The Randomized Block Design

Suppose a researcher is interested in how several treatments affect a continuous response variable (Y). The treatments may be the levels of a single factor or they may be the combinations of levels of several factors. Suppose we have available to us a total of N = nt experimental units to which we are going to apply the different treatments.

The Completely Randomized (CR) design randomly divides the experimental units into t groups of size n and randomly assigns a treatment to each group.

The Randomized Block Design divides the group of experimental units into n homogeneous groups of size t. These homogeneous groups are called blocks. The treatments are then randomly assigned to the experimental units in each block - one treatment to a unit in each block.

The ANOVA table for the Completely Randomized Design SourcedfSum of Squares Treatmentst - 1SS Tr Errort(n – 1)SS Error Totaltn - 1SS Total SourcedfSum of Squares Blocksn - 1SS Blocks Treatmentst - 1SS Tr Error (t – 1) (n – 1) SS Error Totaltn - 1SS Total The ANOVA table for the Randomized Block Design

Comments The ability to detect treatment differences depends on the magnitude of the random error term The error term,, for the Completely Randomized Design models variability in the reponse, y, between experimental units The error term,, for the Completely Block Design models variability in the reponse, y, between experimental units in the same block (hopefully the is considerably smaller than.

Example – Weight gain, diet, source of protein, level of protein (Completely randomized design)

Randomized Block Design

The Anova Table for Diet Experiment

Example 1: Suppose we are interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low). There are a total of t = 3  2 = 6 treatment combinations of the two factors (Beef -High Protein, Cereal-High Protein, Pork-High Protein, Beef -Low Protein, Cereal-Low Protein, and Pork-Low Protein).

Suppose we have available to us a total of N = 60 experimental rats to which we are going to apply the different diets based on the t = 6 treatment combinations. Prior to the experimentation the rats were divided into n = 10 homogeneous groups of size 6. The grouping was based on factors that had previously been ignored (Example - Initial weight size, appetite size etc.) Within each of the 10 blocks a rat is randomly assigned a treatment combination (diet).

The weight gain after a fixed period is measured for each of the test animals and is tabulated on the next slide:

Randomized Block Design

Example 2: The following experiment is interested in comparing the effect four different chemicals (A, B, C and D) in producing water resistance (y) in textiles. A strip of material, randomly selected from each bolt, is cut into four pieces (samples) the pieces are randomly assigned to receive one of the four chemical treatments.

This process is replicated three times producing a Randomized Block (RB) design. Moisture resistance (y) were measured for each of the samples. (Low readings indicate low moisture penetration). The data is given in the diagram and table on the next slide.

Diagram: Blocks (Bolt Samples)

Table Blocks (Bolt Samples) Chemical123 A B C D

The Model for a randomized Block Experiment i = 1,2,…, tj = 1,2,…, b y ij = the observation in the j th block receiving the i th treatment  = overall mean  i = the effect of the i th treatment  j = the effect of the j th Block  ij = random error

The Anova Table for a randomized Block Experiment SourceS.S.d.f.M.S.Fp-value TreatSS T t-1MS T MS T /MS E BlockSS B n-1MS B MS B /MS E ErrorSS E (t-1)(b-1)MS E

A randomized block experiment is assumed to be a two-factor experiment. The factors are blocks and treatments. The is one observation per cell. It is assumed that there is no interaction between blocks and treatments. The degrees of freedom for the interaction is used to estimate error.

The Anova Table for Diet Experiment

The Anova Table forTextile Experiment

If the treatments are defined in terms of two or more factors, the treatment Sum of Squares can be split (partitioned) into: –Main Effects –Interactions

The Anova Table for Diet Experiment terms for the main effects and interactions between Level of Protein and Source of Protein

Repeated Measures Designs

In a Repeated Measures Design We have experimental units that may be grouped according to one or several factors (the grouping factors) Then on each experimental unit we have not a single measurement but a group of measurements (the repeated measures) The repeated measures may be taken at combinations of levels of one or several factors (The repeated measures factors)

Example In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery. The enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for n = 15 cardiac surgical patients.

The data is given in the table below. Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery

The subjects are not grouped (single group). There is one repeated measures factor - Time – with levels –Day 0, –Day 1, –Day 2, –Day 7 This design is the same as a randomized block design with –Blocks = subjects

The Anova Table for Enzyme Experiment The Subject Source of variability is modelling the variability between subjects The ERROR Source of variability is modelling the variability within subjects

Example : (Repeated Measures Design - Grouping Factor) In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery. In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.

The 24 patients were randomly divided into three groups of n= 8 patients. The first group of patients were left untreated as a control group while the second and third group were given drug treatments A and B respectively. Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.

Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery for three treatment groups (control, Drug A, Drug B)

The subjects are grouped by treatment –control, –Drug A, –Drug B There is one repeated measures factor - Time – with levels –Day 0, –Day 1, –Day 2, –Day 7

The Anova Table There are two sources of Error in a repeated measures design: The between subject error – Error 1 and the within subject error – Error 2

Tables of means DrugDay 0Day 1Day 2Day 7Overall Control A B Overall

Example : Repeated Measures Design - Two Grouping Factors In the following example, the researcher was interested in how the levels of Anxiety (high and low) and Tension (none and high) affected error rates in performing a specified task. In addition the researcher was interested in how the error rates also changed over time. Four groups of three subjects diagnosed in the four Anxiety-Tension categories were asked to perform the task at four different times patients in the study.

The number of errors committed at each instance is tabulated below.

The Anova Table

Latin Square Designs

Selected Latin Squares 3 x 34 x 4 A B CA B C DA B C DA B C DA B C D B C AB A D CB C D AB D A CB A D C C A BC D B AC D A BC A D BC D A B D C A BD A B CD C B AD C B A 5 x 56 x 6 A B C D EA B C D E F B A E C DB F D C A E C D A E BC D E F B A D E B A CD A F E C B E C D B AE C A B F D F E B A D C

A Latin Square

Definition A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column. Example - 4 x 4 Latin Square. A B C D B C D A C D A B D A B C

In a Latin square You have three factors: Treatments (t) (letters A, B, C, …) Rows (t) Columns (t) The number of treatments = the number of rows = the number of colums = t. The row-column treatments are represented by cells in a t x t array. The treatments are assigned to row-column combinations using a Latin-square arrangement

Example A courier company is interested in deciding between five brands (D,P,F,C and R) of car for its next purchase of fleet cars. The brands are all comparable in purchase price. The company wants to carry out a study that will enable them to compare the brands with respect to operating costs. For this purpose they select five drivers (Rows). In addition the study will be carried out over a five week period (Columns = weeks).

Each week a driver is assigned to a car using randomization and a Latin Square Design. The average cost per mile is recorded at the end of each week and is tabulated below:

The Model for a Latin Experiment i = 1,2,…, tj = 1,2,…, t y ij(k) = the observation in i th row and the j th column receiving the k th treatment  = overall mean  k = the effect of the i th treatment  i = the effect of the i th row  ij(k) = random error k = 1,2,…, t  j = the effect of the j th column No interaction between rows, columns and treatments

A Latin Square experiment is assumed to be a three-factor experiment. The factors are rows, columns and treatments. It is assumed that there is no interaction between rows, columns and treatments. The degrees of freedom for the interactions is used to estimate error.

The Anova Table for a Latin Square Experiment SourceS.S.d.f.M.S.F p-value TreatSS Tr t-1MS Tr MS Tr /MS E RowsSS Row t-1MS Row MS Row /MS E ColsSS Col t-1MS Col MS Col /MS E ErrorSS E (t-1)(t-2)MS E TotalSS T t 2 - 1

The Anova Table for Example SourceS.S.d.f.M.S.F p-value Week Driver Car Error Total

Example In this Experiment the we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low). There are a total of t = 3 X 2 = 6 treatment combinations of the two factors. Beef -High Protein Cereal-High Protein Pork-High Protein Beef -Low Protein Cereal-Low Protein and Pork-Low Protein

In this example we will consider using a Latin Square design Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories. A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories. A Latin square is then used to assign the 6 diets to the 36 test animals in the study.

In the latin square the letter A represents the high protein-cereal diet B represents the high protein-pork diet C represents the low protein-beef Diet D represents the low protein-cereal diet E represents the low protein-pork diet and F represents the high protein-beef diet.

The weight gain after a fixed period is measured for each of the test animals and is tabulated below:

The Anova Table for Example SourceS.S.d.f.M.S.F p-value Inwt App Diet Error Total

Diet SS partioned into main effects for Source and Level of Protein SourceS.S.d.f.M.S.F p-value Inwt App Source Level SL Error Total

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 

Note: At most (t –1) t x t Latin squares L 1, L 2, …, L t-1 such that any pair are mutually orthogonal. It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L 1, L 2, L 3, L 4, L 5, L 6.

The Greaco-Latin Square Design - An Example A researcher is interested in determining the effect of two factors the percentage of Lysine in the diet and percentage of Protein in the diet have on Milk Production in cows. Previous similar experiments suggest that interaction between the two factors is negligible.

For this reason it is decided to use a Greaco-Latin square design to experimentally determine the two effects of the two factors (Lysine and Protein). Seven levels of each factor is selected 0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E), 0.5(F), and 0.6(G)% for Lysine and 2(a), 4(b), 6(c), 8(d), 10(e), 12(f) and 14(g)% for Protein ). Seven animals (cows) are selected at random for the experiment which is to be carried out over seven three-month periods.

A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (Lysine and Protein) to a period and a cow. The data is tabulated on below:

The Model for a Greaco-Latin Experiment i = 1,2,…, t j = 1,2,…, t y ij(kl) = the observation in i th row and the j th column receiving the k th Latin treatment and the l th Greek treatment k = 1,2,…, t l = 1,2,…, t

 = overall mean  k = the effect of the k th Latin treatment  i = the effect of the i th row  ij(k) = random error  j = the effect of the j th column No interaction between rows, columns, Latin treatments and Greek treatments l = the effect of the l th Greek treatment

A Greaco-Latin Square experiment is assumed to be a four-factor experiment. The factors are rows, columns, Latin treatments and Greek treatments. It is assumed that there is no interaction between rows, columns, Latin treatments and Greek treatments. The degrees of freedom for the interactions is used to estimate error.

The Anova Table for a Greaco-Latin Square Experiment SourceS.S.d.f.M.S.F p-value LatinSS La t-1MS La MS La /MS E GreekSS Gr t-1MS Gr MS Gr /MS E RowsSS Row t-1MS Row MS Row /MS E ColsSS Col t-1MS Col MS Col /MS E ErrorSS E (t-1)(t-3)MS E TotalSS T t 2 - 1

The Anova Table for Example SourceS.S.d.f.M.S.F p-value Protein Lysine Cow Period Error Total