"Margins" After GLM with Offsets

[David Crow]

Dear Statalisters-
I'm estimating some models with offsets (log rate, binomial, etc.) for aggregate data. It appears that "margins" doesn't give the correct predicted values after some of these models, though "predict" does. My question: am I doing something wrong, or does "margins" not work well after models with offsets?

Here's the data:

. input FamAb Sex Rural LivAb F

FamAb Sex Rural LivAb F
1. 0 0 0 11 440
2. 0 0 1 8 205
3. 0 1 0 28 400
4. 0 1 1 24 184
5. 1 0 0 50 397
6. 1 0 1 24 166
7. 1 1 0 105 426
8. 1 1 1 50 173
9. end


The table represents the frequency of respondents who report having lived abroad (LivAb, fourth column), out of the number of people "at risk" of having lived abroad (F, fifth column) in each of eight categories of a full cross-classification of three independent binary variables: having family members abroad (FamAb), sex (Male=1), and rural vs. urban residence (Rural=1).

Here are two possible models for the data: 1) a log rate model with F as the offset term (specified with the "exposure" option) and 2) a binomial regression in which F enters the model as the denominator of a dependent variable specified as a proportion.

Model 1:

glm LivAb FamAbroad Sex Rural, family(poisson) trace exposure(F)

And the results:

	OIM
LivAb	Coef.	Std. Err.	z	P>z	[95% Conf.	Interval]
FamAbroad	1.203989	.1359458	8.86	0.000	.9375405	1.470438
Sex	.7786142	.1249003	6.23	0.000	.5338142	1.023414
Rural	.2590993	.1208217	2.14	0.032	.022293	.4959055
_cons	-3.385296	.1517851	-22.30	0.000	-3.68279	-3.087803
ln(F)	1	(exposure)

The problem is this. When I use "margins" after the model to predict the frequencies of "LivAb", the results are patently wrong:

margins, at(FamAb=(0 1) Sex=(0 1) Rural=(0 1))

Here are the predictions:

------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 10.12218 1.536396 6.59 0.000 7.110899 13.13346
2 | 13.11594 2.138864 6.13 0.000 8.923843 17.30804
3 | 22.05068 2.948346 7.48 0.000 16.27202 27.82933
4 | 28.57244 4.199202 6.80 0.000 20.34215 36.80272
5 | 33.74116 3.959172 8.52 0.000 25.98133 41.501
6 | 43.72053 5.886074 7.43 0.000 32.18404 55.25702
7 | 73.50348 6.376324 11.53 0.000 61.00611 86.00084
8 | 95.24304 10.45466 9.11 0.000 74.75229 115.7338
------------------------------------------------------------------------------


And here are the predictions recovered using "predict"

predict n1

+----------------------------------------------+
| FamAb Sex Rural LivAb F n1 |
|----------------------------------------------|
1. | 0 0 0 11 440 14.90174 |
2. | 0 0 1 8 205 8.996295 |
3. | 0 1 0 28 400 29.51157 |
4. | 0 1 1 24 184 17.59039 |
5. | 1 0 0 50 397 44.81887 |
|----------------------------------------------|
6. | 1 0 1 24 166 24.28309 |
7. | 1 1 0 105 426 104.7678 |
8. | 1 1 1 50 173 55.13023 |
+----------------------------------------------+


It's easy to verify that the predictions with "margins" are wrong, and those with "predict" are right. For example, the predicted frequency for a rural male with family members abroad is 55.13, not the 95.24 predicted by "margins":

display exp(1.203989 + .7786142 + .2590993 - 3.385296 + ln(173))
55.130224

Model 2

The same goes for this binomial regression (the canonical link is "logit", so the model is logit for grouped data).

glm LivAb FamAbroad Sex Rural, family(binomial F)

Again, the predictions obtained with "predict" are correct, and those with "margins," at least how I specified the command (again, "margins, at(FamAb=(0 1) Sex=(0 1) Rural=(0 1))"), wrong. (I spare readers a tedious rehearsal of the details--unless you want one!

So, am I specifying "margins" wrong, or does "margins" not work with offsets?

Thanks,
David

-- Personal Web site:
http://investigadores.cide.edu/crow/

[Jeff Pitblado (StataCorp)]

margins is using all exposure values to compute the predictive margins, not just the observations at which FamAb, Sex, and Rural are equal to the specified at() values.

Here is a version of your code that shows how to compute the first predictive margin by hand.

clear all

input FamAb Sex Rural LivAb F
0 0 0 11 440
0 0 1 8 205
0 1 0 28 400
0 1 1 24 184
1 0 0 50 397
1 0 1 24 166
1 1 0 105 426
1 1 1 50 173
end

glm LivAb FamAb Sex Rural, family(poisson) trace exposure(F)     

predict n1
list FamAb Sex Rural n1

margins, at(FamAb=(0 1) Sex=(0 1) Rural=(0 1))

* first _at case

preserve
replace FamAb = 0
replace Sex = 0
replace Rural = 0
predict n_at_000
sum n_at_000
restore

For this example, the output from margins is

------------------------------------------------------------------------------
             |            Delta-methodal mean
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   10.12218   1.536396     6.59   0.000     7.110899    13.13346
          2  |   13.11594   2.138864     6.13   0.000     8.923843    17.30804
          3  |   22.05068   2.948346     7.48   0.000     16.27202    27.82933
          4  |   28.57244   4.199202     6.80   0.000     20.34215    36.80272
          5  |   33.74116   3.959172     8.52   0.000     25.98133      41.501
          6  |   43.72053   5.886074     7.43   0.000     32.18404    55.25702
          7  |   73.50348   6.376324    11.53   0.000     61.00611    86.00084
          8  |   95.24304   10.45466     9.11   0.000     74.75229    115.7338
------------------------------------------------------------------------------

The the summarize results for n_at_000, which corresponds with the margins output for 1._at is

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
    n_at_000 |         8    10.12218    4.273244   5.622022   14.90174

You can use the over() option if you want the predictive margins within the groups identified by the observed values of FamAb, Sex, and Rural.

. margins, over(FamAb Sex Rural)

Predictive margins                                Number of obs   =          8
Model VCE    : OIM

Expression   : Predicted mean LivAb, predict()
over         : FamAb Sex Rural

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   FamAb#Sex#|
       Rural |
      0 0 0  |   14.90174   2.261863     6.59   0.000     10.46857    19.33491
      0 0 1  |   8.996295   1.467059     6.13   0.000     6.120913    11.87168
      0 1 0  |   29.51157   3.945925     7.48   0.000      21.7777    37.24544
      0 1 1  |   17.59039   2.585205     6.80   0.000     12.52348     22.6573
      1 0 0  |   44.81887   5.259026     8.52   0.000     34.51137    55.12637
      1 0 1  |   24.28309    3.26922     7.43   0.000     17.87553    30.69064
      1 1 0  |   104.7678   9.088461    11.53   0.000     86.95476    122.5809
      1 1 1  |   55.13023   6.051547     9.11   0.000     43.26941    66.99104
------------------------------------------------------------------------------

. list FamAb Sex Rural n1

     +--------------------------------+
     | FamAb   Sex   Rural         n1 |
     |--------------------------------|
  1. |     0     0       0   14.90174 |
  2. |     0     0       1   8.996295 |
  3. |     0     1       0   29.51157 |
  4. |     0     1       1   17.59039 |
  5. |     1     0       0   44.81887 |
     |--------------------------------|
  6. |     1     0       1   24.28309 |
  7. |     1     1       0   104.7678 |
  8. |     1     1       1   55.13023 |
     +--------------------------------+

[David Crow]

Thanks! David