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I want to model a response variable (y) as a function of two explanatory variables (x and z). y are concentration measures of a physiological parameter of the human body with a worldwide accepted method (method A). x is a novel method (method B) that might be alternative since it is much cheaper. However, it is thought that method B is more or less accurate depending on the time (z) that the equipment needed is used. So, my goal is to test the significance of x and z to accurately measure y. I have 6 individuals (ID: A,B,C,D,E,F). Below I show the relationship between x and y:

enter image description here *Note: here I categorized Z for illustration purposes, but z, as x, is numerical, ranging from 1 (hr) to 7 (hr).

Given that my response variable has a sharp non-normal distribution, that the variance increases as x increases, and that I have several individuals but I am not interested in differences among individuals, I though on GLMM with a GAMMA distribution to test the significant effect of x and z for explaining y.

enter image description here

I run GLMMs with a gamma distribution and using the log link.

## Setting random structure ####
mod1<-glmer(Y~ 1 + (1|ID),data = df, family=Gamma(link=log)) 
mod2<-glmer(Y~ 1 + (X|ID),data = df, family=Gamma(link=log)) 
AIC(mod1,mod2)

## Setting fixed structure ####
mod1<-glmer(Y~ 1 + (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod2<-glmer(Y~X + (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod3<-glmer(Y~X + Z + (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod4<-glmer(Y~X + X:Z +  (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 

r.squaredGLMM(mod4)[1,c(1,2)]

mod.list <- list(mod1,mod2,mod3,mod4)
model.sel<-model.sel(mod.list, rank="AIC",extra=c(r2=function(x) round(r.squaredGLMM(x)[1,c(1,2)],2)))

To test if x and z are significant I compared models by AIC.

model.sel

Model selection table 
  (Intr) ns(VDB.V13,3)    n.V13 cnd((Int)) dsp((Int)) cnd(ns(VDB.V13,3)) cnd(n.V13:VDB.V13) r2.R2m r2.R2c    class   control ziformula dispformula     random df   logLik     AIC  delta weight
3 -2.567             + -0.04178                                                               0.66   0.71 glmerMod gC(Nl_Md)                          VD.V1|I  9 2017.580 -4017.2   0.00      1
2 -2.645             +                                                                        0.65   0.70 glmerMod gC(Nl_Md)                          VD.V1|I  8 2006.875 -3997.7  19.41      0
4                                   -2.661          +                  +                  +   0.66   0.76  glmmTMB                  ~0          ~1 c(VD.V1|I)  9 2001.622 -3985.2  31.92      0
1 -1.559                                                                                      0.00   0.36 glmerMod gC(Nl_Md)                          VD.V1|I  5 1682.428 -3354.9 662.31      0
Abbreviations:
control: gC(Nl_Md) = ‘glmerControl(Nelder_Mead)’
Models ranked by AIC(x) 
Random terms: 
VD.V1|I = ‘VeDBA.V13AP | ID’
c(VD.V1|I) = ‘cond(VeDBA.V13AP | ID)’

My problem comes when I observe the residual patterns vs predicted values since I think there are clear patterns. Here I don't show the distribution of the residuals since if I understand correctly, a normal distribution of them is not needed.

enter image description here

Does anyone know what I could do? Any proposal? I did not share the data because is too long (n=2027).

Thanks in advance

Head of my dataframe:

   ID          Y          X Z
1   A 0.34136077 1.55682000 2
2   A 0.05124066 0.05766000 2
3   A 0.05901189 0.05125333 3
4   A 0.05213855 0.05766000 2
5   A 0.05437708 0.05125333 3
6   A 0.08433229 0.05766000 3
7   A 0.03618396 0.04484667 3
8   A 0.03622474 0.05766000 1
9   A 0.18244336 0.05125333 3
10  A 0.03625487 0.03844000 2
11  A 0.03840890 0.04484667 3
12  A 0.04235018 0.03844000 3
13  A 0.03862926 0.03844000 3
14  A 0.03749647 0.02883000 2
15  A 0.04395015 0.03844000 2
16  A 0.04040225 0.04805000 2
17  A 0.04419507 0.05766000 3
18  A 0.33186947 2.53704000 1
19  A 0.31986092 0.74958000 1
20  A 0.08127853 0.05766000 1
")

Procedure followed according to the proposal of JTH

mod1<-glmer(Y~ 1 + (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod2<- glmer(Y~ ns(X, 4)  + (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod3<-glmer(Y~ ns(X, 4) + Z + (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod4<-glmer(Y~ X:Z + ns(X, 4) +  (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
mod5<-glmer(Y~ ns(X, 4):Z +  (X|ID),data = df, family=Gamma(link=log), control = glmerControl(optimizer ="Nelder_Mead")) 
AIC(mod1,mod2,mod3,mod4,mod5)

The results are next:

r.squaredGLMM(mod4)[1,c(1,2)]

mod.list <- list(mod1,mod2,mod3,mod4)
model.sel<-model.sel(mod.list, rank="AIC",extra=c(r2=function(x) round(r.squaredGLMM(x)[1,c(1,2)],2)))

model.sel

        Int ns(X)            Z       Z:X Z:ns(X)  r2m  r2c df   logLik       AIC     delta       weight
4 -2.827029     +              0.1871558         0.65 0.69 10 320.9342 -621.8684   0.00000 1.000000e+00
2 -2.660326     +                                0.48 0.54  9 289.8442 -561.6884  60.17996 8.552394e-14
3 -2.660204     + 0.0001000814                   0.48 0.54 10 289.8443 -559.6886  62.17976 3.146559e-14
5 -2.220731                                    + 0.04 0.72  9 259.3610 -500.7219 121.14643 4.936132e-27
1 -1.377842                                      0.00 0.27  5 246.9520 -483.9039 137.96446 1.100015e-30

The plot of my residuals vs predicted values is this now:

enter image description here

Although the patterns have been sharply reduced, I suspect there is still some heteroscedasticity. However, I can't remove it.

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    $\begingroup$ I would try to add a natural spline to the regression equation to stamp out the pattern in the residuals. possibly something like glmer(y ~ x*z + ns(x, df=4) + (1 + x | ID), data= data, family=Gamma(link='log'). It may take some experimentation. $\endgroup$
    – JTH
    Commented Sep 19, 2020 at 20:24
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    $\begingroup$ You can use AIC to select among these models. m4 looks much better than m5, not only because it has a lower AIC, but because I think the interaction ns(x)*z could be hard to interpret. As for the interpretation, I think that's tricky. I'm already not sure off the dome how to best interpret the coefficients of a simple gamma glmer, adding a spline to the mix makes this harder. Sorry I got DF and knots mixed up. What I meant by experimentation, is that you should try different values for the knots= and possibly try different splines, e.g. bs instead of ns I think (1+X|ID)=(X|ID) $\endgroup$
    – JTH
    Commented Sep 20, 2020 at 13:27
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    $\begingroup$ See stats.stackexchange.com/questions/96972/… for information on coefficient interpretation. Because you have a mixed model, your interpretations are additionally conditional on the predicted values of the random effects. I wouldn't try to interpret the ns coefs, rather try to make plots of the model predictions (use predict function). $\endgroup$
    – JTH
    Commented Sep 20, 2020 at 13:41
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    $\begingroup$ Yes, it does look heteroskedastic. I'm not a expert on deviance residuals, so perhaps it's okay. You might try glmer(y ~ x*z + ns(x, 3) + (x + ns(x,3) | ID), ... ) to fit a different natural spline for each level of ID. As always, plot the data and predictions together to best check the model. $\endgroup$
    – JTH
    Commented Sep 20, 2020 at 21:20
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    $\begingroup$ scaling x and z was a bad idea. collinearity is not that much of an issue (and can be solved differently) and interpretability is affected. most importantly now, we don't know what was the range of that variables and which transformation to reccomend you. $\endgroup$
    – carlo
    Commented Sep 26, 2020 at 7:14

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