多元相关的P值

Question

I have some basic questions concerning the polyserial() {polycor} function.

1. Does a p-value exist for rho, or can it be calculated?
2. For the assumption of a bivariate normal, is the tested null hypothesis "Yes, bivariate normal"? That is, do I want a high or low p-value.

Thanks.

Answer 1

If you form the returned object with:

 polS <- polyserial(x, y, ML=TRUE, std.err=TRUE) # ML estimate

... You should have no difficulty forming a p-value for the hypothesis: rho == 0 using a z-statistic formed by the ratio of a parameter divided by its standard error. But that is not the same as testing the assumption of bivariate normality. For that you need to examine "chisq" component of polS . The print method for objects of class 'polycor' hands that to you in a nice little sentence. You interpret that result in the usual manner: Low p-values are stronger evidence against the null hypothesis (in this case H0: bivariate normality). As a scientist, you do not "want" either result. You want to understand what the data is telling you.

Answer 2

I e-mailed the package author -because I had the same questions) and based on his clarifications, I offer my answers:

First, the easy question: higher p-values (traditionally > 0.05) give you more confidence that the distribution is bivariate normal. Lower p-values indicate a non-normal distribution, BUT, if the sample size is sufficiently large, the maximum likelihood estimate (option ML=TRUE ), non-normality doesn't matter; the correlation is still reliable anyway.

Now, for the harder question: to calculate the p-value, you need to:

1. Execute polyserial with the std.err=TRUE option to have access to more details.
2. From the resulting polyserial object, access the var[1, 1] element. var is the covariance matrix of the parameter estimates, and sqrt(var[1, 1]) is the standard error (which displays in parentheses in the output after the rho result).
3. From the standard error, you can calculate the p-value based on the R code below.

Here's some code to illustrate this with copiable R-code, based on the example code in the polyserial documentation:

library(mvtnorm)
library(polycor)

set.seed(12345)
data <- rmvnorm(1000, c(0, 0), matrix(c(1, .5, .5, 1), 2, 2))
x <- data[,1]
y <- data[,2]
y <- cut(y, c(-Inf, -1, .5, 1.5, Inf))

# 2-step estimate
poly_2step <- polyserial(x, y, std.err=TRUE)  
poly_2step
## 
## Polyserial Correlation, 2-step est. = 0.5085 (0.02413)
## Test of bivariate normality: Chisquare = 8.604, df = 11, p = 0.6584
std.err_2step <- sqrt(poly_2step$var[1, 1])
std.err_2step
## [1] 0.02413489
p_value_2step <- 2 * pnorm(-abs(poly_2step$rho / std.err_2step))
p_value_2step
## [1] 1.529176e-98
# ML estimate
poly_ML <- polyserial(x, y, ML=TRUE, std.err=TRUE) 
poly_ML
## 
## Polyserial Correlation, ML est. = 0.5083 (0.02466)
## Test of bivariate normality: Chisquare = 8.548, df = 11, p = 0.6635
## 
##                  1      2       3
## Threshold -0.98560 0.4812 1.50700
## Std.Err.   0.04408 0.0379 0.05847
std.err_ML <- sqrt(poly_ML$var[1, 1])
std.err_ML
## [1] 0.02465517
p_value_ML <- 2 * pnorm(-abs(poly_ML$rho / std.err_ML))
p_value_ML
##              
## 1.927146e-94

And to answer an important question that you didn't ask: you would want to always use the maximum likelihood version ( ML=TRUE ) because it is more accurate, except if you have a really slow computer, in which case the default 2-step approach is acceptable.