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## LS Means

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**LS Means**web example http://onbiostatistics.blogspot.com/2009/04/least-squares-means-marginal-means-vs.html**Model with only main effects (JMP output): Center**Level Least Sq Mean Mean 1 4.00 4.00 2 6.00 6.00**Model with only main effects (JMP output): Trt**Level Least Sq Mean Mean A 5.00 4.80 B 5.00 5.20**Model with interaction in JMP (the right model to use in**JMP): Center Level Least Sq Mean Mean 1 4.25 4.00 2 6.25 6.00**Model with interaction in JMP (the right model to use in**JMP): Trt Level Least Sq Mean Mean A 5.25 4.80 B 5.25 5.20**How are Means calculated? (from webpage)**• The mean value for Treatment A is simply the summation of all measures divided by the total number of observations: • Mean for treatment A = 24/5 = 4.8) • Mean for treatment B = 26/5 = 5.2. • Mean for treatment A > Mean for treatment B.**How are LS Means calculated? (again, webpage)**• Table 2 shows the calculation of least squares means. • First step is to calculate the means for each cell of treatment and center combination. • The mean 9/3=3 for treatment A and center 1 combination • 7.5 for treatment A and center 2 combination • 5.5 for treatment B and center 1 combination • 5 for treatment B and center 2 combination.**LS Means continued (again from webpage)**• After the mean for each cell is calculated, the least squares means are simply the average of these means. • For treatment A, the LS mean is (3+7.5)/2 = 5.25 • For treatment B, it is (5.5+5)/2=5.25 • The LS Mean for both treatment groups are identical.**Ahh, that’s fine, but what about empty cells?**• First fit a two-way model with least squares • Then estimate the predicted value for that empty cell • Put that value in as though it were “real” data, this is called “imputing” the value • Redo the analysis and voila, you get the LS Means!!! • One can do this iteratively and this is called “multiple imputation”, but that gets used elsewhere in Statistics (i.e. you don’t have to worry about it)**Predicted values for empty cells**• Predicted values for empty cells are obtained with a Regression model. • With Regression, you can obtain “predicted values” even where there is no data point. • You can do the same thing in ANOVA by using Regression with Dummy Variables. • Dummy variables are “indicator variables’ for class variables.