What's the correct way of interpreting the (exponentiated) log odds?

[Toon Zegers]

Hello everyone,


I am trying to translate the following output to a table in a word document. However, I am a bit unsure about how to correctly interpret the output. Primarily when looking at the constant.

I am currently running two tables: one with a baseline of 4 (Green/ecologist parties) and one with 2 (Communist/socialist parties). The dependent variable is political success. In both tables, the output is exponentiated (using ", or"). However, I obviously still want to include both families. How do I do this correctly? Since the output from both tables are vastly different from one another

Table 1 (Green/ecologist parties as baseline)

success	Odds Ratio	Std. Err.	z	P>z	[95% Conf.	Interval]
party_family
Christian democracy	2.429069	.0834748	25.83	0.000	2.27085	2.598312
Communist/Socialist	.7689889	.026734	-7.56	0.000	.7183365	.8232128
Conservative	1.256617	.0455363	6.30	0.000	1.170463	1.349112
Liberal	1.798223	.060369	17.48	0.000	1.68371	1.920523
Right-wing	2.589988	.1875777	13.14	0.000	2.247244	2.985007
Social democracy	2.145219	.1034633	15.83	0.000	1.951724	2.357897
Special issue	.5662629	.0364536	-8.83	0.000	.4991389	.6424137
cabinetnumb	.9163369	.0041901	-19.11	0.000	.9081612	.9245862
seats	.9893395	.00136	-7.80	0.000	.9866775	.9920088
elecsuccess	1.239797	.0307061	8.68	0.000	1.181052	1.301465
governing	10.3113	.3449301	69.75	0.000	9.656935	11.01001
_cons	.8578049	.0282499	-4.66	0.000	.8041853	.9149996

Table 2 (communist/socialist as baseline)

success	Odds Ratio	Std. Err.	z	P>z	[95% Conf.	Interval]
party_family
Christian democracy	3.158783	.0983181	36.95	0.000	2.971843	3.357482
Conservative	1.634116	.0590701	13.59	0.000	1.522347	1.754091
Green/Ecologist	1.300409	.0452088	7.56	0.000	1.214753	1.392105
Liberal	2.338425	.0749027	26.52	0.000	2.196131	2.489938
Right-wing	3.368044	.2450815	16.69	0.000	2.920375	3.884337
Social democracy	2.789661	.1146532	24.96	0.000	2.573758	3.023676
Special issue	.7363734	.0479969	-4.69	0.000	.6480622	.8367188
cabinetnumb	.9163369	.0041901	-19.11	0.000	.9081612	.9245862
seats	.9893395	.00136	-7.80	0.000	.9866775	.9920088
elecsuccess	1.239797	.0307061	8.68	0.000	1.181052	1.301465
governing	10.3113	.3449301	69.75	0.000	9.656935	11.01001
_cons	.6596424	.0258202	-10.63	0.000	.6109283	.7122409

[Marcos Almeida]

The correct way to interpret odds ratio is displayed in several texts. But it seems that the issue here is the reference group. If you change the reference group, the odds ratio of each group will be compared to the reference, hence the difference in the results.

Please see the output below, just changing the reference group of race:

. webuse lbw
(Hosmer & Lemeshow data)

.  logistic low age lwt i.race smoke ptl ht ui

Logistic regression                             Number of obs     =        189
                                                LR chi2(8)        =      33.22
                                                Prob > chi2       =     0.0001
Log likelihood =   -100.724                     Pseudo R2         =     0.1416

------------------------------------------------------------------------------
         low | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .9732636   .0354759    -0.74   0.457     .9061578    1.045339
         lwt |   .9849634   .0068217    -2.19   0.029     .9716834    .9984249
             |
        race |
      black  |   3.534767   1.860737     2.40   0.016     1.259736    9.918406
      other  |   2.368079   1.039949     1.96   0.050     1.001356    5.600207
             |
       smoke |   2.517698    1.00916     2.30   0.021     1.147676    5.523162
         ptl |   1.719161   .5952579     1.56   0.118     .8721455    3.388787
          ht |   6.249602   4.322408     2.65   0.008     1.611152    24.24199
          ui |     2.1351   .9808153     1.65   0.099     .8677528      5.2534
       _cons |   1.586014   1.910496     0.38   0.702     .1496092     16.8134
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.

.  logistic low age lwt ib2.race smoke ptl ht ui

Logistic regression                             Number of obs     =        189
                                                LR chi2(8)        =      33.22
                                                Prob > chi2       =     0.0001
Log likelihood =   -100.724                     Pseudo R2         =     0.1416

------------------------------------------------------------------------------
         low | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .9732636   .0354759    -0.74   0.457     .9061578    1.045339
         lwt |   .9849634   .0068217    -2.19   0.029     .9716834    .9984249
             |
        race |
      white  |   .2829041   .1489236    -2.40   0.016     .1008227     .793817
      other  |   .6699393   .3609569    -0.74   0.457     .2330324    1.925993
             |
       smoke |   2.517698    1.00916     2.30   0.021     1.147676    5.523162
         ptl |   1.719161   .5952579     1.56   0.118     .8721455    3.388787
          ht |   6.249602   4.322408     2.65   0.008     1.611152    24.24199
          ui |     2.1351   .9808153     1.65   0.099     .8677528      5.2534
       _cons |   5.606189   7.070615     1.37   0.172     .4732807    66.40743
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.

.  logistic low age lwt ib3.race smoke ptl ht ui

Logistic regression                             Number of obs     =        189
                                                LR chi2(8)        =      33.22
                                                Prob > chi2       =     0.0001
Log likelihood =   -100.724                     Pseudo R2         =     0.1416

------------------------------------------------------------------------------
         low | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .9732636   .0354759    -0.74   0.457     .9061578    1.045339
         lwt |   .9849634   .0068217    -2.19   0.029     .9716834    .9984249
             |
        race |
      white  |   .4222832    .185447    -1.96   0.050     .1785648    .9986461
      black  |   1.492672   .8042376     0.74   0.457     .5192127     4.29125
             |
       smoke |   2.517698    1.00916     2.30   0.021     1.147676    5.523162
         ptl |   1.719161   .5952579     1.56   0.118     .8721455    3.388787
          ht |   6.249602   4.322408     2.65   0.008     1.611152    24.24199
          ui |     2.1351   .9808153     1.65   0.099     .8677528      5.2534
       _cons |   3.755806   4.149787     1.20   0.231     .4307362    32.74877
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.

Since the constant concerns the "baseline odds", i.e., the odds ratio when all predictors are set to zero, and considering the reference group (zero, so to speak) for race was changed, the constant must change as well

Hopefully that helps

[Bruce Weaver]

Hello Toon. If you are after all pairwise odds ratios, you can get them with -contrast- commands. This follows on from the examples Marcos gave in #2.

clear *
webuse lbw

// ssc describe fre //
fre race
logit low age lwt i.race smoke ptl ht ui
logit, or

pwcompare race, eform // All pairwise ORs

contrast { race 1 -1 0 }, nowald effects eform // White - Black
contrast { race -1 1 0 }, nowald effects eform // Black - White
// From table of coefficients:  OR = 3.534767 (1.259736 to 9.918406)

contrast { race 1 0 -1 }, nowald effects eform // White - Other
contrast { race -1 0 1 }, nowald effects eform // Other - White
// From table of coefficients:  OR = 2.368079 (1.001356 to 5.600207)

contrast { race 0 1 -1 }, nowald effects eform // Black - Other
contrast { race 0 -1 1 }, nowald effects eform // Other - Black

Notice that two of the custom contrasts match the odds ratios (and 95% CIs) for race in the table of coefficients.

HTH.

[Toon Zegers]

I think i was a bit unclear. I know how to interpret the log odds but am more or less unclear about how to interpret them seeing as their values change depending on the baseline. It was a bit unclear to me which value to use. Am I understand this correctly that if i just keep changing the baseline to another party family (1 through 7) that I just can take the constant values?

[Maarten Buis]

It is still unclear to me what is exactly unclear to you. Your output does not contain log odds but odds ratios (for all variables) and odds (for your constant). The fact that odds ratios change when the reference category changes is discussed above, but if I interpret you correctly it seems that it is still unclear to you. Can you tell us in more detail what it is that is unclear to you? Maybe it helps if you explain why the answers above did not help you.

[Toon Zegers]

Hey Maarten thanks for your reply.


Maybe I am just missing something. I want to put the data from the output in a table for a paper. However, having a constant does not really make the data all too easy to interpret. I just gives us the likelihood of party family X being successful in comparison to the baseline. I much rather showcase the likelihood of party family X having success (dependent variable) compared to zero/no success. That's why I wondered in post #4 if I should just run the model for every baseline and just put the baselines in the table.

[Maarten Buis]

Ok I see what you want. You are probably looking for this Stata Tip: https://www.stata-journal.com/articl...article=st0250

As much as I like that, you will need to be very careful in your discussion of the results. Many readers have enough trouble with odds ratios. Now you are mixing odds and odds ratios in your output table. You can do it - I would probably do it - but it takes some careful writing and careful design of the table as most readers are not used to interpreting models without the constant and indicator variables without a reference category. You need to make a decision whether that possible confusion is worth it. Remember that most of your potential readers will not read your article very carefully. The risk is that they will skim the article, see a table with indicator variables without a reference, decide that the author does not know what he is doing, and start reading another article.

An alternative would be to look at the contrast command with the gw. prefix. That way you can compare each party family with the grand mean. This is another way of not having to choose a reference category, but this has the problem of having to imagine what an "average party family" would be.

[Toon Zegers]

Hey Maarten,

Thank you so much for your reply. What you were saying was in fact exactly what I was meaning to do and now I know how. I'll make sure to thoroughly mention how I came to my results. Again, thank you!

[Marcos Almeida]

You may think about "tricking" the regression with a constant equal to 1.

In the example below, you can observe that the margins for all levels are still the same, corroborating the strategy:

 
. webuse lbw
(Hosmer & Lemeshow data)

. logistic low age lwt i.race smoke ptl ht ui

Logistic regression                             Number of obs     =        189
                                                LR chi2(8)        =      33.22
                                                Prob > chi2       =     0.0001
Log likelihood =   -100.724                     Pseudo R2         =     0.1416

------------------------------------------------------------------------------
         low | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .9732636   .0354759    -0.74   0.457     .9061578    1.045339
         lwt |   .9849634   .0068217    -2.19   0.029     .9716834    .9984249
             |
        race |
      black  |   3.534767   1.860737     2.40   0.016     1.259736    9.918406
      other  |   2.368079   1.039949     1.96   0.050     1.001356    5.600207
             |
       smoke |   2.517698    1.00916     2.30   0.021     1.147676    5.523162
         ptl |   1.719161   .5952579     1.56   0.118     .8721455    3.388787
          ht |   6.249602   4.322408     2.65   0.008     1.611152    24.24199
          ui |     2.1351   .9808153     1.65   0.099     .8677528      5.2534
       _cons |   1.586014   1.910496     0.38   0.702     .1496092     16.8134
------------------------------------------------------------------------------
Note: _cons estimates baseline odds.

. margins race

Predictive margins                              Number of obs     =        189
Model VCE    : OIM

Expression   : Pr(low), predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        race |
      white  |   .2258418    .041197     5.48   0.000     .1450972    .3065864
      black  |   .4585359   .0908183     5.05   0.000     .2805353    .6365365
      other  |   .3769422   .0592032     6.37   0.000     .2609061    .4929783
------------------------------------------------------------------------------

. */ not, we may get OR for all levels of race

. gen cons1 = 1

. logistic low age lwt ibn.race smoke ptl ht ui cons1, nocons
note: cons1 omitted because of collinearity

Logistic regression                             Number of obs     =        189
                                                Wald chi2(9)      =      40.58
Log likelihood =   -100.724                     Prob > chi2       =     0.0000

------------------------------------------------------------------------------
         low | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .9732636   .0354759    -0.74   0.457     .9061578    1.045339
         lwt |   .9849634   .0068217    -2.19   0.029     .9716834    .9984249
             |
        race |
      white  |   1.586014   1.910496     0.38   0.702     .1496092     16.8134
      black  |   5.606189   7.070615     1.37   0.172     .4732807    66.40743
      other  |   3.755807   4.149787     1.20   0.231     .4307362    32.74877
             |
       smoke |   2.517698    1.00916     2.30   0.021     1.147676    5.523162
         ptl |   1.719161   .5952579     1.56   0.118     .8721455    3.388787
          ht |   6.249602   4.322408     2.65   0.008     1.611152    24.24199
          ui |     2.1351   .9808153     1.65   0.099     .8677528      5.2534
       cons1 |          1  (omitted)
------------------------------------------------------------------------------

. margins race

Predictive margins                              Number of obs     =        189
Model VCE    : OIM

Expression   : Pr(low), predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        race |
      white  |   .2258418    .041197     5.48   0.000     .1450972    .3065864
      black  |   .4585359   .0908183     5.05   0.000     .2805353    .6365365
      other  |   .3769422   .0592032     6.37   0.000     .2609061    .4929783
------------------------------------------------------------------------------

Hopefully that helps