I’m working with proportion data and have tried several models to address issues of overdispersion and underdispersion. Initially, I used a binomial distribution model which showed overdispersion. After introducing an OLRE (Overdispersed Random Effect), I encountered underdispersion. Then, I fitted the model using beta_family, which resolved the overdispersion and underdispersion issues. However, when checking the residual plot, a pattern appears.

`The variables are scaled, and the model I’m using is as follows:

bdd$prop <- (bdd$tot_paras) / bdd$puesta

y <- cbind(bdd$tot_paras, bdd$nac) 

bdd$prop <- pmax(pmin(bdd$prop, 1 - 1e-5), 1e-5)
y <- pmax(pmin(bdd$prop, 1 - 1e-5), 1e-5)


mpt5 <- glmmTMB(y ~ altsca * coastsca * infestsca * NAO_t1sca + 
                  (1|site) + (altsca|year) + (1|id), 
                family = beta_family(), 
                data = bdd)

Here are the results from the summary of the model:
summary(mpt5)
 Family: beta  ( logit )
Formula:          y ~ altsca * coastsca * infestsca * NAO_t1sca + (1 | sitio) +  
    (altsca | year) + (1 | id)
Data: bdd

     AIC      BIC   logLik deviance df.resid 
 -3046.9  -2910.8   1545.5  -3090.9     3581 

Random effects:

Conditional model:
 Groups Name        Variance Std.Dev. Corr 
 sitio  (Intercept) 0.32707  0.5719        
 year   (Intercept) 0.04375  0.2092        
        altsca      0.02278  0.1509   0.84 
 id     (Intercept) 0.65928  0.8120        
Number of obs: 3603, groups:  sitio, 35; year, 12; id, 3603

Dispersion parameter for beta family (): 7.25 

Conditional model:
                                    Estimate Std. Error z value Pr(>|z|)    
(Intercept)                         -0.85460    0.14358  -5.952 2.65e-09 ***
altsca                              -0.59531    0.13203  -4.509 6.52e-06 ***
coastsca                            -0.09911    0.12997  -0.763 0.445730    
infestsca                            0.15908    0.03603   4.415 1.01e-05 ***
NAO_t1sca                           -0.08945    0.06985  -1.281 0.200322    
altsca:coastsca                     -0.39947    0.17170  -2.327 0.019987 *  
altsca:infestsca                     0.01210    0.04966   0.244 0.807511    
coastsca:infestsca                  -0.10618    0.06121  -1.735 0.082811 .  
altsca:NAO_t1sca                     0.08227    0.05643   1.458 0.144869    
coastsca:NAO_t1sca                  -0.09151    0.03187  -2.871 0.004088 ** 
infestsca:NAO_t1sca                  0.08243    0.02259   3.649 0.000263 ***
altsca:coastsca:infestsca           -0.11711    0.07914  -1.480 0.138915    
altsca:coastsca:NAO_t1sca           -0.08114    0.04026  -2.015 0.043858 *  
altsca:infestsca:NAO_t1sca          -0.01356    0.02488  -0.545 0.585860    
coastsca:infestsca:NAO_t1sca        -0.02923    0.03031  -0.964 0.334825    
altsca:coastsca:infestsca:NAO_t1sca -0.07461    0.03592  -2.077 0.037795 *  

Results from the overdispersion check:

performance::check_overdispersion(mpt5)

Overdispersion test

dispersion ratio = 0.887
p-value = 0.12

No overdispersion detected.

Here is the residual plot showing the pattern:
enter image description here

And here is the plot of observed vs predicted values:
[enter image description here](https://i.sstatic.net/oXP7O4A4.png)