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Experiments with both nested and “crossed” or factorial factors •Example 14-2 Nested and Factorial Factors (Hand insertion of electronic components) •Three assembly fixtures •Two workplace layouts •Four operators for each fixture-layout combination •It’s difficult to use the same (4) operators for each layout because of location • Four operators for layout 1 • Four operators for layout 2 •Assume that fixtures and layouts are fixed, operators are random – gives a mixed model (use restricted form) •Fixtures and layouts: factorial •Operators: nested within layouts 1 Table 14-9 Assembly Time Data for Example 14-2 yijkl i 1,2,3 j 1,2 i j k ( j ) ( )ij ( )ik ( j ) (ijk )l k 1,2,3,4 l 1,2 2 Example 14-2 – Expected Mean Squares 3 Example 14-2 – Minitab Analysis Table 14-12 Minitab Balanced ANOVA Analysis of Example 14-2 Using the Restricted Model 4 The Split-Plot Design • Text reference, Section 14-4 page 540 • The split-plot is a multifactor experiment where it is not possible to completely randomize the order of the runs • Example – paper manufacturing – – – – – Three pulp preparation methods Four different temperatures Each replicate requires 12 runs The experimenters want to use three replicates How many batches of pulp are required? 5 The Split-Plot Design • Pulp preparation methods is a hard-to-change factor • Consider an alternate experimental design: – In replicate 1, select a pulp preparation method, prepare a batch – Divide the batch into four sections or samples, and assign one of the temperature levels to each – Repeat for each pulp preparation method – Conduct replicates 2 and 3 similarly 6 The Split-Plot Design • Each replicate (sometimes called blocks) has been divided into three parts, called the whole plots • Pulp preparation methods is the whole plot treatment • Each whole plot has been divided into four subplots or split-plots • Temperature is the subplot treatment • Generally, the hard-to-change factor is assigned to the whole plots • This design requires only 9 batches of pulp (assuming three replicates) 7 The Split-Plot Design Model and Statistical Analysis yijk i j ( )ij k ( )ik ( ) jk ( )ijk ijk i 1, 2,..., r j 1, 2,..., a k 1, 2,..., b Table 14-15 Expected Mean Square Derivation for Split-Plot Design 8 Split-Plot ANOVA 9 Split-Plot ANOVA Table 14-16 Analysis of Variance for the Split-Plot Design Using Tensile Strength Data from Table 14-14 Calculations follow a three-factor ANOVA with one replicate Note the two different error structures; whole plot and subplot 10 Alternate Model for the Split-Plot yijk i j ( )ij k ( ) jk ijk i 1, 2,..., r j 1, 2,..., a k 1, 2,..., b 11 The Agriculture Heritage of Split-Plot Design • Whole plots: large areas of land • Subplots: smaller areas of land within large areas • Example: Effects of variety, field, and fertilizer on the growth of a crop • One variety is planted in a field (a whole plot) • Each field is divided into subplots with each subplot is treated with one type of fertilizer • Crop varieties: main treatments • Fertilizers: subtreatments 12