MML-WBCORR is an extension and web application implementation of WBCORR, a correlation pattern hypothesis test program developed for Mathematica by Steiger (2004). It can handle raw or correlation data, in one or more samples, with or without equal sample sizes, and with or without the assumption of multivariate normality. MML-WBCORR may be cited in APA style as follows:

Fouladi, R. T., & Serafini, P. E. (2018). MML-WBCORR. Retrieved from

Offline and bootstrap versions of MML-WBCORR are available at its github page.

Input consists of a hypothesis matrix and one or more raw or correlation data files. Input files should be in headerless, comma-separated (.csv) format. The "CSV Generator" featured in the header bar is a web application for creating properly-formatted correlation and hypothesis files.

Here are six lines from a possible raw data file:


Each row is a participant or observation, and each column is a variable. Note the absence of an ID column, which would be interpreted as a variable.

Here is a sample correlation data file:


Data from the upper half of the correlation matrix is optional. In SPSS, you can produce a plain correlation matrix such as this using the following syntax:

  /VARIABLES v1 v2 v3
  /ANALYSIS v1 v2 v3

Here is a sample hypothesis file:


And here is what that hypothesis would look like in the CSV Generator:

Each row of the hypothesis matrix is of the form: group, row, column, parameter tag, fixed value. The first three columns specify a correlation, conventionally from the lower half of the correlation matrix, and the fourth and fifth make an assertion about the value of that correlation. Correlations with the same positive integer in the parameter tag column are hypothesised to be equal, and correlations with a parameter tag of 0 are hypothesised to be equal to the value of their fixed value column. Correlations with different parameter tags are not hypothesised to be unequal. If a parameter tag is assigned to a single correlation, the output will include an estimate of that correlation but the p value for the test will be the same as if it hadn't been included.

Take the first row of the sample hypothesis matrix. The first three columns specify the correlation in correlation matrix 1 at row 2 of column 1 (ρ21 or ρ121). Put differently, they pick out the correlation in group 1 between variable 2 and variable 1. This correlation has a parameter tag of 1, so it is hypothesised to be equal to the correlation picked out by the second row, which also has a parameter tag of 1. On the other hand, since its parameter tag is 0, the correlation in the third row is hypothesised to be equal to 0.2. Finally, the correlation in the fourth row is the only correlation with a parameter tag of 2, so it does not figure into the hypothesis at all. Effectively, the hypothesis says: (ρ21 = ρ31) & (ρ32 = 0.2).

The parameter tag column has a "hole" in it if the value of the greatest parameter tag is not equal to the number of unique non-zero parameter tags. For example, if the parameter tag column were [1, 1, 2, 2, 4, 4], then it would have a hole because 3 is skipped. If the column has a hole then it is renumbered, in thise case to [1, 1, 2, 2, 3, 3]. This does not affect the result of the test, and the amended hypothesis matrix is included in the output.

The estimation methods offered by MML-WBCORR are GLS (generalised least squares), ADF (asymptotically distribution-free), and a "two-step" version of each (TSGLS and TSADF). The practical difference between GLS and ADF is that whereas GLS assumes multivariate normality, ADF does not. However, ADF relies on sample estimates of fourth-order moments, so it requires raw data and is incompatible with pairwise deletion. Also, since these estimates have large standard errors for small to moderate sample sizes, employing ADF may result in a considerable loss of power: it should only be used if the assumption of multivariate normality is untenable. A test of multivariate normality is provided when you use raw data. The relation between TSGLS and TSADF is the same, but that they provide superior estimates compared to their one-step counterparts.

In brief: use TSGLS if multivariate normality is a tenable assumption, and use TSADF otherwise.

Listwise and pairwise deletion are offered to deal with missing data. In pairwise deletion, the sample size is the harmonic mean of the number of observed scores for each variable, rounded to one decimal place. Empty or NA values (and only empty or NA values) are interpreted as missing.

This page will interpret the output for an example data set. The first table in the output is the hypothesis matrix:

Input Hypothesis Matrix
Group Row Column Parameter Tag Fixed Value
1 2 1 1 0
1 3 1 1 0
1 3 2 1 0

The hypothesis asserts that ρ21 = ρ31 = ρ32. Next we see the correlation matrix for each group. In this case there is only one group, so there is only one correlation matrix:

Input Correlation Matrix #1 (N=25)
X1 X2 X3 X4 X5 X6
X1 1 0.109 0.007 -0.074 0.196 -0.116
X2 0.109 1 0.077 -0.067 0.107 -0.239
X3 0.007 0.077 1 0.129 0.15 -0.069
X4 -0.074 -0.067 0.129 1 -0.152 -0.193
X5 0.196 0.107 0.15 -0.152 1 -0.323
X6 -0.116 -0.239 -0.069 -0.193 -0.323 1
And the matrix of OLS estimates for each group:
OLS Estimates Matrix #1 (N=25)
X1 X2 X3 X4 X5 X6
X1 1 0.064 0.064 -0.074 0.196 -0.116
X2 0.064 1 0.064 -0.067 0.107 -0.239
X3 0.064 0.064 1 0.129 0.15 -0.069
X4 -0.074 -0.067 0.129 1 -0.152 -0.193
X5 0.196 0.107 0.15 -0.152 1 -0.323
X6 -0.116 -0.239 -0.069 -0.193 -0.323 1
In this case, since ρ21 = 0.109, ρ31 = 0.007, and ρ32 = 0.077 are equal under the null, the OLS estimate of ρ21 = ρ31 = ρ32 is (0.109 + 0.007 + 0.077)/3 = 0.064. Next we have the parameter estimates:
TSGLS Parameter Estimates
Parameter Tag Estimate Standard Error 95% Confidence Interval
1 0.064 0.124 [-0.339, 0.448]
The TSGLS estimate is ρ21 = ρ31 = ρ32 = 0.064. Note that this is a null estimate! This table also gives confidence intervals on point estimates, using a strict Bonferroni on an alpha of .05 via the Fisher transform method. Next we have the significance of the test as a whole:
Significance Test Results
Chi Square df plevel
0.147 2 0.929

Next, since we used raw data, we have the test of multivariate normality. The specific test will depend on 1) whether there is missing data and 2) the deletion method. Yuan, Lambert & Fouladi's (2004) test is given if pairwise deletion was employed, and Mardia's (1970) is provided otherwise. Yuan, Lambert & Fouladi's (2004) test is only useable if the observed marginals for the incomplete variables do not sit in a restricted range, so a test of that assumption is provided for each variable with missing data. This is only possible if there is at least one variable without missing data.

Since the example data is complete, Mardia's (1970) test is used:

Assessment of Multivariate Skewness
Group Multivariate Skewness Chi Square df plevel
1 27.396 114.151 56 < 0.001
Assessment of Multivariate Kurtosis
Group Multivariate Kurtosis Z statistic plevel (2-tailed)
1 55.168 2.771 0.006

Since the p value is low for both the skewness test and the kurtosis test, multivariate normality is not supported and we should rerun the test using TSADF.

Suppose we remove one value from the data set and choose pairwise deletion. Now Yuan, Lambert, and Fouladi's (2004) test is displayed. This begins with an assessment of the distribution of the observed marginals:

Assessment of the Distribution of the Observed Marginals
Group Variable Missing values ZR plevel (two-tail)
1 1 1 -0.416 0.677

The p value is high, so the observed marginals do not appear to be restricted and we can move on to the assessment of multivariate normality:

Assessment of Multivariate Normality
Group M2 plevel (two-tail)
1 -2.468 0.014

Again, the p value is low, so multivariate normality is not supported and we should re-run the test using TSADF.