John Fox jfox at mcmaster.ca
Sun Apr 23 16:35:24 CEST 2006

```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