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Friday, January 22, 2010

Why are pattern cross-loadings so low in WarpPLS?

I have recently received a few related questions from WarpPLS users. Essentially, they noted that the pattern loadings generated by WarpPLS were very similar to those generated by other PLS-based SEM software. However, they wanted to know why the pattern cross-loadings were so much lower in WarpPLS, compared to other PLS-based SEM software.

Low cross-loadings suggest good discriminant validity; a type of validity that is usually tested via WarpPLS using a separate procedure, involving tabulation of latent variable correlations and average variances extracted.

Nevertheless, low cross-loadings, combined with high loadings, are a "good thing" (generally speaking) in the context of a PLS-based SEM analysis.

The pattern loadings and cross-loadings provided by WarpPLS are from a pattern matrix, which is obtained after the transformation of a structure matrix through an oblique rotation (similar to Promax).

The structure matrix contains the Pearson correlations between indicators and latent variables, which are not particularly meaningful prior to rotation in the context of measurement instrument validation (e.g., validity and reliability assessment).

In an oblique rotation the loadings shown on the pattern matrix are very similar to those on the structure matrix. The latter are the ones that other PLS-based SEM software usually report, which is why the loadings obtained through WarpPLS and other PLS-based SEM software are very similar. The cross-loadings though, can be very different in the pattern (rotated) matrix, as these WarpPLS users noted.

In short, the reason for the comparatively low cross-loadings is the oblique rotation employed by WarpPLS.

Here is a bit more information regarding rotation methods:

Because an oblique rotation is employed by WarpPLS, in some (relatively rare) cases pattern loadings may be higher than 1, which should have no effect on their interpretation. The expectation is that pattern loadings, which are shown within parentheses (on the "View indicator loadings and cross-loadings" option), will be high; and cross-loadings will be low.

The combined loadings and cross-loadings table always shows loadings lower than 1, because that table combines structure loadings with pattern cross-loadings. This obviates the need for a normalization step, which can distort loadings and cross-loadings somewhat.

Also, let me add that the main difference between oblique and orthogonal rotation methods (e.g., Varimax) is that the former assume that there are correlations, some of which may be strong, among latent variables.

Arguably oblique rotation methods are the most appropriate in PLS-based SEM analysis, because by definition latent variables are expected to be correlated. Otherwise, no path coefficient would be significant.

Technically speaking, it is possible that a research study will hypothesize only neutral relationships between latent variables, which could call for an orthogonal rotation. However, this is rarely, if ever, the case.

See the most recent version of the WarpPLS User Manual (linked below) for more details.


Unknown said...

Hi Ned,

I am very glad that I have found your software. It is a piece of excellent work.

I have tried to re-run my model on WarpPLS and compared the results from another PLS software with that from yours. Most of the results are similar, if not the same. However, when it comes to indicator loadings, they are quite different. Please enlighten me on the reasons. It's important for me to know as I am trying to compare the results. Should I change the resample size, algorithm, etc.?

Thanks, Park

Ned Kock said...

Hi Fairview.

The loadings in WarpPLS are usually different because of the oblique rotation that is applied to the original loadings. This usually leads to loadings that are similar to the original values, and low cross-loadings.

This is the way it generally should be anyway, for reflective LVs. It suggests good measurement instrument validity.

Are you getting results that are unexpected?

Unknown said...

Hi Ned,

Thanks a lot for your reply.

The results are not unexpected. It's just that most of the results are the same except the indicator loadings. Since I am trying to verify my research results using different PLS software, it would be helpful if WarpPLS could produce the same results as other software.

Any chance that I could get the same indicator loadings by changing some settings of WarpPLS?

Thanks again!


Ned Kock said...

Hi Fairview.

WarpPLS only provides rotated loadings. These are the most meaningful numbers of indicator-to-LV instrument assessment. Actually, some argue that unrotated loadings are not appropriate for that assessment.

Having said that, I suspect that the loadings you are getting from other software are simply the bivariate correlations between the LV score and the indicators.

Since WarpPLS generates LV scores, it should be easy to calculate those bivariate correlations, even with Excel.

Unknown said...

Hi Ned,

Thanks a lot for your explanation. I understand now.

May I seek your advice on a side issue on Statistics that is bothering me?

There is 1 construct in my research model that i want to compare using Independent Sample Means t-test in SPSS. There are 2 groups of cases. The construct (T) has 3 measurement indicators. I tried running the t-test, but what it did was to compare each indicator one by one across 2 groups of case. What should I do? Should I calculate a mean for each case for 3 measurement indicators first and then use the resulting means for t-test?

Your help is very much appreciated.



Ned Kock said...

There is a much better way of doing that test, with WarpPLS:

- Create a dummy variable (G) with numbers associated with each of the two groups - e.g., 0 for one group, and 1 for the other group. This dummy variable should be implemented as a LV with 1 indicator.

- Define your dependent (or criterion) construct (T) as you would normally do; in this case, I think that would be as a reflective LV with 3 indicators.

- Create a link between G and T, with G pointing at T.

- Estimate the model parameters with WarpPLS; this will calculate the beta and P values for the link. The P value is analogous to the P value you would obtain with a t test.

This type of WarpPLS test has a number of advantages over a standard t test or a one-way ANOVA test (which are essentially the same thing). For example, it allows for the use of LVs as dependent variables, and it is a robust test (which does not require normality).

Unknown said...

Hi Ned,

Thanks a lot! It's so nice to get some help finally.

Before I try out your suggested method, I'd like to ask if the method that I mentioned - "calculate a mean for each case for 3 measurement indicators first and then use the resulting means for t-test" is wrong or not.

Your help is much appreciated! Will now go try out your suggested method.


Ned Kock said...

Hi Fairview.

Averaging indicators usually leads to values that are similar to those generated by WarpPLS if the indicators are highly correlated. You always lose some precision doing that though.

If the indicators are not highly correlated, then there will be a large difference between the average and the LV score. This is particularly true if two (or more) of the indicators are negatively correlated.

Another problem with the t test is that it is fundamentally a parametric test. Yes, you can set the t test algorithm to correct for heteroscedasticity. But that is not the same thing as using a "robust statistics" approach as WarpPLS does.

Ned Kock said...


From version 2.0 of WarpPLS on both rotated and unrotated loadings are provided.

This post was revised to reflect that.

Anonymous said...


I am just wondering if the correlation matrix is based on "the Pearson correlations", wouldn't this cause low loadings in case the relationships are non-linear?

Ned Kock said...

Hi Anon. The answers to this and related issues are available from page 70 (and subsequent pages) of the WarpPLS User Manual: Version 6.0. (link below). Best regards!