Dear lmer people & Dr Bates
  
  I  am a cognitive psychologist who needs to leave the  world of ANOVA and move to lmer. I am looking for advice on coding  random effects in lmer.
  
   My experiment is: 30 subjects judge the laterality of a hand (i.e.. say whether it is a left or right hand) presented onscreen which varies in its 
  (A) Laterality (2 levels, right/left) and 
  (B) Orientation (8 levels, 0 to 315 degs in steps of 45 degs) while holding their own hands in one of 3 postures
  (C) Postures (3 levels, coded as both, minusRight & minusLeft). 
  
  The dependent variable is reaction time (RT). 
  
  Laterality, Orientation & Posture are fixed effects (all  coded as categorical variables), the random effects come from the  subjects i.e., 
  all 30 subjects respond in all possible combinations of the  experimental variables and we need to generalise from them to the  population ... 
  
  My expectation (based on theory & previous studies) is that there will be a sig. main effect of angle, of laterality & possibly a condition by laterality interaction; 
  and graphing shows this.
  
  The traditional way to analyse such data  in psychology, where we typically look at all main effects & possible  interactions as a first pass,  is via a repeated measures 
  (or within-subjects) ANOVA and the  code in R is
  
  aov(RT~Laterality*Angle*Posture+Error(subject/(Laterality*Angle*Posture)),data=RTdata)
  
  moving to lmer, simplifying the model to just look at effects I am interested in, and specifying the random effects as shown in many examples online as follows:
  
  model2a<-lmer(meanRT~(angle+laterality+condition:laterality)+(1|subject),data=RTdata)
  
  I receive the output shown below as OUTPUT 1. Looking at the table of fixed effects I note that the Std. Err. within a specific explanatory variable (e.g, Angle) is constant across all levels of that variable. Obviously I am on the wrong track as this is not an assumption I want to  make. One of the reasons I am moving from ANOVA to lmer is because variance is not constant across the levels of some factors (both angle & laterality) as seen from graph, by running levene's test etc
  
  Rerunning as:
  model2<-lmer(meanRT~(angle+laterality+condition:laterality)+(angle|subject)+(laterality|subject)+(condition:laterality|subject),data=RTdata)
  
  (...and quite honestly, I am generalizing here from how one might specify error in aov....) 
  
  I receive the output shown below as OUTPUT 2. Scrolling down to the fixed effects, the Std., Errs are looking a lot better to me, BUT I am unsure whether I am using the
  syntax correctly
  
  
  can anyone advise?  I  appreciate this may be a very basic question, but I have not found many  examples in my reading except for nested designs (which do not apply here, as least in my understanding of 'nested designs' ), and crossed random effects (which seem more complex than I need, having more than 1 source of random effects) 
  
  thanks in advance, - Nuala
  
  ps - in case the description of the experiment is not clear, I copy data for s1 (aine) at the very end of the email - this pattern will repeat for s2 to s30
  
  OUTPUT 1: summary(model2a)
  Linear mixed model fit by REML 
  Formula: meanRT ~ (angle + laterality + condition:laterality) + (1 | subject) 
     Data: data 
     AIC   BIC logLik deviance REMLdev
   20620 20699 -10295    20700   20590
  Random effects:
   Groups   Name        Variance Std.Dev.
   subject  (Intercept) 124335   352.61  
   Residual              94809   307.91  
  Number of obs: 1440, groups: subject, 30
  
  Fixed effects:
                                      Estimate Std. Error t value
  (Intercept)                          1218.82      70.70  17.240
  angle45                                13.49      32.46   0.416
  angle90                               217.20      32.46   6.692
  angle135                              499.11      32.46  15.378
  angle180                              961.80      32.46  29.633
  angle225                              471.60      32.46  14.530
  angle270                              228.82      32.46   7.050
  angle315                               62.12      32.46   1.914
  lateralityright                      -122.60      28.11  -4.362
  lateralityleft:conditionminusLeft      14.23      28.11   0.506
  lateralityright:conditionminusLeft    -27.28      28.11  -0.971
  lateralityleft:conditionminusRight    -33.77      28.11  -1.201
  lateralityright:conditionminusRight    35.94      28.11   1.279
  
  
  OUTPUT 2: summary(model2)
   Linear mixed model fit by REML 
  Formula: meanRT ~ (angle + laterality + condition:laterality) + (angle | subject) + (laterality | subject) + (condition:laterality |subject) 
     Data: data 
     AIC   BIC logLik deviance REMLdev
   19918 20345  -9878    19867   19756
  Random effects:
   Groups   Name                                Variance   Std.Dev.   Corr  
   subject  (Intercept)                         3.5601e+04 188.681267       
            angle45                             3.5488e+01   5.957191  1.000
            angle90                             1.9515e+04 139.697353  1.000
            angle135                            7.8349e+04 279.909544  0.704
            angle180                            3.3373e+05 577.689525  0.390
            angle225                            7.1096e+04 266.638569  0.462
            angle270                            1.7412e+04 131.954987  0.695
            angle315                            7.1155e+03  84.353226  0.759
   subject  (Intercept)                         2.5444e-04   0.015951       
            lateralityright                     5.2171e-05   0.007223 -1.000
   subject  (Intercept)                         1.4016e+04 118.388702       
            conditionboth:lateralityleft        1.2554e+04 112.046691  0.327
            conditionminusLeft:lateralityleft   3.0653e+04 175.080964  0.257
            conditionminusRight:lateralityleft  1.6837e+04 129.758438 -0.019
            conditionboth:lateralityright       1.0627e+04 103.089657 -0.340
            conditionminusLeft:lateralityright  9.6021e+03  97.990531 -0.822
            conditionminusRight:lateralityright 1.0345e+04 101.711720 -0.453
   Residual                                     4.2251e+04 205.549629       
                                             
                                             
                                             
    1.000                                    
    0.704  0.704                             
    0.390  0.390  0.847                      
    0.462  0.462  0.837  0.889               
    0.695  0.695  0.711  0.609  0.867        
    0.759  0.759  0.324 -0.042  0.021  0.317 
                                             
                                             
                                             
                                             
    0.424                                    
    0.196  0.569                             
    0.428 -0.431 -0.555                      
   -0.112 -0.364 -0.475  0.730               
   -0.451 -0.211  0.622 -0.400 -0.094        
                                             
  Number of obs: 1440, groups: subject, 30
  
  Fixed effects:
                                      Estimate Std. Error t value
  (Intercept)                          1218.82      52.37  23.273
  angle45                                13.49      21.69   0.622
  angle90                               217.20      33.47   6.490
  angle135                              499.11      55.51   8.992
  angle180                              961.80     107.67   8.933
  angle225                              471.60      53.29   8.850
  angle270                              228.82      32.40   7.062
  angle315                               62.12      26.58   2.337
  lateralityright                      -122.60      28.20  -4.348
  lateralityleft:conditionminusLeft      14.23      35.18   0.404
  lateralityright:conditionminusLeft    -27.28      23.13  -1.180
  lateralityleft:conditionminusRight    -33.77      33.79  -0.999
  lateralityright:conditionminusRight    35.94      36.48   0.985
  
  
  Example data for 1 subject
  data[1:48,1:5] - RT is actially mean RT of 18 trails
     subject laterality  posture angle    RT
  1     aine       left       both     0  844.8000
  2     aine       left       both    45  796.4706
  3     aine       left       both    90 1007.5722
  4     aine       left       both   135 1214.7556
  5     aine       left       both   180 1249.9625
  6     aine       left       both   225 1305.0500
  7     aine       left       both   270 1043.8000
  8     aine       left       both   315  814.6833
  9     aine       left  minusLeft     0  817.3778
  10    aine       left  minusLeft    45  951.0588
  11    aine       left  minusLeft    90 1044.5706
  12    aine       left  minusLeft   135 1345.5625
  13    aine       left  minusLeft   180 1482.8333
  14    aine       left  minusLeft   225 1331.3588
  15    aine       left  minusLeft   270  985.1000
  16    aine       left  minusLeft   315  995.2563
  17    aine       left minusRight     0  986.8556
  18    aine       left minusRight    45  903.2176
  19    aine       left minusRight    90  947.8059
  20    aine       left minusRight   135 1453.8750
  21    aine       left minusRight   180 1698.8278
  22    aine       left minusRight   225 1337.1200
  23    aine       left minusRight   270 1109.2467
  24    aine       left minusRight   315  929.0412
  25    aine      right       both     0  913.5944
  26    aine      right       both    45  930.5056
  27    aine      right       both    90 1093.9167
  28    aine      right       both   135 1275.9647
  29    aine      right       both   180 1489.1750
  30    aine      right       both   225 1188.1333
  31    aine      right       both   270 1054.7778
  32    aine      right       both   315  904.8722
  33    aine      right  minusLeft     0  888.9375
  34    aine      right  minusLeft    45  915.0706
  35    aine      right  minusLeft    90 1060.3167
  36    aine      right  minusLeft   135 1240.0867
  37    aine      right  minusLeft   180 1772.4611
  38    aine      right  minusLeft   225 1168.5625
  39    aine      right  minusLeft   270 1093.1889
  40    aine      right  minusLeft   315  842.2667
  41    aine      right minusRight     0  971.3944
  42    aine      right minusRight    45  974.3333
  43    aine      right minusRight    90 1064.9833
  44    aine      right minusRight   135 1389.0059
  45    aine      right minusRight   180 1575.9000
  46    aine      right minusRight   225 1322.6444
  47    aine      right minusRight   270 1053.7389
  48    aine      right minusRight   315 1077.0529
       
  
  Nuala Brady
  School of Psychology
  University College Dublin
  Belfield, D4
  IRELAND
  
  +353 (0)1 716 8247
  nuala.brady@ucd.ie
  


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