# kd: Generate data via the Kaiser-Dickman (1962) algorithm. In semTools: Useful Tools for Structural Equation Modeling

 kd R Documentation

## Generate data via the Kaiser-Dickman (1962) algorithm.

### Description

Given a covariance matrix and sample size, generate raw data that correspond to the covariance matrix. Data can be generated to match the covariance matrix exactly, or to be a sample from the population covariance matrix.

### Usage

```kd(covmat, n, type = c("exact", "sample"))
```

### Arguments

 `covmat` a symmetric, positive definite covariance matrix `n` the sample size for the data that will be generated `type` type of data generation. `exact` generates data that exactly correspond to `covmat`. `sample` treats `covmat` as a poulation covariance matrix, generating a sample of size `n`.

### Details

By default, R's `cov()` function divides by `n`-1. The data generated by this algorithm result in a covariance matrix that matches `covmat`, but you must divide by `n` instead of `n`-1.

### Value

`kd` returns a data matrix of dimension `n` by `nrow(covmat)`.

### Author(s)

Ed Merkle (University of Missouri; merklee@missouri.edu)

### References

Kaiser, H. F. and Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix. Psychometrika, 27(2), 179–182. doi: 10.1007/BF02289635

### Examples

```
#### First Example

## Get data
dat <- HolzingerSwineford1939[ , 7:15]
hs.n <- nrow(dat)

## Covariance matrix divided by n
hscov <- ((hs.n-1)/hs.n) * cov(dat)

## Generate new, raw data corresponding to hscov
newdat <- kd(hscov, hs.n)

## Difference between new covariance matrix and hscov is minimal
newcov <- (hs.n-1)/hs.n * cov(newdat)
summary(as.numeric(hscov - newcov))

## Generate sample data, treating hscov as population matrix
newdat2 <- kd(hscov, hs.n, type = "sample")

#### Another example

## Define a covariance matrix
covmat <- matrix(0, 3, 3)
diag(covmat) <- 1.5
covmat[2:3,1] <- c(1.3, 1.7)
covmat[3,2] <- 2.1
covmat <- covmat + t(covmat)

## Generate data of size 300 that have this covariance matrix
rawdat <- kd(covmat, 300)

## Covariances are exact if we compute sample covariance matrix by
## dividing by n (vs by n - 1)
summary(as.numeric((299/300)*cov(rawdat) - covmat))

## Generate data of size 300 where covmat is the population covariance matrix
rawdat2 <- kd(covmat, 300)

```

semTools documentation built on May 10, 2022, 9:05 a.m.