# Can anyone provide a peer reviewed reference for the calculation of least squares means as implemented in the R package lsmeans?

I am using the lsmeans package from the R programming language for follow up analyses of a linear mixed model. However, my target journal does not generally use these methods and I would like to have strong references to back my general approach as well as a general understanding of what is being calculated in the case of a linear mixed model using participants as a random effect. The documentation for the lsmeans package points to a reference page hosted by SAS. Can anyone point me to a peer-reviewed article or book that validates the procedures described there and used by lsmeans for calculating these values?

Thanks in advance for your assistance. Further thanks to the developers of this great package.

• I'd really be interested if somebody has an answer. That old PROC HARVEY reference is the best I could come up with. Oct 30 '14 at 16:25
• As you are a psychologist I am really interested to know which journal you are talking about. Did the reviewers demand it or are you preemptively trying to avoid criticism? I would be really surprised if reviewers would pick up on this. Nov 5 '14 at 17:54
• @Henrik, I am in psychology and the target is the Journal of Autism and Developmental Disorders. I haven't been asked for the references, but have used analyses like these at other journals only to have them rejected in favour of ANOVA, presumably due to relatively low adoption of these techniques in our discipline thus far. Nov 6 '14 at 21:08
• @rvl: A meta-question: how is it possible that there is no authoritative reference? Is it because you actually developed the techniques implemented in lsmeans and they have never been written up? Or is it because you combined many different existing techniques in one package, so there is no one reference covering it? Sorry if it is a naive question. Nov 7 '14 at 13:46
• In cardiovascular medicine, authors frequently refer to:Verbeke G, Molenberghs G. Linear Mixed Models for Longitudinal Data. New York City, NY, Springer, 2000. I rarely see anyone refer to the exact package being used. E.g: care.diabetesjournals.org/content/37/7/1797.full#ref-20 Nov 9 '14 at 15:37

The history of the least squares mean, its appearance in SAS, and its interpretation is discussed in Searle, Milliken, and Speed (1979).

Some discussion of the concepts around least squares means (population marginal means) is found in Searle, Speed, and Milliken (1980). Earliest mention of the concept that they note is Damon et al (1959). They provide some other references, but I do not have access to the full article.

The initial implementations of the calculation seem to have been worked out explicitly in Harvey (1960) and some subsequent publications, including but probably not limited to Harvey (1977), Goodnight (1979), Harvey (1982), and Goodnight and Harvey (1997).

It looks like the computation routines were first developed as LSML 76 and LSML GP before the user-contributed PROC HARVEY.

References

J.H. Goodnight (1979) A tutorial on the SWEEP operator. The American Statistician 33 (3): 149-159.

J.H. Goodnight and W.R. Harvey (1997) Least squares means in the fixed effects general model. SAS Technical Report R-103. SAS Institute Inc.

W.R. Harvey (1960) Least-squares analysis of data with unequal subclass numbers. USDA National Agricultural Library ARS-20-8.

Harvey, W.R (1977) User's guide for LSML 76. Mixed model least-squares and maximum likelihood computer program. Ohio State Univ., Colarubus (Mimeo).

W.R. Harvey (1982) Mixed model capabilities of LSML76. Journal of Animal Science 54:1279-1285.

S.R. Searle, F.M. Speed, and G.A. Milliken (1980) Population marginal means in the linear model: An alternative to least squares means. The American Statistician 34 (4):216-221.

Searle, S. R., Milliken, G. A., and Speed, F. M. (1979). Expected Marginal Means in the Linear Model. Cornell University Biometrics Unit Technical Reports: Number BU-672-M.

• (+1) I liked Searle & Speed & Milliken (and it can certainly serve as a decent reference, as OP wanted), but they write only about usual fixed effect models. Is there really nothing about mixed models, apart from Harvey's user guides? Amazing. Nov 11 '14 at 22:49
• The concept carries over very naturally to mixed models. We still define the LS means (or PMMs if you prefer) in terms of the fixed effects. The only thing that's different is the covariance matrix. That said, one could talk about generalizations based on predictions at various levels of he random effects, conditional on the random effects up to that level. That'd be interesting, but the lsmeans package does not delve into that, at least at this point. Nov 15 '14 at 19:17
• Also, note the 1982 Harvey reference. Nov 15 '14 at 19:55