Regression
----------

Enjoy this brief demonstration of the regression module

### First we simulate some data

    # Create normal distributed data with mean = 0 and standard deviation = 1
    ## r = 0.5
    data <- MASS::mvrnorm(n=100,
                          mu=c(0, 0),
                          Sigma=matrix(c(1, 0.5, 0.5, 1), 2),
                          empirical=TRUE)
    # Add names
    colnames(data) <- c("x","y")

### Check the correlation and regression results using frequentist methods

    # Correlation
    stats::cor(data)[2]
    #> [1] 0.5

    # Regression
    summary(stats::lm(y ~ x, data=data.frame(data)))
    #> 
    #> Call:
    #> stats::lm(formula = y ~ x, data = data.frame(data))
    #> 
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max 
    #> -2.0482 -0.5273  0.0887  0.5984  1.8607 
    #> 
    #> Coefficients:
    #>                           Estimate             Std. Error t value   Pr(>|t|)    
    #> (Intercept) 0.00000000000000000416 0.08704326862109958152    0.00          1    
    #> x           0.50000000000000022204 0.08748177652797066439    5.72 0.00000012 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> 
    #> Residual standard error: 0.87 on 98 degrees of freedom
    #> Multiple R-squared:  0.25,   Adjusted R-squared:  0.242 
    #> F-statistic: 32.7 on 1 and 98 DF,  p-value: 0.000000118

### Then the regression results using the Bayesian model

    mcmc <- bfw::bfw(project.data = data,
                y = "y",
                x = "x",
                saved.steps = 50000,
                jags.model = "regression",
                jags.seed = 100,
                silent = TRUE)
    # Print the results            
    round(mcmc$summary.MCMC[,3:7],3)
    #>                        Mode   ESS  HDIlo HDIhi   n
    #> beta0[1]: Intercept  -0.008 50000 -0.172 0.173 100
    #> beta[1]: Y vs. X      0.492 51970  0.330 0.674 100
    #> sigma[1]: Y vs. X     0.863 28840  0.760 1.005 100
    #> zbeta0[1]: Intercept -0.008 50000 -0.172 0.173 100
    #> zbeta[1]: Y vs. X     0.492 51970  0.330 0.674 100
    #> zsigma[1]: Y vs. X    0.863 28840  0.760 1.005 100
    #> R^2 (block: 1)        0.246 51970  0.165 0.337 100

### Now we add some noise

    # Create noise with mean = 10 / -10 and sd = 1
    ## r = -1.0
    noise <- MASS::mvrnorm(n=2,
                           mu=c(10, -10),
                           Sigma=matrix(c(1, -1, -1, 1), 2),
                           empirical=TRUE)
    # Combine data
    biased.data <- rbind(data,noise)

### Repeat

    # Correlation
    stats::cor(biased.data)[2]
    #> [1] -0.498

    # Regression
    summary(stats::lm(y ~ x, data=data.frame(biased.data)))
    #> 
    #> Call:
    #> stats::lm(formula = y ~ x, data = data.frame(biased.data))
    #> 
    #> Residuals:
    #>    Min     1Q Median     3Q    Max 
    #> -5.272 -0.815  0.027  0.822  3.108 
    #> 
    #> Coefficients:
    #>             Estimate Std. Error t value    Pr(>|t|)    
    #> (Intercept)  -0.0983     0.1487   -0.66        0.51    
    #> x            -0.4984     0.0867   -5.75 0.000000098 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #> 
    #> Residual standard error: 1.49 on 100 degrees of freedom
    #> Multiple R-squared:  0.248,  Adjusted R-squared:  0.241 
    #> F-statistic: 33.1 on 1 and 100 DF,  p-value: 0.0000000975

### Finally, using Bayesian model with robust estimates

    mcmc.robust <- bfw::bfw(project.data = biased.data,
                y = "y",
                x = "x",
                saved.steps = 50000,
                jags.model = "regression",
                jags.seed = 100,
                run.robust = TRUE,
                silent = TRUE)
    # Print the results            
    round(mcmc.robust$summary.MCMC[,3:7],3)
    #>                        Mode   ESS  HDIlo HDIhi   n
    #> beta0[1]: Intercept  -0.026 29844 -0.204 0.141 102
    #> beta[1]: Y vs. X      0.430 29549  0.265 0.604 102
    #> sigma[1]: Y vs. X     0.671 16716  0.530 0.842 102
    #> zbeta0[1]: Intercept  0.138 28442  0.042 0.244 102
    #> zbeta[1]: Y vs. X     0.430 29549  0.265 0.604 102
    #> zsigma[1]: Y vs. X    0.392 16716  0.310 0.492 102
    #> R^2 (block: 1)        0.236 29549  0.145 0.331 102
