John Fox jfox at mcmaster.ca
Mon Apr 24 02:52:10 CEST 2006

```Dear Adrian and Ian (and r-helpers),

To answer my own question: The difference is simply due to rounding; that
is, there are actually small differences in the R^2 values that account for
the differences in the BICs.

Sorry for the false alarm,
John

> -----Original Message-----
> From: John Fox [mailto:jfox at mcmaster.ca]
> Sent: Sunday, April 23, 2006 9:35 AM
> To: 'Adrian Raftery'; 'Ian Painter'
> Cc: 'r-help at stat.math.ethz.ch'
>
> Dear Adrian and Ian (and r-helpers),
>
> I encountered a curious result in developing an example using
> the bicreg function in the BMA package: I noticed that pairs
> of models with equal R^2 and equal numbers of predictors had
> nevertheless different BIC values.
> Looking at the bicreg function, the definition of BIC appears
> to be the usual one, or close to it [bic <- n * log(1 -
> r2/100) + (size - 1) * log(n)], though in the quick look I
> took, I didn't quite see where "size"
> was coming from.
>
> In any event, I've appended some output. The data are the
> 1987 baseball-salary data (with which you might be familiar);
> I can send you the data set, if that proves useful, but at
> this point simply wanted to run the question by you.
>
> Any help would be appreciated,
>
> John
>
> ---------------- snip -------------------
>
>
> > br <- bicreg(X, log.salary, maxCol=33)
> >
> > summary(br)
>
> Call:
> bicreg(x = X, y = log.salary, maxCol = 33)
>
>
>   65  models were selected
>  Best  5  models (cumulative posterior probability =  0.2022 ):
>
>                 p!=0    EV         SD         model 1
> model 2     model
> 3     model 4     model 5
> Intercept       100.0  -5.712e-01  0.6720850  -9.097e-01  -9.099e-01
> -1.001e+00   1.610e-01  -1.146e+00
> atbats            1.2  -7.598e-05  0.0006955       .
>  .           .
> .           .
> hits              7.9   4.280e-04  0.0023605       .
>  .           .
> .           .
> homers            2.3  -1.562e-04  0.0011977       .
>  .           .
> .           .
> runs              0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> rbis              0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> walks            93.4   6.686e-03  0.0025198   7.512e-03   7.753e-03
> 8.239e-03   6.739e-03   6.247e-03
> years             0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> career.atbats   100.0   8.316e-01  0.1272297   9.004e-01   9.014e-01
> 9.151e-01   6.858e-01   9.426e-01
> career.hits     100.0  -1.098e-03  0.0004007  -1.284e-03  -1.269e-03
> -1.651e-03  -6.317e-04  -1.557e-03
> career.homers     2.4   4.541e-05  0.0003012       .
>  .           .
> .           .
> career.runs      54.6   8.680e-04  0.0008855   1.704e-03   1.688e-03
> 1.529e-03       .       1.614e-03
> career.rbis      36.2   3.764e-04  0.0005293       .           .
> 9.876e-04       .       1.051e-03
> career.walks     59.5  -6.551e-04  0.0006580  -8.521e-04  -8.894e-04
> -9.950e-04       .      -1.583e-03
> putouts           0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> assists           8.3  -4.809e-05  0.0001787       .
>  .           .
> .           .
> errors            0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> middle.infield   44.2   7.835e-02  0.1082097   1.475e-01
>  .           .
> .           .
> center.field      1.1  -1.619e-03  0.0177267       .
>  .           .
> .           .
> catcher          71.8   1.537e-01  0.1182395   2.406e-01   2.123e-01
> 2.009e-01       .       2.107e-01
> dh                3.4  -6.842e-03  0.0425172       .
>  .           .
> .           .
> atbats.year      98.8  -7.200e-03  0.0014719  -7.737e-03  -7.159e-03
> -7.116e-03  -6.448e-03  -7.556e-03
> hits.year        98.8   2.608e-02  0.0050614   2.572e-02   2.470e-02
> 2.808e-02   2.457e-02   2.732e-02
> homers.year      11.1   1.966e-03  0.0058173       .
>  .           .
> .           .
> runs.year         4.3   4.312e-04  0.0022034       .
>  .           .
> .           .
> rbis.year        50.7   4.702e-03  0.0049808   1.010e-02
> 8.065e-03       .
> 7.805e-03       .
> walks.year       20.5   2.386e-03  0.0052883       .
>  .           .
> .       8.710e-03
> average           0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> career.average    0.0   0.000e+00  0.0000000       .
>  .           .
> .           .
> onbase            1.2  -9.736e-02  0.8998396       .
>  .           .
> .           .
> career.onbase     1.2   7.627e-02  0.6997039       .
>  .           .
> .           .
> arbitration      41.1   1.099e-01  0.1527324       .
>  .           .
> 2.982e-01       .
> freeagent       100.0   3.991e-01  0.2066829   2.520e-01   2.525e-01
> 2.693e-01   6.452e-01   2.560e-01
>
>
> nVar                                             11
> 10          10
> 8           11
> r2                                             0.844
> 0.841       0.841
> 0.833       0.844
> BIC                                           -4.261e+02  -4.259e+02
> -4.255e+02  -4.251e+02  -4.250e+02
> post prob                                      0.053
> 0.049       0.038
> 0.032       0.030
>
> --------------------------------
> John Fox
> Department of Sociology
> McMaster University
> Hamilton, Ontario