lect9 - IE410 Design of Experiments Notes for Lecture 9:...

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IE410 Design of Experiments Notes for Lecture 9: Remedial Measures What do we do if the assumptions are not met? Depending on which assumptions are violated, one has a few different options.
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Often, a single cause (reason for the assumptions not being valid) may show up in several diagnostics. For example, non-constant variance often results in the residuals appearing non- Normal as well. Thus one remedial measure may cure more than one problem.
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Transforms Variance stabilizing transforms are probably the most useful remedial measure, perhaps because non-constant variance is also quite common, indeed it is often quite natural. The most common phenomenon is that the variance increases with the mean.
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Example 1. Use a 50 foot long tape measure to measure: i) something 1 inch long n=5 times ii) something 49 feet long n=5 times From each sample of n=5, calculate the sample variance
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Would you not expect σ to be bigger for the 49 foot measure than for the 1 inch measure? (If not, then remember that tapes can stretch).
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Example 2 . We are testing two paper airplanes. Each plane is flown n=5 times. We measure Y= time aloft Suppose plane one averages 5 seconds aloft, while plane 2 averages 5 minutes aloft. Over the 5 runs, which do you expect will exhibit more variation?
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A Common Occurrence: We have in general σ Y = f( μ Y ) in particular, suppose σ Y μ Y α Now consider transforming the data as follows:
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Y = original data Y * = transformed data using the transform: Y * = Y λ for λ 0 Y * = log(Y) for λ = 0 We see soon that: σ Y μ Y λ + α -1
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Thus setting : λ = 1 - α will yield Y * with constant variance.
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Mathematical Details Suppose σ Y 2 = g( μ Y ) and we propose the transform Y * = H(Y) Do a first order Taylor series expansion about the point Y= μ :
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Y * = H( μ ) + (Y- μ )H'( μ ) where H'( μ ) = d/dy H(Y) evaluated at Y= μ So V(Y * ) = V[H( μ )]+V[(Y- μ )H'( μ )] + Rem = 0 + [H'( μ )] 2 V(Y)
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lect9 - IE410 Design of Experiments Notes for Lecture 9:...

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