regression-set: Add regression association to latent variable model

In lava: Latent Variable Models

Description

Define regression association between variables in a lvm-object and define linear constraints between model equations.

Usage

## S3 method for class 'lvm'
regression(object = lvm(), to, from, fn = NA,
silent = lava.options()$silent, additive=TRUE, ...)
## S3 replacement method for class 'lvm'
regression(object, to=NULL, quick=FALSE, ...) <- value

Arguments

object	lvm-object.
value	A formula specifying the linear constraints or if to=NULL a list of parameter values.
to	Character vector of outcome(s) or formula object.
from	Character vector of predictor(s).
fn	Real function defining the functional form of predictors (for simulation only).
silent	Logical variable which indicates whether messages are turned on/off.
additive	If FALSE and predictor is categorical a non-additive effect is assumed
quick	Faster implementation without parameter constraints
...	Additional arguments to be passed to the low level functions

Details

The regression function is used to specify linear associations between variables of a latent variable model, and offers formula syntax resembling the model specification of e.g. lm.

For instance, to add the following linear regression model, to the lvm-object, m:

E(Y|X_1,X_2) = β_1 X_1 + β_2 X_2

We can write

regression(m) <- y ~ x1 + x2

Multivariate models can be specified by successive calls with regression, but multivariate formulas are also supported, e.g.

regression(m) <- c(y1,y2) ~ x1 + x2

defines

E(Y_i|X_1,X_2) = β_{1i} X_1 + β_{2i} X_2

The special function, f, can be used in the model specification to specify linear constraints. E.g. to fix β_1=β_2 , we could write

regression(m) <- y ~ f(x1,beta) + f(x2,beta)

The second argument of f can also be a number (e.g. defining an offset) or be set to NA in order to clear any previously defined linear constraints.

Alternatively, a more straight forward notation can be used:

regression(m) <- y ~ beta*x1 + beta*x2

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

regression(m,y1~x1+x2) <- list("a1","b1")

defines E(Y_1|X_1,X_2) = a1 X_1 + b1 X_2. The rhs argument can be a mixture of character and numeric values (and NA's).

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

For backward compatibility the "$"-symbol can be used to fix parameters at a given value. E.g. to add a linear relationship between y and x with slope 2 to the model m, we can write regression(m,"y") <- "x$2". Similarily we can use the "@"-symbol to name parameters. E.g. in a multiple regression we can force the parameters to be equal: regression(m,"y") <- c("x1@b","x2@b"). Fixed parameters can be reset by fixing (with \$) them to NA.

Value

A lvm-object

Note

Variables will be added to the model if not already present.

Author(s)

Klaus K. Holst

See Also

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

Examples

m <- lvm() ## Initialize empty lvm-object
### E(y1|z,v) = beta1*z + beta2*v
regression(m) <- y1 ~ z + v
### E(y2|x,z,v) = beta*x + beta*z + 2*v + beta3*u
regression(m) <- y2 ~ f(x,beta) + f(z,beta)  + f(v,2) + u
### Clear restriction on association between y and
### fix slope coefficient of u to beta
regression(m, y2 ~ v+u) <- list(NA,"beta")

regression(m) ## Examine current linear parameter constraints

## ## A multivariate model, E(yi|x1,x2) = beta[1i]*x1 + beta[2i]*x2:
m2 <- lvm(c(y1,y2) ~ x1+x2)

Example output

Regression parameters:
      y1 z    v y2 x    u   
   y1    *    *             
   y2    beta *    beta beta

lava documentation built on May 2, 2019, 4:49 p.m.