galamm-latent-covariates-interaction.md
In galamm: Generalized Additive Latent and Mixed Models

Interaction between latent and observed covariates works

Code
  print(summary(mod), digits = 2)
Output
  GALAMM fit by maximum marginal likelihood.
  Formula: formula
     Data: data

       AIC      BIC   logLik deviance df.resid 
     138.3    177.9    -60.2    120.3      591

  Scaled residuals: 
     Min     1Q Median     3Q    Max 
   -2.20  -0.53  -0.03   0.51   3.50

  Lambda:
            loading    SE
  lambda1      1.00     .
  lambda2      1.30 0.013
  lambda3     -0.32 0.016
  lambda4_x    0.23 0.029

  Random effects:
   Groups   Name    Variance Std.Dev.
   id       loading 0.982    0.99    
   Residual         0.012    0.11    
  Number of obs: 600, groups:  id, 200

  Fixed effects:
                   Estimate Std. Error t value Pr(>|t|)
  (Intercept)       -0.0106      0.070  -0.150  8.8e-01
  typemeasurement2  -0.0022      0.024  -0.091  9.3e-01
  typeresponse       0.0340      0.094   0.361  7.2e-01
  x:response         0.4625      0.033  14.016  1.3e-44

Code
  print(summary(modq), digits = 2)
Output
  GALAMM fit by maximum marginal likelihood.
  Formula: formula
     Data: data

       AIC      BIC   logLik deviance df.resid 
     140.3    184.2    -60.1    120.3      590

  Scaled residuals: 
     Min     1Q Median     3Q    Max 
   -2.20  -0.52  -0.02   0.52   3.50

  Lambda:
                 loading    SE
  lambda1          1.000     .
  lambda2          1.303 0.013
  lambda3         -0.314 0.024
  lambda4_x        0.209 0.113
  lambda5_I(x^2)   0.025 0.111

  Random effects:
   Groups   Name    Variance Std.Dev.
   id       loading 0.982    0.99    
   Residual         0.012    0.11    
  Number of obs: 600, groups:  id, 200

  Fixed effects:
                   Estimate Std. Error t value Pr(>|t|)
  (Intercept)       -0.0099      0.071  -0.141  8.9e-01
  typemeasurement2  -0.0020      0.024  -0.083  9.3e-01
  typeresponse       0.0333      0.094   0.353  7.2e-01
  x:response         0.4622      0.033  13.984  2.0e-44

Code
  print(anova(modq, mod), digits = 3)
Output
  Data: data
  Models:
  mod: formula
  modq: formula
       npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
  mod     9 138 178  -60.2      120                    
  modq   10 140 184  -60.1      120  0.05  1       0.83

Crossed latent-observed interaction models work

Code
  print(summary(mod), digits = 2)
Output
  GALAMM fit by maximum marginal likelihood.
  Formula: formula
     Data: data

       AIC      BIC   logLik deviance df.resid 
      63.2    103.0    -21.6     43.2      386

  Scaled residuals: 
     Min     1Q Median     3Q    Max 
   -3.27  -0.52  -0.04   0.50   3.80

  Lambda:
            loading    SE
  lambda1      1.00     .
  lambda2      1.31 0.024
  lambda3     -0.37 0.023
  lambda4_x    0.31 0.040

  Random effects:
   Groups   Name     Variance Std.Dev.
   id       loading  0.904    0.95    
   id.1     response 0.000    0.00    
   Residual          0.019    0.14    
  Number of obs: 396, groups:  id, 99

  Fixed effects:
                   Estimate Std. Error t value Pr(>|t|)
  (Intercept)        -0.033      0.097   -0.34  7.3e-01
  typemeasurement2   -0.012      0.036   -0.35  7.3e-01
  typeresponse        0.073      0.134    0.55  5.8e-01
  x:response          0.432      0.048    8.98  2.7e-19