GLM Fractional Response coefficient interpretation

[notsavy]

Hi, I'm trying to interpret the output for the glm command but I'm struggling with it. Could you tell me what is the transformation I should apply to the coefficients given by Stata into the units of the dependent variable? (The independent variable is between 0 and 1). I understand this is a silly question but I cannot figure it out. Thanks for your help.

[Maarten Buis]

I like what we in the help-file of betafit (available from SSC) called relative proportion ratios. Consider the example below:

. use http://fmwww.bc.edu/repec/bocode/c/citybudget.dta, clear
(Spending on different categories by Dutch cities in 2005)

. sum popdens if !missing(minorityleft, noleft, houseval, popdens), meanonly

. gen cpopdens = popdens - r(mean)
(1 missing value generated)

.     
. sum houseval if !missing(minorityleft, noleft, houseval, popdens), meanonly

. gen chouseval = houseval - r(mean)
(1 missing value generated)

. gen byte pol = minorityleft + 2*noleft

. label define pol 0 "majority left" ///
>                  1 "minority left" ///
>                  2 "right"

. label val pol pol

.     
. glm governing i.pol chouseval cpopdens, ///
>     link(logit) family(binomial) vce(robust) ///
>     eform base nolog
note: governing has noninteger values

Generalized linear models                          No. of obs      =       394
Optimization     : ML                              Residual df     =       389
                                                   Scale parameter =         1
Deviance         =  6.133626519                    (1/df) Deviance =  .0157677
Pearson          =  6.425762774                    (1/df) Pearson  =  .0165187

Variance function: V(u) = u*(1-u/1)                [Binomial]
Link function    : g(u) = ln(u/(1-u))              [Logit]

                                                   AIC             =  .4846608
Log pseudolikelihood = -90.47818714                BIC             = -2318.667

--------------------------------------------------------------------------------
               |               Robust
     governing | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
           pol |
majority left  |          1  (base)
minority left  |   .8456308   .0632362    -2.24   0.025      .730345    .9791146
        right  |    1.06247    .083297     0.77   0.440     .9111359    1.238941
               |
     chouseval |   1.395905   .0920578     5.06   0.000      1.22665    1.588515
      cpopdens |    .881338   .0227379    -4.90   0.000     .8378805    .9270494
         _cons |   .1081369   .0074875   -32.12   0.000     .0944138    .1238547
--------------------------------------------------------------------------------

In this example I model the proportion of a city-budget spent on each city's own organization. In that case the relative proportion is the proportion spent on governing divided by the proportion spent on useful stuff. As the total budget size drops out of this ratio, we can also say that this is the number of euros spent on governing per euro spent on productive stuff. Notice that glm sees the link(logit) and eform options, so by default calls the coefficients odds ratios, which in this case is not quite correct. It would be safer to call them relative proportion ratios. If you decide to use them, you'll need to explain to your audience what they are, as it is not as standard terminology as odds ratios or marginal effects.

It is useful to see the baseline relative proportion, that is, the relative proportion when all covariates are equal to zero. This is the exponentiated constant. In the example above, a city with a city government consisting of majority left-leaning members, an average population and house value can expect to spent about 11 cents on governing per euro spent on productive stuff. This ratio decreases by 15% (i.e. [1-.85]*100% = -15%) if it is governed by a minority left government, and it increases by a non-significant 6% when no left parties are represented in the city government. A 100,000 euro increase in average house value will lead to an 40% increase in the relative proportion and an extra 1000 persons per square kilometre will lead to an 12% decrease in the relative proportion.

Alternatively, you can also compute average marginal effect using the margins command. Continuing the example above:

. margins, dydx(*)

Average marginal effects                          Number of obs   =        394
Model VCE    : Robust

Expression   : Predicted mean governing, predict()
dy/dx w.r.t. : 1.pol 2.pol chouseval cpopdens

--------------------------------------------------------------------------------
               |            Delta-method
               |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
           pol |
minority left  |  -.0139847   .0065599    -2.13   0.033    -.0268419   -.0011276
        right  |   .0055353    .007059     0.78   0.433    -.0083001    .0193707
               |
     chouseval |   .0287157   .0058056     4.95   0.000      .017337    .0400944
      cpopdens |  -.0108747   .0022165    -4.91   0.000     -.015219   -.0065305
--------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

In this case the minority left goverment spents about 1.3 percentage points less on governing compared with majority left governments, while right governments spent a non-significant 0.6 percentage points more on governing compared to majority left governments. A 100,000 euro increase in average house value will lead to a 2.9 percentage point increase in the proportion spent on governing and an extra 1000 persons per square kilometre will lead to an 1.1 percentage point decrease in the proportion spent on governing.

Notice the difference between percentage point changes (in the section on marginal effects) and percentage changes (in the section on relative proportion ratios). If we start with a baseline value of 1% and change by 1 percentage point, then the result will be 1 + 1 = 2%. If we change the baseline value by 1%, the result will be 1 * 1.01 = 1.01%.

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[Ritika Khurana]

Hi Maarten,

Continuing from the above example, had the 'chousval' be in log form, the coefficient would have been interpreted as 100,000 euro increase in average house value will lead to a 0.029 percentage point increase in the proportion spent on governing ?

I'm checking if my interpretation is correct as my independent variable is in log form and dependent variable is a proportion.

Looking forward to hearing from you.

Thank you.