lect17

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:
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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
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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.
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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.
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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)
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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
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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))
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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
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Consider the following: Example: A Company deals with 3 suppliers of chemicals. Each Company ships chemicals in Batches.
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lect17 - IE 410 Lecture 17: More on Factorial Experiments...

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