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 means should NOT be done as they are meaningless. 2) Should instead perform multiple comparisons among all factorial means of interest.

Multiple Comparisons in Factorial Experiments In addition, interactions must be decomposed to determine what they mean A significant interaction between two variables means that one factor value changes as a function of the other, but gives no specific information The most simple and common method of interpreting interactions is to look at a graph

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

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 B Factorials design The 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 A Nested design

Agricultural Field Trial Investigate the yield of a new variety of crop Factors Insecticides Fertilizers Experimental Units Farms Fields within farms Experimental Design ? Fertilizers can be applied to individual fields; Insecticides must be applied to an entire farm from an airplane

Agricultural Field Trial Farms Insecticides applied to farms One-factor ANOVA Main effect: Insecticides MSE: Farm-to-farm variability

Agricultural Field Trial Fertilizers applied to fields One-factor ANOVA Main Effect: Fertilizers MSE: Field-to-field variability Fields

Agricultural Field Trial Farms Fields Insecticides applied to farms, fertilizers to fields Two sources of variability Insecticides subject to farm-to-farm variability Fertilizers and insecticides x fertilizers subject to field-to-field variability

Nested Design Factorial design when the levels of one factor (B) are similar, but not identical to each other at different levels of another factor (A). b1 b3 a1 a2 b2 b4

Nested Design

Nested Design A 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

Examples - Nested

Examples - Nested

Examples - Crossed

Examples - Crossed

Examples - Nested

Two-Stage Nested Design Statistical Model and ANOVA

Two-Stage Nested Design Statistical Model and ANOVA

Residual Analysis Calculation of residuals.

m-Stage Nested Design

m-Stage Nested Design Test statistics depend on the type of factors and the expected mean squares. Random. Fixed.

Expected Mean Squares Assume that fixtures and layouts are fixed, operators are random – gives a mixed model (use restricted form).

Alternative Analysis If 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.

Alternative Analysis

Further phenomena in Experimental Design In a single factor experiment has different features, such as: Multi-locations Repeated measurements Factorial experiment can have either of these features: Two hierarchically nested factors, with additional crossed factors occurring within levels of the nested factor Two sizes of experimental units, one nested within the other, with crossed factors applied to the smaller units Split-Plot Design

Split-plot Design There 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.

Split-Plot Design Whole-Plot Experiment : Whole-Plot Factor = A Level a1 Level a2 Level a2 Level a1

Split Plot Designs Analysis of Variance Table

Split-Plot Design Split-Plot Experiment : Split-Plot Factor = B Level a1 Level a2 Level a2 Level a1

Split Plot Designs Analysis of Variance Table

Agricultural Field Trial

Agricultural Field Trial Insecticide 2 Insecticide 1 Insecticide 2 Insecticide 2 Insecticide 1 Insecticide 1

Agricultural Field Trial Insecticide 2 Insecticide 1 Insecticide 2 Fert A Fert B Fert A Fert B Fert B Fert B Fert A Fert A Fert B Fert A Fert B Fert A Fert A Insecticide 2 Fert B Fert A Fert A Fert B Fert A Fert B Fert B Fert B Fert B Fert A Fert A Fert B Fert B Fert A Fert B Fert A Fert B Fert A Fert A Fert A Fert A Fert A Fert B Fert B Fert B Fert A Fert B Insecticide 1 Insecticide 1

Agricultural Field Trial Whole Plots = Farms Large Experimental Units Split Plots = Fields Small Experimental Units

Agricultural Field Trial Whole Plots = Farms Large Experimental Units Whole-Plot Factor = Insecticide Whole-Plot Error = Whole-Plot Replicates Split Plots = Fields Small Experimental Units Split-Plot Factor = Fertilizer Split-Plot Error = Split-Plot Replicates

The Split-Plot Design a multifactor experiment where it is not practical to completely randomize the order of the runs. Example – paper manufacturing Three pulp preparation methods. Four different temperatures. The experimenters want to use three replicates. How many batches of pulp are required?

The Split-Plot Design Pulp 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.

The Split-Plot Design Each 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).

The Split-Plot Design

The Split-Plot Design There are two levels of randomization restriction. Two levels of experimentation

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.

The Split-Plot Design Another way to view a split-plot design is a RCBD with replication. Inferences on the blocking factor can be made with data from replications.

The Split-Plot Design Model and Statistical Analysis Sum of squares are computed as for a three factor factorial design without replication.

RCBD Model

The Split-Plot Design Model and Statistical Analysis There are two error structures; the whole-plot error and the subplot error

Split-Plot Design Whole-Plot Experiment : Whole-Plot Factor = A Level a1 Level a2 Level a2 Level a1

Split-Plot Design Split-Plot Experiment : Split-Plot Factor = B Level a1 Level a2 Level a2 Level a1

Split-Plot Design Split-Plot Experiment : Split-Plot Factor = B Level a1 Level a2 Level a2 Level a1