Completely Randomized Design 9 17

Completely Randomized Design 1. Experimental Units (Subjects) Are Assigned Randomly to Treatments Subjects are Assumed Homogeneous 2. One Factor or Independent Variable 2 or More Treatment Levels or Classifications 3. Analyzed by One-Way ANOVA

Randomized Design Example Are the mean training times the same for 3 different methods? 9 subjects 3 methods (factor levels)    78

The Linier Model Xij =μ + ti +εij i = 1,2,…, t j = 1,2,…, r Xij = the observation in ith treatment and the jth replication m = overall mean t i = the effect of the ith treatment eij = random error

One-Way ANOVA F-Test 1. Tests the Equality of 2 or More (t) Population Means 2. Variables One Nominal Scaled Independent Variable 2 or More (t) Treatment Levels or Classifications One Interval or Ratio Scaled Dependent Variable 3. Used to Analyze Completely Randomized Experimental Designs Note: There is one dependent variable in the ANOVA model. MANOVA has more than one dependent variable. Ask, what are nominal & interval scales?

Assumptions 1. Randomness & Independence of Errors 2. Normality Independent Random Samples are Drawn for each condition 2. Normality Populations (for each condition) are Normally Distributed 3. Homogeneity of Variance Populations (for each condition) have Equal Variances

Hypotheses H0: 1 = 2 = 3 = ... = t Ha: Not All i Are Equal All Population Means are Equal No Treatment Effect Ha: Not All i Are Equal At Least 1 Pop. Mean is Different Treatment Effect NOT 1  2  ...  t

Hypotheses f(X) X  =  =  f(X) X  =   H0: 1 = 2 = 3 = ... = t All Population Means are Equal No Treatment Effect Ha: Not All i Are Equal At Least 1 Pop. Mean is Different Treatment Effect NOT 1  2  ...  t f(X) X  =  =  1 2 3 f(X) X  =   1 2 3

One-Way ANOVA Basic Idea 1. Compares 2 Types of Variation to Test Equality of Means 2. Comparison Basis Is Ratio of Variances 3. If Treatment Variation Is Significantly Greater Than Random Variation then Means Are Not Equal 4. Variation Measures Are Obtained by ‘Partitioning’ Total Variation

One-Way ANOVA Partitions Total Variation Variation due to Random Sampling are due to Individual Differences Within Groups. 84

One-Way ANOVA Partitions Total Variation Variation due to Random Sampling are due to Individual Differences Within Groups. 85

One-Way ANOVA Partitions Total Variation Variation due to Random Sampling are due to Individual Differences Within Groups. Variation due to treatment 86

One-Way ANOVA Partitions Total Variation Variation due to Random Sampling are due to Individual Differences Within Groups. Variation due to treatment Variation due to random sampling 87

One-Way ANOVA Partitions Total Variation Variation due to Random Sampling are due to Individual Differences Within Groups. Variation due to treatment Variation due to random sampling Sum of Squares Among Sum of Squares Between Sum of Squares Treatment Among Groups Variation 88

One-Way ANOVA Partitions Total Variation Variation due to Random Sampling are due to Individual Differences Within Groups. Variation due to treatment Variation due to random sampling Sum of Squares Among Sum of Squares Between Sum of Squares Treatment (SST) Among Groups Variation Sum of Squares Within Sum of Squares Error (SSE) Within Groups Variation 89

Total Variation Response, X X Group 1 Group 2 Group 3

Treatment Variation Response, X X3 X X2 X1 Group 1 Group 2 Group 3

Random (Error) Variation Response, X X3 X2 X1 Group 1 Group 2 Group 3

SStotal=SSE+SST

But

Thus, SStotal=SSE+SST

One-Way ANOVA F-Test Test Statistic F = MST / MSE MST Is Mean Square for Treatment MSE Is Mean Square for Error 2. Degrees of Freedom 1 = t -1 2 = tr - t t = # Populations, Groups, or Levels tr = Total Sample Size

One-Way ANOVA Summary Table Source of Degrees Sum of Mean F Variation of Squares Square Freedom (Variance) n = sum of sample sizes of all populations. c = number of factor levels All values are positive. Why? (squared terms) Degrees of Freedom & Sum of Squares are additive; Mean Square is NOT. Treatment t - 1 SST MST = MST SST/(t - 1) MSE Error tr - t SSE MSE = SSE/(tr - t) Total tr - 1 SS(Total) = SST+SSE

ANOVA Table for a Completely Randomized Design Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Treatments SST t - 1 SST/t-1 MST/MSE Error SSE tr - t SSE/tr-t Total SSTot tr - 1

The F distribution  F  Two parameters increasing either one decreases F-alpha (except for v2<3) I.e., the distribution gets smashed to the left  F F  ( v1 , v2 )

One-Way ANOVA F-Test Critical Value If means are equal, F = MST / MSE  1. Only reject large F! Reject H  Do Not Reject H F F a ( t  1 , tr -t) Always One-Tail! © 1984-1994 T/Maker Co.

Example: Home Products, Inc. Completely Randomized Design Home Products, Inc. is considering marketing a long-lasting car wax. Three different waxes (Type 1, Type 2, and Type 3) have been developed. In order to test the durability of these waxes, 5 new cars were waxed with Type 1, 5 with Type 2, and 5 with Type 3. Each car was then repeatedly run through an automatic carwash until the wax coating showed signs of deterioration. The number of times each car went through the carwash is shown on the next slide. Home Products, Inc. must decide which wax to market. Are the three waxes equally effective?

Example: Home Products, Inc. Wax Wax Wax Observation Type 1 Type 2 Type 3 1 27 33 29 2 30 28 28 3 29 31 30 4 28 30 32 5 31 30 31 Sample Mean 29.0 30.4 30.0 Sample Variance 2.5 3.3 2.5

Example: Home Products, Inc. Hypotheses H0: 1=2=3 Ha: Not all the means are equal where: 1 = mean number of washes for Type 1 wax 2 = mean number of washes for Type 2 wax 3 = mean number of washes for Type 3 wax

Example: Home Products, Inc. Mean Square Between Treatments Since the sample sizes are all equal: μ= (x1 + x2 + x3)/3 = (29 + 30.4 + 30)/3 = 29.8 SSTR= 5(29–29.8)2+ 5(30.4–29.8)2+ 5(30–29.8)2= 5.2 MSTR = 5.2/(3 - 1) = 2.6 Mean Square Error SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2 MSE = 33.2/(15 - 3) = 2.77 _ _ _ =

Example: Home Products, Inc. Rejection Rule Using test statistic: Reject H0 if F > 3.89 Using p-value: Reject H0 if p-value < .05 where F.05 = 3.89 is based on an F distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom

Example: Home Products, Inc. Test Statistic F = MST/MSE = 2.6/2.77 = .939 Conclusion Since F = .939 < F.05 = 3.89, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

Example: Home Products, Inc. ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Treatments 5.2 2 2.60 .9398 Error 33.2 12 2.77 Total 38.4 14

Using Excel’s ANOVA: Single Factor Tool Value Worksheet (top portion)

Using Excel’s ANOVA: Single Factor Tool Value Worksheet (bottom portion)

Using Excel’s ANOVA: Single Factor Tool Conclusion Using the p-Value The value worksheet shows a p-value of .418 The rejection rule is “Reject H0 if p-value < .05” Because .418 > .05, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

(Randomized Complete Block Design) RCBD (Randomized Complete Block Design)

Randomized Complete Block Design An experimental design in which there is one independent variable, and a second variable known as a blocking variable, that is used to control for confounding or concomitant variables. It is used when the experimental unit or material are heterogeneous There is a way to block the experimental units or materials to keep the variability among within a block as small as possible and to maximize differences among block The block (group) should consists units or materials which are as uniform as possible

Randomized Complete Block Design Confounding or concomitant variable are not being controlled by the analyst but can have an effect on the outcome of the treatment being studied Blocking variable is a variable that the analyst wants to control but is not the treatment variable of interest. Repeated measures design is a randomized block design in which each block level is an individual item or person, and that person or item is measured across all treatments.

The Blocking Principle Blocking is a technique for dealing with nuisance factors A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units Many industrial experiments involve blocking (or should) Failure to block is a common flaw in designing an experiment

The Blocking Principle If the nuisance variable is known and controllable, we use blocking If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to statistically remove the effect of the nuisance factor from the analysis If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

Partitioning the Total Sum of Squares in the Randomized Block Design SStotal (total sum of squares) SSE (error sum of squares) SST (treatment sum of squares) SSB (sum of squares blocks) SSE’ (sum of squares error)

A Randomized Block Design Individual observations . Single Independent Variable Blocking Variable 30

The Linier Model i = 1,2,…, t j = 1,2,…,r yij = the observation in ith treatment in the jth block m = overall mean ti = the effect of the ith treatment No interaction between blocks and treatments rj = the effect of the jth block eij = random error

Extension of the ANOVA to the RCBD ANOVA partitioning of total variability:

Extension of the ANOVA to the RCBD The degrees of freedom for the sums of squares in are as follows: Ratios of sums of squares to their degrees of freedom result in mean squares, and The ratio of the mean square for treatments to the error mean square is an F statistic  used to test the hypothesis of equal treatment means

ANOVA Procedure The ANOVA procedure for the randomized block design requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error. The formula for this partitioning is SSTot = SSTreatment + SSBlock + SSE The total degrees of freedom, tr - 1, are partitioned such that t - 1 degrees of freedom go to treatments, r - 1 go to blocks, and (t - 1)(r - 1) go to the error term.

ANOVA Table for a Randomized Block Design Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Treatments SST t – 1 SST/t-1 MST/MSE Blocks SSB r - 1 Error SSE (t - 1)(r - 1) SSE/(t-1)(r-1) Total SSTot tr - 1

Example: Eastern Oil Co. Randomized Block Design Eastern Oil has developed three new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends. Five automobiles have been tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide.

Example: Eastern Oil Co. Automobile Type of Gasoline (Treatment) Blocks (Block) Blend X Blend Y Blend Z Means 1 32 31 30 31 2 29 28 27 28 3 30 30 27 29 4 32 31 30 31 5 27 25 26 26 Treatment Means 30 29 28 29

Example: Eastern Oil Co. Mean Square Due to Treatments The overall sample mean is 29. Thus, SSTreatment = 5[(30-29)2+ (29-29)2+ (28-29)2]= 10.00 MSTreatment = 10/(3 - 1) = 5 Mean Square Due to Blocks SSBlock = 3[(31-29)2 + . . . + (26-29)2] = 54 MSBlock= 54/(5 - 1) = 13.5 Mean Square Due to Error CF = (435)2 /(3x5) =12615 SStotal = (32)2+ …+ (26)2 – 12615 = 62 SSE = 68 - 10 – 54 = 4 MSE = 4/[(3 - 1)(5 - 1)] = .5

Example: Eastern Oil Co. Rejection Rule Using test statistic: Reject H0 if F > 4.46 Assuming α = .05, F.05 = 4.46 (2 d.f. numerator and 8 d.f. denominator)

Example: Eastern Oil Co. Test Statistic F = MSTreatment/MSE = 5/.5 = 10 Conclusion Since 10.00 > 4.46, we reject H0. There is sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends.

Using Excel’s Anova Step 1 Select the Tools pull-down menu Step 2 Choose the Data Analysis option Step 3 Choose Anova: Two Factor Without Replication from the list of Analysis Tools

Using Excel’s Anova Step 4 When the Anova: Two Factor Without Replication dialog box appears: Enter A1:D6 in the Input Range box Select Labels Enter .05 in the Alpha box Select Output Range Enter A8 (your choice) in the Output Range box Click OK

Using Excel’s Anova: Two-Factor Without Replication Tool Value Worksheet (top portion)

Using Excel’s Anova: Two-Factor Without Replication Tool Value Worksheet (middle portion)

Using Excel’s Anova

Using Excel’s Anova: Two-Factor Without Replication Tool Conclusion Using F table The value worksheet shows that F table for column is 4.459 The rejection rule is “Reject H0 if F calculated > F Table” Thus, we reject H0 because F calculated > F Table for a = .05 There is sufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends

Similarities and differences between CRD and RCBD: Procedures RCBD: Every level of “treatment” encountered by each experimental unit; CRD: Just one level each Descriptive statistics and graphical display: the same as CRD Model adequacy checking procedure: the same except: specifically, NO Block x Treatment Interaction ANOVA: Inclusion of the Block effect; dferror change from t(r – 1) to (t – 1)(r – 1)

Latin Square Design

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  

The Latin Square Design This design is used to simultaneously control (or eliminate) two sources of nuisance variability (confounding variables) It is called “Latin” because we usually specify the treatment by the Latin letters “Square” because it always has the same number of levels (t) for the row and column nuisance factors A significant assumption is that the three factors (treatments and two nuisance factors) do not interact More restrictive than the RCBD Each treatment appears once and only once in each row and column If you can block on two (perpendicular) sources of variation (rows x columns) you can reduce experimental error when compared to the RCBD A B C D

Useful in Animal Nutrition Studies Suppose you had four feeds you wanted to test on dairy cows. The feeds would be tested over time during the lactation period This experiment would require 4 animals (think of these as the rows) There would be 4 feeding periods at even intervals during the lactation period beginning early in lactation (these would be the columns) The treatments would be the four feeds. Each animal receives each treatment one time only. Usually done this way for economic reasons (limited number of cows). One potential problem is carryover from one treatment to the next on the same cow.

The “Latin Square” Cow Mid Early Late This is a simple type of ‘crossover’ design, commonly used in human and animal subjects.

Uses in Field Experiments When two sources of variation must be controlled Slope and fertility Furrow irrigation and shading If you must plant your plots perpendicular to a linear gradient B C D A ‘Row’ 1 2 3 4 1 2 3 4 ‘Column’ With four treatments, there would only be 6 df remaining for the error term With five treatments, dfe = 24-4-4-4=12 Practically speaking, use only when you have more than four but fewer than ten treatments a minimum of 12 df for error

Randomization First row in alphabetical order A B C D E Subsequent rows - shift letters one position 4 3 5 1 2 A B C D E 2 C D E A B A B D C E B C D E A 4 A B C D E D E B A C C D E A B 1 D E A B C B C E D A D E A B C 3 B C D E A E A C B D E A B C D 5 E A B C D C D A E B Randomize the order of the rows: 2 4 1 3 5 Finally, randomize the order of the columns: 4 3 5 1 2

Analysis Set up a two-way table and compute the row and column totals Compute a table of treatment totals and means Set up an ANOVA table divided into sources of variation Rows Columns Treatments Error Significance tests FT tests difference among treatment means FR and FC test if row and column groupings are effective

Advantages and Disadvantages Allows the experimenter to control two sources of variation Disadvantages: Error degree of freedom (df) is small if there are only a few treatments The experiment becomes very large if the number of treatments is large The statistical analysis is complicated by missing plots and mis-assigned treatments Another variation – use a smaller Latin Square (3x3 or 4x4), but repeat it. Analysis can then be combined across the two replicates.

Selected Latin Squares Latin Square Designs Selected Latin Squares 3 x 3 4 x 4 A B C A B C D A B C D A B C D A B C D B C A B A D C B C D A B D A C B A D C C A B C D B A C D A B C A D B C D A B D C A B D A B C D C B A D C B A   5 x 5 6 x 6 A B C D E A B C D E F B A E C D B F D C A E C D A E B C D E F B A D E B A C D A F E C B E C D B A F E B A D C

The row-column treatments are represented by cells in a t x t array. In a Latin square, there are three factors: Treatments (t) (letters A, B, C, …) Rows (t) Columns (t) The number of treatments = the number of rows = the number of columns = 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 

The Linier Model i = 1,2,…, t j = 1,2,…, t k = 1,2,…, t yij(k) = the observation in ith row and the jth column receiving the kth treatment m = overall mean tk = the effect of the kth treatment No interaction between rows, columns and treatments ri = the effect of the ith row gj = the effect of the jth column eij(k) = random error

Latin Square 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 for a Latin Square Source S.S. d.f. M.S. Fcal Ftable Treat SST t-1 MST MST /MSE Fα, (t-1); (t-1)(t-2) Rows SSRow MSRow Cols SSCol MSCol Error SSE (t-1)(t-2) MSE Total t2 - 1

Example 1. 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 Anova Table Source S.S. d.f. M.S. Fcalc Total Ftable Week Driver 50.0 4 12.5 Driver 80.0 20.0 Car 55.2 13.8 12.94 3.26 Error 12.8 12 1.07 Total 198 24

Example 2. 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 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 F represents the high protein-beef diet.

The weight gain after a fixed period is measured for each of the test animals

The Anova Table Source S.S. d.f. M.S. Fcalc Total Ftable Inwt 1767.0836 5 353.41673 111.1 2.71 App 2195.4331 439.08662 138.03 Diet 4183.9132 836.78263 263.06 Error 63.61999 20 3.181 Total 8210.0499 35

Diet SS partioned into main effects for Source and Level of Protein d.f. M.S. Fcalc Ftable Inwt 1767.0836 5 353.41673 111.1 App 2195.4331 439.08662 138.03 631.22173 2 315.61087 99.22 0.0000 Level 2611.2097 1 820.88 SL 941.48172 470.74086 147.99 Error 63.61999 20 3.181 Total 8210.0499 35

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 a, b, c, …) 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 Aa Be Cb Df Ec Fg Gd Bb Cf Dc Eg Fd Ga Ae Cc Dg Ed Fa Ge Ab Bf Dd Ea Fe Gb Af Bc Cg Ee Fb Gf Ac Bg Cd Da Ff Gc Ag Bd Ca De Eb Gg Ad Ba Ce Db Ef Fc

The Graeco-Latin Square Design This design is used to simultaneously control (or eliminate) three sources of nuisance variability It is called “Graeco-Latin” because we usually specify the third nuisance factor, represented by the Greek letters, orthogonal to the Latin letters A significant assumption is that the four factors (treatments, nuisance factors) do not interact If this assumption is violated, as with the Latin square design, it will not produce valid results Graeco-Latin squares exist for all t ≥ 3 except t = 6

Note: At most (t –1) t x t Latin squares L1, L2, …, Lt-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 L1, L2, L3, L4, L5, L6 .

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 Linear Model j = 1,2,…, t i = 1,2,…, t k = 1,2,…, t l = 1,2,…, t yij(kl) = the observation in ith row and the jth column receiving the kth Latin treatment and the lth Greek treatment

tk = the effect of the kth Latin treatment m = overall mean tk = the effect of the kth Latin treatment ll = the effect of the lth Greek treatment ri = the effect of the ith row gj = the effect of the jth column eij(k) = random error No interaction between rows, columns, Latin treatments and Greek treatments

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.