Bartlett-Sphericity {REdaS}

Bartlett's Test of Sphericity

Description

Implements Barlett's Test of Sphericity which tests whether a matrix is significantly different from an identity matrix.

Usage

bart_spher(x, use = c("everything", "all.obs", "complete.obs",
                      "na.or.complete", "pairwise.complete.obs"))

## S3 method for class 'bart_spher'
print(x, ...)

Arguments

x

a data matrix or the object to be printed.

use

defines the method to use if missing values are present (see Examples and cor).

...

further arguments for the print method.

Details

The test statistic X^2 as defined in Eq. (3) in Bartlett (1951) is X^2=-[(n-1)-(2k+5)/6]\cdot\log(\left|\mathbf{R}\right|) where n is the number of observations, k the number of variables, and \mathbf{R} the correlation matrix of the data supplied in x. \left|\mathbf{R}\right| is the determinant of \mathbf{R}.

Bartlett's X^2 is asymptotically \chi^2-distributed with \mathit{df}=k(k-1)/2 under the null hypothesis.

Note that, because the bias-corrected correlation matrix is used, (n-1) is employed instead of n, as in the paper.

Treatment of Missing Values

If no missing values are present in the data matrix x, use will work with any setting and no adjustments are necessary. In this case, n is the number of rows in x.

For listwise deletion (use = "complete.obs" or "na.or.complete"), n is the number of remaining rows in x.

When use = "pairwise.complete.obs", n is approximated as the sum of relative non-missing responses for all observations with 2 or more valid responses.

If listwise/pairwise methods are used to compute the correlation matrix and the test statistic, a warning will be issued when printing the object.

Value

A list object of class 'bart_spher'

call

the issued function call

x

the original data

cormat

the correlation matrix computed from the data

use

treatment of NAs

n

the number of used observations

k

the number of variables/items

X2

the computed X^2 value

df

degrees of freedom

p.value

the p-value

warn

logical value indicating whether a warning regarding missing values will be issued (see Details)

Author(s)

Marco J. Maier

References

Bartlett, M. S. (1951). The Effect of Standardization on a \chi^2 Approximation in Factor Analysis. Biometrika 38(3/4), 337–344.

See Also

cor() and KMOS()

Examples

# generate a data frame with 3 variables and 100 observations
set.seed(5L)
datamatrix <- data.frame("A" = rnorm(100), "B" = rnorm(100), "C" = rnorm(100))
head(datamatrix)

# correlation matrix
cor(datamatrix)


# bartlett's test
bart_spher(datamatrix)


# effects of missing observations on correlations: to illustrate this, the first
# observation on variable A is set to NA
datamatrix[1, 1] <- NA
head(datamatrix)

# "everything" (the default) causes all correlations involving a variable with
# missing values to be NA (in this case, all pairwise correlations with the
# variable "A")
cor(datamatrix)

# "all.obs" generates an error if missing values are present.
## Not run: 
cor(datamatrix, use = "all.obs")
## End(Not run)

# "complete.obs" and "na.or.complete" delete complete observations if there are
# NA (in this case, the first case would be deleted). If there are no complete
# cases left after the listwise deletion, "complete.obs" results in an error
# while "na.or.complete" returns a matrix with all elements being NA.
cor(datamatrix, use = "complete.obs")
cor(datamatrix, use = "na.or.complete")

# "pairwise.complete.obs" uses all non-missing pairwise values. If there are no
# non-missing value pairs in two variables, the results will be NA.
# It is possible that correlation matrices are not positive semi-definite.
cor(datamatrix, use = "pairwise.complete.obs")


# with the missing value in the first cell, the test does not work anymore:
## Not run: 
bart_spher(datamatrix)
## End(Not run)

# deleting the whole first observation (listwise) gives
bart_spher(datamatrix, use = "na.or.complete")

# using pairwise-correlation, the result is
bart_spher(datamatrix, use = "pairwise.complete.obs")