# [R] comparing glm models - lower AIC but insignificant coefficients

Constantinos Antoniou antoniou at central.ntua.gr
Mon May 23 20:59:11 CEST 2005

```Hello,

I am a new R user and I am trying to estimate some generalized linear
models (glm). I am trying to compare a model with a gaussian
distribution and an identity link function, and a poisson model with
a log link function. My problem is that while the gaussian model has
significantly lower (i.e. "better") AIC (Akaike Information
Criterion) most of the coefficients are not significant. On the other
hand, the poisson model has a higher (i.e. "worse") AIC, but almost
all the coefficients are extremely significant (expect for one that
still has p=0.07).

Summary output of the two models follows... [sorry for the large
number of independent variables, but the issue is less pronounced
with fewer covariates].

My question is two-fold:
- AIC supposedly can be used to compare non-nested models (although
there are concerns and I have also seen a couple in this list's
archives). Is this a case where AIC is not a good measure to compare
the two models? If so, is there another measure (besides choosing the
model with the significant coefficients)? [These are time-series
data, so I am also looking at acf/pacf of the residuals].
- Could the very high significance of the coefficients in the poisson
model hint at some issue?

Costas

+++++++++++++++++++++++
+++++++++++++++++++++++

Call:
glm(formula = TotalDeadInjured[3:48] ~ -1 + Month[3:48] + sin(pi *
Month[3:48]/6) + cos(pi * Month[3:48]/6) + sin(pi * Month[3:48]/
12) +
cos(pi * Month[3:48]/12) + ThousandCars[3:48] + monthcycle[3:48] +

Deviance Residuals:
Min       1Q   Median       3Q      Max
-3.6900  -1.1901  -0.1847   0.9477   4.3967

Coefficients:
Estimate Std. Error z value Pr(>|z|)
Month[3:48]                   -7.712e-02  5.530e-03 -13.947  < 2e-16 ***
sin(pi * Month[3:48]/6)       -1.419e-01  2.759e-02  -5.144 2.68e-07 ***
cos(pi * Month[3:48]/6)       -8.407e-02  1.799e-02  -4.672 2.99e-06 ***
sin(pi * Month[3:48]/12)      -2.776e-02  1.558e-02  -1.782 0.074702 .
cos(pi * Month[3:48]/12)       5.195e-02  1.608e-02   3.232 0.001231 **
ThousandCars[3:48]             2.733e-02  2.255e-03  12.118  < 2e-16 ***
monthcycle[3:48]               6.307e-02  6.546e-03   9.635  < 2e-16 ***
TotalDeadInjured[1:46]        -2.925e-02  8.460e-03  -3.457 0.000546 ***
I((TotalDeadInjured[1:46])^2)  1.218e-04  3.613e-05   3.370 0.000750 ***
I((TotalDeadInjured[1:46])^3) -1.640e-07  4.961e-08  -3.306 0.000946 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 78694.70  on 46  degrees of freedom
Residual deviance:   130.03  on 36  degrees of freedom
AIC: 476.08

Number of Fisher Scoring iterations: 4

+++++++++++++++++++++++++
GAUSSIAN
++++++++++++++++++++++++++

Call:
glm(formula = TotalDeadInjured[3:48] ~ -1 + Month[3:48] + sin(pi *
Month[3:48]/6) + cos(pi * Month[3:48]/6) + sin(pi * Month[3:48]/
12) +
cos(pi * Month[3:48]/12) + ThousandCars[3:48] + monthcycle[3:48] +

Deviance Residuals:
Min       1Q   Median       3Q      Max
-61.326  -12.012   -1.756   14.204   78.991

Coefficients:
Estimate Std. Error t value Pr(>|t|)
Month[3:48]                   -8.111e+00  2.115e+00  -3.835 0.000487 ***
sin(pi * Month[3:48]/6)       -2.639e+01  1.095e+01  -2.409 0.021246 *
cos(pi * Month[3:48]/6)       -1.700e+01  7.138e+00  -2.382 0.022629 *
sin(pi * Month[3:48]/12)       2.392e-01  6.524e+00   0.037 0.970956
cos(pi * Month[3:48]/12)       8.785e+00  6.317e+00   1.391 0.172835
ThousandCars[3:48]             2.219e+00  8.604e-01   2.579 0.014146 *
monthcycle[3:48]               5.364e+00  2.494e+00   2.151 0.038301 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for gaussian family taken to be 831.6357)

Null deviance: 1927714  on 46  degrees of freedom
Residual deviance:   29939  on 36  degrees of freedom
AIC: 450.54

Number of Fisher Scoring iterations: 2

```