} $$

(11.12)

In the presence of more than one risk factor, due to the absence of data covering different combinations of risk factors, we assume conditional independence. For example, to estimate the lifestyle effects CPT P(E|X ₁, X ₂), assuming smoking (X ₁) and alcohol (X ₂) are conditionally independent, the CPT is quantified as,

$$\displaystyle \begin{aligned} P(E | X_1, X_2) &= \frac{P(E, X_1, X_2)}{P(X_1, X_2)} \\ &= \frac{P(E,X_1,X_2)}{P(X_1)P(X_2)} \\ &= \frac{P(X_1|E,X_2)P(E,X_2)}{P(X_1)P(X_2)} \\ &= \frac{P(X_1|E)P(E,X_2)}{P(X_1)P(X_2)} \\ &= \frac{P(E|X_1)P(X_1)P(E,X_2)}{P(E)P(X_1)P(X_2)} \\ &= \frac{P(E|X_1)P(E,X_2)}{P(E)P(X_2)} \\ &= \frac{P(E|X_1)P(E|X_2)P(X_2)}{P(E)P(X_2)} \\ P(E | X_1, X_2) &= \frac{P(E | X_1)P(E | X_2)}{P(E)} {} \end{aligned} $$

(11.13)

where P(E|X ₁) and P(E|X ₂) can be evaluated as per Eq. 11.12. From Eq. 11.9 where we have the presence of atleast one risk factor, P(E) is,

$$\displaystyle \begin{aligned} P(E) &= P(Y, X_{1}, X_{2}) + P(Y, X_{1}, X^{c}_{2}) + P(Y, X^{c}_{1}, X_{2}) {} \end{aligned} $$

(11.14)

In the QPP data source, we derived an OR for early onset given pesticide exposure using logistic regression. We assume this is approximately equal to the OR for early onset given OCP, OR(Y |OCP) where OCP represents exposure to OCPs. Thus, we apply Eq. 11.12 to estimate P(Y |E _OCP) where E _OCP represents the effect of OCP exposure conducive to an early age at onset.

To estimate the conditional probabilities of the terminal node P(Y |E _L, E _M, E _OCP), we assume that E _L, E _M and E _OCP are conditionally independent of each other due to the lack of existing studies on the combined effects of lifestyle, medical history and OCP exposure on an early age at PD onset. Here, E _L represents lifestyle effect and E _M represents medical history effect.

$$\displaystyle \begin{aligned} P(Y | E_L, E_M, E_{OCP}) &= \frac{P(Y,E_L,E_M,E_{OCP})}{P(E_L,E_M,E_{OCP})} \\ &= \frac{P(E_L|Y,E_M,E_{OCP})P(Y,E_M,E_{OCP})}{P(E_L)P(E_M)P(E_{OCP})} \\ &= \frac{P(E_L|Y)P(Y,E_M,E_{OCP})}{P(E_L)P(E_M)P(E_{OCP})} \\ &= \frac{P(E_L|Y)P(E_M|Y)P(Y,E_{OCP})}{P(E_L)P(E_M)P(E_{OCP})} \\ &= \frac{P(E_L|Y)P(E_M|Y)P(Y|E_{OCP})}{P(E_L)P(E_M)} \\ &= \frac{P(E_L,Y)P(E_M,Y)P(Y|E_{OCP})}{P(E_L)P(E_M)P(Y)^2} \\ P(Y | E_L, E_M, E_{OCP}) &= \frac{P(Y | E_L)P(Y | E_M) P(Y | E_{OCP})}{P(Y)^2} {} \end{aligned} $$

(11.15)

We can obtain P(Y |E _L) and equivalently P(Y |E _M) based on their relevant risk factors as informed by the QPP data source.

$$\displaystyle \begin{aligned} P(Y|E_L) = P(Y|S \vee A) = \frac{P(Y,S,A) + P(Y,S,A^c) + P(Y,S^c,A)}{P(S,A) + P(S,A^c) + P(S^c,A)} {} \end{aligned} $$

(11.16)

where S and A represent smoking and alcohol respectively and S ^c and A ^c represent the absence of smoking and alcohol respectively.