.. _risk_factor:

.. _modelling:

Risk factors
===============

Modelling Framework
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Main Principles
---------------
There are six risk factors included in the model – alcohol consumption (:ref:`alcohol_rf`), blood pressure (:ref:`bp_rf`), BMI (:ref:`bmi_rf`), 
physical activity (:ref:`pa_rf`), diabetes (:ref:`diabetes_rf`) and sedentary behaviours (:ref:`sed_rf`). 
Consistent with the literature, for each independent risk factor, an individual 
is assigned permanently into one of four quantiles (highest quantile representing individuals with the 
highest risk factor). Therefore, each individual maintains throughout his/her life a fixed position in 
the cohort-specific distribution of a given risk factor; with cohorts defined by age, gender and 
country. In practical terms, this means that if in year X a person living in country K, with age Y 
and of gender Z is in the 32nd highest percentile for body-mass distribution, in year X+10, the 
same person, now aged Y+10 will maintain his/her position in the 30th highest percentile of the 
distribution for that cohort. So, while in relative terms the BMI of the person would not have 
changed, in absolute terms the person may have now increased or decreased depending on the new 
BMI distribution at year X+10.

.. admonition:: Fixed quantile, using uniform distributions to sample arbitrary distribution.

   The “fixed quantile” approach is based on the mathematical method that uses the uniform distribution for sampling
   from arbitrary distributions.

   The probability integral transform states that if :math:`X` is a continuous random variable with cumulative distribution
   function :math:`F_X` then the random variable :math:`Y=F_X (X)` has a uniform distribution on :math:`[0, 1]`.
   The inverse probability integral transform is just the inverse of this: specifically, if Y has a uniform distribution
   on :math:`[0,1]` and if :math:`X` has a cumulative distribution :math:`F^{-1}_X`, then the random variable
   :math:`F_X^{-1} (Y)` has the same distribution as :math:`X` :cite:p:`Devroye1986`.


The numeric value assigned to each risk factor evolves over time and by gender following changes 
to the risk factor distribution. A fixed quantile approach is utilised to maintain longitudinal 
consistency. The key drawback of this method is that it cannot model longitudinal impacts. For 
instance, a change in the prevalence rate of obesity amongst children does not automatically affect 
the distribution of obesity in adults 30 years in the future. 

Impact of risk factors on health (relative risks)
-------------------------------------------------
Risk factors (RF) affect disease incidence through ‘relative risks’ (RRs). Each RR is adjusted for 
age, gender and other potential confounding variables. 

For each RF/disease combination, a ‘baseline risk’ is computed. Essentially, the baseline risk 
combines the prevalence of different risk categories and their associated RR. The 
incidence of developing a disease, I, given a specific RF for a disease, d, is equal to:  

.. math::
    I|Risk_d=RR_d* BaselineRisk_d * I_d

As an example, for generic risk factors entailing two states: 1) with the RF (:math:`RF`); and 
2) without the risk factor  (:math:`\bar{RF}`), the model uses the following formulae: 

.. math::
   \begin{eqnarray}
      I_d |RF &=& RR_d* BaselineRisk_d * I_d \\
      I_d |\bar{RF} &=& BaselineRisk_d * I_d
   \end{eqnarray}

Relative risks combine ‘multiplicatively’, that is, the model multiplies both relative 
risk and baseline risk to get the specific incidence of the global risk profile.  We illustrate the 
formula below with harmful alcohol consumption (HAC) and BMI. 

.. math::
   \begin{eqnarray}
      I_d|(HAC \& BMI=3) &= & RR(HAC){BaselineRisk}_{d}(HAC)\\
      && *{RR}_{d}(BMI=30){BaselineRisk}_{d} * {I}_{d}
   \end{eqnarray}

Only the current value of a RF is used to evaluate the probability of an individual developing a 
disease. That is, the past trajectory of a risk factor is not considered. Using the example above, 
this means that a 30-year old man who began drinking heavily at age 30 will have the same associated 
disease risk as a person who has been heavily drinking from the age of 20 and who has lost weight so 
that their BMI dropped from 40 to 30. In real life, the latter individual would be at greater risk of 
developing any one disease compared to the former. While this assumption matters for individual 
persons, it does not affect results at the population level. 

Conservatively, risk factors only affect disease incidence, and are not associated with fatality 
directly. 

Specific Models
``````````````````````

.. toctree::
   :maxdepth: 1

   alcohol
   bloodpression
   bmi
   PA
   Diabetes
   sedbeh