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Lec 05b - Sig with Reps

# Lec 05b - Sig with Reps - WHAT EFFECTS ARE SIGNIFICANT in...

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WHAT EFFECTS ARE SIGNIFICANT in 2 k Factorial Experiments? Model Y b b x b x b x N o Y = + + + + + + 1 1 2 2 12 12 2 0 ... ... ( , ) ε ε σ Significance of b’s rests on estimate of σ Y 2 Which b’s are significantly different than 0? Two Cases one replicate two or more replicates

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Determining Which Effects are Significant when there are two or more replicates Model & experiment Y b b x b x b x N o Y = + + + + + + 1 1 2 2 12 12 2 0 ... ... ( , ) ε ε σ m= 2 k treatments; n replicates; N = m n runs Estimate σ Y 2 with S p 2 1. Compute estimate of variance for treatment i = 1…m ( 29 s i i i i n i i i n Y Y n where Y Y n 2 2 1 1 1 = - - = = = 2. Pool the m estimates S = S m p 2 i 2 i=1 m 2
Assumption Assume factors only affect the mean of the outcome – not the variance In notation: for treatment i = 1…m σ σ Y Y i 2 2 = 3

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Example: a pooled estimated of σ y 2 2 2 Factorial Experiment with 2 replicates trt # I x 1 x 2 x 1 x 2 Y i1 Y i2 Y i S i 2 1 -1 -1 +1 .50 .46 .480 .0008 2 +1 -1 -1 .42 .37 .395 .00125 3 -1 +1 -1 .33 .30 .315 .00045 4 +1 +1 +1 .11 .16 .135 .00125 Compute S i 2 , a mini estimate of σ Y 2 ex i=1 [(.50 - .48) 2 + (.46 - .48) 2 ] / (2-1) = .0008 S 1 = .028 Compute S p 2 , an estimate for σ Y 2 S p 2 = [ .0008 + .00125 + .00045 + .00125]/4 4
= .0009375 and S p = .031 5

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Use S p 2 to Compute Confidence Intervals for b 0 = Y m= 2 k treatments; n replicates; N = m n runs estimate b Y 0 = V Y N S N Y p ( ) = σ 2 2 estimate by 100(1- α )% Confidence Interval for b 0 N S t Y Y SD t Y p ) ( , 2 / ( ) ( , 2 / ( ) ( 1)m - n = d.f.
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