Predicted values from glm() when linear predictor is NA.

Yes, I am familiar with linear algebra. I nod along with you until you get to "hence" and then you make a leap. The user made the decision (or accepted the default) that the rank problem be ignored... they have the option to reduce their model, but they wanted this coefficient (communicated via the model and the parameter) to behave as if it were zero. The issue is that that decision is getting hidden behind this additional implementation decision to exclude certain columns in the model matrix _even after the model has been solved_. 

If they try to use the coefficients themselves then they have to apply this odd interpretation of NA that is embedded in predict.glm but not apparent in their model instead of using a simple newdata %*% coefs.

It may be easier to implement summary.glm with an NA there as a hint to the rank deficiency, but at the cost of mathematical inconsistency in the reporting of coefficients. IMO either the NA should always be presented to the user as if it were a zero or the rank deficiency should be recorded separately.

On July 28, 2022 8:46:13 AM PDT, John Fox <jfox at mcmaster.ca> wrote:

Dear Jeff,

On 2022-07-28 11:12 a.m., Jeff Newmiller wrote:

No, in this case I think I needed the "obvious" breakdown. Still digesting, though... I would prefer that if an arbitrary selection had been made that it be explicit .. the NA should be replaced with zero if the singular.ok argument is TRUE, rather than making that interpretation in predict.glm.

That's one way to think about, but another is that the model matrix X has 10 columns but is of rank 9. Thus 9 basis vectors are needed to span the column space of X, and a simple way to provide a basis is to eliminate a redundant column, hence the NA. The fitted values y-hat in a linear model are the orthogonal projection of y onto the space spanned by the columns of X, and are thus independent of the basis chosen. A GLM is a little more complicated, but it's still the column space of X that's important.

Best,
John

On July 28, 2022 5:45:35 AM PDT, John Fox <jfox at mcmaster.ca> wrote:

Dear Jeff,

On 2022-07-28 1:31 a.m., Jeff Newmiller wrote:

But "disappearing" is not what NA is supposed to do normally. Why is it being treated that way here?

NA has a different meaning here than in data.

By default, in glm() the argument singular.ok is TRUE, and so estimates are provided even when there are singularities, and even though the singularities are resolved arbitrarily.

In this model, the columns of the model matrix labelled LifestageL1 and TrtTime:LifestageL1 are perfectly collinear -- the second is 12 times the first (both have 0s in the same rows and either 1 or 12 in three of the rows) -- and thus both can't be estimated simultaneously, but the model can be estimated by eliminating one or the other (effectively setting its coefficient to 0), or by taking any linear combination of the two regressors (i.e., using any regressor with 0s and some other value). The fitted values under the model are invariant with respect to this arbitrary choice.

My apologies if I'm stating the obvious and misunderstand your objection.

Best,
John

On July 27, 2022 7:04:20 PM PDT, John Fox <jfox at mcmaster.ca> wrote:

Dear Rolf,

The coefficient of TrtTime:LifestageL1 isn't estimable (as you explain) and by setting it to NA, glm() effectively removes it from the model. An equivalent model is therefore

fit2 <- glm(cbind(Dead,Alive) ~ TrtTime + Lifestage +

+               I((Lifestage == "Egg + L1")*TrtTime) +
+               I((Lifestage == "L1 + L2")*TrtTime) +
+               I((Lifestage == "L3")*TrtTime),
+             family=binomial, data=demoDat)
Warning message:
glm.fit: fitted probabilities numerically 0 or 1 occurred

cbind(coef(fit, complete=FALSE), coef(fit2))

                                    [,1]         [,2]
(Intercept)                -0.91718302  -0.91718302
TrtTime                     0.88846195   0.88846195
LifestageEgg + L1         -45.36420974 -45.36420974
LifestageL1                14.27570572  14.27570572
LifestageL1 + L2           -0.30332697  -0.30332697
LifestageL3                -3.58672631  -3.58672631
TrtTime:LifestageEgg + L1   8.10482459   8.10482459
TrtTime:LifestageL1 + L2    0.05662651   0.05662651
TrtTime:LifestageL3         1.66743472   1.66743472

There is no problem computing fitted values for the model, specified either way. That the fitted values when Lifestage == "L1" all round to 1 on the probability scale is coincidental -- that is, a consequence of the data.

I hope this helps,
John

On 2022-07-27 8:26 p.m., Rolf Turner wrote:

I have a data frame with a numeric ("TrtTime") and a categorical
("Lifestage") predictor.

Level "L1" of Lifestage occurs only with a single value of TrtTime,
explicitly 12, whence it is not possible to estimate a TrtTime "slope"
when Lifestage is "L1".

Indeed, when I fitted the model

        fit <- glm(cbind(Dead,Alive) ~ TrtTime*Lifestage, family=binomial,
                   data=demoDat)

I got:

as.matrix(coef(fit))
                                      [,1]
(Intercept)                -0.91718302
TrtTime                     0.88846195
LifestageEgg + L1         -45.36420974
LifestageL1                14.27570572
LifestageL1 + L2           -0.30332697
LifestageL3                -3.58672631
TrtTime:LifestageEgg + L1   8.10482459
TrtTime:LifestageL1                 NA
TrtTime:LifestageL1 + L2    0.05662651
TrtTime:LifestageL3         1.66743472

That is, TrtTime:LifestageL1 is NA, as expected.

I would have thought that fitted or predicted values corresponding to
Lifestage = "L1" would thereby be NA, but this is not the case:

predict(fit)[demoDat$Lifestage=="L1"]
          26       65      131
24.02007 24.02007 24.02007

fitted(fit)[demoDat$Lifestage=="L1"]
     26  65 131
      1   1   1

That is, the predicted values on the scale of the linear predictor are
large and positive, rather than being NA.

What this amounts to, it seems to me, is saying that if the linear
predictor in a Binomial glm is NA, then "success" is a certainty.
This strikes me as being a dubious proposition.  My gut feeling is that
misleading results could be produced.

Can anyone explain to me a rationale for this behaviour pattern?
Is there some justification for it that I am not currently seeing?
Any other comments?  (Please omit comments to the effect of "You are as
thick as two short planks!". :-) )

I have attached the example data set in a file "demoDat.txt", should
anyone want to experiment with it.  The file was created using dput() so
you should access it (if you wish to do so) via something like

        demoDat <- dget("demoDat.txt")

Thanks for any enlightenment.

cheers,

Rolf Turner