Presentation on theme: "Multiple Comparisons in Factorial Experiments"— Presentation transcript:
1 Multiple Comparisons in Factorial Experiments If Main Effects are significant AND Interactions are NOT significant:Use multiple comparisons on factor main effects (factor means).If Interactions ARE significant:1) Multiple comparisons on main effect level meansshould NOT be done as they are meaningless.2) Should instead perform multiple comparisons amongall factorial means of interest.
2 Multiple Comparisons in Factorial Experiments In addition, interactions must be decomposed to determine what they meanA significant interaction between two variables means that one factor value changes as a function of the other, but gives no specific informationThe most simple and common method of interpreting interactions is to look at a graph
3 Problems in factorial experiment In some two-factor experiments the level of one factor, say B, is not really similar with the other factor.There are multifactor experiments that address common economic and practical constraints encountered in experimentation with real systems.There is no link from any sites on one area to any sites on another area.Nested and Split-plot design
4 Cross and nested Factorials design The levels of factor A are said to be crossed with the level of factor B if every level of A occurs in combinations with every level of BFactorials designThe levels of factor B are said to be nested within the level of factor A if the levels of B can be divided into subsets (nests) such that every level in any given subset occurs with exactly one level of ANested design
5 Agricultural Field Trial Investigate the yield of a new variety of cropFactorsInsecticidesFertilizersExperimental UnitsFarmsFields within farmsExperimental Design ?Fertilizers can be applied to individual fields;Insecticides must be applied to an entire farmfrom an airplane
6 Agricultural Field Trial FarmsInsecticides applied to farmsOne-factor ANOVAMain effect: InsecticidesMSE: Farm-to-farm variability
7 Agricultural Field Trial Fertilizers applied to fieldsOne-factor ANOVAMain Effect: FertilizersMSE: Field-to-field variabilityFields
8 Agricultural Field Trial FarmsFieldsInsecticides applied to farms, fertilizers to fieldsTwo sources of variabilityInsecticides subject to farm-to-farm variabilityFertilizers and insecticides x fertilizers subject to field-to-field variability
9 Nested DesignFactorial design when the levels of one factor (B) are similar, but not identical to each other at different levels of another factor (A).b1b3a1a2b2b4
11 Nested DesignA factor B is considered nested in another factor, A if the levels of factor B differ for different levels of factor A.The levels of B are different for different levels of A.Synonyms indicating nesting:Hierarchical, depends on, different for, within, in, each
21 m-Stage Nested DesignTest statistics depend on the type of factors and the expected mean squares.Random.Fixed.
22 Expected Mean SquaresAssume that fixtures and layouts are fixed, operators are random – gives a mixed model (use restricted form).
23 Alternative AnalysisIf the need detailed analysis is not available, start with multi-factor ANOVA and then combine sum of squares and degrees of freedom.Applicable to experiments with only nested factors as well as experiments with crossed and nested factors.Sum of squares from interactions are combined with the sum of squares for a nested factor – no interaction can be determined from the nested factor.
25 Further phenomena in Experimental Design In a single factor experiment has different features, such as:Multi-locationsRepeated measurementsFactorial experiment can have either of these features:Two hierarchically nested factors, with additional crossed factors occurring within levels of the nested factorTwo sizes of experimental units, one nested within the other, with crossed factors applied to the smaller unitsSplit-Plot Design
26 Split-plot DesignThere are numerous types of split-plot designs, including the Latin square split plot design, in which the assignment of the main treatments to the main plots is based on a Latin square design.A split-plot design can be conceptualized as consisting of two designs: a main plot design and a subplot design.The main plot design is the protocol used to assign the main treatment to the main units. In a completely randomized split-plot design, the main plot design is a completely randomized design, in a randomized complete block design, by contrast, the main plot design is a RCBD.The subplot design in a split-plot experiment is a collection of a RCBD, where a is the number of main treatment. Each of these RCBDs has b treatments arranged in r blocks (main plots), where b is the number of sub treatment.
36 The Split-Plot Designa multifactor experiment where it is not practical to completely randomize the order of the runs.Example – paper manufacturingThree pulp preparation methods.Four different temperatures.The experimenters want to use three replicates.How many batches of pulp are required?
37 The Split-Plot DesignPulp preparation method 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.
38 The Split-Plot DesignEach replicate 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 9 batches of pulp (assuming three replicates).
40 The Split-Plot DesignThere are two levels of randomization restriction.Two levels of experimentation
41 Experimental Units in Split Plot Designs Possibilities for executing the example split plot design.Run separate replicates. Each pulp prep method (randomly selected) is tested at four temperatures (randomly selected).Large experimental unit is four pulp samples.Smaller experimental unit is a an individual sample.If temperature is hard to vary select a temperature at random and then run (in random order) tests with the three pulp preparation methods.Large experimental unit is three pulp samples.
42 The Split-Plot DesignAnother way to view a split-plot design is a RCBD with replication.Inferences on the blocking factor can be made with data from replications.
43 The Split-Plot Design Model and Statistical Analysis Sum of squares are computed as for a three factor factorial design without replication.