# lect17 - IE 410 Lecture 17 More on Factorial Experiments...

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IE 410 Lecture 17: More on Factorial Experiments Our last topic is "unbalanced data" in factorial designs That is, unequal numbers of observations in cells

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Case 1. "Proportional data" In some cases in which the number of observations in each cell satisfies the following constraint: n(ij) = n(i.)n(.j)/n(. .) Then SST can still be decomposed. Examples are:
n(ij) 4 4 2 | 10 2 2 1 | 5 n(.j) 2 2 1 | 5 ------------ 8 8 4 n..=20 n(i.)

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Equations for SS are quite similar to the usual one's except that n(ij)'s appear instead of n's. Also the df for SSE must be appropriately calculated
Case 2. Missing Data Estimating missing values is easy in factorial designs since the estimate which minimizes SSE is clearly the cell average Y (ij.) This will be ok if only a couple of values are missing.

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Don't forget to subtract an error df for each missing value.
Case 3. Setting data aside Suppose you had five data points in every cell except for one cell that had 6 data points. You could set aside one of the points from that cell. But which one? Pick the smallest obs. to set aside right?

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Clearly not as this can bias the results. You should randomly pick which point to set aside.
Case 4. Yate's method of unweighted means. This method works in more general cases of unbalanced data It is an approximate method. It is really cool method.

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1. Take the average of each cell. 2. Do calculations on this "averaged" data as if it was a factorial design with one obs. per cell. (recall we can get everything but SSE which has zero df)
3. Estimate MSE as i j k [ Y(ijk) - Y (ij.) ] 2 / (N. . - ab) MSE estimates V(Yijk). However we did not use Yijk data in our "averaged" analysis.

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Rather we used Y (ij.) values to get the MSA MSB and MSAB
Now V( Y (ij.)) = σ 2 /n(ij) thus we could use MSE/n(ij) for testing. The only problem is that n(ij) varies from cell to cell.

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Thus we shall use the average variance instead. That is: AveVar( Y (ij.)) = i j [ σ 2 /n(ij)]/ab or MSE' = (MSE/ab)* i j (1/n(ij))
Thus our approximate test is: F 0 = MSA/MSE' compared to F α , a-1, n. .-ab

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I think we are a little ahead of the schedule, and thus I would now like to cover the following two topics, albeit quite quickly: Nested and Hierarchical Designs Rules for Expected Mean Squares
Consider the following: Example: A Company deals with 3 suppliers of chemicals. Each Company ships chemicals in Batches.

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## This note was uploaded on 08/06/2008 for the course IE 410 taught by Professor Storer during the Fall '04 term at Lehigh University .

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lect17 - IE 410 Lecture 17 More on Factorial Experiments...

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