title: Multiple random effects without interaction in nlme::lme
body: <p>Let consider this simple model consisting in two independent random effects:</p>
  
  <pre><code>$$y_{ijk} = \mu + \delta1_i + \delta2_j + \epsilon_{i,j,k}$$
  </code></pre>
  
  <p>where <code>\delta1_i</code> and <code>\delta2_j</code> are independent random effect (i.e. <code>\delta1_i \sim N(0,\sigma_1^2)</code> iid, <code>\delta2_i \sim N(0,\sigma_1^2)</code> iid, both independent).</p>
  
  <p>Defining and inferring the variance components of such a model is straightforward using lme4::lmer (sorry the example below is likely to have no physical meaning) :</p>
  
  <pre><code>library(lme4)
  library(nlme)
  data(Orthodont,package="nlme")   
  
  lmer(data=Orthodont,distance~1+(1|Sex)+(1|Subject))
  #Linear mixed model fit by REML ['lmerMod']
  #Formula: distance ~ 1 + (1 | Sex) + (1 | Subject)
  #   Data: Orthodont
  #REML criterion at convergence: 510.3937
  #Random effects:
  # Groups   Name        Std.Dev.
  # Subject  (Intercept) 1.596   
  # Sex      (Intercept) 1.550   
  # Residual             2.220   
  #Number of obs: 108, groups:  Subject, 27; Sex, 2
  #Fixed Effects:
  #(Intercept)  
  #      23.83 
  </code></pre>
  
  <p>How can this be modeled easily using nlme::lme ? Specifying the random effect using a list of random effect implicitly introduces interaction depending from the ordering of the random effects , which can be verified here (changing order of specification, change results):</p>
  
  <pre><code>VarCorr(lme(data=Orthodont,distance~1,random=list(~1|Subject,~1|Sex)))
  #            Variance     StdDev  
  #Subject =   pdLogChol(1)         
  #(Intercept) 1.875987     1.369667
  #Sex =       pdLogChol(1)         
  #(Intercept) 1.875987     1.369667
  #Residual    4.929784     2.220312
  
  VarCorr(lme(data=Orthodont,distance~1,random=list(~1|Sex,~1|Subject)))
  #            Variance     StdDev  
  #Sex =       pdLogChol(1)         
  #(Intercept) 2.403699     1.550387
  #Subject =   pdLogChol(1)         
  #(Intercept) 2.546702     1.595839
  #Residual    4.929784     2.220312
  </code></pre>