Dear Dr. Guerin,

Yes, after reading my own mail I can see its confusing too. Sorry for 
this. I will try to do it better to clarify such amount of information:

0.-lme4 version: 1.0.4

1.-Experimental design:

a-response variable: catch comparison of two gears (test and reference) fishing in paralell  prop=nfish.test/(nfish.test+nfish.reference)
b-Experimental design: two different test *gear_setups* (test_a and test_b) and fishes length class *length_fish* to be considered as fixed effects
c-Sampling effort: 7  parallel fishing (hauls) with setup a (test_a, reference) and 7 hauls with setup b (test_b, reference). Caught fishes were measured to the half cm    *length_fish*.
d-modeling approach: Longitudinal modeling
e-Target fixed effects: *length_fish* (the xaxis of the longitudinal model setup and *gear_setups* as main factor.
f-Potential random effects: haul(random intercept),  and *length_fish* (random slope)


2.-Modeling aims:

a-To assess if test gear has different catch efficiency than reference gear (invLogit(prop)=0.5==equal catch efficiency)
b-To assess if *length_fish* significantly condition the response
c-To assess if the *gear_setup* significantly condition the response

3.-Descriptive analysis:

a.-ploting the data as (using lattice language) prop~length_fish|haul and estimating a standard glm inverse logit curve glm(prop~length_fish
) showed that prop was below 0.5 (more catch efficiency in reference codend) in all hauls, being the intercepts quite similar but with quite variable intercepts.


4.- random intercept modeling

mod0: prop ~ 1 + (1 | haul)
mod1: prop ~ length_fish + (1 | haul)
mod2: prop ~ length_fish + I(length_fish^2) + (1 | haul)
mod3: prop ~ gear_setups + length_fish + (1 | haul)
mod4: prop ~ gear_setups * length_fish + (1 | haul)
mod5: prop ~ gear_setups:length_fish + (1 | haul)


Best model (based on anova test) was mod4

      Df    AIC    BIC  logLik deviance   Chisq Chi Df Pr(>Chisq)
mod0  2 2131.7 2139.8 -1063.8   2127.7
mod1  3 2072.8 2085.0 -1033.4   2066.8 60.8450      1  6.175e-15 ***
mod2  4 2074.8 2091.0 -1033.4   2066.8  0.0254      1  0.8734745
mod3  4 2074.7 2091.0 -1033.4   2066.7  0.0603      0  < 2.2e-16 ***
mod5  4 2067.4 2083.6 -1029.7   2059.4  7.3644      0  < 2.2e-16 ***
mod4  5 2056.1 2076.4 -1023.0   2046.1 13.2588      1  0.0002713 ***



5.- random slope modeling

modRS5: prop ~ 1 + (length_fish | haul)
modRS2: prop ~ l + (length_fish | haul)
modRS4: prop ~ gear_setups + (length_fish | haul)
modRS3: prop ~ gear_setups + length_fish + (length_fish | haul)
modRS1: prop ~ gear_setups * length_fish + (length_fish | haul)


Best model (based on anova test) was modRS5


modRS5  4 1876.7 1892.9 -934.35   1868.7
modRS2  5 1878.6 1898.9 -934.29   1868.6 0.1163      1     0.7331
modRS4  5 1878.4 1898.7 -934.20   1868.4 0.1673      0     <2e-16 ***
modRS3  6 1880.3 1904.7 -934.15   1868.3 0.1035      1     0.7476
modRS1  7 1880.2 1908.6 -933.10   1866.2 2.0967      1     0.1476


6.-Predictions based on modRS5

The "response" predictions using the fixef errors (fvar<-sqrt(diag(model.matrix %*% tcrossprod(vcov(modRS5),model.matrix))) is:

  hat.prop.1= 0.15 (0.18-0.23)

When adding the error from the random effect component sqrt(fvar+VarCorr(modRS5)$haul) the prediction turns:

   
hat.prop.2= 0.15 (0.01-0.97)


7.-My understanding of the output:

Altough during the experimental results  shows a significant catch reduction (below 0.5) in test gear (0.18[0.15-0.23 ]), this catch reduction is
not longer significant if we raise the inference into a "population of hauls" perpective (0.15[ 0.001-0.97]).



8.-My questions:

     1.- Is my interpretation correct?

     2.- Why hat.prop.2 produce such big CI band, since the individual haul assessment (using glm(prop~1))shows that none of the hauls has an intercept crossing the 0.5 value?

inv.logit(list.glm.haul$beta0)

  [1] 0.17488076 0.21137393 0.11544715 0.14787431 0.38698916 0.20431328 0.20926518 0.19069240 0.21739130 0.16785911 0.06284153
[12] 0.24972253 0.24489796 0.16738817



All the best for all,

Juan



  



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