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# proc mixed: negative AIC values

When modelling in proc mixed, I get negative values for my fit statistics, e.g. AIC.
How to interpret?
Which model is best: the more negative the better?
How can this happen?

Best regards,
Bart
Regular Contributor
Posts: 169

## Re: proc mixed: negative AIC values

AIC is computed as -2LL + 2p where LL is the log-likelihood for the fitted model summed over all observations and p is the number of parameters in the model. The value 2p must be positive, so a negative value for a fit statistic like AIC is due to a negative value for the -2LL part of the equation.

So, how do we get a negative value for the -2LL part of the equation. Consider a standard normal variable X ~ N(0,1). Given X=x, the log-likelihood of the parameters mu=0 and V=1 is

ll = -0.5 * (log(2*pi) + log(V) + (x-0)**2/V)
or
ll = -0.5 * (log(2*pi) + log(1) + (x-0)**2)

Now, suppose that we rescaled x by multiplying by some factor. In particular, let's compute y = x*M where M=(10**i) for i=-5,-4,-3,...,3,4,5. The mean will stay the same, but the variance V of the observed value will be M**2. When we compute the log likelihood of the mean and V, we have

LL = -0.5 * (log(2*pi) + log(M**2) + (y-0)**2/(M**2) )
or
LL = -0.5 * (log(2*pi) + log(M**2) + (x*M - 0)**2/(M**2) )

The variance cancels out in the last added term because (x*M-0)**2 = (x**2)*(M**2), so that the log likelihood can be written as

LL = -0.5 * (log(2*pi) + log(M**2) + x**2)

The only part of this expression which varies according to the multiplier is -0.5*log(M**2) = -0.5*log(V). As log(V[i}) decreases, the value of LL increases. Identically, as log(V) decreases, the value of -2LL will decreases.

Here is an experiment for you to perform. Take your observed data and multiply every observation by 1000. Fit the same model as you did that gave you negative values for the fit statistics. Parameter estimates and their standard errors should all be scaled by the multiplier 1000. Test statistics (F-test and t-test values) should all be the same. (There could be very small differences in any of these statistics due to the problem of trying to perform base 10 arithmetic employing a computer which employs base 2 computations). But the value of -2LL and the fit statistics will all increase. My guess is that they will all be positive at the end of this experiment.

HTH
Regular Contributor
Posts: 169

## Re: proc mixed: negative AIC values

This is essentially an identical post to the one above. I had not anticipated that when I subscripted variables with "left bracket"i'right bracket" that this would turn on italics and that the subscripting would not be recognized. This repost uses instead of "left bracket"i"right bracket" so that italics are not produced and so that subscripting is indicated where it is needed.

AIC is computed as -2LL + 2p where LL is the log-likelihood for the fitted model summed over all observations and p is the number of parameters in the model. The value 2p must be positive, so a negative value for a fit statistic like AIC is due to a negative value for the -2LL part of the equation.

So, how do we get a negative value for the -2LL part of the equation. Consider a standard normal variable X ~ N(0,1). Given X=x, the log-likelihood of the parameters mu=0 and V=1 is

ll = -0.5 * (log(2*pi) + log(V) + (x-0)**2/V)
or
ll = -0.5 * (log(2*pi) + log(1) + (x-0)**2)

Now, suppose that we rescaled x by multiplying by some factor. In particular, let's compute y = x*M where M=(10**i) for j=-5,-4,-3,...,3,4,5. The mean will stay the same, but the variance V of the observed value will be M**2. When we compute the log likelihood of the mean and V, we have

LL = -0.5 * (log(2*pi) + log(M**2) + (y-0)**2/(M**2) )
or
LL = -0.5 * (log(2*pi) + log(M**2) + (x*M - 0)**2/(M**2) )

The variance cancels out in the last added term because (x*M-0)**2 = (x**2)*(M**2), so that the log likelihood can be written as

LL = -0.5 * (log(2*pi) + log(M**2) + x**2)

The only part of this expression which varies according to the multiplier is -0.5*log(M**2) = -0.5*log(V). As log(V[j}) decreases, the value of LL increases. Identically, as log(V) decreases, the value of -2LL will decreases.

Here is an experiment for you to perform. Take your observed data and multiply every observation by 1000. Fit the same model as you did that gave you negative values for the fit statistics. Parameter estimates and their standard errors should all be scaled by the multiplier 1000. Test statistics (F-test and t-test values) should all be the same. (There could be very small differences in any of these statistics due to the problem of trying to perform base 10 arithmetic employing a computer which employs base 2 computations). But the value of -2LL and the fit statistics will all increase. My guess is that they will all be positive at the end of this experiment.

HTH
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## Re: proc mixed: negative AIC values

Thanks a lot. It works out. So nothing to worry about negative AIC values.

Best regards,
Bart
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