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 http://members.psyc.sfu.ca/labs/mml
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:
1.24,2.61,0.260,1.05,7.55,0.2 0.280,2.36,1.19,1.66,0.470,0.460 0.370,0.820,3.23,0.120,4.57,1.84 0.110,5.73,3.04,1.04,0.380,0.350 0.750,2.66,2.62,0.290,4.12,0.380
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:
1.0 0.2,1.0 0.3,0.4,1.0
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:
FACTOR /VARIABLES v1 v2 v3 /MISSING LISTWISE /ANALYSIS v1 v2 v3 /PRINT CORRELATION /ROTATION NOROTATE /METHOD=CORRELATION
Here is a sample hypothesis file:
1,2,1,1,0 1,3,1,1,0 1,3,2,0,0.2 1,4,1,2,0
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 (ρ_{21} or ρ_{121}). 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: (ρ_{21} = ρ_{31}) & (ρ_{32} = 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 ρ_{21} = ρ_{31} = ρ_{32}. 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) | ||||||
X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | |
---|---|---|---|---|---|---|
X_{1} | 1 | 0.109 | 0.007 | -0.074 | 0.196 | -0.116 |
X_{2} | 0.109 | 1 | 0.077 | -0.067 | 0.107 | -0.239 |
X_{3} | 0.007 | 0.077 | 1 | 0.129 | 0.15 | -0.069 |
X_{4} | -0.074 | -0.067 | 0.129 | 1 | -0.152 | -0.193 |
X_{5} | 0.196 | 0.107 | 0.15 | -0.152 | 1 | -0.323 |
X_{6} | -0.116 | -0.239 | -0.069 | -0.193 | -0.323 | 1 |
OLS Estimates Matrix #1 (N=25) | ||||||
X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | |
---|---|---|---|---|---|---|
X_{1} | 1 | 0.064 | 0.064 | -0.074 | 0.196 | -0.116 |
X_{2} | 0.064 | 1 | 0.064 | -0.067 | 0.107 | -0.239 |
X_{3} | 0.064 | 0.064 | 1 | 0.129 | 0.15 | -0.069 |
X_{4} | -0.074 | -0.067 | 0.129 | 1 | -0.152 | -0.193 |
X_{5} | 0.196 | 0.107 | 0.15 | -0.152 | 1 | -0.323 |
X_{6} | -0.116 | -0.239 | -0.069 | -0.193 | -0.323 | 1 |
TSGLS Parameter Estimates | |||
Parameter Tag | Estimate | Standard Error | 95% Confidence Interval |
---|---|---|---|
1 | 0.064 | 0.124 | [-0.339, 0.448] |
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 | Z_{R} | 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 | M_{2} | 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.