This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Unit 2: Experiments with a Single Factor: One Way ANOVA Sources : Sections 1.6 to 1.9, additional materials (in these notes) on random effects . Oneway layout with fixed effects. Multiple comparisons. Quantitative factors and orthogonal polynomials. Residual analysis. Oneway layout with random effects. 1 Oneway layout and ANOVA: An Example Reflectance data in pulp experiment: each of four operators made five pulp sheets; reflectance was read for each sheet using a brightness tester. Randomization : assignment of 20 containers of pulp to operators and order of reading. Table 1: Reflectance Data, Pulp Experiment Operator A B C D 59.8 59.8 60.7 61.0 60.0 60.2 60.7 60.8 60.8 60.4 60.5 60.6 60.8 59.9 60.9 60.5 59.8 60.0 60.3 60.5 Objective : determine if there are differences among operators in making sheets and reading brightness. 2 Model and ANOVA Model : y i j = + i + i j , i = 1 ,..., k ; j = 1 ,..., n i , where y i j = j th observation with treatment i , i = i th treatment effect, i j = error, independent N ( , 2 ) . Model fit: y i j = + i + r i j = y +( y i y )+( y i j y i ) , where . means average over the particular subscript. ANOVA Decomposition : k i = 1 n i j = 1 ( y i j y ) 2 = k i = 1 n i ( y i y ) 2 + k i = 1 n i j = 1 ( y i j y i ) 2 . 3 FTest ANOVA Table Degrees of Sum of Mean Source Freedom ( d f ) Squares Squares treatment k 1 SST = k i = 1 n i ( y i y ) 2 MST = SST / d f residual N k SSE = k i = 1 n i j = 1 ( y i j y i ) 2 MSE = SSE / d f total N 1 k i = 1 n i j = 1 ( y i j y ) 2 The F statistic for the null hypothesis that there is no difference between the treatments, i.e., H : 1 = = k , is F = k i = 1 n i ( y i y ) 2 / ( k 1 ) k i = 1 n i j = 1 ( y i j y i ) 2 / ( N k ) = MST MSE , which has an F distribution with parameters k 1 and N k . 4 ANOVA for Pulp Experiment Degrees of Sum of Mean Source Freedom ( d f ) Squares Squares F operator 3 1.34 0.447 4.20 residual 16 1.70 0.106 total 19 3.04 Prob ( F 3 , 16 &gt; 4 . 20 ) = 0.02 = p value, thus declaring a significant operatortooperator difference at level 0.02. Further question: among the 6 = ( 4 2 ) pairs of operators, what pairs show significant difference? Answer: Need to use multiple comparisons. 5 Multiple Comparisons For one pair of treatments, it is common to use the t test and the t statistic t i j = y j y i 1 / n j + 1 / n i , where n i = number of observations for treatment i , 2 = RSS/df in ANOVA; declare treatments i and j different at level if  t i j  &gt; t N k , / 2 ....
View
Full
Document
 Spring '08
 peter

Click to edit the document details