Heterogeneous Correlation Matrix

Description

hetcor computes a heterogenous correlation matrix, consisting of Pearson product-moment correlations between numeric variables, polyserial correlations between numeric and ordinal variables, and polychoric correlations between ordinal variables.

The detectCores function is imported from the parallel package and re-exported.

Usage

hetcor(data, ..., ML = FALSE, std.err = TRUE, 
  use=c("complete.obs", "pairwise.complete.obs"), 
  bins=4, pd=TRUE, parallel=FALSE, ncores=detectCores(logical=FALSE),
  thresholds=FALSE)
## S3 method for class 'data.frame'
hetcor(data, ML = FALSE, std.err = TRUE, 
  use = c("complete.obs", "pairwise.complete.obs"), 
  bins=4, pd=TRUE, parallel=FALSE, ncores=detectCores(logical=FALSE), 
  thresholds=FALSE, ...)
## Default S3 method:
hetcor(data, ..., ML = FALSE, std.err = TRUE, 
  use=c("complete.obs", "pairwise.complete.obs"), 
  bins=4, pd=TRUE, parallel=FALSE, ncores=detectCores(logical=FALSE),
  thresholds=FALSE)
## S3 method for class 'hetcor'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'hetcor'
as.matrix(x, ...)
detectCores(all.tests=FALSE, logical=TRUE) 

Arguments

Value

hetcor returns an object of class "hetcor" with the following components:

Warning

Be careful with character variables (as opposed to factors), the values of which are ordered alphabetically. Thus, e.g., the values "disagree", "neutral", "agree" are ordered "agree", "disagree", "neutral".

Note

Although the function reports standard errors for product-moment correlations, transformations (the most well known is Fisher's z-transformation) are available that make the approach to asymptotic normality much more rapid.

Author(s)

John Fox jfox@mcmaster.ca

References

Drasgow, F. (1986) Polychoric and polyserial correlations. Pp. 68-74 in S. Kotz and N. Johnson, eds., The Encyclopedia of Statistics, Volume 7. Wiley.

Olsson, U. (1979) Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika 44, 443-460.

Rodriguez, R.N. (1982) Correlation. Pp. 193-204 in S. Kotz and N. Johnson, eds., The Encyclopedia of Statistics, Volume 2. Wiley.

Ghosh, B.K. (1966) Asymptotic expansion for the moments of the distribution of correlation coefficient. Biometrika 53, 258-262.

Olkin, I., and Pratt, J.W. (1958) Unbiased estimation of certain correlation coefficients. Annals of Mathematical Statistics 29, 201-211.

See Also

polychor, polyserial, nearPD, detectCores

Examples

if(require(mvtnorm)){
    set.seed(12345)
    R <- matrix(0, 4, 4)
    R[upper.tri(R)] <- runif(6)
    diag(R) <- 1
    R <- cov2cor(t(R) %*% R)
    round(R, 4)  # population correlations
    data <- rmvnorm(1000, rep(0, 4), R)
    round(cor(data), 4)   # sample correlations
    }
if(require(mvtnorm)){
    x1 <- data[,1]
    x2 <- data[,2]
    y1 <- cut(data[,3], c(-Inf, .75, Inf))
    y2 <- cut(data[,4], c(-Inf, -1, .5, 1.5, Inf))
    data <- data.frame(x1, x2, y1, y2)
    hetcor(data)  # Pearson, polychoric, and polyserial correlations, 2-step est.
    }
if(require(mvtnorm)){
    hetcor(x1, x2, y1, y2, ML=TRUE) # Pearson, polychoric, polyserial correlations, ML est.
    }

## Not run: 
    hc <- hetcor(data, ML=TRUE)
      # parallel computation:
    hc.m <- hetcor(data, ML=TRUE, parallel=TRUE,
                   ncores=min(2, detectCores()))
    hc.m
    all.equal(hc, hc.m)
    
      # error handling:
    data$y1[data$y2 == "(0.5,1.5]"] <- NA
    hetcor(data)
    
## End(Not run)

Polychoric Correlation

Description

Computes the polychoric correlation (and its standard error) between two ordinal variables or from their contingency table, under the assumption that the ordinal variables dissect continuous latent variables that are bivariate normal. Either the maximum-likelihood estimator or a (possibly much) quicker “two-step” approximation is available. For the ML estimator, the estimates of the thresholds and the covariance matrix of the estimates are also available.

Usage

polychor(x, y, ML = FALSE, control = list(), 
  std.err = FALSE, maxcor=.9999, start, thresholds=FALSE)

Arguments

Details

The ML estimator is computed by maximizing the bivariate-normal likelihood with respect to the thresholds for the two variables (\tau^{x}_i, i = 1,\ldots, r - 1; \tau^{y}_j, j = 1,\ldots, c - 1) and the population correlation (\rho). Here, r and c are respectively the number of levels of x and y. The likelihood is maximized numerically using the optim function, and the covariance matrix of the estimated parameters is based on the numerical Hessian computed by optim.

The two-step estimator is computed by first estimating the thresholds (\tau^{x}_i, i = 1,\ldots, r - 1 and \tau^{y}_j, i = j,\ldots, c - 1) separately from the marginal distribution of each variable. Then the one-dimensional likelihood for \rho is maximized numerically, using optim if standard errors are requested, or optimise if they are not. The standard error computed treats the thresholds as fixed.

Value

If std.err or thresholds is TRUE, returns an object of class "polycor" with the following components:

Othewise, returns the polychoric correlation.

Author(s)

John Fox jfox@mcmaster.ca

References

Drasgow, F. (1986) Polychoric and polyserial correlations. Pp. 68–74 in S. Kotz and N. Johnson, eds., The Encyclopedia of Statistics, Volume 7. Wiley.

Olsson, U. (1979) Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika 44, 443-460.

See Also

hetcor, polyserial, print.polycor, optim

Examples

if(require(mvtnorm)){
    set.seed(12345)
    data <- rmvnorm(1000, c(0, 0), matrix(c(1, .5, .5, 1), 2, 2))
    x <- data[,1]
    y <- data[,2]
    cor(x, y)  # sample correlation
    }
if(require(mvtnorm)){
    x <- cut(x, c(-Inf, .75, Inf))
    y <- cut(y, c(-Inf, -1, .5, 1.5, Inf))
    polychor(x, y)  # 2-step estimate
    }
if(require(mvtnorm)){
    polychor(x, y, ML=TRUE, std.err=TRUE)  # ML estimate
    }

Polyserial Correlation

Description

Computes the polyserial correlation (and its standard error) between a quantitative variable and an ordinal variable, based on the assumption that the joint distribution of the quantitative variable and a latent continuous variable underlying the ordinal variable is bivariate normal. Either the maximum-likelihood estimator or a quicker “two-step” approximation is available. For the ML estimator the estimates of the thresholds and the covariance matrix of the estimates are also available.

Usage

polyserial(x, y, ML = FALSE, control = list(), 
  std.err = FALSE, maxcor=.9999, bins=4, start, thresholds=FALSE)

Arguments

Details

The ML estimator is computed by maximizing the bivariate-normal likelihood with respect to the thresholds for y (\tau^{y}_j, i = 1,\ldots, c - 1) and the population correlation (\rho). The likelihood is maximized numerically using the optim function, and the covariance matrix of the estimated parameters is based on the numerical Hessian computed by optim.

The two-step estimator is computed by first estimating the thresholds (\tau^{y}_j, i = 1,\ldots, c - 1) from the marginal distribution of y. Then if the standard error of \hat{\rho} is requested, the one-dimensional likelihood for \rho is maximized numerically, using optim if standard errors are requested; the standard error computed treats the thresholds as fixed. If the standard error isn't request, \hat{\rho} is computed directly.

Value

If std.err or thresholds is TRUE, returns an object of class "polycor" with the following components:

Othewise, returns the polyserial correlation.

Author(s)

John Fox jfox@mcmaster.ca

References

Drasgow, F. (1986) Polychoric and polyserial correlations. Pp. 68–74 in S. Kotz and N. Johnson, eds., The Encyclopedia of Statistics, Volume 7. Wiley.

See Also

hetcor, polychor, print.polycor, optim

Examples

if(require(mvtnorm)){
    set.seed(12345)
    data <- rmvnorm(1000, c(0, 0), matrix(c(1, .5, .5, 1), 2, 2))
    x <- data[,1]
    y <- data[,2]
    cor(x, y)  # sample correlation
    }
if(require(mvtnorm)){
    y <- cut(y, c(-Inf, -1, .5, 1.5, Inf))
    polyserial(x, y)  # 2-step estimate
    }
if(require(mvtnorm)){
    polyserial(x, y, ML=TRUE, std.err=TRUE) # ML estimate
    }

Standard Methods for polycor Objects

Description

Some standard methods for objects of class polycor, produced by polychor and polyserial, including print, summary, coef, and vcov. The summary method simply invokes the print method.

Usage

## S3 method for class 'polycor'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'polycor'
summary(object, ...)
## S3 method for class 'polycor'
coef(object, correlation=TRUE, thresholds=TRUE, ...)
## S3 method for class 'polycor'
vcov(object, correlation=TRUE, thresholds=TRUE, ...)

Arguments

Author(s)

John Fox jfox@mcmaster.ca

See Also

polychor, polyserial

Examples

if(require(mvtnorm)){
    set.seed(12345)
    data <- rmvnorm(1000, c(0, 0), matrix(c(1, .5, .5, 1), 2, 2))
    x <- data[,1]
    y <- data[,2]
    cor(x, y) # sample correlation
    }
if(require(mvtnorm)){
    x <- cut(x, c(-Inf, .75, Inf))
    y <- cut(y, c(-Inf, -1, .5, 1.5, Inf))
    print(polychor(x, y, ML=TRUE, std.err=TRUE), 
      digits=3)  # polychoric correlation, ML estimate
    }
if(require(mvtnorm)){
    coef(polychor(x, y, ML=TRUE, std.err=TRUE))
    }
if(require(mvtnorm)){
    vcov(polychor(x, y, ML=TRUE, std.err=TRUE))
    }