WarpPLS

New PLS-based SEM email distribution list

2012-02-05T14:20:00.000-08:00

A new email distribution list is available for those who share a common interest in partial least squares (PLS) regression and its use in structural equation modeling (SEM). To check it out click here.

Version 3.0 of WarpPLS is coming soon, with several new features

2012-01-23T07:19:00.000-08:00

Version 3.0 of WarpPLS is currently undergoing a battery of tests, and will be made available soon. Among the new features is the calculation of indirect and total effects, which are exemplified in this health data analysis post based on the China Study II dataset. Here is a comprehensive list of new features in this version:

- Addition of latent variables as indicators. Users now have the option of adding latent variable scores to the set of standardized indicators used in an SEM analysis. This option is useful in the removal of outliers, through the use of restricted ranges for latent variable scores, particularly for outliers that are clearly visible on the plots depicting associations among latent variables. This option is also useful in hierarchical analysis, where users define second-order (and higher order) latent variables, and then conduct analyses with different models including latent variables of different orders.

- Blindfolding. Users now have the option of using a third resampling algorithm, namely blindfolding, in addition to bootstrapping and jackknifing. Blindfolding is a resampling algorithm that creates a number of resamples (a number that can be selected by the user), where each resample has a certain number of rows replaced with the means of the respective columns. The number of rows modified in this way in each resample equals the sample size divided by the number of resamples. For example, if the sample size is 200 and the number of resamples selected is 100, then each resample will have 2 rows modified. If a user chooses a number of resamples that is greater than the sample size, the number of resamples is automatically set to the sample size (as with jackknifing).

- Effect sizes. Cohen’s (1988) f-squared effect size coefficients are now calculated and shown for all path coefficients. These are calculated as the absolute values of the individual contributions of the corresponding predictor latent variables to the R-square coefficients of the criterion latent variable in each latent variable block. With these effect sizes users can ascertain whether the effects indicated by path coefficients are small, medium, or large. The values usually recommended are 0.02, 0.15, and 0.35; respectively (Cohen, 1988). Values below 0.02 suggest effects that are too weak to be considered relevant from a practical point of view, even when the corresponding P values are statistically significant; a situation that may occur with large sample sizes.

- Full colinearity VIFs. VIFs are now shown for all latent variables, separately from the VIFs calculated for predictor latent variables in individual latent variable blocks. These new VIFs are calculated based on a full colinearity test, which identifies not only vertical but also lateral colinearity, and allows for a test of colinearity involving all latent variables in a model. Vertical, or classic, colinearity is predictor-predictor latent variable colinearity in individual blocks. Lateral colinearity is a new term that refers to predictor-criterion latent variable colinearity; a type of colinearity that can lead to particularly misleading results. Full colinearity VIFs can also be used for common method (Lindell & Whitney, 2001) bias tests that are more conservative than, and arguably superior to, the traditionally used tests relying on exploratory factor analyses.

- Incremental code optimization. At several points the code was optimized for speed, which led to incremental gains even as a significant number of new features were added. Several of these new features required new and complex calculations, mostly to generate coefficients that were not available before.

- Indirect and total effects. Indirect and total effects are now calculated and shown, together with the corresponding P values, standard errors, and effect sizes. The calculation of indirect and total effects can be critical in the evaluation of downstream effects of latent variables that are mediated by other latent variables, especially in complex models with multiple mediating effects in concurrent paths. Indirect effects also allow for direct estimations, via resampling, of the P values associated with mediating effects that have traditionally relied on time-consuming and not fully automated calculations based on linear (Preacher & Hayes, 2004) and nonlinear (Hayes & Preacher, 2010) assumptions.

- P values for all weights and loadings. P values are now shown for all weights and loadings, including those associated with indicators that make up moderating variables. With these P values, users can check whether moderating latent variables satisfy validity and reliability criteria for either reflective or formative measurement. This can help users demonstrate validity and reliability in hierarchical analyses involving moderating effects, where double, triple etc. moderating effects are tested. For instance, moderating latent variables can be created, added to the model as standardized indicators, and then their effects modeled as being moderated by other latent variables; an example of double moderation.

- Predictive validity. Stone-Geisser Q-squared coefficients (Geisser, 1974; Stone, 1974) are now calculated and shown for each endogenous variable in an SEM model. The Q-squared coefficient is a nonparametric measure traditionally calculated via blindfolding. It is used for the assessment of the predictive validity (or relevance) associated with each latent variable block in the model, through the endogenous latent variable that is the criterion variable in the block. Sometimes referred to as a resampling analog of the R-squared, it is often similar in value to that measure; even though, unlike the R-squared coefficient, the Q-squared coefficient can assume negative values. Acceptable predictive validity in connection with an endogenous latent variable is suggested by a Q-squared coefficient greater than zero.

- Ranked data. Users can now select an option to conduct their analyses with only ranked data, whereby all the data is automatically ranked prior to the SEM analysis (the original data is retained in unranked format). When data is ranked, typically the value distances that typify outliers are significantly reduced, effectively eliminating outliers without any decrease in sample size. A concomitant increase in colinearity is usually observed, but not to the point of threatening the credibility of the results. This option can be very useful in assessments of whether the presence of outliers significantly affects path coefficients and respective P values, especially when outliers are not believed to be due to measurement error.

- Restricted ranges. Users can now run their analyses with subsamples defined by a range restriction variable, which may be standardized or unstandardized. This option is useful in multi-group analyses, whereby separate analyses are conducted for each subsample and the results then compared with one another. One example would be a multi-country analysis, with each country being treated as a subsample, but without separate datasets for each country having to be provided as inputs. This range restriction feature is also useful in situations where outliers are causing instability in a resample set, which can lead to abnormally high standard errors and thus inflated P values. Users can remove outliers by restricting the values assumed by a variable to a range that excludes the outliers, without having to modify and re-read a dataset.

- Standard errors for all weights and loadings. Standard errors are now shown for all loadings and weights. Among other purposes, these standard errors can be used in multi-group analyses, with the same model but different subsamples. In these cases, users may want to compare the measurement models to ascertain equivalence, using a multi-group comparison technique such as the one documented by Keil et al. (2000), and thus ensure that any observed differences in structural model coefficients are not due to measurement model differences.

- VIFs for all indicators. VIFs are now shown for all indicators, including those associated with moderating latent variables. With these VIFs, users can check whether moderating latent variables satisfy criteria for formative measurement, in case they do not satisfy validity and reliability criteria for reflective measurement. This can be particularly helpful in hierarchical analyses involving moderating effects, where formative latent variables are frequently employed, including cases where double, triple etc. moderating effects are tested. Here moderating latent variables can be created, added to the model as standardized indicators, and then their effects modeled as being moderated by other latent variables; with this process being repeated at different levels.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Lawrence Erlbaum.

Geisser, S. (1974). A predictive approach to the random effects model. Biometrika, 61(1), 101-107.

Hayes, A. F., & Preacher, K. J. (2010). Quantifying and testing indirect effects in simple mediation models when the constituent paths are nonlinear. Multivariate Behavioral Research, 45(4), 627-660.

Keil, M., Tan, B.C., Wei, K.-K., Saarinen, T., Tuunainen, V., & Wassenaar, A. (2000). A cross-cultural study on escalation of commitment behavior in software projects. MIS Quarterly, 24(2), 299–325.

Lindell, M., & Whitney, D. (2001). Accounting for common method variance in cross-sectional research designs. Journal of Applied Psychology, 86(1), 114-121.

Preacher, K.J., & Hayes, A.F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. Behavior Research Methods, Instruments, & Computers, 36 (4), 717-731.

Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society, Series B, 36(1), 111–147.

Hands-On Workshop on WarpPLS; 6-7 January 2012; San Antonio, Texas

2011-10-10T13:57:00.000-07:00

Two-Day Hands-On Workshop on WarpPLS: SEM Fundamentals with Linear and Nonlinear Applications

*** Registration and additional details ***

http://bit.ly/oqoG5C

or

http://scriptwarp.com/warppls/prjs/2012_WarpPLSwkshp_Jan_SanAntonio

*** Instructor ***

Ned Kock, Ph.D.
WarpPLS Developer
http://nedkock.com

*** Location and dates ***

Our Lady of the Lake University
San Antonio, Texas
6-7 January 2012 (Fri-Sat), 8 am–5 pm

*** Workshop program at a glance ***

The main goal of this workshop is to give participants a practical understanding of how to use the software WarpPLS to conduct variance-based structural equation modeling (SEM). The workshop is very hands-on and covers linear and nonlinear applications.

Day 1 of workshop
• Overview of workshop and formation of teams
• Overview of web resources: Video clips, blog, publications, spreadsheets, and templates
• Overview of steps 1 to 5 of a complete SEM analysis
• Hands-on exercise: Steps 1 to 5 of a complete SEM analysis
• Resampling as shuffling multiple decks of cards
• Choosing the right resampling method
• Hands-on exercise: Changing the resampling method
• Choosing the right warping (i.e., nonlinear) algorithm
• Viewing plots of linear and nonlinear relationships
• Hands-on exercise: Changing the warping algorithm and viewing plots
• Charting non-standardized data
• Hands-on exercise: Charting non-standardized data
• Reading discussion: Kock (2011) – WarpPLS 2.0 User Manual

Day 2 of workshop
• Testing a mediating effect using the Baron & Kenny approach
• Hands-on exercise: Testing a mediating effect using the Baron & Kenny approach
• Testing a mediating effect using the Preacher & Hayes approach
• Hands-on exercise: Testing a mediating effect using the Preacher & Hayes approach
• Reading discussion: Kock et al. (2009) – Communication flow orientation article
• Testing a moderating effect
• Hands-on exercise: Testing a moderating effect
• Adding control variables into an analysis
• Conducting a multi-group analysis
• Conducting a full collinearity test
• Reading discussion: Zhang et al. (2010) – Organizing software testing article
• Hands-on exercise: Team project using participant’s own data
• Presentation of results from team project

Using WarpPLS in E-Collaboration Studies: Mediating Effects, Control and Second Order Variables, and Algorithm Choices

2011-08-29T08:25:00.000-07:00

A new article discussing WarpPLS is available. The article is titled “Using WarpPLS in E-Collaboration Studies: Mediating Effects, Control and Second Order Variables, and Algorithm Choices”. It has been recently published in the International Journal of e-Collaboration. A full text version of the article is available here as a PDF file. Below is the abstract of the article.

This is a follow-up on two previous articles on WarpPLS and e-collaboration. The first discussed the five main steps through which a variance-based nonlinear structural equation modeling analysis could be conducted with the software WarpPLS (Kock, 2010b). The second covered specific features related to grouped descriptive statistics, viewing and changing analysis algorithm and resampling settings, and viewing and saving various results (Kock, 2011). This and the previous articles use data from the same e-collaboration study as a basis for the discussion of important WarpPLS features. Unlike the previous articles, the focus here is on a brief discussion of more advanced issues, such as: testing the significance of mediating effects, including control variables in an analysis, using second order latent variables, choosing the right warping algorithm, and using bootstrapping and jackknifing in combination.

WarpPLS workshop at Fundação Getúlio Vargas in June 2011: Details and some photos

2011-07-18T09:48:00.000-07:00

Below are some photos from the June 2011 WarpPLS workshop at Fundação Getúlio Vargas, one of the highest ranked and most prestigious universities in Brazil in its main areas of focus. FGV’s main foci are business management and public administration. The workshop was in Rio de Janeiro. The workshop participants included faculty, doctoral students, and masters’ students at FGV.

This was a very hands-on workshop, as the participants had taken a course in structural equation modeling prior to it. They used Amos in that course, which was great because the workshop then highlighted the power of WarpPLS vis-à-vis a well established and also very useful tool for multivariate analyses with latent variables (Amos). We had about 15 contact hours for this workshop. Activities included commentaries based on video clips, live demonstrations, discussions of selected readings, and practical assignments focusing on linear and nonlinear empirical data analyses.

About 30 percent of the workshop was set aside for “free data analyses”, building on data that the participants brought into the workshop. That is, the participants had time to analyze their own data, and solve specific problems with my help. (There are always issues that are specific to a given dataset; e.g., problems with indicator loadings and interpretation of nonlinear results.) There was also a team workshop project, where participant teams presented an independent empirical study with analyses employing WarpPLS.

Some of the participants were faculty members from other universities in Rio de Janeiro, as well as employees of a few major research and training organizations in Brazil. Among these organizations were Fundação Oswaldo Cruz (a.k.a. FIOCRUZ), and the Escola de Comando e Estado Maior do Exército (ECEME). FIOCRUZ is one of the world’s foremost public health organizations, known for its strengths in various related areas, including epidemiological research. ECEME is an education institution that prepares officers of the Brazilian Army to take up command positions at the rank of General.

WarpPLS’ treatment of formative latent variables: More in line with Wold than Lohmöller

2011-06-28T06:59:00.000-07:00

WarpPLS uses what is often referred to as Wold’s original “PLS regression” algorithm to calculate indicator weights, for both formative and reflective variables. PLS regression was developed by Wold, and is slightly different from the modification of Wold’s algorithm developed by Lohmöller, which is the one normally used in other publicly available PLS-based structural equation modeling software.

Generally speaking, the PLS regression algorithm generates coefficients that are more stable and robust – i.e., reliable for hypothesis testing. It also tends to minimize collinearity. On the other hand, it is more computing intensive, which is probably one of the reasons why Lohmöller developed a modified version, with sub-versions called “modes” – see Lohmöller (1989) for more details. Personal computers were not that powerful in the 1980s.

Moreover, the type of nonlinear treatment employed by WarpPLS cannot be properly performed with Lohmöller’s algorithm. The problem is that with Lohmöller’s algorithm, as a model changes, the weights and loadings also change, even if the latent variables do not change. That is, with Lohmöller’s algorithm, two models with the same latent variables but different structures (i.e., links among latent variables) will have different weights and loadings.

The weights of formative latent variables will be essentially the same in WarpPLS as they would be if the variables were defined as reflective. That is, they will be obtained by an iterative algorithm that stops when two conditions are met: (a) the weights between indicators and latent variable are standardized partial regression coefficients calculated with the indicators as independent variables and the latent variable as the dependent variable; and (b) the regression equation expressing the latent variable as a combination of the indicators has an error term of zero.

So why should the user define a latent variable as formative or reflective? The reason are the interpretations of the outputs generated by the software. When a latent variable is formative, both the P values for the weights and the variance inflation factors for the indicators should be generally low; ideally below 0.05 and 2.5, respectively.

True formative variables are fundamentally different from true reflective variables; there are cases that can be seen as “in between” formative and reflective. True formative and reflective variables behave differently, whether the software treats them differently or not. For example, with true formative variables you would expect indicators to be significantly associated with the scores of their respective latent variable; which is indicated by low P values for their weights. However, you would not normally expect the indicators to be redundant; which is indicated by low variance inflation factors for the indicators.

The way formative variables are treated in Lohmöller’s approach leads to unstable weights, with the signs of weights frequently changing in the resample set. See Temme et al. (2006) for a discussion on this phenomenon. Lohmöller’s approach also leads to “lateral” collinearity; or collinearity between predictor and criteria latent variables. This “stealth” type of collinearity often leads to inflated path coefficients for links involving formative latent variables.

References

Lohmöller, J.-B. (1989). Latent variable path modeling with partial least squares. Heidelberg, Germany: Physica-Verlag.

Temme, D., Kreis, H., & Hildebrandt, L. (2006). PLS path modeling – A software review. Berlin, Germany: Institute of Marketing, Humboldt University Berlin.

Dealing with country-specific number punctuation systems

2011-06-25T05:15:00.000-07:00

WarpPLS users in countries that adopt number punctuation systems different from that adopted in the USA may have problems when using Excel to manipulate WarpPLS files.

For instance, in Brazil a comma is used to separate the integer from the fractional part of a real number (e.g., 1,431), whereas in the USA a period is used for that purpose (e.g., 1.431).

Because of that, a coefficient calculated by WarpPLS and exported into a .txt file as “1.431” may be read by a Brazilian version of Excel as one thousand four hundred and thirty-one, and not as one plus the 431/1000 fraction.

This tends to happen in certain types of analyses, such as second order latent variable analyses, where WarpPLS outputs are used as inputs after manipulation with country-specific versions of Excel.

A simple way to solve this problem is to use Excel, Notepad, or another simple text editing tool and replace the offending punctuation items, all points with commas (or vice-versa) for example, before using the inputs for other purposes.

Testing the significance of mediating effects with WarpPLS using the Preacher & Hayes approach

2011-06-18T14:18:00.000-07:00

Previously I discussed on this blog the classic approach proposed by Baron & Kenny (1986) to test the significance of mediating effects with WarpPLS.

A more modern approach, with increasingly wider acceptance, has been proposed by Preacher & Hayes (2004) to test the significance of mediating effects. This approach has been further extended by Hayes & Preacher (2010) for nonlinear relationships.

These more modern approaches are implemented through an Excel spreadsheet available from the “Resources” area of the WarpPLS.com site, under “Excel files”.

The Excel spreadsheet above takes as inputs coefficients generated by WarpPLS, including path coefficients and their standard errors. The outputs are Sobel’s standard errors, product path coefficients, as well as T and P values, for mediating effects.

References

Baron, R.M., & Kenny, D.A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality & Social Psychology, 51(6), 1173-1182.

Hayes, A.F., & Preacher, K.J. (2010). Quantifying and testing indirect effects in simple mediation models when the constituent paths are nonlinear. Multivariate Behavioral Research, 45(4), 627-660.

Preacher, K.J., & Hayes, A.F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. Behavior Research Methods, Instruments, & Computers, 36 (4), 717-731.

Multi-group analysis with WarpPLS: Comparing path coefficients for two or more group samples

2011-06-18T14:00:00.000-07:00

I previously discussed on this post multi-group analysis with WarpPLS from the perspective of comparing means of two or more groups.

A different type of multi-group analysis would be one in which the same model is analyzed for two or more different samples, where each sample refers to a data group.

For example, a researcher could test the same model with data from the USA and Mexico. In this case, two project files would be used, and the goal of the multi-group analysis would be to assess whether the path coefficients differ significantly across groups.

Dr. Wynne Chin discussed an approach to conduct this type of multi-group analysis. This approach is implemented through an Excel spreadsheet available from the “Resources” area of the WarpPLS.com site, under “Excel files”.

The Excel spreadsheet above takes as inputs coefficients generated by WarpPLS, including path coefficients and their standard errors. The outputs are T and P values for each pair of coefficients being compared.

Using WarpPLS in E-Collaboration Studies: Descriptive Statistics, Settings, and Key Analysis Results

2011-04-22T09:26:00.000-07:00

A new article discussing WarpPLS is available. The article is titled “Using WarpPLS in E-Collaboration Studies: Descriptive Statistics, Settings, and Key Analysis Results”. It has been recently published in the International Journal of e-Collaboration. A full text version of the article is available here as a PDF file. Below is the abstract of the article.

This is a follow-up on a previous article (Kock, 2010b) discussing the five main steps through which a nonlinear structural equation modeling analysis could be conducted with the software WarpPLS (warppls.com). Both this and the previous article use data from the same e-collaboration study as a basis for the discussion of important WarpPLS features. Unlike in the previous article, the focus here is on specific features related to saving and analyzing grouped descriptive statistics, viewing and changing analysis algorithm and resampling settings, and viewing and saving the various minor and major results of the analysis. Even though its focus is on an e-collaboration study this article contributes to the broad literature on multivariate analysis methods, in addition to the more specific research literature on e-collaboration. The reason for this is that the vast majority of relationships between variables, in investigations of both natural and behavioral phenomena, are nonlinear; usually taking the form of U and S curves. Structural equation modeling software tools, whether variance- or covariance-based, typically do not estimate coefficients of association based on nonlinear analysis algorithms. WarpPLS is an exception in this respect. Without taking nonlinearity into consideration, the results can be misleading; especially in complex and multi-factorial situations such as those stemming from e-collaboration in virtual teams.

Transitioning from WarpPLS 1.0 to 2.0

2011-04-20T08:28:00.000-07:00

Transitioning from version 1.0 to 2.0 of WarpPLS is very easy. Even though both can be installed and ran on the same computer, and should not interfere with each other, I recommend using only the latest version.

There is no need to uninstall version 1.0, but if you want to do so you can follow the instructions at the beginning of the User Manual for version 1.0.

Version 1.0 users can enter the same license information as for version 1.0; it will work for version 2.0 for the remainder of their license periods.

Project files generated with version 1.0 can be used with version 2.0, but only after running Step 5 again. This is needed because version 2.0 generates additional estimates.

Enjoy!

Two new WarpPLS workshops in April and May of 2011

2011-04-08T08:08:00.001-07:00

PLS-SEM.com will conduct two new online workshops on WarpPLS in April and May of 2011!

For more information on these and other WarpPLS workshops please visit:

http://pls-sem.com

Version 2.0 of WarpPLS is now available!

2011-04-06T08:38:00.001-07:00

Version 2.0 of WarpPLS is now available!

Download and install it for a free trial from:

http://warppls.com

Enjoy!

Version 2.0 of WarpPLS will be available soon!

2011-03-11T09:13:00.000-08:00

Version 2.0 of WarpPLS is currently being tested, and will be available soon, barring any unexpected problems.

Here is a list of new features in this version:

- APC fit index. The algorithm that calculates the average path coefficient (APC) was modified to correct a problem that was leading it to be underestimated for some models.

- Drag-and-drop user interface. Latent variables can now be moved around, during the model creation/editing step, via drag-and-drop actions.

- Help menu options. Most menus now provide one-click access to Web resources, including Web videos and the WarpPLS blog.

- Incremental code optimization. At several points the code was optimized for speed.

- Loadings and cross-loadings. The software now provides the following tables in connection with loadings and cross-loadings, from a confirmatory factor analysis, on both the screen and model estimates .txt file: combined loadings and cross-loadings, pattern loadings and cross-loadings, and structure loadings and cross-loadings.

- Moderating relationship visualization. The software now shows two plots for moderating relationships, referring to low and high values of the moderating variable. They can be viewed through the “View/plot linear and nonlinear relationships among latent variables” option, under “View and save results”.

- Saving model diagrams as .jpg files. The software now allows for model diagrams, with and without results, to be saved into .jpg files for later inclusion in reports.

- Standard errors for path coefficients. The software now reports standard errors for path coefficients, on both the screen and the model estimates .txt file. These allow for newer mediating effect tests to be conducted, in addition to Baron & Kenny’s (1986) test. The newer tests include those proposed by Preacher & Hayes (2004), and Hayes & Preacher (2010). These citations are fully referenced in the newest version of the User Manual.

- Support for .xlsx Excel files. The software now reads new .xlsx Excel files. Excel workbooks with multiple sheets can also be used, in which case the sheet with the data to be analyzed must be the first in the workbook.

- System.wrp file. The extension of the “System.sys” file was changed, and the content of the file was modified to store a few additional parameters necessary for code optimization. The file is now called “System.wrp”.

- VIFs for formative indicators. The software now calculates variance inflation factors (VIFs) for the indicators of formative latent variables, which can be used for indicator redundancy assessment.

Stay tuned!

Error writing on System.sys at WarpPLS 1.0 start

2010-12-06T14:41:00.000-08:00

(Note: This post refers to version 1.0 of WarpPLS only. The System.sys file has been renamed System.wrp in version 2.0 to avoid this problem.)

Some users may receive an error message on the WarpPLS command prompt window indicating that the System.sys file cannot be updated. This issue occurs when WarpPLS is launched, and seems to happen with a few Windows 7 users.

Even with this error, WarpPLS runs normally on the trial version. However, it does not allow a user to update the license information.

To solve this problem on Windows 7, follow the steps below. Similar steps related to granting users full control over a file should be followed with other Windows operating systems.

1) Go to the folder containing the WarpPLS program. Typically this will be:

C:\Program Files\WarpPLS 1.0

2) Right-click on the file System.sys, and then choose properties.

3) Click on the Security tab.

4) Click on the Users group, and then on the button to change permissions.

5) Check the option to give the group Users full control on the System.sys file.

This problem occurs because Windows 7 does not seem to grant all computer administrator rights to new users, even when those users are included in the Administrators group.

By following the instructions on this post, you will essentially be allowing any user to change the System.sys file while running WarpPLS. This should not be a problem, as long as no user decides to manually delete the System.sys file.

Using WarpPLS in e-collaboration studies: An overview of five main analysis steps

2010-11-24T16:08:00.000-08:00

A new article discussing WarpPLS is available. The article is titled “Using WarpPLS in e-collaboration studies: An overview of five main analysis steps”. It has been recently published in the International Journal of e-Collaboration. A full text version of the article is available as a PDF file. Below is the abstract of the article.

Most relationships between variables describing natural and behavioral phenomena seem to be nonlinear, with U-curve and S-curve relationships being particularly common. Yet, structural equation modeling software tools do not typically estimate coefficients of association taking nonlinear relationships between latent variables into consideration. This can lead to misleading results, particularly in multivariate and complex phenomena such as those related to e-collaboration. One notable exception is WarpPLS (available from: warppls.com), a new structural equation modeling software currently available in its first release, version 1.0. The discussion presented here contributes to the literature on e-collaboration research methods by providing a description of the main features of WarpPLS, in the context of an e-collaboration study. The focus of this discussion is on the software’s features and their use, and not on the e-collaboration study itself. Particular emphasis is placed on the five steps through which a structural equation modeling analysis is conducted through WarpPLS.

Incompatibility between WarpPLS and XLSTAT

2010-09-26T07:41:00.000-07:00

Most users seem to have no problems installing and running WarpPLS. A minority have problems with the MATLAB Compiler Runtime, which can be addressed by following the instructions on this post.

A very small number of users seem to be unable to properly install and run WarpPLS, even after following the instructions above. A common, but not exclusive, error message in this case is: “An application has made an attempt load the C runtime library incorrectly”.

One common characteristic among these users is that they have the software XLSTAT installed on their computers. I have already received a few reports suggesting XLSTAT changes operating system settings in such a way as to prevent WarpPLS from properly running.

When those users removed XLSTAT from their computers, they were able to run WarpPLS without problems.

There is no need for two-way arrows in WarpPLS

2010-09-15T15:49:00.001-07:00

In covariance-based structural equation modeling (SEM) software tools, often one has to explicitly model correlations between predictor latent variables (LVs). In WarpPLS, correlations between predictor LVs are automatically taken into consideration in the calculation of path coefficients.

That is, the path coefficients calculated by WarpPLS are true standardized partial regression coefficients, of the same type as those calculated through multiple regression analysis. The difference is, of course, that in WarpPLS the model variables are LVs, which are usually measured through more than one indicator. With multiple regression, only one measure (or indicator) is used for each variable in the model.

Testing the significance of mediating effects with WarpPLS using the Baron & Kenny approach

2010-07-26T09:44:00.000-07:00

Using WarpPLS, one can test the significance of a mediating effect of a variable M, which is hypothesized to mediate the relationship between two other variables X and Y, by using Baron & Kenny’s (1986) criteria. (Full reference at the end of this post.) The procedure is outlined below. It can be easily adapted to test multiple mediating effects, and more complex mediating effects (e.g., with multiple mediators). Please note that we are not referring to moderating effects here; these can be tested directly with WarpPLS, by adding moderating links to a model.

First two models must be built. The first model should have X pointing at Y, without M being included in the model. (You can have the variable in the WarpPLS model, but there should be no links from or to it.) The second model should have X pointing at Y, X pointing at M, and M pointing at Y. This is a “triangle”-looking model. A WarpPLS analysis must be conducted with both models, which may be saved in two different project files; this analysis may use linear or nonlinear analysis algorithms. The mediating effect will be significant if the three following criteria are met:

- In the first model, the path between X and Y is significant (e.g., P < 0.05, if this is the significance level used).

- In the second model, the path between X and M is significant.

- In the second model, the path between M and Y is significant.

Note that, in the second model, the path between M and Y controls for the effect of X. That is the way it should be. Also note that the effect of X on Y in the second model is irrelevant for this mediation significance test. Nevertheless, if the effect of X on Y in the second model is insignificant (i.e., indistinguishable from zero, statistically speaking), one can say that the case is one of “perfect” mediation. On the other hand, if the effect of X on Y in the second model is significant, one can say that the case is one of “partial” mediation. This of course assumes that the three criteria are met.

Generally, the lower the effect of X on Y in the second model, the more “perfect” the mediation is, if the three criteria for mediating effect significance are met.

Reference

Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality & Social Psychology, 51(6), 1173-1182.

Use formative latent variables with caution

2010-07-23T14:49:00.000-07:00

One should use formative latent variables (LVs) with caution in structural equation modeling analyses using WarpPLS. It is not uncommon to see formative LVs being created simply by casually aggregating indicators, without much concern about the indicators being actually facets of the same construct. See this post for more details.

It is also important to stress that formative LVs are better assessed when included as part of a model. This is preferable to analyzing formative LVs individually; that is, as “models” that include one single LV. The loadings and cross-loadings table takes into consideration both formative and reflective LVs in its calculation, and may suggest that some indicators do not “belong” to a formative LV.

Also, certain model parameters may sometimes become unstable due to colinearity. High colinearity among indicators is to be expected in reflective LV measurement, but not in formative LV measurement. In the context of formative LV assessment, colinearity may be reflected in unstable weights, where unexpected P values (usually statistically insignificant) are associated with weights.

In formative LVs, indicators are expected to measure different facets of the LV, not the same thing. If two (or more) indicators are collinear in a formative LV, it may be a good idea to collapse them into one indicator. This can be done by defining second order LVs (a two-step, somewhat complex procedure), averaging the indicators, or simply eliminating one of the indicators from the analysis.

Using WarpPLS in an exploratory path analysis of health-related data

2010-07-22T09:14:00.000-07:00

There has been quite a lot of debate lately on the findings of a study known as the China Study. One of the key hypotheses of that study is that animal protein consumption (e.g., meat, dairy) causes various types of cancer, including colorectal cancer. Total cholesterol has been proposed as one of the intervening variables in connection with this effect. Given that, I decided to take a look at some of the data from the China Study and do a couple of multivariate data analysis on it using WarpPLS.

First I built a model that explores relationships with the goal of testing the assumption that the consumption of animal protein causes colorectal cancer, via an intermediate effect on total cholesterol. I built the model with various hypothesized associations to explore several relationships simultaneously, including some commonsense ones. Including commonsense relationships is usually a good idea in exploratory multivariate analyses.

The model is shown on the graph below, with the results. (Click on it to enlarge. Use the "CRTL" and "+" keys to zoom in, and CRTL" and "-" to zoom out.) The arrows explore causative associations between variables. The variables are shown within ovals. The meaning of each variable is the following: aprotein = animal protein consumption; pprotein = plant protein consumption; cholest = total cholesterol; crcancer = colorectal cancer.

The “(R)1i” below the variable names simply means that each of the variables is measured through a single indicator. This characterizes this analysis as a path analysis, rather than a true structural equation modeling (SEM) analysis. The P values were calculated through jackknifing. Like bootstrapping and other nonparametric resampling techniques, jackknifing does not require the assumption that the data be normally distributed. This is good, because I checked the data, and it does not look like it is normally distributed. So what does the model above tell us? It tells us that:

- As animal protein consumption increases, colorectal cancer decreases, but not in a statistically significant way (beta=-0.13; P=0.11).

- As animal protein consumption increases, plant protein consumption decreases significantly (beta=-0.19; P<0.01).

- As plant protein consumption increases, colorectal cancer increases significantly (beta=0.30; P=0.03). This is statistically significant because the P is lower than 0.05.

- As animal protein consumption increases, total cholesterol increases significantly (beta=0.20; P<0.01).

- As plant protein consumption increases, total cholesterol decreases significantly (beta=-0.23; P=0.02).

- As total cholesterol increases, colorectal cancer increases significantly (beta=0.45; P<0.01). Big surprise here!

Why the big surprise with the apparently strong relationship between total cholesterol and colorectal cancer? The reason is that it does not make sense, because animal protein consumption seems to increase total cholesterol, and yet animal protein consumption seems to decrease colorectal cancer.

When something like this happens in a multivariate analysis, it may be due to the model not incorporating a variable that has important relationships with the other variables. In other words, the model is incomplete, hence the nonsensical results. Relationships among variables that are implied by coefficients of association must also make sense to be credible.

Now, it has been pointed out that the missing variable here possibly is schistosomiasis infection. The dataset from the China Study included that variable, even though there were some missing values (about 28 percent of the data for that variable was missing), so I added it to the model in a way that seems to make sense. The new model is shown on the graph below. In the model, schisto = schistosomiasis infection.

So what does this new, and more complete, model tell us? It tells us some of the things that the previous model told us, but a few new things, which make a lot more sense. Note that this model fits the data much better than the previous one, particularly regarding the overall effect on colorectal cancer, which is indicated by the high R-squared value for that variable (R-squared=0.73). Most notably, this new model tells us that:

- As schistosomiasis infection increases, colorectal cancer increases significantly (beta=0.83; P<0.01). This is a MUCH STRONGER relationship than the previous one between total cholesterol and colorectal cancer; even though some data on schistosomiasis infection for a few counties is missing (the relationship might have been even stronger with a complete dataset). And this strong relationship makes sense, because schistosomiasis infection is indeed associated with increased cancer rates. More information on schistosomiasis infections can be found here.

- Schistosomiasis infection has no significant relationship with these variables: animal protein consumption, plant protein consumption, or total cholesterol. This makes sense, as the infection is caused by a worm that is not normally present in plant or animal food, and the infection itself is not specifically associated with abnormalities that would lead one to expect major increases in total cholesterol.

- Total cholesterol has no significant relationship with colorectal cancer (beta=0.24; P=0.11). The beta here is nontrivial, but too low to be significant; i.e., we cannot discard chance within the context of this relatively small dataset.

- Animal protein consumption has no significant relationship with colorectal cancer. The beta here is very low, and negative (beta=-0.03).

- Plant protein consumption has no significant relationship with colorectal cancer. The beta for this association is positive and nontrivial (beta=0.15), but the P value is too high (P=0.20) for us to discard chance within the context of this dataset.

Below is the plot showing the relationship between schistosomiasis infection and colorectal cancer. The values are standardized, which means that the zero on the horizontal axis is the mean of the schistosomiasis infection numbers in the dataset. The shape of the plot is the same as the one with the unstandardized data. As you can see, the data points are very close to a line, which suggests a very strong linear association.

In summary, an exploratory path analysis with WarpPLS can shed light on data patterns that would look rather “mysterious” otherwise. Still, one has to use commonsense, good theory, and past empirical results to derive conclusions.

Using WarpPLS for multiple regression analyses

2010-07-13T09:17:00.000-07:00

Why should someone use WarpPLS for multiple regression analysis?

There are two main advantages of using WarpPLS to conduct a multiple regression analysis. The advantages are over a traditional multiple regression analysis, where the independent and dependent variables are measured through single indicators. With WarpPLS, this would be implemented through the creation of "latent" variables that would each be associated with a single indicator; which means that they would not be true latent variables in the sense normally assumed in structural equation modeling.

The first advantage is that the calculation of P values with WarpPLS is based on a nonparametric algorithm, resampling, and thus does not require that the variables be normally distributed. A traditional multiple regression analysis, on the other hand, requires that the variables be normally distributed. In this sense, WarpPLS can be seen as conducting a robust, or nonparametric, multiple regression analysis. This first advantage assumes that all one is doing is a plain linear analysis with WarpPLS, for which one would use the algorithms PLS Regression or Robust Path Analysis. See the software's User Manual for more details.

The second advantage is that WarpPLS allows for nonlinear relationships between the independent and dependent variables to be analyzed. This provides a much richer view of the associations between variables, and sometimes leads to path coefficients that are different from (often higher than) those obtained through a linear analysis (as in a traditional multiple regression analysis). The nonlinear analysis algorithms available are Warp3 PLS Regression (which yields S curves) and Warp2 PLS Regression (which yields U curves). Again, see the software's User Manual for more details.

Second order latent variables in WarpPLS: YouTube videos by Jaime León

2010-06-20T10:09:00.000-07:00

The YouTube videos below have been created by WarpPLS user and blog commenter Jaime León. They illustrate how steps 1 and 2, described in this post, can be implemented in WarpPLS. The goal of those steps is to use second order latent variables (LVs) in an SEM analysis. Latent variable (LV) scores are generated, saved, and then used in a subsequent SEM analysis.

Step 1: YouTube video 1.

Step 2: YouTube video 2.

In the first video Jaime includes only LVs in the model, without any links among them, and then runs the SEM analysis. This generates the LV scores for the LVs, which Jaime then saves into a .txt file. The LV scores generated are then combined with indicators from the original dataset.

Note that Jaime does not set the LVs in the first video as formative before generating the scores. That is okay if the LVs are reflective; that is, if the indicators of the LVs are highly correlated. (In reflective LVs the loadings are expected to be all high, ideally greater than .7, and significant.) If not, then the LVs should be set as formative.

Also, note that Jaime combined the LV scores in standardized format with indicator data from the original dataset, which were not standardized. That is fine because WarpPLS always standardizes the raw data before proceeding to an SEM analysis. Standardized data, when used as input, will not be affected by standardization (since they are already standardized).

In the second video Jaime creates a model with new LVs, some of which include the previously generated LV scores as indicators. These are frequently referred to as second order LVs. (Although sometimes the original LVs, shown in the first video, are the ones called second order LVs.) Jaime then builds a model by creating several direct links among the LVs.

Cool example, with a Bob Marley song in the background; thanks Jaime!

Multi-group analysis with WarpPLS: Comparing means of two or more groups

2010-06-17T08:16:00.000-07:00

Comparing means of two groups is something that behavioral researchers often do. In fact, this is the most widely used quantitative analysis approach. This is also a form of multi-group analysis, where the number of groups is two. Common tests for comparing means are the t and and one-way ANOVA tests.

There is an arguably much better way of doing comparison of means test, with WarpPLS. Follow the steps below. This is a two-group test. Multi-group tests can be done in a similar way, through multiple two-group tests where conditions (i.e., groups) are compared pair by pair.

- 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 that the dependent variables be normally distributed.

Multi-group tests, with more than two groups, can also be conducted by assigning different values to each of the groups. The key here is to decide what values to assign to each group. This choice is often somewhat arbitrary in exploratory analyses. The “Save grouped descriptive statistics into a tab-delimited .txt file” option may be helpful in this respect. This is a special option that allows you to save descriptive statistics (means and standard deviations) organized by groups defined based on certain parameters. For more details, see the user manual for WarpPLS, which is available from Warppls.com.

Using second order latent variables in WarpPLS

2010-06-15T08:14:00.000-07:00

Second order latent variables (LVs) can be implemented in WarpPLS 1.0 through two steps. These steps are referred to as Step 1 and Step 2 in the paragraphs below. Higher order LVs can also be implemented, following a similar procedure, but with additional steps.

With second order LVs, a set of LV scores are used as indicators of a LV. Often second order LVs are decompositions of a formative LV into a few reflective LVs. The scores of the component reflective LVs are used as indicators of the original formative LV.

In Step 1, you will create models that relate LVs to their indicators. Only the LVs and their indicators should be included. No links between LVs should be created. This will allow you to calculate the LV scores for the LVs, based on the indicators. You will then save the LV scores using the option “Save factor scores into a tab-delimited .txt file”, available from the “Save” option of the “View and save results” window menu.

In Step 2, you will create a new model where the saved LV scores are indicators of a new LV. This LV is usually called the second order LV, although sometimes the indicators (component LVs) are referred to as second order LVs. The rest of the data will be the same. Note that you will have to create and read the raw data used in the SEM analysis again, for this second step.

See this post for a couple of YouTube videos illustrating these 2 steps.

Selecting among different WarpPLS analysis algorithms

2010-05-18T13:22:00.000-07:00

First a quick recap of some issues already discussed in previous posts. WarpPLS offers the following analysis algorithms: Warp3 PLS Regression, Warp2 PLS Regression, PLS Regression, and Robust Path Analysis.

Many relationships in nature, including relationships involving behavioral variables, are nonlinear and follow a pattern known as U-curve (or inverted U-curve). In this pattern a variable affects another in a way that leads to a maximum or minimum value, where the effect is either maximized or minimized, respectively. This type of relationship is also referred to as a J-curve pattern; a term that is more commonly used in economics and the health sciences.

The Warp2 PLS Regression algorithm tries to identify a U-curve relationship between latent variables, and, if that relationship exists, the algorithm transforms (or “warps”) the scores of the predictor latent variables so as to better reflect the U-curve relationship in the estimated path coefficients in the model. The Warp3 PLS Regression algorithm, the default algorithm used by the software, tries to identify a relationship defined by a function whose first derivative is a U-curve. This type of relationship follows a pattern that is more similar to an S-curve (or a somewhat distorted S-curve), and can be seen as a combination of two connected U-curves, one of which is inverted.

The PLS Regression algorithm does not perform any warping of relationships. It is essentially a standard PLS regression algorithm, whereby indicators’ weights, loadings and factor scores (a.k.a. latent variable scores) are calculated based on a least squares minimization sub-algorithm, after which path coefficients are estimated using a robust path analysis algorithm. A key criterion for the calculation of the weights, observed in virtually all PLS-based algorithms, is that the regression equation expressing the relationship between the indicators and the factor scores has an error term that equals zero. In other words, the factor scores are calculated as exact linear combinations of their indicators. PLS regression is the underlying weight calculation algorithm used in both Warp3 and Warp2 PLS Regression. The warping takes place during the estimation of path coefficients, and after the estimation of all weights and loadings in the model. The weights and loadings of a model with latent variables make up what is often referred to as outer model, whereas the path coefficients among latent variables make up what is often called the inner model.

Finally, the Robust Path Analysis algorithm is a simplified algorithm in which factor scores are calculated by averaging all of the indicators associated with a latent variable; that is, in this algorithm weights are not estimated through PLS regression. This algorithm is called “Robust” Path Analysis, because, as with most robust statistics methods, the P values are calculated through resampling. If all latent variables are measured with single indicators, the Robust Path Analysis and the PLS Regression algorithms will yield identical results.

Okay, so what algorithm should you use?

Generally it will be one of these: Warp3 PLS Regression, Warp2 PLS Regression, PLS Regression. Only in a small number of instances, quite rare, will the Robust Path Analysis algorithm be the best choice.

If you analyze your dataset using different algorithms (e.g., Warp3 PLS Regression, Warp2 PLS Regression, and PLS Regression), usually the “best” algorithm will be the one leading to the most stable path coefficients. The most stable path coefficients are the ones with the lowest P values, whether the P values are obtained through bootstrapping or jackknifing. The best algorithm will also be the one leading to the highest average R-squared (ARS).

Another important consideration is theory. Does the theory underlying a hypothesized relationship between latent variables support the expectation of a U-curve or S-curve relationship? If the theory supports the expectation of a U-curve relationship, but not of an S-curve relationship, then you should favor Warp2 PLS Regression over Warp3 PLS Regression, even if the latter leads to the most stable path coefficients (i.e., with the lowest P values).

Naming of latent variable indicators in WarpPLS

2010-04-30T07:37:00.000-07:00

The indicator names are the headings of your raw data file, which are also shown on a few software screens during Step 2. That is the step where you read the raw data used in the SEM analysis. The indicator names are displayed at the top of the table on the figure below, a screen from Step 3; click on it to enlarge.

The software does not force users to restrict the names of indicators to a set number of characters (e.g., seven), or to exclude certain types of characters (e.g., blank spaces) from them. Nevertheless, it is usually a good idea to follow a few simple rules when naming indicators:

- Make the indicator names 7 characters in length, or less. This will ensure that they will not take up too much space on various screens and reports. In fact, in reports saved in text files, any character after the seventh is removed by the software. Otherwise the text file will become difficult to read without manual editing.

- Name the indicators using their latent variable name as a reference, and using sequential numbers at the end. For example, indicators that refer to the latent variable “ECU” should be named: “ECU1”, “ECU2”, “ECU3”, and so on.

- Do not use blank spaces in the indicator names. If you do, and an indicator name is long, the software may show the indicator split on two different rows on the model screen (Step 4), when you choose to show indicators on the model.

Interpreting the U and S curves generated by WarpPLS

2010-03-26T19:13:00.000-07:00

Linear relationships between pairs of latent variables, that is, those relationships best described by a line, are relatively easy to interpret. They suggest that an increase in one variable either leads to an increase (if the slope of the line is positive) or decrease (if the slope is negative) in the other variable.

Nonlinear relationships provide a much more nuanced view of the data, but at the same time are much more difficult to interpret correctly. The graph below (click on it to enlarge) shows an S curve that is fitted to the data represented by the dots (or small circles) plotted in a scattered way on the graph. The latent variables are “ECU”, the extent to which electronic communication media are used by a team charged with developing a new product; and “Effi”, the efficiency of the team.

As you can see, the S curve is actually a combination of two U curves, one straight and the other inverted, connected at an inflection point. The inflection point is the point at the curve where the curvature, or second derivative of the S curve, changes direction. On the graph shown above, the inflection point is located at around minus 1 standard deviations from the "ECU" mean. That mean is at the zero mark on the horizontal axis.

Because an S curve is a combination of two U curves, we can interpret each U curve section separately. A straight U curve, like the one shown on the left side of the graph, before the inflection point, can be interpreted as follows.

The first half of the U curve goes from approximately minus 3.4 to minus 1.9 standard deviations from the mean, at which point the lowest team efficiency value is reached for the U curve. In that first half of the U curve, an increase in electronic communication media use leads to a decrease in team efficiency. After that first half, an increase in electronic communication media use leads to an increase in team efficiency.

One interpretation is that the first half of the U curve refers to novice users of electronic communication media. That is, novice users struggling to use more and more intensely communication media that they are not familiar with end up leading to efficiency losses for the team. At a certain point, around minus 1.9 standard deviations, that situation changes, and the teams start to really benefit from the use of electronic communication media, possibly because the second half of the U curve refers to users with more experience in using the media.

The interpretation of the second, inverted U curve, should be done in a similar fashion. As you can see, it is not easy to interpret nonlinear relationships. But the apparent simplicity of linear estimations of nonlinear relationships, which is usually what is done by other structural equation modeling software, is nothing but a mirage.

Field studies, small samples, and WarpPLS

2010-03-22T17:10:00.000-07:00

Let us assume that a researcher wants to evaluate the effectiveness of new management method by conducting an intervention study in one single organization.

In this example, the researcher facilitates the use of a new management method by 20 managers in the organization, and then measures their degree of adoption of the method and their effectiveness.

The above is an example of a field study. Often field studies will yield small datasets, which will not conform to parametric analysis (e.g., ANOVA and ordinary multiple regression) pre-conditions. For example, the data will not typically be normally distributed.

WarpPLS can be very useful in the analysis of this type of data.

One reason is that, with small sample sizes, it may be difficult to identify linear relationships that are strong enough to be statistically significant (at P lower than 0.05, or less). Since WarpPLS implements nonlinear analysis algorithms, it can be very useful in the analysis of small samples.

Another reason is that P values are calculated through resampling, a nonparametric approach to statistical significance estimation. For small samples (i.e., lower than 100), jackknifing is the recommended resampling approach. Bootstrapping is recommended only for sample sizes greater than 100.

Standard deviation is not the same as range of variation

2010-03-15T23:16:00.000-07:00

Means and standard deviations can be generated and saved through the “Save grouped descriptive statistics into a tab-delimited .txt file” option of WarpPLS. You can choose a grouping variable, number of groups, and the variables to be grouped. This option is useful if one wants to conduct a comparison of means analysis using the software, where one variable (the grouping variable) is the predictor, and one or more variables are the criteria (the variables to be grouped).

In comparisons of means analyses, research results are normally expressed in means and standard deviations. For example, in the study reviewed in this post, it is stated that the weight of participants in a 12-week weight loss study varied from: 87.9 plus or minus 15.4 kg (at baseline, or before the 12-week intervention) to 81.7 plus or minus 16.2 kg (after the 12-week intervention).

The 87.9 and 81.7 are the average weights (a.k.a. “mean” weights), in kilograms, before and after the 12-week intervention. However, the 15.4 and 16.2 are NOT the range of variation in weights around the means before and after 12-week intervention. They are actually the ranges around the means encompassing approximately 68 percent of all of the values measured (see figure below, from www.electrical-res.com).

In the figure above, the minus and plus 15.4 and 16.2 values would be the “mean(x) – s” and “mean(x) + s” points on the horizontal axis of histograms of weights plotted before and after the 12-week intervention. This assumes that the distributions of weights are normal, or quasi-normal (i.e., similar to a bell-shaped, or normal, curve); a common assumption in this type of research.

The larger the standard deviation, the wider is the variation in the measures, and the flatter is the associated histogram (the bell-shaped curve). This property has a number of interesting implications, some of which will be discussed in other posts.

Sometimes another measure of dispersion, the variance, is reported instead of the standard deviation. The variance is the standard deviation squared.

The reason why standard deviations are reported instead of ranges of variation is that outliers (unusually high or low values) can dramatically widen the ranges. The standard deviation coefficient is much less sensitive to outliers.

Geographically distributed collaborative SEM analysis using WarpPLS

2010-03-02T15:42:00.000-08:00

I am currently conducting a geographically distributed collaborative SEM analysis using WarpPLS. The analysis involves a few people in different states of the USA, and two people outside the country. The collaborators are not only separated by large distances, but also operate in different time zones.

Yet, we have no problems collaborating. The collaboration is asynchronous – one person does some work one day, and shares it with the others, who review the work in the next few days and respond.

Since we all have WarpPLS installed on our computers, we exchange different versions of a WarpPLS project file (extension “.prj”) with the same dataset. This way we can do analyses in turns, and discuss the results on emails.

Each slightly different project file is saved with a different name – e.g., W3J_InfoOvld_2010_03_02.prj, W3B_InfoOvld_2010_03_02.prj, W2J_InfoOvld_2010_03_02.prj etc.

In the examples above, the first three letters indicate the SEM algorithm used (W3 = Warp3 PLS Regression; W2 = Warp2 PLS Regression), and the resampling method used (J = jackknifing; B = bootstrapping). The second part of the name describes the dataset, and the final part the date.

This is just one way of naming files. It works for our particular project, but more elaborate file names can be used in more complex collaborative SEM analyses.

This geographically distributed collaborative SEM analysis highlights one of the advantages of WarpPLS over other SEM software: all that is needed for the analysis is contained in one single project file.

Moreover, the project file will typically be only a few hundred kilobytes in size. In spite of its small size, the file includes the original data, and all of the results of the analysis.

One member of our team asked me how the project file can be so small. The reason is that all of the SEM analysis results are stored in a format that allows for their rendering every time they are viewed.

Plots of nonlinear relationships, for example, are not stored as bitmaps, but as equations that allow WarpPLS to re-create those plots at the time of viewing.

What are the inner and outer models in SEM?

2010-02-21T09:38:00.000-08:00

In a structural equation modeling (SEM) analysis, the inner model is the part of the model that describes the relationships between the latent variables that make up the model. The outer model is the part of the model that describes the relationships between the latent variables and their indicators.

In this sense, the path coefficients are inner model parameter estimates. The weights and loadings are outer model parameter estimates. The inner and outer models are also frequently referred to as the structural and measurement models, respectively.

More precisely, the mathematical equations that express the relationships among latent variables are referred to as the structural model. The equations that express the relationships between latent variables and indicators are referred to as the measurement model.

The term structural equation model is used to refer to both the structural and measurement model, combined.

Nonlinearity and type I and II errors in SEM analysis

2010-02-21T09:19:00.000-08:00

Many relationships between variables studied in the natural and behavioral sciences seem to be nonlinear, often following a J-curve pattern (a.k.a. U-curve pattern).

Yet, the vast majority of statistical analysis methods used in the natural and behavioral sciences, from simple correlation analysis to structural equation modeling, assume relationships to be linear in the estimation of coefficients of association (e.g., Pearson correlations, standardized partial regression coefficients).

This may significantly distort results, especially in multivariate analyses, increasing the likelihood that researchers will commit type I and II errors in the same study. A type I error occurs in SEM analysis when an insignificant association is estimated as being significant (i.e., a “false positive”); a type II error occurs when a significant association is estimated as being insignificant (i.e., an existing association is “missed”).

The figure below shows a distribution of points typical of a J-curve pattern involving two variables, disrupted by uncorrelated error. The pattern, however, is modeled as a linear relationship. The line passing through the points is the best linear approximation of the distribution of points. It yields a correlation coefficient of .582. In this situation, the variable on the horizontal axis explains 33.9 percent of the variance of the variable on the vertical axis.

The figure below shows the same J-curve scatter plot pattern, but this time modeled as a nonlinear relationship. The curve passing through the points is the best nonlinear approximation of the distribution of the underlying J-curve, and excludes the uncorrelated error. That is, the curve does not attempt to model the uncorrelated error, only the underlying nonlinear relationship. It yields a correlation coefficient of .983. Here the variable on the horizontal axis explains 96.7 percent of the variance of the variable on the vertical axis.

WarpPLS transforms (or “warps”) J-curve relationship patterns like the one above BEFORE the corresponding path coefficients between each pair of variables are calculated. In multivariate analyses, this may significantly change the values of the path coefficients, reducing the risk that researchers will commit type I and II errors.

The risk of committing type I and II errors is particularly high when: (a) a block of latent variables includes multiple predictor variables pointing and the same criterion variable; (b) one or more relationships between latent variables are significantly nonlinear; and (c) the predictor latent variables are correlated, even if they clearly measure different constructs (suggested by low variance inflation factors).

How do I control for the effects of one or more demographic variables in an SEM analysis?

2010-02-14T07:21:00.000-08:00

As part of an SEM analysis using WarpPLS, a researcher may want to control for the effects of one ore more variables. This is typically the case with what are called “demographic variables”, or variables that measure attributes of a given unit of analysis that are (usually) not expected to influence the results of the SEM analysis.

For example, let us assume that one wants to assess the effect of a technology, whose intensity of use is measured by a latent variable T, on a behavioral variable measured by B. The unit of analysis for B is the individual user; that is, each row in the dataset refers to an individual user of the technology. The researcher hypothesizes that the association between T and B is significant, so a direct link between T and B is included in the model.

If the researcher wants to control for age (A) and gender (G), which have also been collected for each individual, in relation to B, all that is needed is to include the variables A and G in the model, with direct links pointing at B. No hypotheses are made. For that to work, gender (G) has to be included in the dataset as a numeric variable. For example, the gender "male" may be replaced with 1 and "female" with 2, in which case the variable G will essentially measure the "degree of femaleness" of each individual. Sounds odd, but works.

After the analysis is conducted, let us assume that the path coefficient between T and B is found to be statistically significant, with the variables A and G included in the model as described above. In this case, the researcher can say that the association between T and B is significant, “regardless of A and G” or “when the effects of A and G are controlled for”.

In other words, the technology (T) affects behavior (B) in the hypothesized way regardless of age (A) and gender (B). This conclusion would remain the same whether the path coefficients between A and/or G and B were significant, because the focus of the analysis is on B, the main dependent variable of the model.

Some special considerations and related analysis decisions usually have to be made in more complex models, with multiple endogenous variables, and also regarding the fit indices. These will be discussed in future posts.

Variance inflation factors: What they are and what they contribute to SEM analysis

2010-02-09T15:14:00.001-08:00

Variance inflation factors are provided in table format by WarpPLS for each latent variable that has two or more predictors. Each variance inflation factor is associated with one predictor, and relates to the link between that predictor and its latent variable criterion. (Or criteria, when one predictor latent variable points at two or more different latent variables in the model.)

A variance inflation factor is a measure of the degree of multicolinearity among the latent variables that are hypothesized to affect another latent variable. For example, let us assume that there is a block of latent variables in a model, with three latent variables A, B, and C (predictors) pointing at latent variable D. In this case, variance inflation factors are calculated for A, B, and C, and are estimates of the multicolinearity among these predictor latent variables.

Two criteria, one more conservative and one more relaxed, are recommended in connection with variance inflation factors. More conservatively, it is recommended that variance inflation factors be lower than 5; a more relaxed criterion is that they be lower than 10 (Hair et al., 1987; Kline, 1998). High variance inflation factors usually occur for pairs of predictor latent variables, and suggest that the latent variables measure the same thing. This problem can be solved through the removal of one of the offending latent variables from the block.

References:

Hair, J.F., Anderson, R.E., & Tatham, R.L. (1987). Multivariate data analysis. New York, NY: Macmillan.

Kline, R.B. (1998). Principles and practice of structural equation modeling. New York, NY: The Guilford Press.

Two new WarpPLS workshops in March and April 2010

2010-02-09T06:10:00.001-08:00

PLS-SEM.com will conduct two new online workshops on WarpPLS in March and April 2010!

For more information on these and other WarpPLS workshops please visit:

http://pls-sem.com/cgi-bin/p/awtp-custom.cgi?d=plssem&page=10403

Reading data into WarpPLS: An easy and flexible step

2010-02-04T16:08:00.000-08:00

Through Step 2, you will read the raw data used in the SEM analysis. While this should be a relatively trivial step, it is in fact one of the steps where users have the most problems with other SEM software. Often an SEM software application will abort, or freeze, if the raw data is not in the exact format required by the SEM software, or if there are any problems with the data, such as missing values (empty cells).

WarpPLS employs an import wizard that avoids most data reading problems, even if it does not entirely eliminate the possibility that a problem will occur. Click only on the “Next” and “Finish” buttons of the file import wizard, and let the wizard to the rest. Soon after the raw data is imported, it will be shown on the screen, and you will be given the opportunity to accept or reject it. If there are problems with the data, such as missing column names, simply click “No” when asked if the data looks correct.

Raw data can be read directly from Excel files, with extension “.xls”, or text files where the data is tab-delimited or comma-delimited. When reading from an “.xls” file, make sure that the spreadsheet file has only one worksheet – the worksheet that contains the data. If the spreadsheet has multiple worksheets, Step 2 will most likely fail. Raw data files, whether Excel or text files, must have indicator names in the first row, and numeric data in the following rows. They may contain empty cells, or missing values; these will be automatically replaced with column averages in a later step.

One simple test can be used to try to find out if there are problems with a raw data file. Try to open it with a spreadsheet software (e.g., Excel), if it is originally a text file; or try to create a tab-delimited text file with it, if it is originally a spreadsheet file. If you try to do either of these things, and the data looks messed up (e.g., corrupted, or missing column names), then it is likely that the original file has problems, which may be hidden from view. For example, a spreadsheet file may be corrupted, but that may not be evident based on a simple visual inspection of the contents of the file.

Project files in WarpPLS: Small but information-rich

2010-01-31T07:16:00.000-08:00

Project files in WarpPLS are saved with the “.prj” extension, and contain all of the elements needed to perform an SEM analysis. That is, they contain the original data used in the analysis, the graphical model, the inner and outer model structures, and the results.

Once an original data file is read into a project file, the original data file can be deleted without effect on the project file. The project file will store the original location and file name of the data file, but it will no longer use it.

Project files may be created with one name, and then renamed using Windows Explorer or another file management tool. Upon reading a project file that has been renamed in this fashion, the software will detect that the original name is different from the file name, and will adjust the name of the project file accordingly.

Different users of this software can easily exchange project files electronically if they are collaborating on a SEM analysis project. This way they will have access to all of the original data, intermediate data, and SEM analysis results in one single file.

Project files are relatively small. For example, a complete project file of a model containing 5 latent variables and 32 indicators will typically be only approximately 200 KB in size. Simpler models may be stored in project files as small as 50 KB.

Reflective and formative latent variable measurement in WarpPLS

2010-01-30T08:07:00.000-08:00

A reflective latent variable is one in which all the indicators are expected to be highly correlated with the latent variable score. For example, the answers to certain question-statements by a group of people, measured on a 1 to 7 scale (1=strongly disagree; 7 strongly agree) and answered after a meal, are expected to be highly correlated with the latent variable “satisfaction with a meal”. The question-statements are: “I am satisfied with this meal”, and “After this meal, I feel full”. Therefore, the latent variable “satisfaction with a meal”, can be said to be reflectively measured through two indicators. Those indicators store answers to the two question-statements. This latent variable could be represented in a model graph as “Satisf”, and the indicators as “Satisf1” and “Satisf2”.

A formative latent variable is one in which the indicators are expected to measure certain attributes of the latent variable, but the indicators are not expected to be highly correlated with the latent variable score, because they (i.e., the indicators) are not expected to be correlated with each other. For example, let us assume that the latent variable “Satisf” (“satisfaction with a meal”) is now measured using the two following question-statements: “I am satisfied with the main course” and “I am satisfied with the dessert”. Here, the meal comprises the main course, say, filet mignon; and a dessert, a fruit salad. Both main course and dessert make up the meal (i.e., they are part of the same meal) but their satistisfaction indicators are not expected to be highly correlated with each other. The reason is that some people may like the main course very much, and not like the dessert. Conversely, other people may be vegetarians and hate the main course, but may like the dessert very much.

If the indicators are not expected to be highly correlated with each other, they cannot be expected to be highly correlated with their latent variable’s score. So here is a general rule of thumb that can be used to decide if a latent variable is reflectively or formatively measured. If the indicators are expected to be highly correlated, then the measurement model should be set as reflective in WarpPLS. If the indicators are not expected to be highly correlated, even though they clearly refer to the same latent variable, then the measurement model should be set as formative.

Bootstrapping or jackknifing (or both) in WarpPLS?

2010-01-28T07:19:00.000-08:00

Arguably jackknifing does a better job at addressing problems associated with the presence of outliers due to errors in data collection. Generally speaking, jackknifing tends to generate more stable resample path coefficients (and thus more reliable P values) with small sample sizes (lower than 100), and with samples containing outliers. In these cases, outlier data points do not appear more than once in the set of resamples, which accounts for the better performance of jackknifing (see, e.g., Chiquoine & Hjalmarsson, 2009).

Bootstrapping tends to generate more stable resample path coefficients (and thus more reliable P values) with larger samples and with samples where the data points are evenly distributed on a scatter plot. The use of bootstrapping with small sample sizes (lower than 100) has been discouraged (Nevitt & Hancock, 2001).

Since the warping algorithms are also sensitive to the presence of outliers, in many cases it is a good idea to estimate P values with both bootstrapping and jackknifing, and use the P values associated with the most stable coefficients. An indication of instability is a high P value (i.e., statistically insignificant) associated with path coefficients that could be reasonably expected to have low P values. For example, with a sample size of 100, a path coefficient of .2 could be reasonably expected to yield a P value that is statistically significant at the .05 level. If that is not the case, there may be a stability problem. Another indication of instability is a marked difference between the P values estimated through bootstrapping and jackknifing.

P values can be easily estimated using both resampling methods, bootstrapping and jackknifing, by following this simple procedure. Run an SEM analysis of the desired model, using one of the resampling methods, and save the project. Then save the project again, this time with a different name, change the resampling method, and run the SEM analysis again. Then save the second project again. Each project file will now have results that refer to one of the two resampling methods. The P values can then be compared, and the most stable ones used in a research report on the SEM analysis.

References:

Chiquoine, B., & Hjalmarsson, E. (2009). Jackknifing stock return predictions. Journal of Empirical Finance, 16(5), 793-803.

Nevitt, J., & Hancock, G.R. (2001). Performance of bootstrapping approaches to model test statistics and parameter standard error estimation in structural equation modeling. Structural Equation Modeling, 8(3), 353-377.

How many resamples to use in bootstrapping?

2010-01-28T07:12:00.000-08:00

The default number of resamples is 100 for bootstrapping in WarpPLS. This setting can be modified by entering a different number in the appropriate edit box. (Please note that we are talking about the number of resamples here, not the original data sample size.)

Leaving the number of resamples for bootstrapping as 100 is recommended because it has been shown that higher numbers of resamples lead to negligible improvements in the reliability of P values; in fact, even setting the number of resamples at 50 is likely to lead to fairly reliable P value estimates (Efron et al., 2004).

Conversely, increasing the number of resamples well beyond 100 leads to a higher computation load on the software, making the software look like it is having a hard time coming up with the results. In very complex models, a high number of resamples may make the software run very slowly.

Some researchers have suggested in the past that a large number of resamples can address problems with the data, such as the presence of outliers due to errors in data collection. This opinion is not shared by the original developer of the bootstrapping method, Bradley Efron (see, e.g., Efron et al., 2004).

Reference:

Efron, B., Rogosa, D., & Tibshirani, R. (2004). Resampling methods of estimation. In N.J. Smelser, & P.B. Baltes (Eds.). International Encyclopedia of the Social & Behavioral Sciences (pp. 13216-13220). New York, NY: Elsevier.

Viewing and changing settings in WarpPLS

2010-01-28T07:05:00.000-08:00

The view or change settings window (see figure below, click on it to enlarge) allows you to select an algorithm for the SEM analysis, select a resampling method, and select the number of resamples used, if the resampling method selected was bootstrapping. The analysis algorithms available are Warp3 PLS Regression, Warp2 PLS Regression, PLS Regression, and Robust Path Analysis.

Many relationships in nature, including relationships involving behavioral variables, are nonlinear and follow a pattern known as U-curve (or inverted U-curve). In this pattern a variable affects another in a way that leads to a maximum or minimum value, where the effect is either maximized or minimized, respectively. This type of relationship is also referred to as a J-curve pattern; a term that is more commonly used in economics and the health sciences.

The Warp2 PLS Regression algorithm tries to identify a U-curve relationship between latent variables, and, if that relationship exists, the algorithm transforms (or “warps”) the scores of the predictor latent variables so as to better reflect the U-curve relationship in the estimated path coefficients in the model. The Warp3 PLS Regression algorithm, the default algorithm used by the software, tries to identify a relationship defined by a function whose first derivative is a U-curve. This type of relationship follows a pattern that is more similar to an S-curve (or a somewhat distorted S-curve), and can be seen as a combination of two connected U-curves, one of which is inverted.

The PLS Regression algorithm does not perform any warping of relationships. It is essentially a standard PLS regression algorithm, whereby indicators’ weights, loadings and factor scores (a.k.a. latent variable scores) are calculated based on a least squares minimization sub-algorithm, after which path coefficients are estimated using a robust path analysis algorithm. A key criterion for the calculation of the weights, observed in virtually all PLS-based algorithms, is that the regression equation expressing the relationship between the indicators and the factor scores has an error term that equals zero. In other words, the factor scores are calculated as exact linear combinations of their indicators. PLS regression is the underlying weight calculation algorithm used in both Warp3 and Warp2 PLS Regression. The warping takes place during the estimation of path coefficients, and after the estimation of all weights and loadings in the model. The weights and loadings of a model with latent variables make up what is often referred to as outer model, whereas the path coefficients among latent variables make up what is often called the inner model.

Finally, the Robust Path Analysis algorithm is a simplified algorithm in which factor scores are calculated by averaging all of the indicators associated with a latent variable; that is, in this algorithm weights are not estimated through PLS regression. This algorithm is called “Robust” Path Analysis, because, as with most robust statistics methods, the P values are calculated through resampling. If all latent variables are measured with single indicators, the Robust Path Analysis and the PLS Regression algorithms will yield identical results.

One of two resampling methods may be selected: bootstrapping or jackknifing. Bootstrapping, the software’s default, is a resampling algorithm that creates a number of resamples (a number that can be selected by the user), by a method known as “resampling with replacement”. This means that each resample contains a random arrangement of the rows of the original dataset, where some rows may be repeated. (The commonly used analogy of a deck of cards being reshuffled, leading to many resample decks, is a good one, but not entirely correct because in bootstrapping the same card may appear more than once in each of the resample decks.) Jacknifing, on the other hand, creates a number of resamples that equals the original sample size, and each resample has one row removed. That is, the sample size of each resample is the original sample size minus 1. Thus, the choice of number of resamples has no effect on jackknifing, and is only relevant in the context of bootstrapping.

Saving and using grouped descriptive statistics in WarpPLS

2010-01-28T06:58:00.000-08:00

When the “Save grouped descriptive statistics into a tab-delimited .txt file” option is selected, a data entry window is displayed. There you can choose a grouping variable, number of groups, and the variables to be grouped. This option is useful if one wants to conduct a comparison of means analysis using the software, where one variable (the grouping variable) is the predictor, and one or more variables are the criteria (the variables to be grouped).

The figure below (click on it to enlarge) shows the grouped statistics data saved through the “Save grouped descriptive statistics into a tab-delimited .txt file” option. The tab-delimited .txt file was opened with a spreadsheet program, and contained the data on the left part of the figure.

That data on the left part of the figure was organized as shown above the bar chart; next the bar chart was created using the spreadsheet program’s charting feature. If a simple comparison of means analysis using this software had been conducted in which the grouping variable (in this case, an indicator called “ECU1”) was the predictor, and the criterion was the indicator called “Effe1”, those two variables would have been connected through a path in a simple path model with only one path. Assuming that the path coefficient was statistically significant, the bar chart displayed in the figure, or a similar bar chart, could be added to a report describing the analysis.

Some may think that it is an overkill to conduct a comparison of means analysis using an SEM software package such as this, but there are advantages in doing so. One of those advantages is that this software calculates P values using a nonparametric class of estimation techniques, namely resampling estimation techniques. (These are sometimes referred to as bootstrapping techniques, which may lead to confusion since bootstrapping is also the name of a type of resampling technique.) Nonparametric estimation techniques do not require the data to be normally distributed, which is a requirement of other comparison of means techniques (e.g., ANOVA).

Another advantage of conducting a comparison of means analysis using this software is that the analysis can be significantly more elaborate. For example, the analysis may include control variables (or covariates), which would make it equivalent to an ANCOVA test. Finally, the comparison of means analysis may include latent variables, as either predictors or criteria. This is not usually possible with ANOVA or commonly used nonparametric comparison of means tests (e.g., the Mann-Whitney U test).

How is the warping done in WarpPLS?

2010-01-23T08:40:00.000-08:00

WarpPLS does linear and nonlinear analyses. That is, users can set WarpPLS to estimate parameters based on a standard linear algorithm, and without any warping. They can also choose one of two nonlinear algorithms, thus taking advantage of the warping capabilities of the software.

In nonlinear analyses, what WarpPLS does is relatively simple at a conceptual level. It identifies a set of functions F1(LVp1), F2(LVp2) … that relate blocks of latent variable predictors (LVp1, LVp2 ...) to a criterion latent variable (LVc) in this way:

LVc = p1*F1(LVp1) + p2*F2(LVp2) + … + E.

In the equation above, p1, p2 ... are path coefficients, and E is the error term of the equation. All variables are standardized. Any model can be decomposed into a set of blocks relating latent variable predictors and criteria in this way.

In the Warp2 mode, the functions F1(LVp1), F2(LVp2) ... take the form of U curves (also known as J curves); defaulting to lines, if the relationships are linear.

In the Warp3 mode, the functions F1(LVp1), F2(LVp2) ... take the form of S curves; defaulting to U curves or lines, if the relationships follow U-curve patterns or are linear, respectively.

S curves are curves whose first derivative is a U curve. Similarly, U curves are curves whose first derivative is a line. U curves seem to be the most commonly found in natural and behavioral phenomena. S curves are also found, but apparently not as frequently as U curves.

U curves can be used to model most of the commonly seen functions in natural and behavioral studies, such as logarithmic, exponential, and hyperbolic decay functions. For these common types of functions, S-curve approximations will usually default to U curves.

Other types of curves beyond S curves might be found in specific types of situations, and require specialized analysis methods that are typically outside the scope of structural equation modeling. Examples are time series and Fourier analyses. Therefore these are beyond the scope of application of WarpPLS.

Typically, the more the functions F1(LVp1), F2(LVp2) ... look like curves, and unlike lines, the greater is the difference between the path coefficients p1, p2 ... and those that would have been obtained through a strictly linear analysis.

So, what WarpPLS does is not unlike what a researcher would do if he or she modified predictor latent variables prior to the calculation of path coefficients using a function like the logarithmic function. For example, as in the equation below, where a log transformation is applied to LVp1.

LVc = p1*log(LVp1) + p2*LVp2 + … + E.

However, WarpPLS does that automatically, and for a much wider range of functions, since a fairly wide range of functions can be modeled as U or S curves. Exceptions are complex trigonometric functions, where the dataset comprises many cycles. These require different methods to be properly modeled, such as the Fourier analyses methods mentioned above, and are usually outside the scope of structural equation modeling (SEM; which is the analysis method that WarpPLS automates).

Often the path coefficients p1, p2 ... will go up in value due to warped analysis, but that may not always be the case. Given the nature of multivariate analysis, an increase in a path coefficient may lead to a decrease in a path coefficient for an arrow pointing at the same criterion latent variable, because each path coefficient in a block is calculated in a way that controls for the effects of the other predictor latent variables.

How are the model fit indices calculated by WarpPLS?

2010-01-23T08:11:00.001-08:00

The model fit indices calculated by WarpPLS are the following: average path coefficient (APC), average R-squared (ARS), and average variance inflation factor (AVIF).

They are discussed in the WarpPLS User Manual, which is available separately from the software, as a standalone document, on the WarpPLS web site.

The fit indices are calculated as their name implies, that is, as averages of: the (absolute values of the ) path coefficients in the model, the R-squared values in the model, and the variance inflation factors in the model. All of these are also provided individually by the software.

The P values for APC and ARS are calculated through re-sampling. A correction is made to account for the fact that these indices are calculated based on other parameters, which leads to a biasing effect – a variance reduction effect associated with the central limit theorem.

Typically the addition of new latent variables into a model will increase the ARS, even if those latent variables are weakly associated with the existing latent variables in the model. However, that will generally lead to a decrease in APC, since the path coefficients associated with the new latent variables will be low. Thus, the APC and ARS will counterbalance each other, and will only increase together if the latent variables that are added to the model enhance the overall predictive and explanatory quality of the model.

The AVIF index will increase if new latent variables are added to the model in such a way as to add multicolinearity to the model, which may result from the inclusion of new latent variables that overlap in meaning with existing latent variables. It is generally undesirable to have different latent variables in the same model that measure the same thing; those should be combined into one single latent variable. Thus, the AVIF brings in a new dimension that adds to a comprehensive assessment of a model’s overall predictive and explanatory quality.

As a final note, I would like to point out that the interpretation of the model fit indices depends on the goal of the SEM analysis. If the goal is to test hypotheses, where each arrow represents a hypothesis, then the model fit indices are of little importance. However, if the goal is to find out whether one model has a better fit with the original data than another, then the model fit indices are a useful set of measures related to model quality.

Why are pattern cross-loadings so low in WarpPLS?

2010-01-22T06:10:00.000-08:00

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 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.

References:

Kock, N. (2010). WarpPLS 1.0 User Manual. Laredo, Texas: ScriptWarp Systems.

Kock, N. (2011). WarpPLS 2.0 User Manual. Laredo, Texas: ScriptWarp Systems.

WarpPLS running on a Mac? Sure!

2010-01-21T10:59:00.000-08:00

When WarpPLS was first made available, I told a colleague of mine that it would probably run on a Mac without problems. Without trying, he said: No way Jose!

Then he really tried (using virtualization software, more below); it worked, and his response: Maybe u wuz royt eh!?

I have since received a few emails from WarpPLS users who own Mac computers. They run WarpPLS on those computers, without problems, even though WarpPLS was designed to be used with Windows.

How is that possible?

Those users have virtualization (a.k.a., virtual machine) software installed on their computers, which allow them to run WarpPLS on different types of computers, including Mac computers.

The virtualization software actually allows them to run the Windows operating system (typically the XP or 7 versions) on a Mac computer. They then install WarpPLS on the Windows virtual machine created by the virtualization software.

It seems that VMware is one of the most popular virtualization software systems in this respect.

March 2010 online workshop on WarpPLS

2010-01-11T17:07:00.001-08:00

PLS-SEM.com will conduct an online workshop on WarpPLS in March 2010!

The direct link to the workshop site is:

http://www.regonline.com/builder/site/Default.aspx?eventid=811252

The list of upcoming workshops is on:

http://pls-sem.com/cgi-bin/p/awtp-custom.cgi?d=plssem&page=10403

Possible installation problems and the MATLAB Compiler Runtime

2010-01-07T17:38:00.000-08:00

Most WarpPLS users do not seem to be having installation problems, but some users do. The likely cause is an incompatibility between the MATLAB Compiler Runtime and their computer's operating system setup.

(Update: Another possible cause of installation problems is one or more software applications that interfere with the proper running of WarpPLS. There have been reports from users suggesting that the following software application may do that: XLSTAT.)

The MATLAB Compiler Runtime is for MATLAB programs what the Java Runtime is for Java programs, and what the Microsoft .NET Framework is for .NET-based programs. That is, it is a set of executable modules that are called by executable files compiled using MATLAB.

WarpPLS is an executable file compiled using MATLAB, and thus requires the MATLAB Compiler Runtime (version 7.14) to run properly. Like many other runtime libraries, the MATLAB Compiler Runtime has originally been developed in C and C++.

In theory, the MATLAB Compiler Runtime should allow for a “compile once, run everywhere” approach to programming. That is, code that uses the MATLAB Compiler Runtime would be developed on one operating system, compiled, and then deployed, together with the MATLAB Compiler Runtime, to computers running any operating system.

This approach works well in theory, but not always in practice. This comment applies not only to MATLAB but also to Java and .NET applications – you are probably well aware of this if you are a Java or .NET programmer.

It is possible that a specific user’s computer configuration will prevent the proper installation of the MATLAB Compiler Runtime, by blocking certain operating system configuration changes (e.g., Windows registry changes), as a security measure.

Also, a user may not have administrator rights on a computer, or have limited administrator/power user rights, which may prevent certain operating system configuration changes necessary for the proper installation of the MATLAB Compiler Runtime.

Here are a few steps to take if you are having problems installing and running WarpPLS on a Windows computer:

1) Check the "Program Files" and the "Program Files (x86)" directories (assuming that the MATLAB Compiler Runtime is installed on the C drive), to see if one of the following folders is there.

C:\Program Files\MATLAB\MATLAB Compiler Runtime\v714\runtime\win32

C:\Program Files (x86)\MATLAB\MATLAB Compiler Runtime\v714\runtime\win32

If not, make sure that you are logged into your computer with full administrator rights, and reinstall the MATLAB Compiler Runtime. You can do that running the self-installing .exe file (177 MB) for WarpPLS, which includes the MATLAB Compiler Runtime. Or, contact your local IT support, and ask them to help you do so.

2) Go to the Command Prompt and type “PATH”, to see if one of the following paths shows on the list provided.

C:\Program Files\MATLAB\MATLAB Compiler Runtime\v714\runtime\win32

C:\Program Files (x86)\MATLAB\MATLAB Compiler Runtime\v714\runtime\win32

If not, on the Command Prompt, type one of the following commands, depending on the folder in which the MATLAB Compiler Runtime is installed:

set PATH=C:\Program Files\MATLAB\MATLAB Compiler Runtime\v714\runtime\win32;%PATH%

set PATH=C:\Program Files (x86)\MATLAB\MATLAB Compiler Runtime\v714\runtime\win32;%PATH%

Then type “PATH” again, and make sure that the new path has been added. This will change the Windows registry; a minor and pretty harmless change. If you are concerned about making registry changes yourself, or cannot do that due to limited rights or any other reason, please contact your local IT support, and ask them to help you do so.

3) Try to install WarpPLS on a different computer, and see if it runs well there.

This last step is annoying but important because there are certain computer configuration setups, or even malware, that may prevent the MATLAB Compiler Runtime from properly installing. This is rare, but does happen sometimes.

If you can install and run WarpPLS on one computer, but not on another, there may be a computer configuration or malware problem that is preventing you from doing so. If you have access to good-quality local IT support, I would suggest you contact your local IT support, and ask them to help you identify and solve the problem.

Solve colinearity problems in WarpPLS: YouTube video

2010-01-02T08:37:00.000-08:00

A new YouTube video for WarpPLS is available; please see link below.

http://www.youtube.com/watch?v=avPWO324E0g

The video shows how problems associated with latent variable colinearity, suggested by unstable path coefficients and high variance inflation factors, can be solved in a structural equation modeling (SEM) analysis using the software WarpPLS.

Note that this type of problem is different from problems related to indicators having low loadings and high cross-loadings. The problem here is associated with colinearity among latent variables.

Having said that, in many cases these two types of problems happen together: latent variable colinearity (often referred to as multicolinearity) and poor loadings/cross-loadings.

Happy New Year!

Solve indicator problems in WarpPLS: YouTube video

2009-12-29T17:55:00.000-08:00

A new WarpPLS YouTube video is available:

http://www.youtube.com/watch?v=G49aIm-14kU

This video shows how problems with indicators that load poorly on their latent variables, and that have high cross-loadings, can be solved in a structural equation modeling (SEM) analysis using the software WarpPLS.

Enjoy!

Warped paths become significant in WarpPLS: YouTube video

2009-12-24T18:48:00.000-08:00

Yet another new YouTube video is available for WarpPLS:

http://www.youtube.com/watch?v=_NQnVckeBb8

This video shows how path coefficients sometimes go up, and P values become significant, when warping takes place in a structural equation modeling (SEM) analysis using the software WarpPLS.

Since path coefficients are typically associated with hypotheses, which are supported if the paths are found to be significant, this will likely be music to many researchers' ears.

However, it is important to make two important points regarding this effect:

1. Path coefficients are not artificially inflated. They increase simply because the software is taking the nonlinear associations between latent variables into account when estimating path coefficients. Much like a researcher would apply a log(X) transformation to a predictor, before an ordinary regression analysis, if he or she knew that the predictor's relationship with a criterion variable Y was of the type Y=log(X).

2. Path coefficients do not always increase. Due to the nature of standardized partial regression coefficient calculation (the path coefficients, or betas, are standardized partial regression coefficients), when several predictor latent variables (LVs) point a one criterion LV, if one of the predictor LVs increases, some of the others predictor LVs may decrease as a result. In a sense, the predictor LVs "compete" for pieces of the space of variance explained for the criterion LV; if the predictor LVs are correlated, they tend "steal" variance space from each other (so to speak).

View nonlinear relationships in WarpPLS: YouTube video

2009-12-24T18:31:00.000-08:00

A new Youtube video is available for WarpPLS:

http://www.youtube.com/watch?v=lHrTxWmM43A

This video shows how one can view nonlinear and linear relationships estimated through a structural equation modeling (SEM) analysis using the software WarpPLS.

The video also highlights one fact that makes software like WarpPLS particularly useful - most relationships in nature are nonlinear. This includes relationships in biology, business, sociology, physics etc.

As you will see in this video, the software shows a table with the types of relationships, warped or linear, between latent variables that are linked in the model. The term “warped” is used for relationships that are clearly nonlinear, and the term “linear” for linear or quasi-linear relationships. Quasi-linear relationships are slightly nonlinear relationships, which look linear upon visual inspection on plots of the regression curves that best approximate the relationships.

Plots with the points as well as the regression curves that best approximate the relationships can be viewed by clicking on a cell containing a relationship type description. These cells are the same as those that contain path coefficients, in the path coefficients table.

The plots of relationships between pairs of latent variables provide a much more nuanced view of how each pair of latent variables is related. However, caution must be taken in the interpretation of these plots, especially when the distribution of data points is very uneven.

An extreme example would be a warped plot in which all of the data points would be concentrated on the right part of the plot, with only one data point on the far left part of the plot. That single data point, called an outlier, would influence the shape of the nonlinear relationship. In these cases, the researcher must decide whether the outlier is “good” data that should be allowed to shape the relationship, or is simply “bad” data resulting from a data collection error.

Change resampling method in WarpPLS: YouTube video

2009-12-23T09:40:00.000-08:00

A new Youtube video is available for WarpPLS:

http://www.youtube.com/watch?v=Hf-t70r7NKo

This video shows how one can conduct an SEM analysis using WarpPLS, save that analysis with a different project name, change the resampling method (from bootstrapping to jackknifing), and then redo the analysis.

At the end, the user has two project files, one with all of the P values calculated through bootstrapping, and the other with all of the P values calculated through jackknifing.

As noted in the WarpPLS User Manual, bootstrapping and jackknifing provide a good complement to each other in the context of warped PLS-based SEM.

Thus, some users may want to run two analyses of the same model, one with each resampling method, and use the results that are associated with the most stable resample path coefficients. These will typically be the ones with the lowest P values, since P values go up as the standard errors in the resample set go up. High resample standard errors are associated with instability. The instability itself often comes from outliers, which may drastically change the shape of a warped relationship in each resample.

Well, moving from statspeach to plain English, there are good theoretical reasons to recommend that users choose the most stable results (i.e., with the lowest P values) as the results that they will use in research reports, whether they are obtained with bootstrapping or jackknifing. The choice may be made individually, for each path coefficient. This should be disclosed to the readers of the report; a sentence like this would probably be enough: "Both bootstrapping or jackknifing were used in the analyses. The results reported here are those associated with the most stable resample estimates."

Windows Vista issue with WarpPLS 1.0: mclmcrrt710.dll was not found

2009-12-23T09:18:00.000-08:00

(Note: This post refers to version 1.0 of WarpPLS only.)

Some Windows Vista users have reported a problem installing and running WarpPLS 1.0. The error message looks like this:

"WarpPLS_1_0.exe - Unable To Locate Component: This application has failed to start because mclmcrrt710.dll was not found. Re-installing the application may fix this problem."

This seems to be a problem of incompatibility between the MATLAB Compiler Runtime and Windows Vista. The problem appears to occur in some computer configurations, but not all of them.

The solution below has worked so far for all users that contacted me with this problem:

- Check your "Program Files" directory, and make sure that you have this folder there: C:\Program Files\MATLAB\MATLAB Compiler Runtime\v710\runtime\win32.

- If yes (usually the case when this problem occurs), go to the Windows command prompt, and enter this:

set PATH=C:\Program Files\MATLAB\MATLAB Compiler Runtime\v710\runtime\win32;%PATH%

(See screen snapshot below).

- Then restart the computer, and try to run WarpPLS again.

- If only restarting the computer does not work; reinstall WarpPLS, and then try to run WarpPLS again.

Structural equation modeling made easy with WarpPLS: YouTube video

2009-12-06T06:34:00.000-08:00

Conducting a basic structural equation modeling (SEM) analysis using WarpPLS is relatively easy. The software takes the user through 5 steps, from project file creation to model building (using a graphical user interface) and viewing the results of the analysis. Take a look at the Youtube video below.

Choose the high quality (HQ) option for viewing the video clip above, if it is available (usually at the bottom of the video screen), and expand it to the full screen mode.

As you'll see at the end of the video, the project file is quite small, and it contains everything that is needed for the analysis. The file can be copied into a separate file, which the user can then open and change, by modifying the model for example, to conduct a different analysis.

Welcome to the WarpPLS blog!

2009-12-04T09:40:00.000-08:00

WarpPLS is a powerful new structural equation modeling (SEM) software. WarpPLS is commercialized by ScriptWarp Systems: www.scriptwarp.com.

Among other things, WarpPLS identifies nonlinear (or “warped”, hence the name of the software) relationships among latent variables and corrects the values of path coefficients accordingly. WarpPLS is arguably the first SEM software to do this.

Since most relationships between numeric variables are nonlinear, one could argue that WarpPLS finds the "real" relationships between latent variables in an SEM analysis. Typically path coefficients are increased, in some cases going from non-significant to significant at the P lower than 1 percent level.

The underlying algorithm employed by WarpPLS is partial least squares (PLS) regression, whose main characteristic is its ability to minimize multicollinearity among latent variables (even in the presence of overlapping manifest variables, or indicators).

Additionally, WarpPLS offers the following features, which are largely absent from most, if not all, PLS-based SEM software packages available today:

It estimates P values for path coefficients automatically, instead of providing only standard errors or T values, and leaving the user to figure out what the corresponding P values are.
It estimates several model fit indices, which have been designed to be meaningful in the context of PLS-based SEM analyses.
It automatically builds the indicators’ product structure underlying moderating relationships, and goes a little further. It shows those moderating relationships, related path coefficients, and related P values in a model graph as they should be shown – that is, as links between latent variables and direct links. The latter connect pairs of latent variables, while the former connect latent variables and direct links between pairs of latent variables.
It allows users to view scatter plots of each of the relationships among latent variables (when they are connected through arrows in the model), together with the regression curves that best approximate those relationships, and save those plots as .jpg files for inclusion in research reports.
It calculates variance inflation factor (VIF) coefficients for latent variable predictors associated with each latent variable criterion. This allows users to check whether some predictors should be removed due to multicolinearity (this feature is particularly useful with latent variables that are measured based on only 1 or a few indicators).

These are only a few of the new features offered by WarpPLS.

Ned Kock

WarpPLS developer