Polychoric Correlation

Estimates the polychoric correlation between two ordinal variables based on a contingency table. Both a two-step estimator and full maximum likelihood (ml) estimation are supported.

Usage

corPolychor(
  x,
  y = NULL,
  method = c("two-step", "ml"),
  se = FALSE,
  control = list(),
  maxcor = 0.9999,
  ...
)

Arguments

x

a contingency table or an ordinal vector

y

optional second ordinal vector. If supplied, a contingency table is constructed via table(x, y, \dots).

method

character string specifying the estimation method:

"two-step"

two-step estimator (default, fast)

"ml"

full maximum likelihood estimation

se

logical; if TRUE, standard errors are computed via the Hessian matrix. This requires ml estimation, so it is an error to combine it with method = "two-step".

control

a list of control parameters passed to stats::optim()

maxcor

numeric; maximum absolute correlation allowed (default 0.9999) to avoid numerical issues near the boundary

...

further arguments passed to table() when y is supplied, for example useNA

Value

if se = FALSE, a numeric value giving the estimated correlation.

If se = TRUE, a list with components:

type

type of correlation

rho

estimated polychoric correlation

rowCuts

estimated row thresholds

colCuts

estimated column thresholds

var

variance-covariance matrix of the estimates, on the scale of rho and the thresholds. The optimiser works on atanh(rho), so the leading row and column are transformed back with the delta method, factor \((1 - \rho^2)\).

n

total sample size

chisq

likelihood-ratio test statistic

df

degrees of freedom

method

estimation method actually used

The returned object has class "Polychor".

Details

The polychoric correlation estimates the correlation between two latent normally distributed variables underlying observed ordinal variables.

The likelihood is based on a discretized bivariate normal distribution, evaluated via mvtnorm::pmvnorm().

For numerical stability:

• The correlation parameter is internally transformed using tanh() to enforce \(|\rho| < 1\). The search range on that scale is derived from maxcor, so the estimate is free to approach the documented boundary.

• Cell probabilities are bounded away from zero to avoid log(0).

Empty rows or columns in the contingency table are removed with a warning.

References

Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44(4), 443–460.

Fox, J. (2016). Applied Regression Analysis and Generalized Linear Models.

See also

mvtnorm::pmvnorm(), stats::optim()

Other assoc.continuous: corPart(), findCorrX(), hoeffdingD(), keepSig(), pearsonCor(), spearmanCor()

Examples

# Example with ordinal variables
set.seed(1)
z <- rnorm(200)
x <- factor(cut(z + rnorm(200, sd = 0.6), 3), ordered = TRUE)
y <- factor(cut(z + rnorm(200, sd = 0.6), 3), ordered = TRUE)

# Two-step estimate
corPolychor(x, y)
#> [1] 0.6300061

# ml estimate
corPolychor(x, y, method = "ml")
#> [1] 0.6290183

# With standard errors
res <- corPolychor(x, y, method = "ml", se = TRUE)
res$rho
#> [1] 0.6290183