A Bayesian Outlier Detection Technique

1 The OLS fit

> summary(olsfit _lm(LogRadon ~ Floor + Clay + Goodhue,data=radon))
Call: lm(formula = LogRadon ~ Floor + Clay + Goodhue, data = radon)
Residuals:
    Min      1Q   Median     3Q   Max
 -1.653 -0.5572 -0.01608 0.5933 1.855
Coefficients:
              Value Std. Error t value Pr(>|t|)
(Intercept)  1.9561  0.2158     9.0663  0.0000
      Floor -0.3283  0.3155    -1.0409  0.3047
       Clay -0.0798  0.3018    -0.2644  0.7929
    Goodhue -0.0379  0.3041    -0.1248  0.9014
Residual standard error: 0.7895 on 37 degrees of freedom
Multiple R-Squared: 0.0332
F-statistic: 0.4235 on 3 and 37 degrees of freedom, the p-value is 0.7373
Correlation of Coefficients:
        (Intercept)   Floor    Clay
  Floor -0.2089
   Clay -0.6526     -0.1493
Goodhue -0.6762     -0.0114  0.4868

2 The model

For each i, let

z_{i} = \{ \begin{matrix}1, if obs i is an outlier; \\ 2, if obs i is not an outlier. \end{matrix}

Then our model is

y_{i} ∼ \{ \begin{matrix}\underline{x}_{i}^{T} \underline{β} + \sqrt{(ν − 2)/ν} · σ · t_{ν}, if z_{i} = 1; \\ \underline{x}_{i}^{T} \underline{β} + σ · N(0, 1), if z_{i} = 2. \end{matrix}

z_{i} = \{ \begin{matrix}1, with probability p_{1}; \\ 2, with probability p_{2}. \end{matrix}
\underline{β} ∼ flat
log(σ) ∼ flat
p_{1} ∼ Beta(a, b)
p_{2} = 1 − p_{1}

-bb-error = = Figure 1: Standard Splus diagnostics for the OLS fit. Note that observations X, Y and Z, have maximum infleunce and Z and X seem to be outliers.

3 The BUGS code

4 The output