Model

Using a Bayesian regression model, and the formula \[Risk∼β0​+β1​×BMI\]

Model Coefficients
Characteristic Beta SE 95% CI Lower 95% CI Upper
b_Intercept −3.71 0.41 −4.53 −2.93
b_BMI 0.09 0.01 0.07 0.12
Intercept −0.70 0.08 −0.85 −0.54
lprior −6.45 0.00 −6.45 −6.44
lp__ −467.79 1.01 −470.57 −466.83

1. Model Formula

Logistic Regression Model:

[ (p) = _0 + _1 ]

  • ( p ): Probability of the outcome (e.g., having diabetes).

  • logit(p): The log-odds of the probability ( p ), which is defined as:

    [ (p) = () ]

    This transformation converts probabilities (ranging from 0 to 1) into a scale from (-) to (+).

  • **( _0 )**: The intercept, representing the log-odds of the outcome when all predictors (e.g., BMI) are zero.

  • **( _1 )**: The coefficient for BMI, indicating how the log-odds of the outcome change with a one-unit increase in BMI.

2. Likelihood Function

Binary Outcome:

[ P(Y | _0, _1) ]

  • ( Y ): The binary outcome variable (0 or 1).

  • ( p ): The probability of ( Y = 1 ) given BMI, modeled by the logistic function:

    [ p = ]

This function maps the log-odds back to a probability between 0 and 1.

3. Priors

In Bayesian regression, prior distributions are specified for the model parameters:

  • **Intercept (( _0 ))**:
    ( _0 (0, 10) )
    A normal distribution with mean 0 and standard deviation 10, allowing a wide range of intercept values.

  • **Coefficient (( _1 ))**:
    ( _1 (0, 10) )
    Similarly, a normal distribution with mean 0 and standard deviation 10 for the BMI coefficient.

Summary

  • Logit Function: Converts probability ( p ) to log-odds.
  • Logistic Function: Converts log-odds back to probability.
  • Likelihood: Describes the distribution of observed data ( Y ) given the parameters and predictors.
  • Priors: Reflect beliefs about the parameters before observing the data.

The Bayesian logistic regression model updates the priors based on observed data to estimate the parameters ( _0 ) and ( _1 ).