Definition of SSE and SSO when calculating Q-Square

Frequently asked questions about PLS path modeling.
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schuelerfabian
PLS Junior User
Posts: 1
Joined: Mon Aug 17, 2020 9:38 am
Real name and title: Fabian Schüler - PhD Candidate

Definition of SSE and SSO when calculating Q-Square

Post by schuelerfabian »

Dear SmartPLS community members,

while writing the documentation of my model evaluation with SmartPLS I have some problems with the calculation and definition of SSO and SSE in SmartPLS 3.3.0.
While looking at the formula for SSO in different papers I find that SSO should be the sum of squares of observations for each variable. When looking at the SmartPLS output I find that SSO is always calculated as: Sample Size * Number of Indicators. (For example in the SmartPLS Book (German Version Page 188) SSO for ATTR is 1032,00 which equals 3 * 344.
So is it still true that SSO is the sum of squares of observations? Finally this leads to another problem, when calculating Q-Square. In my understanding Q-Square represents a comparison of the prediction errors of the model and the prediction errors of a simple estimation by the mean values? Where in the formula is the representation of the mean value estimation, when we calculate Q-Square = (1 - SSE/SSO)?
Can someone please explain to me if I miss any intermediate steps or how SSO is exactly calculated so that it is exactly 1 for every data point of the latent variable (Sample Size * Number of Indicators)?

I would be very grateful for every explanation.

Best regards
Fabian
jmbecker
SmartPLS Developer
Posts: 1129
Joined: Tue Mar 28, 2006 11:09 am
Real name and title: Dr. Jan-Michael Becker

Re: Definition of SSE and SSO when calculating Q-Square

Post by jmbecker »

1) Yes, SSO is equal to the number of observations * indicators, because we have standardized data in PLS which is used by the blindfolding procedure. For standardized data the SSO is equal to the number of observations. Every observation is a scaled deviation from the mean (scaled by the standard deviation). Accordingly, it does not imply that the SSO is exactly 1 for each observation. It depends on the original value and its difference from the mean. Therefore, the SSO represents the mean value prediction.

2) In contrast, the SSE is the prediction error when using the model prediction.
Dr. Jan-Michael Becker, University of Cologne, SmartPLS Developer
Researchgate: https://www.researchgate.net/profile/Jan_Michael_Becker
GoogleScholar: http://scholar.google.de/citations?user ... AAAJ&hl=de
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