How do I check goodness of fit when using "xtmlogit"?

[Hayoung Lee]

Hello,
I'm trying to use "xtmlogit", a newly introduced command in stata17.
How do I check the goodness of fit (e.g., Hosmer-Lemeshow statistics) when using "xtmlogit"?
It would be very helpful if you could let me know.

[Andrew Musau]

The concept of goodness of fit is difficult to define for nonlinear models. Only OLS sets out to minimize the sum of squared errors which coincides with maximizing \(R^2\). However, the output of xtmlogit includes both the maximized log-likelihood from the estimated model \((L_1)\) and the log-likelihood from the model with intercept only \((L_0)\), so you can calculate McFadden's Pseudo \(R^2\) using the following formula:

$$\text{McFadden's Pseudo}\; R^{2}= 1 - \frac{L_{1}}{L_{0}}$$

use https://www.stata-press.com/data/r17/estatus.dta, clear
xtset id year
xtmlogit estatus i.hhchild age hhincome i.hhsigno i.bwinner, fe
ereturn list
display "Pseudo R2= `:di %5.4f `=1-(`e(ll)'/`e(ll_0)')''"

Res.:

. xtmlogit estatus i.hhchild age hhincome i.hhsigno i.bwinner, fe
note: 80 groups (451 obs) omitted because of no variation in the outcome variable over time.

Computing initial values ...

Setting up 26,168 permutations:
....10%....20%....30%....40%....50%....60%....70%....80%....90%....100%

Fitting full model:

Iteration 0:   log likelihood = -2136.5919  
Iteration 1:   log likelihood = -2136.2728  
Iteration 2:   log likelihood = -2136.2728  

Fixed-effects multinomial logistic regression        Number of obs    =  4,310
Group variable: id                                   Number of groups =    720

                                                     Obs per group:
                                                                  min =      5
                                                                  avg =    6.0
                                                                  max =      7

                                                     LR chi2(10)      = 103.29
Log likelihood = -2136.2728                          Prob > chi2      = 0.0000

------------------------------------------------------------------------------------
           estatus | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------------+----------------------------------------------------------------
Out_of_labor_force |
           hhchild |
              Yes  |   .5881852   .1258696     4.67   0.000     .3414854     .834885
               age |  -.0003842   .0147741    -0.03   0.979    -.0293409    .0285725
          hhincome |  -.0122043   .0088464    -1.38   0.168     -.029543    .0051344
                   |
           hhsigno |
              Yes  |   .5090034    .100111     5.08   0.000     .3127893    .7052174
                   |
           bwinner |
              Yes  |  -.4655745   .0782841    -5.95   0.000    -.6190085   -.3121406
-------------------+----------------------------------------------------------------
Unemployed         |
           hhchild |
              Yes  |    .163612   .1638934     1.00   0.318    -.1576132    .4848372
               age |   .0063355    .019404     0.33   0.744    -.0316957    .0443667
          hhincome |   -.029742   .0120031    -2.48   0.013    -.0532676   -.0062164
                   |
           hhsigno |
              Yes  |   .1173192   .1301364     0.90   0.367    -.1377435    .3723819
                   |
           bwinner |
              Yes  |  -.2489958   .1030027    -2.42   0.016    -.4508773   -.0471142
-------------------+----------------------------------------------------------------
Employed           |  (base outcome)
------------------------------------------------------------------------------------



ereturn list

scalars:
               e(rank) =  10
                 e(ic) =  2
                  e(k) =  16
               e(k_dv) =  1
          e(converged) =  1
                 e(rc) =  0
                 e(ll) =  -2136.272777563178
               e(ll_0) =  -2187.918
               e(df_m) =  10
               e(chi2) =  103.2904448736444
                  e(p) =  1.19385874005e-17
       e(N_group_drop) =  80
             e(N_drop) =  451
                e(N_g) =  720
                  e(N) =  4310
         

. 
. display "Pseudo R2= `:di %5.4f `=1-(`e(ll)'/`e(ll_0)')''"
Pseudo R2= 0.0236