Experimental Designs The objective of Experimental design is to reduce the magnitude of random error resulting in more powerful tests to detect experimental effects

The Completely Randomizes Design Treats 123…t Experimental units randomly assigned to treatments

Randomized Block Design Blocks All treats appear once in each block

Latin Square Designs

The Latin square Design All treats appear once in each row and each column Columns Rows

Graeco-Latin Square Designs Mutually orthogonal Squares

Definition A Greaco-Latin square consists of two latin squares (one using the letters A, B, C, … the other using greek letters , , , …) such that when the two latin square are supper imposed on each other the letters of one square appear once and only once with the letters of the other square. The two Latin squares are called mutually orthogonal. Example: a 7 x 7 Greaco-Latin Square A  B  C  D  E  F  G  B  C  D  E  F  G  A  C  D  E  F  G  A  B  D  E  F  G  A  B  C  E  F  G  A  B  C  D  F  G  A  B  C  D  E  G  A  B  C  D  E  F 

Incomplete Block Designs

Comments The within block variability generally increases with block size. The larger the block size the larger the within block variability. For a larger number of treatments, t, it may not be appropriate or feasible to require the block size, k, to be equal to the number of treatments. If the block size, k, is less than the number of treatments (k < t)then all treatments can not appear in each block. The design is called an Incomplete Block Design.

Balanced Incomplete Block Design. 1.if all treatments appear in exactly r blocks. This ensures that each treatment is estimated with the same precision The value of is the same for each treatment pair. 2.if all treatment pairs i and i* appear together in exactly blocks. This ensures that each treatment difference is estimated with the same precision. The value of is the same for each treatment pair.

Some Identities Let b = the number of blocks. t = the number of treatments k = the block size r = the number of times a treatment appears in the experiment. = the number of times a pair of treatment appears together in the same block 1.bk = rt Both sides of this equation are found by counting the total number of experimental units in the experiment. 2.r(k-1) = (t – 1) Both sides of this equation are found by counting the total number of experimental units that appear with a specific treatment in the experiment.

BIB Design A Balanced Incomplete Block Design (b = 15, k = 4, t = 6, r = 10, = 6)

An Example A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) of cereal. For this purpose: subjects will be asked to taste and compare these cereals scoring them on a scale of For practical reasons it is decided that each subject should be asked to taste and compare at most four of the six cereals. For this reason it is decided to use b = 15 subjects and a balanced incomplete block design to assess the differences in taste of the six brands of cereal.

The design and the data is tabulated below:

Analysis for the Incomplete Block Design Recall that the parameters of the design where b = 15, k = 4, t = 6, r = 10, = 6 denotes summation over all blocks j containing treatment i.

Anova Table for Incomplete Block Designs Sums of Squares  y ij 2 =  B j 2 /k =  Q i 2 = Anova Sums of Squares SS total =  y ij 2 –G 2 /bk = SS Blocks =  B j 2 /k – G 2 /bk = SS Tr = (  Q i 2 )/(r – 1) = SS Error = SS total - SS Blocks - SS Tr =

Anova Table for Incomplete Block Designs

Designs for Estimating Carry-over (or Residual) Effects of Treatments

The Cross-over or Simple Reversal Design An Example A clinical psychologist wanted to test two drugs, A and B, which are intended to increase reaction time to a certain stimulus. He has decided to use n = 8 subjects selected at random and randomly divided into two groups of four. –The first group will receive drug A first then B, while –the second group will receive drug B first then A.

To conduct the trial he administered a drug to the individual, waited 15 minutes for absorption, applied the stimulus and then measured reaction time. The data and the design is tabulated below:

The Switch-back or Double Reversal Design An Example A following study was interested in the effect of concentrate type on the daily production of fat-corrected milk (FCM). Two concentrates were used: –A - high fat; and –B - low fat. Five test animals were then selected for each of the two sequence groups –( A-B-A and B-A-B) in a switch-back design.

The data and the design is tabulated below: One animal in the first group developed mastitis and was removed from the study.

The Incomplete Block Switch-back Design An Example An insurance company was interested in buying a quantity of word processing machines for use by secretaries in the stenographic pool. The selection was narrowed down to three models (A, B, and C). A study was to be carried out, where the time to process a test document would be determined for a group of secretaries on each of the word processing models. For various reasons the company decided to use an incomplete block switch back design using n = 6 secretaries from the secretarial pool.

The data and the design is tabulated below: BIB incomplete block design with t = 3 treatments – A, B and block size k = 2. ABAB ACAC BCBC

The Latin Square Change-Over (or Round Robin) Design Selected Latin Squares Change-Over Designs (Balanced for Residual Effects) Period = RowsColumns = Subjects

Four Treatments

An Example An experimental psychologist wanted to determine the effect of three new drugs (A, B and C) on the time for laboratory rats to work their way through a maze. A sample of n= 12 test animals were used in the experiment. It was decided to use a Latin square Change-Over experimental design.

The data and the design is tabulated below:

Orthogonal Linear Contrasts This is a technique for partitioning ANOVA sum of squares into individual degrees of freedom

Definition Let x 1, x 2,..., x p denote p numerical quantities computed from the data. These could be statistics or the raw observations. A linear combination of x 1, x 2,..., x p is defined to be a quantity,L,computed in the following manner: L = c 1 x 1 + c 2 x c p x p where the coefficients c 1, c 2,..., c p are predetermined numerical values:

Definition If the coefficients c 1, c 2,..., c p satisfy: c 1 + c c p = 0, Then the linear combination L = c 1 x 1 + c 2 x c p x p is called a linear contrast.

Examples L = x x 2 + 6x x 4 + x 5 = (1)x 1 + (-4)x 2 + (6)x 3 + (-4)x 4 + (1)x 5 A linear combination A linear contrast

Definition Let A = a 1 x 1 + a 2 x a p x p and B= b 1 x 1 + b 2 x b p x p be two linear contrasts of the quantities x 1, x 2,..., x p. Then A and B are c called Orthogonal Linear Contrasts if in addition to: a 1 + a a p = 0 and b 1 + b b p = 0, it is also true that: a 1 b 1 + a 2 b a p b p = 0..

Example Let Note:

Definition Let A = a 1 x 1 + a 2 x a p x p, B= b 1 x 1 + b 2 x b p x p,..., and L= l 1 x 1 + l 2 x l p x p be a set linear contrasts of the quantities x 1, x 2,..., x p. Then the set is called a set of Mutually Orthogonal Linear Contrasts if each linear contrast in the set is orthogonal to any other linear contrast..

Theorem: The maximum number of linear contrasts in a set of Mutually Orthogonal Linear Contrasts of the quantities x 1, x 2,..., x p is p - 1. p - 1 is called the degrees of freedom (d.f.) for comparing quantities x 1, x 2,..., x p.

Comments 1.Linear contrasts are making comparisons amongst the p values x 1, x 2,..., x p 2.Orthogonal Linear Contrasts are making independent comparisons amongst the p values x 1, x 2,..., x p. 3.The number of independent comparisons amongst the p values x 1, x 2,..., x p is p – 1.

Definition denotes a linear contrast of the p means If each mean,, is calculated from n observations then: The Sum of Squares for testing the Linear Contrast L, is defined to be:

the degrees of freedom (df) for testing the Linear Contrast L, is defined to be the F-ratio for testing the Linear Contrast L, is defined to be:

Theorem: Let L 1, L 2,..., L p-1 denote p-1 mutually orthogonal Linear contrasts for comparing the p means. Then the Sum of Squares for comparing the p means based on p – 1 degrees of freedom, SS Between, satisfies:

Comment Defining a set of Orthogonal Linear Contrasts for comparing the p means allows the researcher to "break apart" the Sum of Squares for comparing the p means, SS Between, and make individual tests of each the Linear Contrast.

The Diet-Weight Gain example The sum of Squares for comparing the 6 means is given in the Anova Table:

Five mutually orthogonal contrasts are given below (together with a description of the purpose of these contrasts) : (A comparison of the High protein diets with Low protein diets) (A comparison of the Beef source of protein with the Pork source of protein)

(A comparison of the Meat (Beef - Pork) source of protein with the Cereal source of protein) (A comparison representing interaction between Level of protein and Source of protein for the Meat source of Protein) (A comparison representing interaction between Level of protein with the Cereal source of Protein)

The Anova Table for Testing these contrasts is given below: The Mutually Orthogonal contrasts that are eventually selected should be determine prior to observing the data and should be determined by the objectives of the experiment

Another Five mutually orthogonal contrasts are given below (together with a description of the purpose of these contrasts) : (A comparison of the High protein diets with Low protein diets) (A comparison of the Beef source of protein with the Pork source of protein)

(A comparison of the high and low protein diets for the Beef source of protein) (A comparison of the high and low protein diets for the Cereal source of protein) (A comparison of the high and low protein diets for the Pork source of protein)

The Anova Table for Testing these contrasts is given below:

Orthogonal Linear Contrasts Polynomial Regression

Orthogonal Linear Contrasts for Polynomial Regression

Example In this example we are measuring the “Life” of an electronic component and how it depends on the temperature on activation

The Anova Table SourceSSdfMSF Treat Linear Quadratic Cubic Quartic Error Total73014 L = 25.00Q 2 = C = 0.00Q 4 = 30.00

The Anova Tables for Determining degree of polynomial Testing for effect of the factor

Testing for departure from Linear

Testing for departure from Quadratic

Post-hoc Tests Multiple Comparison Tests

Suppose we have p means An F-test has revealed that there are significant differences amongst the p means We want to perform an analysis to determine precisely where the differences exist.

Tukey’s Multiple Comparison Test

Let Tukey's Critical Differences Two means are declared significant if they differ by more than this amount. denote the standard error of each = the tabled value for Tukey’s studentized range p = no. of means, = df for Error

Scheffe’s Multiple Comparison Test

Scheffe's Critical Differences (for Linear contrasts) A linear contrast is declared significant if it exceeds this amount. = the tabled value for F distribution (p -1 = df for comparing p means, = df for Error)

Scheffe's Critical Differences (for comparing two means) Two means are declared significant if they differ by more than this amount.