• Identification of factors and prediction of the risks of diseases are important for public policies and disease diagnosis in healthcare.

• The biomedical literature suggests that much of an individual’s later life health outcomes is programmed at early stages of life. The programming is strongly modulated by the epigentic inputs created by the environment in mother’s womb at prenatal stage and also by the environment at early postnatal ages.

• The most important epigenetic factor is stress of any kind – psychological, financial, social and chemical. Other significant factors are medical care, diets, smoking, substance use, and exercising. These modulating factors are important throughout life, with stronger effects imparted in early stages of life.

• I use the popular statistical model – mulinomial logit model – and the neural network model – multi-layer deep learning model – for identification of important factors and prediction of the risks of various diseases for individuals at their middle ages. I use the Health and Retirement Studies (HRS) data to that end.

• I compare their predictive powers using confusion matrix indicator and discuss various criteria used by these two methods to identify the important factors of the risks. I discuss pros and cons of these two types of models from prediction and inference viewpoint.

• Let \(y\) denote a random variable denoting an individual’s health status in middle age. The set of health statuses is {1,2,3,4,5}, where 1 = normal health, 2 = Cardiovascular disease, 3 = Cancer, 4 = Other single disease, 5 = comorbid diseases.

• \(X\) is a vector of individual characteristics such as demographic information, education, medical measurements, known as regressors.

• Data on individuals, \((y_i, X_i)\), \(1 = 1,2, ...,n\).

• Goal:
• To find a model \(y = f(X)\) that best explains data in the sample
• To predict the health status \(y\) of an individual not in the sample given his/her measurements \(X\) – i.e., out of sample prediction.
• To have inference about which factors in \(X\) are most important.
  
• How to judge which model is best or how to choose between models.
• This problem is known as classification problem. \(y\) could be a probability mass function. There is an equivalent regression problem.

• Two approaches to get a model:

1. Statistical Models – Multinomial Logistic Regression model in particular.
2. Machine Learning or Algorithms such as Multi Layer Neural Network model also known as Deep Learning model or Deep Neural Network Model, Tree based methods (such as CART, Random Forest etc), Support Vector Machines.

• I will only briefly explain the Multinomial Logistic Regression model and the Multi Layer Neural Network model.

• Assume that data \((y, X)\) is generated from a known probability distribution with the conditional probability distribution \(f(y|X,\beta)\) belonging to a known family of distributions with parameters \(\beta\) such as the exponential family.

• Multinomial Logistic Regression model comes down to the following probability model. \[ log \frac{Prob~(Y=k | X)}{Prob~(Y = 1|X)} = X'\beta ,~~ k=2,...,5 \]

• Neural network is a highly parameterized universal function approximator of the form \(\hat y = f(x;w)\), \(x\) is a set of inputs, and \(w\) is a vector of parameters– same form as a statistical model. The problem is to design a neural network architecture of the approximating function \(y = f(x,w)\) and find a suitable learning algorithm to learn the parameter values \(w\) of the network using a training set of examples. This trained network can then be used to predict \(y\) for an individual given his characteristics \(x\).

• The popularity and wide applicability of neural network lies in the fact that it designs the approximator in a hierarchy of functions, of the form: \[ \hat y = f(x;w) \equiv f^L_{w^L} \circ ....\circ f^1_{w_1} (x). \] Each function corresponds to a layer of artificial neurons.

Consider a simple neural network architecture It has three layers – layer 0: input layer, layer 1: hidden layer, and layer 2: output layer. Last layer is denoted as \(L\). Layer \(0\) has three input neurons. The second layer has 4 neurons. and the last layer has two neurons corresponding to the two output levels, in our case probability of two events, giving the hierarchical function specification of the form:

\[ f(x;w)=\sigma ^{2}\left( z^{2}\left( \sigma ^{1}\left( z^{1}(x,w^{1})\right) ,w^{2} \right) \right) \equiv f_{w^{2}}^{2}\circ f_{w^{1}}^{1}(x). \]

Function \(z^{i}(a^{i},w^{i})=w^{i}\cdot a^{i-1}\) at each layer \(i\) is a linear aggregator. The function \(\sigma ^i\) is a squashing function of the same dimension as \(z^i\), known as activation function.

A neural network is called deep neural network model or deep learning model because it can have more than one hidden layer to get better approximation compared to the original neural network models with only one hidden layer.

• Usefulness for Policy Analysis.

• Statistical models have well developed theories for inference about which inputs are important and other properties like how good the estimates are. But very poor performance at prediction. Some nonlinearity can be introduced to get better predictive performance, but it does not come close to the performance of the machine learning algorithms.

• Machine Learning models are good at prediction, but poor at identifying which inputs or regressors are important. Recently there are attempts to do so such as using Shapley Value criterion, but they are not useful in the multi outcome scenarios such as in the present case.