title: How to force all treatments to have same intercept in lme4?
body: <p>This is my first time posting here, so I hope my question isn't foolish, although I fear it has no answer.</p>
  
  <p>I ran a feeding trial where I fed snails one of five different algae species, then measured individual length over time. I built a LMM using R's <code>lme4</code>:</p>
  
  <pre><code>model&lt;-lmer(log(length)~treatment*day+(1+day|individual), data=real, REML=F)
  </code></pre>
  
  <p>In my model, one of the treatments has a very different intercept than the others:</p>
  
  <p><a href="https://i.stack.imgur.com/uXEIW.png" rel="nofollow noreferrer">Plot of my model, showing the treatment with outlier intercept, and other treatments with roughly matching intercepts.</a></p>
  
  <p>I measured length 4 times over 61 days (day 7, 20, 42, 61), but unfortunately I didn't measure it on day 0. All of the snails came from the same batch, so each treatment had the same mean length at day 0. Is there any way to tell R that this is true, even if I don't know WHAT the mean length was on day 0? In theory, I could just adjust the intercepts so they match, but it seems that the slope of the line would also change if the model took into account this information.</p>
  
  <p>I know that the model is written so that all the individuals have random intercepts. However, writing the model with the term <code>(0+day|individual)</code> so that individuals have fixed intercepts does not solve the problem, because the intercepts of individuals are the same only within treatments, but are different between treatments. It may be that I'm asking R to do the impossible or unethical, i.e. fabricate data.</p>
  
  <p>Thanks in advance for any suggestions!</p>
  
  <p>Edit for a reproducible example:
  The same problem happens if using the R "ChickWeights" dataset (these data are not a perfectly linear trend, but good enough as an example)</p>
  
  <pre><code>model.chix&lt;-lmer(weight~Diet*Time+(Time|Chick), data=ChickWeight, REML=FALSE)
  
  &gt; summary(model.chix)
  Linear mixed model fit by maximum likelihood  ['lmerMod']
  Formula: weight ~ Diet * Time + (Time | Chick)
     Data: ChickWeight
  
       AIC      BIC   logLik deviance df.resid 
    4824.2   4876.5  -2400.1   4800.2      566 
  
  Scaled residuals: 
      Min      1Q  Median      3Q     Max 
  -2.7508 -0.5693 -0.0401  0.4694  3.5415 
  
  Random effects:
   Groups   Name        Variance Std.Dev. Corr 
   Chick    (Intercept) 103.61   10.179        
            Time         10.01    3.165   -0.99
   Residual             163.36   12.781        
  Number of obs: 578, groups:  Chick, 50
  
  Fixed effects:
              Estimate Std. Error t value
  (Intercept)  33.6541     2.8023  12.009
  Diet2        -5.0205     4.8072  -1.044
  Diet3       -15.4038     4.8072  -3.204
  Diet4        -1.7475     4.8145  -0.363
  Time          6.2799     0.7304   8.598
  Diet2:Time    2.3293     1.2508   1.862
  Diet3:Time    5.1430     1.2508   4.112
  Diet4:Time    3.2528     1.2515   2.599
  
  Correlation of Fixed Effects:
             (Intr) Diet2  Diet3  Diet4  Time   Dt2:Tm Dt3:Tm
  Diet2      -0.583                                          
  Diet3      -0.583  0.340                                   
  Diet4      -0.582  0.339  0.339                            
  Time       -0.881  0.513  0.513  0.513                     
  Diet2:Time  0.514 -0.882 -0.300 -0.299 -0.584              
  Diet3:Time  0.514 -0.300 -0.882 -0.299 -0.584  0.341       
  Diet4:Time  0.514 -0.300 -0.300 -0.882 -0.584  0.341  0.341
  </code></pre>
  
  <p>The intercept estimates predict chick weights at Day 0.  Chicks on Diet1 are predicted to weigh 33.7 g, while chicks on Diet3 are predicted to weigh 15.4 g less than that.  Wouldn't the model be a better fit if it forced all chick weights to be the same on Day 0, to match reality of the situation?  </p>