web example
http://onbiostatistics.blogspot.com/2009/04/leastsquares-means-marginal-means-vs.html
Data
Model with only main effects (JMP output):
Center
Level
1
2
Least Sq Mean
4.00
6.00
Mean
4.00
6.00
Model with only main effects (JMP output): Trt
Level
A
B
Least Sq Mean
5.00
5.00
Mean
4.80
5.20
Model with interaction in JMP (the right model to
use in JMP): Center
Level
1
2
Least Sq Mean
4.25
6.25
Mean
4.00
6.00
Model with interaction in JMP (the right model to
use in JMP): Trt
Level
A
B
Least Sq Mean
5.25
5.25
Mean
4.80
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.
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