Ch 13 scaled deviance and AIC

[rlsrachaellouisesmith]

Good evening
I’m just trying to suss our definitions on pages 41 and 44 of chapter 13.
The scaled deviance I understand to be twice the difference between the log likelihood of saturated model and the log likelihood of current model.
There is then a further definition of deviance over phi (the scale parameter). So I understand this to be dependent on the familu of the distribution.
However what is the deviance of the model and how is this calculated?

Secondly the AIC is defined as deviance plus twice parameter. Is this the deviance only, or the scaled deviance?

Thank you

 

[Bev.Butler]

Hi there
The definition of the scaled deviance is as you have said. The deviance is the scaled deviance multiplied by phi, so as you said this depends on the distribution (for example, the Poisson distribution has phi of 1 so the deviance and scaled deviance is the same).
The formula for the AIC in Core Reading isn't right. It should be -2lnL + 2x No of parameters. You substitute the mles for the parameters into the log likelihood to get the numerical value of the AIC.
Bev

 

[rlsrachaellouisesmith]

Thank you this is really helpful.

I now need a bit of help with definitions of the pearson and deviance residuals.

In the core reading pearson residual is y-mu_hat divided by square root of var of mu_hat. Does it follow that for any distribution the pearson residual is just y-mean of distribution divided by standard deviation of the distribution? so in the case of poisson y-lambda/root(lambda)?

With the deviance residual it is y-mu_hat divided by root of phi, this is phi as defined in the exponential family form, is mu_hat = b'(mu) from the exponential family form, and if it is this means the numerator of both pearson and deviance residuals are always equal?

Thank you

 

[John Lee]

In the core reading pearson residual is y-mu_hat divided by square root of var of mu_hat. Does it follow that for any distribution the pearson residual is just y-mean of distribution divided by standard deviation of the distribution? so in the case of poisson y-lambda/root(lambda)?

Yes, and we use the estimated mean in the standard deviation.

With the deviance residual it is y-mu_hat divided by root of phi, this is phi as defined in the exponential family form, is mu_hat = b'(mu) from the exponential family form, and if it is this means the numerator of both pearson and deviance residuals are always equal?

The deviance residual is \(sign(y_i- \hat \mu) d_i\) where \(\sum d_i^2 =\) scaled deviance.

The Pearson and deviance residuals are only equal for the Normal distribution.

 

[rlsrachaellouisesmith]

Yes, and we use the estimated mean in the standard deviation.



The deviance residual is \(sign(y_i- \hat \mu) d_i\) where \(\sum d_i^2 =\) scaled deviance.

The Pearson and deviance residuals are only equal for the Normal distribution.

Thank you.

I’m trying to check whether the expressions I’ve obtained for the Pearson and deviance residuals where underlying population is:
A) Poisson
B) binomial
C) gamma
D) beta

Are we likely to be asked about all of these underlying distributions? Could we be asked to find deviance/Pearson residuals for others?

Thank you

 

[John Lee]

We only do this for members of the exponential family - so Poisson, normal, gamma, exponential and binomial.