How can I interpret non-significance in models built with subsetted data?

Question

I am working on revisions, and it was requested that I re-run my models separately for males and females. When I do this, I lose statistical significance in some tests in females but not in males, but how can I determine if this is due to the lower power or truly an insignificant effect size? I thought I could run a post-hoc power analysis, but it seems that doesn't provide more information than the testing I have already done (https://stat.uiowa.edu/sites/stat.uiowa.edu/files/techrep/tr378.pdf), so I am not sure where to go. The number of men and women in the cohort is different for all levels of disease.state, in particular, there are less women in the control group yet more women in most disease groups and I'm not sure how statistical power would be impacted.

Here is an example of the full linear mixed model I was running:

lmer(y ~ disease.state + sex + (1 | id), dt)

In the new models, I have subset the data by sex and dropped it as a fixed effect

lmer(y ~ disease.state + (1 | id), dt[sex == "Male"])
lmer(y ~ disease.state + (1 | id), dt[sex == "Female"])

Excerpts from the fixed effects tables, note the difference in disease.stateIJ and KL between the three models.

Full model 
                    Estimate Std. Error         df t value Pr(>|t|)    
(Intercept)         1.727e+00  6.836e-03  5.281e+03 252.592  < 2e-16 ***
disease.stateAB    -5.219e-02  7.495e-03  4.193e+03  -6.963 3.85e-12 ***
disease.stateCD    -4.389e-02  8.012e-03  3.867e+03  -5.478 4.58e-08 ***
disease.stateEF    -7.440e-03  9.951e-03  2.928e+03  -0.748 0.454733    
disease.stateGH    -4.768e-03  9.950e-03  2.772e+03  -0.479 0.631822    
disease.stateIJ    -2.365e-02  8.991e-03  3.217e+03  -2.631 0.008553 ** 
disease.stateKL    -2.743e-02  7.820e-03  3.956e+03  -3.507 0.000458 ***
disease.stateMN    -2.407e-02  1.890e-02  2.309e+03  -1.273 0.203073    


Female subset
                     Estimate Std. Error         df t value Pr(>|t|)    
(Intercept)         1.715e+00  1.036e-02  3.778e+03 165.575  < 2e-16 ***
disease.stateAB    -3.119e-02  1.149e-02  2.833e+03  -2.714  0.00669 ** 
disease.stateCD    -5.219e-02  1.246e-02  2.490e+03  -4.190 2.88e-05 ***
disease.stateEF     4.034e-03  1.677e-02  1.710e+03   0.241  0.80995    
disease.stateGH     5.971e-03  1.446e-02  1.834e+03   0.413  0.67974    
disease.stateIJ    -1.061e-02  1.268e-02  2.334e+03  -0.837  0.40263    
disease.stateKL    -1.214e-02  1.150e-02  2.853e+03  -1.056  0.29126    
disease.stateMN    -1.036e-02  2.138e-02  1.545e+03  -0.485  0.62784   

Male subset
Fixed effects:
                     Estimate Std. Error         df t value Pr(>|t|)    
(Intercept)         1.763e+00  8.168e-03  2.422e+03 215.803  < 2e-16 ***
disease.stateAB    -7.123e-02  1.009e-02  1.531e+03  -7.059 2.53e-12 ***
disease.stateCD    -3.409e-02  1.056e-02  1.484e+03  -3.228 0.001272 ** 
disease.stateEF    -1.437e-02  1.249e-02  1.205e+03  -1.150 0.250397    
disease.stateGH    -1.061e-02  1.405e-02  1.057e+03  -0.756 0.450089    
disease.stateIJ    -3.359e-02  1.391e-02  1.101e+03  -2.414 0.015937 *  
disease.stateKL    -4.284e-02  1.152e-02  1.321e+03  -3.719 0.000208 ***
disease.stateMN    -4.120e-02  4.776e-02  7.962e+02  -0.863 0.388624   

Answer 1

Your intercept values (estimates of outcome at reference conditions) are approximately 1.74 for all your models (1.73, 1.72, 1.76), while the largest disease.state coefficient has a magnitude of only about 0.07. That's a pretty small difference from the intercept.

The degrees of freedom (df) values suggest that you have thousands of individuals in each group. Thus you were able to distinguish "statistically significant" disease.state coefficients down to a magnitude of 0.03 or so. For your disease.stateIJ and disease.stateKL values, the Main Model values are pretty close to the averages of the values from the Male and Female subset models; the Male values had magnitudes greater than 0.03 while the Female values were below 0.02 in magnitude.

The above suggests that the Male-Female differences in some coefficient values are real. A more efficient way to test the Male-Female differences is to use an interaction model rather than separate subset analyses, for example:

lmer(y ~ disease.state * sex + (1 | id), dt)

This has the advantage of using information from all cases in a single model. The interaction coefficients would then demonstrate directly the disease.state values for which there was a "statistically significant" difference between Males and Females. You need to apply your understanding of the subject matter to determine whether those differences might be important in practice.