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ANOVA - use the least squares means

By Nina Julius | Mar 01, 2017 08:43AM CET

Dear XLSTAT-team,

regarding a statistical analysis of my sensory data I have a question.
I had twelve panelists evaluating a lots of products (49) regarding 17 attributes. Due to the fact that this took several weeks until all products had been evaluated, I hadn't always the same panelists and amount of panelists. Now I'm trying to compare the products (not all 49 in one group, but e.g. 9 products in one group, 4 in another group,....). When coming to the "multiple comparisons sheet" I found the option "use least squares means". I thought that this is an option when having not always the same panelists evaluating the products.
Is this right to choose it in this (my) situation? And what is mathematically behind this?
Thanks for your reply!
Kind regards,




By Sébastien | Mar 01, 2017 10:50AM CET | XLSTAT Agent

Hi Nina,

Least squares means play an important role when you are dealing with unbalanced multi-way ANOVA’s. In the case of a 1-way ANOVA, least squares means and observed means are the same.

To illustrate things simply, consider you have 2 judges testing an attribute on 2 products, with the following number of replicates:

Judge 1 x Product A: 5 replicates
Judge 1 x Product B: 4 replicates
Judge 2 x Product A: 6 replicates
Judge 2 x Product B: 3 replicates

This is an unbalanced design, as the number of replicates is not the same across the category combinations.

Using the regular observed mean:
Mean of Product A is the mean of the 11 Product A observations

Using the least squares mean:
Mean of Product A is the mean of two numbers:
1) The mean of the 5 replicates of Product A tested by Judge 1
2) The mean of the 6 replicates of Product A tested by Judge 2

This latter estimation is often assumed to be closer to reality as it gives the same weight to both judges, whereas the first estimation gives a weight of 5 for judge 1 and a weight of 6 to judge 2.

In balanced designs, observed means and least squares means are the same.

I hope this helps.


Jean Paul

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