\[X_{j} = \eta_{1} + \epsilon_{j}^{x}, \quad j=1,2,3\] \[Y_{j} = \eta_{2} + \epsilon_{j}^{y}, \quad j=1,2,3\] and with a structural model given by \[\eta_{2} = f(\eta_{1}) + Z + \zeta_{2}\label{ex:eta2}\] \[\eta_{1} = Z + \zeta_{1}\label{ex:eta1}\] with iid measurement errors \(\epsilon_{j}^{x},\epsilon_{j}^{y},\zeta_{1},\zeta_{2}\sim\mathcal{N}(0,1), j=1,2,3.\) and standard normal distributed covariate \(Z\). To simulate from this model we use the following syntax:

Estimation

To estimate the parameters using the two-stage estimator described in ^{holst_budtzjorgensen_2020}, the first step is now to specify the measurement models

m1 <- lvm(x1+x2+x3 ~ eta1, eta1 ~ z, latent=~eta1)
m2 <- lvm(y1+y2+y3 ~ eta2, eta2 ~ z, latent=~eta2)

Next, we specify a quadratic relationship between the two latent variables

nonlinear(m2, type="quadratic") <- eta2 ~ eta1

and the model can then be estimated using the two-stage estimator

e1 <- twostage(m1, m2, data=d)
e1
#>                     Estimate Std. Error  Z-value  P-value
#> Measurements:                                            
#>    y2~eta2           0.97686    0.08406 11.62146   <1e-12
#>    y3~eta2           1.04485    0.04531 23.06093   <1e-12
#> Regressions:                                             
#>    eta2~z            0.88513   16.11882  0.05491   0.9562
#>    eta2~eta1_1       1.14072   18.98292  0.06009   0.9521
#>    eta2~eta1_2      -0.45055    9.16379 -0.04917   0.9608
#> Intercepts:                                              
#>    y2               -0.12198    0.10935 -1.11549   0.2646
#>    y3               -0.09879    0.10569 -0.93471   0.3499
#>    eta2              0.67814   15.00825  0.04518    0.964
#> Residual Variances:                                      
#>    y1                1.30730    0.48572  2.69149         
#>    y2                1.11056    1.07092  1.03702         
#>    y3                0.80961    0.76919  1.05255         
#>    eta2              2.08483    7.57087  0.27537

We see a clear statistically significant effect of the second order term (eta2~eta1_2). For comparison we can also estimate the full MLE of the linear model:

e0 <- estimate(regression(m1%++%m2, eta2~eta1), d)
estimate(e0,keep="^eta2~[a-z]",regex=TRUE) ## Extract coef. matching reg.ex.
#>           Estimate Std.Err    2.5% 97.5%   P-value
#> eta2~eta1   1.4140  0.2261 0.97083 1.857 4.014e-10
#> eta2~z      0.6374  0.2778 0.09291 1.182 2.177e-02

Next, we calculate predictions from the quadratic model using the estimated parameter coefficients \[ \mathbb{E}_{\widehat{\theta}_{2}}(\eta_{2} \mid \eta_{1}, Z=0), \]

newd <- expand.grid(eta1=seq(-4, 4, by=0.1), z=0)
pred1 <- predict(e1, newdata=newd, x=TRUE)
head(pred1)
#>              y1         y2         y3       eta2
#> [1,] -11.093569 -10.958869 -11.689950 -11.093569
#> [2,] -10.623561 -10.499736 -11.198861 -10.623561
#> [3,] -10.162565 -10.049406 -10.717187 -10.162565
#> [4,]  -9.710579  -9.607878 -10.244928  -9.710579
#> [5,]  -9.267605  -9.175153  -9.782084  -9.267605
#> [6,]  -8.833641  -8.751230  -9.328656  -8.833641

To obtain a potential better fit we next proceed with a natural cubic spline

kn <- seq(-3,3,length.out=5)
nonlinear(m2, type="spline", knots=kn) <- eta2 ~ eta1
e2 <- twostage(m1, m2, data=d)
e2
#>                     Estimate Std. Error  Z-value  P-value
#> Measurements:                                            
#>    y2~eta2           0.97752    0.09601 10.18184   <1e-12
#>    y3~eta2           1.04508    0.08418 12.41415   <1e-12
#> Regressions:                                             
#>    eta2~z            0.86729   15.75435  0.05505   0.9561
#>    eta2~eta1_1       2.86231   21.41244  0.13368   0.8937
#>    eta2~eta1_2       0.00344    3.46230  0.00099   0.9992
#>    eta2~eta1_3      -0.26270   11.47891 -0.02289   0.9817
#>    eta2~eta1_4       0.50778   14.09762  0.03602   0.9713
#> Intercepts:                                              
#>    y2               -0.12185    0.10980 -1.10974   0.2671
#>    y3               -0.09874    0.10627 -0.92918   0.3528
#>    eta2              1.83814   69.73236  0.02636    0.979
#> Residual Variances:                                      
#>    y1                1.31286    0.49428  2.65613         
#>    y2                1.10412    1.03272  1.06914         
#>    y3                0.81124    0.72916  1.11257         
#>    eta2              1.99404    6.88395  0.28966

Confidence limits can be obtained via the Delta method using the estimate method:

p <- cbind(eta1=newd$eta1,
      estimate(e2,f=function(p) predict(e2,p=p,newdata=newd))$coefmat)
head(p)
#>    eta1  Estimate   Std.Err      2.5%    97.5%   P-value
#> p1 -4.0 -9.611119 106.92610 -219.1824 199.9602 0.9283781
#> p2 -3.9 -9.324887 105.30961 -215.7279 197.0782 0.9294416
#> p3 -3.8 -9.038656 103.71213 -212.3107 194.2334 0.9305512
#> p4 -3.7 -8.752425 102.13457 -208.9325 191.4276 0.9317089
#> p5 -3.6 -8.466193 100.57784 -205.5951 188.6628 0.9329169
#> p6 -3.5 -8.179962  99.04294 -202.3006 185.9406 0.9341775

The fitted function can be obtained with the following code:

plot(I(eta2-z) ~ eta1, data=d, col=Col("black",0.5), pch=16,
     xlab=expression(eta[1]), ylab=expression(eta[2]), xlim=c(-4,4))
lines(Estimate ~ eta1, data=as.data.frame(p), col="darkblue", lwd=5)
confband(p[,1], lower=p[,4], upper=p[,5], polygon=TRUE,
     border=NA, col=Col("darkblue",0.2))

Cross-validation

A more formal comparison of the different models can be obtained by cross-validation. Here we specify linear, quadratic and cubic spline models with 4 and 9 degrees of freedom.

m2a <- nonlinear(m2, type="linear", eta2~eta1)
m2b <- nonlinear(m2, type="quadratic", eta2~eta1)
kn1 <- seq(-3,3,length.out=5)
kn2 <- seq(-3,3,length.out=8)
m2c <- nonlinear(m2, type="spline", knots=kn1, eta2~eta1)
m2d <- nonlinear(m2, type="spline", knots=kn2, eta2~eta1)

To assess the model fit average RMSE is estimated with 5-fold cross-validation repeated two times

## Scale models in stage 2 to allow for a fair RMSE comparison
d0 <- d
for (i in endogenous(m2))
    d0[,i] <- scale(d0[,i],center=TRUE,scale=TRUE)
## Repeated 5-fold cross-validation:
ff <- lapply(list(linear=m2a,quadratic=m2b,spline4=m2c,spline6=m2d),
        function(m) function(data,...) twostage(m1,m,data=data,stderr=FALSE,control=list(start=coef(e0),contrain=TRUE)))
fit.cv <- cv(ff,data=d,K=5,rep=2,mc.cores=parallel::detectCores(),seed=1)

summary(fit.cv)
#>       Length Class  Mode     
#> cv    40     -none- numeric  
#> call   7     -none- call     
#> names  4     -none- character
#> rep    1     -none- numeric  
#> folds  1     -none- numeric

Here the RMSE is in favour of the splines model with 4 degrees of freedom:

fit <- lapply(list(m2a,m2b,m2c,m2d),
         function(x) {
         e <- twostage(m1,x,data=d)
         pr <- cbind(eta1=newd$eta1,predict(e,newdata=newd$eta1,x=TRUE))
         return(list(estimate=e,predict=as.data.frame(pr)))
         })

plot(I(eta2-z) ~ eta1, data=d, col=Col("black",0.5), pch=16,
     xlab=expression(eta[1]), ylab=expression(eta[2]), xlim=c(-4,4))
col <- c("orange","darkred","darkgreen","darkblue")
lty <- c(3,4,1,5)
for (i in seq_along(fit)) {
    with(fit[[i]]$pr, lines(eta2 ~ eta1, col=col[i], lwd=4, lty=lty[i]))
}
legend("bottomright",
      c("linear","quadratic","spline(df=4)","spline(df=6)"),
      col=col, lty=lty, lwd=3)

For convenience, the function twostageCV can be used to do the cross-validation (also for choosing the mixture distribution via the ``nmix`` argument, see the section below). For example,

selmod <- twostageCV(m1, m2, data=d, df=2:4, nmix=1:2,
        nfolds=2, rep=1, mc.cores=parallel::detectCores())

applies cross-validation (here just 2 folds for simplicity) to select the best splines with degrees of freedom varying from from 1-3 (the linear model is automatically included)

selmod
#>        Length Class               Mode   
#> model1 11     summary.lvm.mixture list   
#> AIC1    2     -none-              numeric
#> cv      4     -none-              numeric
#> knots   4     -none-              list   
#> model2 11     summary.lvmfit      list

Specification of general functional forms

Next, we show how to specify a general functional relation of multiple different latent or exogenous variables. This is achieved via the predict.fun argument. To illustrate this we include interactions between the latent variable \(\eta_{1}\) and a dichotomized version of the covariate \(z\)

d$g <- (d$z<0)*1 ## Group variable
mm1 <- regression(m1, ~g)  # Add grouping variable as exogenous variable (effect specified via 'predict.fun')
mm2 <- regression(m2, eta2~ u1+u2+u1:g+u2:g+z)
pred <- function(mu,var,data,...) {
    cbind("u1"=mu[,1],"u2"=mu[,1]^2+var[1],
      "u1:g"=mu[,1]*data[,"g"],"u2:g"=(mu[,1]^2+var[1])*data[,"g"])
}
ee1 <- twostage(mm1, model2=mm2, data=d, predict.fun=pred)
estimate(ee1,keep="eta2~u",regex=TRUE)
#>           Estimate Std.Err   2.5% 97.5% P-value
#> eta2~u1     0.9891  17.805 -33.91 35.89  0.9557
#> eta2~u2    -0.3962   9.971 -19.94 19.15  0.9683
#> eta2~u1:g   0.4487  13.679 -26.36 27.26  0.9738
#> eta2~u2:g   0.0441   7.178 -14.02 14.11  0.9951

A formal test show no statistically significant effect of this interaction

summary(estimate(ee1,keep="(:g)", regex=TRUE))
#> Call: estimate.default(x = ee1, keep = "(:g)", regex = TRUE)
#> __________________________________________________
#>           Estimate Std.Err   2.5% 97.5% P-value
#> eta2~u1:g   0.4487  13.679 -26.36 27.26  0.9738
#> eta2~u2:g   0.0441   7.178 -14.02 14.11  0.9951
#> 
#>  Null Hypothesis: 
#>   [eta2~u1:g] = 0
#>   [eta2~u2:g] = 0 
#>  
#> chisq = 0.0016, df = 2, p-value = 0.9992

Mixture models

Lastly, we demonstrate how the distributional assumptions of stage 1 model can be relaxed by letting the conditional distribution of the latent variable given covariates follow a Gaussian mixture distribution. The following code explictly defines the parameter constraints of the model by setting the intercept of the first indicator variable, \(x_{1}\), to zero and the factor loading parameter of the same variable to one.

m1 <- baptize(m1)  ## Label all parameters
intercept(m1, ~x1+eta1) <- list(0,NA) ## Set intercept of x1 to zero. Remove the label of eta1
regression(m1,x1~eta1) <- 1 ## Factor loading fixed to 1

The mixture model may then be estimated using the mixture method (note, this requires the mets package to be installed), where the Parameter names shared across the different mixture components given in the list will be constrained to be identical in the mixture model. Thus, only the intercept of \(\eta_{1}\) is allowed to vary between the mixtures.

set.seed(1)
em0 <- mixture(m1, k=2, data=d)

To decrease the risk of using a local maximizer of the likelihood we can rerun the estimation with different random starting values

em0 <- NULL
ll <- c()
for (i in 1:5) {
    set.seed(i)
    em <- mixture(m1, k=2, data=d, control=list(trace=0))
    ll <- c(ll,logLik(em))
    if (is.null(em0) || logLik(em0)<tail(ll,1))
    em0 <- em
}

summary(em0)
#> Cluster 1 (n=162, Prior=0.776):
#> --------------------------------------------------
#>                     Estimate Std. Error Z value  Pr(>|z|)
#> Measurements:                                            
#>    x1~eta1           1.00000                             
#>    x2~eta1           0.99581  0.07940   12.54099   <1e-12
#>    x3~eta1           1.06345  0.08436   12.60541   <1e-12
#> Regressions:                                             
#>    eta1~z            1.06675  0.08527   12.50989   <1e-12
#> Intercepts:                                              
#>    x1                0.00000                             
#>    x2                0.03845  0.09890    0.38883 0.6974  
#>    x3               -0.02549  0.10333   -0.24667 0.8052  
#>    eta1              0.20925  0.13162    1.58984 0.1119  
#> Residual Variances:                                      
#>    x1                0.98540  0.13316    7.40025         
#>    x2                0.97180  0.13156    7.38695         
#>    x3                1.01316  0.14294    7.08815         
#>    eta1              0.29046  0.11129    2.61004         
#> 
#> Cluster 2 (n=38, Prior=0.224):
#> --------------------------------------------------
#>                     Estimate Std. Error Z value  Pr(>|z|) 
#> Measurements:                                             
#>    x1~eta1           1.00000                              
#>    x2~eta1           0.99581  0.07940   12.54099   <1e-12 
#>    x3~eta1           1.06345  0.08436   12.60541   <1e-12 
#> Regressions:                                              
#>    eta1~z            1.06675  0.08527   12.50989   <1e-12 
#> Intercepts:                                               
#>    x1                0.00000                              
#>    x2                0.03845  0.09890    0.38883 0.6974   
#>    x3               -0.02549  0.10333   -0.24667 0.8052   
#>    eta1             -1.44290  0.25867   -5.57812 2.431e-08
#> Residual Variances:                                       
#>    x1                0.98540  0.13316    7.40025          
#>    x2                0.97180  0.13156    7.38695          
#>    x3                1.01316  0.14294    7.08815          
#>    eta1              0.29046  0.11129    2.61004          
#> --------------------------------------------------
#> AIC= 1958.803 
#> ||score||^2= 8.81839e-06

Measured by AIC there is a slight improvement in the model fit using the mixture model

e0 <- estimate(m1,data=d)
AIC(e0,em0)
#>     df      AIC
#> e0  10 1961.839
#> em0 12 1958.803

The spline model may then be estimated as before with the two-stage method

em2 <- twostage(em0,m2,data=d)
em2
#>                      Estimate Std. Error   Z-value   P-value
#> Measurements:                                               
#>    y2~eta2            0.97823    0.16298   6.00228 1.946e-09
#>    y3~eta2            1.04530    0.10922   9.57068    <1e-12
#> Regressions:                                                
#>    eta2~z             1.02884   20.08620   0.05122    0.9591
#>    eta2~eta1_1        2.80413   44.84857   0.06252    0.9501
#>    eta2~eta1_2       -0.02249    6.57553  -0.00342    0.9973
#>    eta2~eta1_3       -0.17333   19.50620  -0.00889    0.9929
#>    eta2~eta1_4        0.38672   22.12310   0.01748    0.9861
#> Intercepts:                                                 
#>    y2                -0.12171    0.11148  -1.09174    0.2749
#>    y3                -0.09870    0.10580  -0.93292    0.3509
#>    eta2               2.12372  112.74655   0.01884     0.985
#> Residual Variances:                                         
#>    y1                 1.31872    0.77923   1.69233          
#>    y2                 1.09690    1.23378   0.88906          
#>    y3                 0.81345    1.13831   0.71461          
#>    eta2               1.99590    9.18508   0.21730

In this example the results are very similar to the Gaussian model:

plot(I(eta2-z) ~ eta1, data=d, col=Col("black",0.5), pch=16,
     xlab=expression(eta[1]), ylab=expression(eta[2]))

lines(Estimate ~ eta1, data=as.data.frame(p), col="darkblue", lwd=5)
confband(p[,1], lower=p[,4], upper=p[,5], polygon=TRUE,
     border=NA, col=Col("darkblue",0.2))

pm <- cbind(eta1=newd$eta1,
        estimate(em2, f=function(p) predict(e2,p=p,newdata=newd))$coefmat)
lines(Estimate ~ eta1, data=as.data.frame(pm), col="darkred", lwd=5)
confband(pm[,1], lower=pm[,4], upper=pm[,5], polygon=TRUE,
     border=NA, col=Col("darkred",0.2))
legend("bottomright", c("Gaussian","Mixture"),
       col=c("darkblue","darkred"), lwd=2, bty="n")

Non-linear latent variable models and error-in-variable models

Klaus Kähler Holst

2023-01-04

Estimation

Cross-validation

Specification of general functional forms

Mixture models

Bibliography