Topic 23 - Randomized Complete Block Designs (RCBD)

Transcription

1 Topc 3 ANOVA (III) 3-1 Topc 3 - Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn (CRD) that we recently learned. In ths desgn, blocks of expermental unts are chosen where the unts wthn are block are more smlar to each other (homogeneous) than to unts n other blocks. In a complete block desgn, there are at least t expermental unts n each block where t s the number of treatments n the factor(s) of nterest. Examples of blocks: 1) a ltter of anmals could be consdered a block snce they all have smlar genetc structure, smlar prenatal/parental care, etc. ) a feld or pasture that can be dvded nto quadrants snce sol propertes, envronmental condtons, etc are smlar wthn a feld 3) a greenhouse wth multple benches snce envronmental condtons are usually more smlar wthn a greenhouse than between greenhouses 4) a year n whch the experment s performed snce envronmental condtons are smlar wthn a year Example of a RCBD: A nutrtonst s nterested n comparng the effect of three dets on weght gan n pglets. In order to perform the experment, the researcher chooses 10 ltters, each wth at least three healthy and smlarly szed pglets that have just been weaned. In each ltter, three pglets are selected and one treatment s randomly assgned to each pglet. Dets are labeled A, B or C.

2 Topc 3 ANOVA (III) 3- Ltter Pglet A C B B C A 10 C B A In a desgn wthout blockng, the researcher would pck 30 pglets from dfferent ltters and randomly assgn treatments to them. Ths s known as unrestrcted randomzaton. Blockng desgns have restrcted randomzaton snce the treatments are randomly assgned WITHIN each block. Another Example of a RCBD: An anmal behavorst s nterested n habtat use by gopher tortoses. There are eleven conservaton management areas wthn the dstrbuton of the speces that have the four habtats of nterest for comparatve study. The response varable s the densty of actve burrows n each habtat. In ths example, the blocks are the conservaton management areas and each has all four levels of the treatment (habtat). Here we do not assgn treatments to ndvdual expermental unts. The unts are the habtat areas wthn each block. An RCBD has two factors: the factor of nterest that ncludes the treatments to be studed and the Blockng Factor that dentfes the blocks used n the experment. There are several forms of Blockng Desgns: 1) the RCBD that we wll study

3 Topc 3 ANOVA (III) 3-3 ) ncomplete block desgns n whch not every block has t expermental unts 3) block desgns n whch the blocks have more than t expermental unts that are used n the experment 4) Latn square desgns whch have very specfc forms of randomzaton of treatments wthn blocks (example s usually relates to tme orderng of treatments) Assumptons of the RCBD: 1) Samplng: a. The blocks are ndependently chosen b. The treatments are randomly assgned to the expermental unts wthn a block. ) Homogeneous Varance: the treatments all have the same varablty,.e. they all have the same varance 3) Approxmate Normalty: each populaton s normally dstrbuted Hypotheses As we wll see, the blockng factor s ncluded n the study only as a way of explanng some of the varaton n responses (Y) of the expermental unts. As such, we are not nterested n testng hypotheses about the blockng factor. Instead, just lke n a one-way ANOVA, we restrct our attenton to the other factor ( research factor). So, hypothess testng proceeds smlar to the technques we learned for the one-way ANOVA. The bg dfferences are 1) we won t test for blockng effect and ) the varablty assgned to the error term s

4 Topc 3 ANOVA (III) 3-4 broken down nto parts, varablty among blocks and the left-over unexplaned varablty (stll assocated wth the error term). Hence the MSE from a 1-way ANOVA can be decomposed nto two parts: calculaton of a new error varance (MSE) and a calculaton of the effect of the blockng factor (MSB). Notaton t b N y j y y j y the number of treatments of nterest n the research factor the number of blocks contanng exactly t expermental unts = t b, the total sample sze observed value for the expermental unt n the j th block assgned to the th treatment, j = 1,,,b and = 1,,,t b j= b y j = 1, the sample mean of the th treatment t = t y j = 1, the sample mean of the j th block t b = j= tb y j = 1 1, the overall sample mean of the combned treatments

5 Topc 3 ANOVA (III) 3-5 Example: pglet det experment wth three ltters Det Block Ltter A B C Mean 1 y A1 = 54.3 y B1 = 53.1 y C1 = 59.7 y = y A = 53.6 y B = 5.4 y C = 59.7 y = y A3 = 55. y B3 = 57.1 y C3 = 67. y 3 = 6. Treatment Mean y A = 54.4 y B = 55. y C = Grand Mean y = 56.9 Model: Yj = μ + α + β + ε j j where μ s the overall (grand) mean, α s the effect due to the th treatment, β j s the effect due to the j th block, and, ε j s the error term where the error terms, are ndependent observatons from an approxmately Normal dstrbuton wth mean = 0 and constant varance Total varablty of all of the Y j, s σ ε TSS = ( y j y ) whch can be broken up nto three parts: TSS = SST + SSB + SSE j

6 Topc 3 ANOVA (III) 3-6 SST SSB SSE b ( y y ) = b ˆ α s the sum of squares treatments = t y j y = t ˆ ( ) β s the sum of squares blocks = j yj y y j + y ) = = j j j j j ( ˆ ε s the sum of squares error. Lke before, we are nterested n the Mean Squares: MST SST =, the Mean Square Treatments t 1 MSB SSB =, the Mean Square Blocks b 1 MSE = SSE, the Mean Square Error ( t 1)( b 1) α σ ε and Here E( MST) = + b ( t 1) ( MSE) = σ ε E. ANOVA Table for a Randomzed Complete Block Desgn Source Sum of Degrees of Mean F-stat Squares Freedom Square Treatment SST t 1 MST F*=MST/MSE Block SSB b 1 MSB Error SSE (t 1)(b 1) MSE Total TSS tb 1

7 Topc 3 ANOVA (III) 3-7 Agan, the test of a treatment effect H 0 : μ 1 = μ = = μ t H A : at least one mean dffers uses the statstc F*=MST/MSE. If the null hypothess s true, F* has an F-Dstrbuton on numerator degrees of freedom t 1 and denomnator degs of freedom (t 1)(b 1). In addton to the smlarty of the F-test of equalty of treatment means, the tests and comparsons of treatment means are done exactly the same as before as well. Example: pglet experment. data pgsblocked; nput ltter det$ datalnes; 1 I 54.3 I I II 53.1 II II III 59.7 III III 67. run; ttle1 RCBD Model wth blockng factor ncluded; proc glm data=pgsblocked; class det ltter; model gan = det ltter; lsmeans det / pdff adjust = tukey; qut;

8 Topc 3 ANOVA (III) 3-8 ttle1 CRD Model wth blockng factor excluded; proc glm data=pgsblocked; class det ltter; model gan = det; lsmeans det / pdff adjust = tukey; qut; RCBD Model wth blockng factor ncluded 13 The GLM Procedure Class Level Informaton Class Levels Values det 3 I II III ltter Number of Observatons Read 9 Number of Observatons Used 9 Dependent Varable: gan Sum of Source DF Squares Mean Square F Value Pr > F Model Error CTotal R-Square Coeff Var Root MSE gan Mean Source DF Type I SS Mean Square F Value Pr > F det ltter Source DF Type III SS Mean Square F Value Pr > F det ltter

9 Topc 3 ANOVA (III) 3-9 Least Squares Means Adjustment for Multple Comparsons: Tukey det gan LSMEAN Number I II III Least Squares Means for effect det Pr > t for H0: LSMean()=LSMean(j) Dependent Varable: gan /j CRD Model wth blockng factor excluded The GLM Procedure Class Level Informaton Class Levels Values det 3 I II III ltter Number of Observatons Read 9 Number of Observatons Used 9 Dependent Varable: gan Sum of Source DF Squares Mean Square F Value Pr > F Model Error CTotal R-Square Coeff Var Root MSE gan Mean Source DF Type I SS Mean Square F Value Pr > F det

10 Topc 3 ANOVA (III) 3-10 Source DF Type III SS Mean Square F Value Pr > F det Least Squares Means Adjustment for Multple Comparsons: Tukey det gan LSMEAN Number I II III Least Squares Means for effect det Pr > t for H0: LSMean()=LSMean(j) Dependent Varable: gan /j Advantages of ths form of the RCBD as compared to the CRD: 1) reduce the error varance by explanng or dentfyng one source of some of the varablty n the observatons a. book refers to ths as flterng out some of the varaton ) the desgn s easy to construct,.e. when there are natural or obvous blocks wth at least t expermental unts, the restrcted randomzaton s easy to acheve Dsadvantages 1) need homogeneous blocks n order for the blockng factor to be effectve ) the effect of the treatments n the Factor under study must be the same n every block,.e. the effect of a treatment cannot depend on whch block t s beng appled to.

11 Topc 3 ANOVA (III) 3-11 Example: experment to compare the unused red lght tme for fve dfferent traffc lght sgnal sequences durng mornng rush hour. Traffc engneer chose several ntersectons and performed the dfferent sequences at each ntersecton n random order. Suppose the effect of a partcular sequence depends on whch ntersecton you are studyng, e.g. n ntersectons wth heavy traffc, the average unused red lght tme s greater than the average tme at ntersectons wth lghter traffc maybe. Ths s known as nteracton of factors. Choosng Varables On Whch To Block: We want expermental unts wthn each block to be as smlar as possble to each other wth respect to any characterstc whch can affect or nfluence the response varable (Y). So, f a study relates to weght gan, we want each block to have smlar characterstcs wth respect to growth such as startng weght, metabolc rates, etc. Whch s better, a RCBD or a CRD? Can check usng Relatve Effcency whch compares the varance of the estmate of the th treatment mean μˆ = y under the two dfferent experment desgns. Effcency s calculated as the number of observatons that would be requred f the experment had been conducted as a CRD wthout any blockng. MSE RE( RCBD, CRD) = MSE ( SSB = CRD RCBD RCBD + SSE MSE RCBD RCBD ) / t( b 1)

12 Topc 3 ANOVA (III) 3-1 If the blockng was not helpful, then the relatve effcency equals 1. The larger the relatve effcency s, the more effcent the blockng was at reducng the error varance. The value can be nterpreted as the rato b r where r s the number of expermental unts that would have to be assgned to each treatment f a CRD had been performed nstead of a RCBD. Example: n the pglet experment, SSB RCBD = 9.30, SSE RCBD = 3.61, t = 3, b = 3, MSE RCBD = 5.90 ( ) / 3() 8.8 RE ( RCBD, CRD) = = = Ths mples that t would have taken 1.5 tmes as many expermental unts/treatment to get the same MSE as we got usng the ltters as blocks. That s, we would have needed approxmately 15 ( 1.5*10) pglets per treatment n a CRD experment testng the three dets.