实验设计与方差分析 Chap11_handoutnew

Chapter 11 Repeated Measures and Related Jingyuan Liu SOE and WISE Xiamen University DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Outline of this Chapter Repeated measures and related designs Overview of repeated measures designs Single factor experiments with repeated measures Two factor experiments with repeated measures on one factor Two factor experiments with repeated measures on both factors Split plot designs Ref Ch 27 in Applied Linear Statistical Models by Kutner et al DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Overview of Repeated Measures Description In a Repeated measures design the same subject is measured repeatedly to study treatment eff ects For each subject a repeated measures study may involve 1 several treatments with random order E g 1 or 2 a single treatment evaluated at diff erent time points E g 2 In this course we focus on 1 These designs are widely used in the behavioral and life sciences where the subjects include persons households observers experimental animals At other times the subjects can be stores test markets cities and plants DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Overview of Repeated Measures Description What is the diff erence between repeated measures design and designs with replications repeated observations In repeated measures several or all treatments are applied to the same subject The design with replications means several observations are made for a given treatment applied to diff erent subjects or say sampling units DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Overview of Repeated Measures Examples Example 1 15 test markets are used to study 2 advertising campaigns In each test market the order of the two campaigns are randomized with a suffi cient time lapse between the two campaigns so that the eff ects of the initial campaign will not carry over into the second DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Overview of Repeated Measures Examples Example 2 100 overweight persons are to be given the same diet and their weights measured at the end of each week for 12 weeks to assess the weight loss over time Here the subjects are the overweight persons who are observed repeatedly to study the eff ects of a single treatment diet over time DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Overview of Repeated Measures Pros and Cons Pros Good precision for comparing treatments because variability between subjects are excluded from experimental error It economizes on subjects particularly important when only a few subjects are available Desirable when the shape of the learning curve is to be studied Cons Order eff ect E g rating tends to go higher in the end Randomization Carryover eff ect later observations might be aff ected by the preceding treatment Suffi cient washout time DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Let us fi rst consider the repeated measures designs where the treatments are based on a single factor as in Example 1 Question It seems that this repeated measures design is just an RCBD with block factor subject Is this true Often subject is indeed viewed as a factor random eff ect The randomization technique the layout of data structure and the model are the same for this design and an RCBD DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Randomization For each subject a random permutation is used to defi ne the treatment order and independent permutations are selected for the diff erent subjects DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Model When treatment eff ects are fi xed a mixed eff ect additive two way factorial ANOVA model is often appropriate yij i j ij i 1 s j 1 r Notations and constraints y ij jth observation in the ith subject overall mean constant i eff ect of subject i i d N 0 2 j fi xed treatment eff ect subject to P j 0 ij i i d N 0 2 independent of i As mentioned before the model is formally the same as RCBD with single observation per cell DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs ANOVA As you can imagine the ANOVA table is also the same as RCBD with single observation Interactions are confounded with random error here Sometimes SSSis called between subject variability SSTR SSTR Sare called within subject variability DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Example Example In a wine judging competition 4 wines were judged by 6 experienced judges How to design the experiment Each judge tastes the wines in a blind fashion The order of the wine presentation is randomized independently for each judge The judges do not drink any wines and rinse their mouths thoroughly between tastings DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Example Each wine was scored on a 40 point scale The data structure is DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Example All the analyses part can be done in the same fashion as RCBD E g the following graph is often used DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Single factor Repeated Measures Designs Example And the ANOVA table can be easily obtained Then help yourself such as treatment eff ect test model assumption check multiple comparison etc DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures Example Study the eff ects of two types of incentives factor A on a person s ability to solve problems Also two types of problem factor B abstract and concrete are studied Each subject could be asked to do each type of problem but could not be exposed to more than one type of incentive stimulus because of potential interference eff ects How to randomize The level of the nonrepeated factor A are randomly assigned to the subjects The order of levels of the repeated factor B are randomized independently for all subjects DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures For factor A it is a CRD for factor B it is a RCBD with block factor subject Thus this design is called the two factor experiment with repeated measures on factor B DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures Comments This example can be roughly viewed as a incomplete block design with the 4 treatments A1B1 A1B2 A2B1and A2B2 But it is not BIBD Why When the factor on which repeated measures are taken in time this factor becomes observational and randomization of its levels is impossible DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures Model Again assume subject is random and treatment factors A and B are fi xed Notice subject is nested within A Why yijk i j j k jk ijk Notations and constraints overall mean constant i j eff ect of subject i i d N 0 2 j fi xed treatment eff ect of factor A subject to P j 0 k fi xed treatment eff ect of factor B subject to P k 0 jk fi xed interaction eff ect between A and B subject to P j jk P k jk 0 ijk i i d N 0 2 independent of i j DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures ANOVA table DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures Example A national retail chain wanted to study the eff ects of two advertising campaigns factor A A1 A2 on the sales volume of shoes over time factor B B1 B2 B3 based on 10 similar test markets subject S Thus only the fi rst layer factor A involves randomization The data structure is as follows DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures B1 two weeks prior to campaign B2 two weeks during which campaign occurred B3 two weeks after campaign was concluded From the graph there is no evidence of any interactions between the test markets and the treatments In general sales tended to increase during each advertising campaign and then declined to previous or lower levels than just before the campaign DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs with One factor Repeated Measures DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Now let us study the two factor experiments with repeated measures on both factors This case is actually easier than the previous case The order of the treatments is randomized within each subject DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures From the experimental setting subject random factor A fi xed and factor B fi xed are all factorial crossed Thus the model is DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures And the ANOVA table is DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Example A clinician studied the eff ects of two drugs A and B on the blood fl ow in human subjects 12 randomly chosen healthy males participated in the study They received each treatment in independently randomized orders for successive days The washout period is ok because the eff ect of each drug is short lived The response is the increase in blood fl ow from before to shortly after the treatment DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Example Cont d The data structure is as follows DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Example Cont d The ANOVA table DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Example Cont d The SAS output The SAS output does not provide the subject eff ect DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Example Cont d The analysis part is nothing new For instance the clinician expected the two drugs to interact in increasing the blood fl ow The test is H0 all jk 0 v s H1 not all jkequal 0 F MSAB MSABS 147 1 1364 129 36 F0 01 1 11 9 65 thus we conclude that the interaction eff ects exist DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Two factor Designs Both with Repeated Measures Comments In economics and econometrics repeated measurement data over time are commonly referred to as panel data Another area of application for repeated measurement data is the growth curve model analysis Here separate regression models are appropriate to each subject over time DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Split plot Designs Now let us move on to a new design related to repeated measures split plot design So how to split the plots The two factor designs with repeated measures on one factor is one type of the split plot designs DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Split plot Designs Now let us move on to a new design related to repeated measures split plot design So how to split the plots The two factor designs with repeated measures on one factor is one type of the split plot designs DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Split plot Designs Split plot designs were originally developed for agricultural experiments and they are frequently used in fi eld laboratory industrial and social science experiments First consider the simplest split plot design Example Study the eff ects of two irrigation methods factor A and two fertilizers factor B on yield of a crop using 4 available fi elds as the experimental units In a CRD 4 treatments A1B1 A1B2 A2B1and A2B2 are assigned to the 4 fi elds thus there will be no replicates hence no degrees of freedom for estimation of error DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Split plot Designs Construction of Design Example Cont d Suppose now the fi elds could be subdivided into smaller experimental units such that replicates may be obtained and error variances could be estimated Unfortunately it is not possible to apply diff erent irrigation methods A in areas smaller than a fi eld although diff erent fertilizers B could be So what do we do split plot design 1 Each irrigation method is randomly assigned to two of the four fi elds which are called whole plots 2 Each whole plot is then subdivided into two or more smaller areas called split plots or subplots and the two fertilizers are then randomly assigned to the split plots within each whole plot DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Split plot Designs Construction of Design Remarks The key feature of split plot designs is the use of two or more distinct levels of randomization Here two levels whole plot randomization and subplot randomization Split plot designs are useful in industrial experiments when one factor requires larger experimental units than another The layout for this agricultural example is conceptually identical to that for the aforementioned two factor repeated measures design This design is sometimes referred to as completely randomized split plot design DOE and ANOVA Chapter11Chapter 11 Repeated Measures and Related Split plot Designs Model The ANOVA model and analyses are the same as the two factor repeated measures design yijk i j j k jk ijk j is the main eff ect of the jth irrigation method jth whole plot treatment and k is the main eff ect of the kth fertilizer type kth subplot treatment i j is the eff ect of the ith whole plot nested within the jth level of factor A DOE and ANOVA Chap