Add covariance structure to Latent Variable Model

Define covariances between residual terms in a lvm-object.

# S3 method for lvm
covariance(object, var1=NULL, var2=NULL, constrain=FALSE, pairwise=FALSE,...) <- value

Arguments

object

lvm-object

...

Additional arguments to be passed to the low level functions

var1

Vector of variables names (or formula)

var2

Vector of variables names (or formula) defining pairwise covariance between var1 and var2)

constrain

Define non-linear parameter constraints to ensure positive definite structure

pairwise

If TRUE and var2 is omitted then pairwise correlation is added between all variables in var1

value

List of parameter values or (if var1 is unspecified)

Value

A lvm-object

Details

The covariance function is used to specify correlation structure between residual terms of a latent variable model, using a formula syntax.

For instance, a multivariate model with three response variables,

$$Y_1 = \mu_1 + \epsilon_1$$

$$Y_2 = \mu_2 + \epsilon_2$$

$$Y_3 = \mu_3 + \epsilon_3$$

can be specified as

m <- lvm(~y1+y2+y3)

Pr. default the two variables are assumed to be independent. To add a covariance parameter \(r = cov(\epsilon_1,\epsilon_2)\), we execute the following code

covariance(m) <- y1 ~ f(y2,r)

The special function f and its second argument could be omitted thus assigning an unique parameter the covariance between y1 and y2.

Similarily the marginal variance of the two response variables can be fixed to be identical (\(var(Y_i)=v\)) via

covariance(m) <- c(y1,y2,y3) ~ f(v)

To specify a completely unstructured covariance structure, we can call

covariance(m) <- ~y1+y2+y3

All the parameter values of the linear constraints can be given as the right handside expression of the assigment function covariance<- if the first (and possibly second) argument is defined as well. E.g:

covariance(m,y1~y1+y2) <- list("a1","b1")

covariance(m,~y2+y3) <- list("a2",2)

Defines

$$var(\epsilon_1) = a1$$

$$var(\epsilon_2) = a2$$

$$var(\epsilon_3) = 2$$

$$cov(\epsilon_1,\epsilon_2) = b1$$

Parameter constraints can be cleared by fixing the relevant parameters to NA (see also the regression method).

The function covariance (called without additional arguments) can be used to inspect the covariance constraints of a lvm-object.

See also

regression<-, intercept<-, constrain<- parameter<-, latent<-, cancel<-, kill<-

Author

Klaus K. Holst

Examples

m <- lvm()
### Define covariance between residuals terms of y1 and y2
covariance(m) <- y1~y2
covariance(m) <- c(y1,y2)~f(v) ## Same marginal variance
covariance(m) ## Examine covariance structure
#> Covariance parameters:
#>       y1 y2
#>    y1 v  * 
#>    y2 *  v