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Included new results after implementing recommended changes
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Charlie Glez
Charlie Glez
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Updated results after changing Year to cYear

Random effects:
 Groups        Name        Variance  Std.Dev. Corr
 Firm:Industry (Intercept) 5.840e-01 0.764205     
               cYear       5.546e-03 0.074474 0.38
 Industry      (Intercept) 8.243e-05 0.009079     
               cYear       2.011e-05 0.004485 0.54
Number of obs: 436, groups:  Firm:Industry, 109; Industry, 37

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -2.87753    0.47970  -5.999 1.99e-09 ***
cYear       -0.02303    0.02778  -0.829    0.407    
X            1.32358    0.33632   3.935 8.30e-05 ***
Xsquared    -0.27628    0.14217  -1.943    0.052 .  
FirmSize     0.31789    0.04989   6.371 1.87e-10 ***

Updated results after changing Year to cYear

Random effects:
 Groups        Name        Variance  Std.Dev. Corr
 Firm:Industry (Intercept) 5.840e-01 0.764205     
               cYear       5.546e-03 0.074474 0.38
 Industry      (Intercept) 8.243e-05 0.009079     
               cYear       2.011e-05 0.004485 0.54
Number of obs: 436, groups:  Firm:Industry, 109; Industry, 37

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -2.87753    0.47970  -5.999 1.99e-09 ***
cYear       -0.02303    0.02778  -0.829    0.407    
X            1.32358    0.33632   3.935 8.30e-05 ***
Xsquared    -0.27628    0.14217  -1.943    0.052 .  
FirmSize     0.31789    0.04989   6.371 1.87e-10 ***
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Charlie Glez
Charlie Glez
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How to interpret squared term in GLMER model

I'm trying to explain Y (a count variable), in terms of X, Xsquared, and Size, with random effects for Industry/Firm, by year. My hypothesis is that there is a positive relationship between Y and X, and that this relationship has a inverted-U shape (thus the Xsquared term). I normally do this with normal regression models, but I want to know if it would mean the same in a GLMER one (I'm new with these models)

Do my results, here below, support my inverted-U hypothesis, or something else? Any tips on how to show this inverted-U effect graphically in r? Lastly, how can I interpret the random effects part?

My GLMER model looks like this:

model <- glmer(Y ~ Year + X + Xsquared + Size + (1 + Year|Industry/Firm), data = mydata, family = poisson)

Results:

Random effects:
 Groups      Name        Variance    Std.Dev.  Corr 
 Firm:Industry (Intercept) 1.787e+04 1.337e+02      
             Year          4.434e-03 6.659e-02 -1.00
 Industry      (Intercept) 7.749e-01 8.803e-01      
             Year          1.923e-07 4.385e-04 -1.00
Number of obs: 436, groups:  Firm:Industry, 109; Industry, 37

Fixed effects:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) 51.697639   9.178129   5.633 1.77e-08 ***
Year        -0.027132   0.004535  -5.983 2.19e-09 ***
X            1.322702   0.334092   3.959 7.52e-05 ***
Xsquared    -0.277335   0.141129  -1.965   0.0494 *  
Size         0.321026   0.043825   7.325 2.39e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
         (Intr) Year   DOI    DOI2  
Year     -0.999                     
DOI       0.004 -0.019              
DOI2     -0.009  0.020 -0.953       
LNAssets -0.162  0.119  0.003 -0.007