Heart

Model Set Up

Determine the effect that the independent variables biking and smoking have on the dependent variable heart disease using a multiple linear regression model.
Link to the dataset
Unfortunately, the database original source does not report the units on each variable.

import pandas as pd

data = pd.read_csv(r'data/data.csv')

Set up a multiple linear regression model, considering biking and smoking as regressors and heart disease as the response variable. Use non-informative priors for regressors and variance:

from baypy.model import LinearModel
import baypy as bp

model = LinearModel()

model.data = data
model.response_variable = 'heart disease'
model.priors = {'intercept': {'mean': 0, 'variance': 1e6},
                'biking': {'mean': 0, 'variance': 1e9},
                'smoking': {'mean': 0, 'variance': 1e9},
                'variance': {'shape': 1, 'scale': 1e-9}}

Sampling

Run the regression sampling on 3 Markov chains, with 500 iterations per each chain and discarding the first 50 burn-in draws:

from baypy.regression import LinearRegression

LinearRegression.sample(model = model, n_iterations = 500,
                        burn_in_iterations = 50, n_chains = 3, seed = 137)

Convergence Diagnostics

Asses the model convergence diagnostics:

bp.diagnostics.effective_sample_size(posteriors = model.posteriors)

                       intercept   biking  smoking  variance
Effective Sample Size    1389.56  1449.73  1362.26   1426.75

bp.diagnostics.autocorrelation_summary(posteriors = model.posteriors)

        intercept    biking   smoking  variance
Lag 0    1.000000  1.000000  1.000000  1.000000
Lag 1   -0.025015 -0.021166  0.009275 -0.021082
Lag 5    0.027681 -0.007564  0.046201  0.030989
Lag 10   0.015334  0.014290  0.043676 -0.057992
Lag 30  -0.041058 -0.008922 -0.013752 -0.040056

bp.diagnostics.autocorrelation_plot(posteriors = model.posteriors)

All diagnostics show a low correlation, indicating the chains converged to the stationary distribution.

Posteriors Analysis

Asses posterior analysis:

bp.analysis.trace_plot(posteriors = model.posteriors)

Traces are quite good, incidating draws from the stationary distribution.

bp.analysis.residuals_plot(model = model)

Also the residuals plot is good: no evidence for patterns, shapes or outliers.

bp.analysis.summary(posteriors = model.posteriors)

Number of chains:           3
Sample size per chian:    500

Empirical mean, standard deviation, 95% HPD interval for each variable:

                Mean        SD    HPD min    HPD max
intercept  14.985169  0.079494  14.811145  15.126328
biking     -0.200122  0.001387  -0.203015  -0.197531
smoking     0.178261  0.003535   0.171384   0.185280
variance    0.427870  0.027745   0.374502   0.480325

Quantiles for each variable:

                2.5%        25%        50%        75%      97.5%
intercept  14.822835  14.933689  14.986087  15.039621  15.141583
biking     -0.202909  -0.201028  -0.200086  -0.199219  -0.197334
smoking     0.171140   0.175893   0.178261   0.180621   0.185169
variance    0.380265   0.408345   0.426025   0.446627   0.488800

The summary reports a statistical evidence for:

• negative effect of biking: \(1\) point increase in biking would result in \(0.2\) points decrease in heart disease

• positive effect of smoking: \(1\) point increase in smoking would result \(0.18\) points increase in heart disease