What is the difference between "Unweighted", "Weighted" and "Least Squares" Means? These concepts have always confused me. In terms of DOE, I think the problem arises when the design is "unbalanced". So, 


*

*what are their differences? 

*Why do you choose one over the other/reasons?

*In one-way ANOVA, does using (can you?) LS means makes sense or LS means is only for two-way ANOVA and onwards?
For ex: In a simple study, say, I am comparing if there is a diff. in height between M and F. I have an unbalanced design, say, I have more M than F (I selected equal sample, but some ppl died and this is what I have to use)  Does having LS means means anything in this scenario? Or LS means= unweighted in this case?
Can anyone please clarify in a simple and intuitive manner?
 A: This question is very large in scope. Here is a short answer:


*

*Unweighted least squares minimizes the mean squared error of the residuals using a linear combination of covariates to estimate the conditional mean of the response. Weighted least squares is an extension of least squares which minimizes the weighted residuals. As a discrepancy, the weighted least squares procedure is somewhat different than in most software where one simply applies a vector of weights; the weights are actually a matrix. This accounts not only for increasing or decreasing the influence of specific observations, but also their pairwise influence. For instance, if two observations are given an off diagonal (probability) weight of 1, then both of these observations are effectively being averaged together. Least Squares Means, commonly called the LSMeans procedure in SAS, is just a method for obtaining contrasts or model parameters in a least squares regression model (weighted or unweighted). 

*Unweighted least squares is appropriate when the sample is obtained by simple random sampling (SRS) from a population of interest, or when the residuals are verifiably independent and identically distributed. What the predictions and coefficients estimate is a population averaged mean and mean difference respectively. Weighting is used to control for non-independence or to standardize the data to a population of interest when the data are not obtained via SRS. Weighting can control for correlated data (family / twin studies) or confounding (inverse probability or propensity score weighting) or missing data and or stratified sampling (Horvitz Thompson Estimator). 

*You can absolutely use LS means for 1 way ANOVA. The analogous procedure for a linear regression is simply a single categorical exposure regressed on a continuous outcome. A downside of this approach is that each coefficient summarizes the mean differences in terms of a difference from a "reference" group, when the specification of such a reference group may be nonsensical. For instance, in a study of education as a predictor of happiness, participants may be grouped by lowest to highest levels of education. The prevalence of non-high school graduates may be very low in prevalence, so such a referent group is not in any intuitive sense "referent". LS means allows you to estimate a variety of contrasts, such as the ANOVA contrast which estimates a coefficient for each education group as a difference from the grand mean. 
