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Help with interpreting one fixed-effect coefficient

9 messages · Juho Kristian Ruohonen, Stuart Luppescu, Fernando Pedro Bruna Quintas +1 more

Original
Simon Harmel
# 
Dear Colleagues,

Apologies for crossposting (https://stats.stackexchange.com/q/545975/284623).

I've two categorical moderators i.e., students' ***sex*** (`boys`,
`girls`) and the ***school-gender system*** (`boy-only`, `girl-only`,
`mixed`) in a model like: `y ~ sex + schoolgend`.

My coefs are below. I can interpret three of the coefs but wonder how
to interpret the third one from the top (.175)?

Assume "intrcpt" represents the boys' mean in mixed schools.

                         Estimate
(Intercept)             -0.189
schgendboy-only   0.180
schgendgirl-only    0.175
sexgirls                  0.168

My interpretations of the coefficients are as follows:

            "(Intercept)": mean of y for boys in mixed schools = -.189
 "schgendboy-only": diff. bet. boys in boy-only vs. mixed schools = +.180
  "schgendgirl-only": diff. bet. ???????????????????????????? = +.175
                "sexgirls": diff. bet. girls vs. boys in mixed schools = +.168

If my interpretation logic for all other coefs is correct, then, this
third coef. must mean:

diff. bet. boys in girl-only vs. mixed schools = +.175! (which makes no sense!)

ps. I know I will end-up interpreting +1.75 as: diff. bet. girls in
girl-only vs. mixed schools BUT this doesn't follow the interpretation
logic for other coefs PLUS there are no labels in the output to show
what's what!

Many thanks,
Simon
Juho Kristian Ruohonen
# 
Fellow student commenting here...

As you suggest, schgendgirl-only can only ever apply to female students.
Strictly speaking, it's the estimated mean difference between a student of
any sex in a girls-only school and a similar student in a mixed school. But
since such comparisons are only observed between girls, the estimate is
necessarily informed by girl data only. So your intended interpretation of
the coefficient is correct.


su 26. syysk. 2021 klo 0.27 Simon Harmel (sim.harmel at gmail.com) kirjoitti:

  
  


  


      

    

    

      
      

        


  

        

      Simon Harmel
    

    

      
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Dear Juho and other List Members,

My problem is the logic of interpretation. Assuming no interaction, a
categorical-predictors-only model, and aside from the intercept which
captures the mean for reference categories (in this case, boys in the
mixed schools), I have learned to interpret any main effect coef for a
categorical predictor by thinking of that coef. as something that can
differ from its reference category to affect "y" ***holding any other
categorical predictor in the model at its reference category***.

By this logic, "schgendboy-only" main effect coef should mean diff.
bet. boys (held constant at the reference category) in boy-only vs.
mixed schools (which shows "schgendboy-only" can differ from its
reference category i.e, mixed schools).

By this logic, "sexgirls" main effect coef should mean diff. bet.
girls vs. boys (which shows "sexgirls" can differ from its reference
category i.e, boys) in mixed schools (held constant at the reference
category).

Therefore, by this logic, "schgendgirl-only" main effect coef should
mean diff. bet. boys (held constant at the reference category) in
girl-only vs. mixed schools (which shows "schgendgirl-only" can differ
from its reference category i.e, mixed schools).

My question is that is my logic of interpretation incorrect? Or are
there exceptions to my logic of interpretation of which interpreting
"schgendgirl-only" coef is one?

Thank you very much,
Simon

On Sun, Sep 26, 2021 at 12:00 AM Juho Kristian Ruohonen

        
<juho.kristian.ruohonen at gmail.com> wrote:

            

      
      
      
    

        

      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Juho Kristian Ruohonen
    

    

      
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In my view, your logic is slightly oversimplified (i.e. incorrect).
Regression models do not estimate coefficients by holding predictors
constant exclusively at the reference category. They do something more
general, namely estimate coefficients by holding predictors constant at any
value at which variation is observed in the values of the other predictors.

su 26. syysk. 2021 klo 9.03 Simon Harmel (sim.harmel at gmail.com) kirjoitti:

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

  
  


  


      

    

    

      
      

        


  

        

      Simon Harmel
    

    

      
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Could you please apply your logic step by step to the three coefficients?



On Sun, Sep 26, 2021 at 1:39 AM Juho Kristian Ruohonen

        
<juho.kristian.ruohonen at gmail.com> wrote:

            

      
      
      
    

        

      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Juho Kristian Ruohonen
    

    

      
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(Intercept): with all predictors at zero, an average student is estimated
to have a response value of -0.18.
schgendboy-only: on average, students in boys-only schools are estimated to
have response values 0.180 units higher than otherwise comparable students
in a mixed-sex schools.
schgendgirl-only: on average, students in girls-only schools are estimated
to have response values  0.175 units higher than otherwise comparable
students in a mixed-sex schools.
sexgirl: on average, female students are estimated to have response values
0.168 units higher than otherwise comparable male students.

su 26. syysk. 2021 klo 9.57 Simon Harmel (sim.harmel at gmail.com) kirjoitti:

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

  
  


  


      

    

    

      
      

        


  

        

      Stuart Luppescu
    

    

      
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On 9/26/21 06:26, Simon Harmel wrote:

            

      
      
      
    

        
I don't get this. Why is this posted to a mixed-effects model list when 
there is no random effect? That said, this really is a hierarchical 
model, since the sex predictor is an individual-level predictor, and 
school-gender-system is a school-level predictor. In a case like this, 
you're getting messed up trying to conceptualize the cross-level 
interaction. My head is all messed up just trying to figure it out.

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Fernando Pedro Bruna Quintas
    

    

      
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It is a good question for a list on mixed models, though the interpretation would be common to traditional linear models. There is no mess in the questions. There are not interactions. The question is about the interpretation of the parameters in the following equation:

Yij = Cij + B1 x Xij + B2 x Zj

The special characteristics of the question is about the gender composition of the variables Xij and Zj.

Best,

Fernando

        

      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Simon Harmel
    

    

      
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Dear Juho,

Sure. However, all major resources disagree with your interpretation
(e.g., https://onlinelibrary.wiley.com/doi/book/10.1002/9781118625590).
This is because the goal of fixed-effect coefs is to predict/estimate
the mean of "y"'s population distribution conditional on specific
values/levels of a combination of predictors in the model. This means
that the coefs. themselves must represent specific values/levels of a
combination of predictors in the model to make such conditional
predictions possible.

Applying this logic to your interpretations, your interpretations
simply ignore specifying which specific values/levels of a combination
of predictors in the model each coef. represents. Therefore, it
logically doesn't agree with the goal of fixed-effects coefficients.

Thanks,
Simon

On Sun, Sep 26, 2021 at 2:43 AM Juho Kristian Ruohonen

        
<juho.kristian.ruohonen at gmail.com> wrote: