The Big PictureElements of Experimental DesignFixed and Random EffectsCRD One-Way ANOVAModelBalanced DesignsRandomizationWhy randomize?Randomization in RRandom EffectsMotivationModelsCompletely Randomized Design and Random EffectsBret LargetDepartments of Botany and of StatisticsUniversity of Wisconsin—MadisonMarch 8, 2007Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 1 / 11The Big Picture Elements of Experimental DesignThe Big PictureThe primary goal of experimental designs is to compare differenttreatments.Experimental units are individuals to which treatments are applied.Elements of design include:Replication — to assess variability.Randomization — to control selection bias.Blo cking — to control known sources of variability.In a completely randomized design, experimental units from a singlehomogeneous group are assigned at random to treatments.In a randomized complete block design, experimental units aregrouped with similar units into blocks and then assigned at random totreatments within blocks.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 2 / 11The Big Picture Fixed and Random EffectsFixed and Random EffectsWhen a blocking variable (or in general, a categorical variable)consists of all groups of interest, it is appropriate to model the effectsof this variable as fixed.When a blocking variable (or in general, a categorical variable) isthought of as a sample of groups from some larger population ofpossible groups, it is appropriate to model the effects of this variableas random.Random effects models involve multiple sources of random variation.We will begin with a review of a fixed effects models and then showhow the model changes with random effects.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 3 / 11CRD One-Way ANOVA ModelDataA categorical variable has k levels (or treatments).There are niobservations in the ith level for i = 1, . . . , k.The jth observation in the ith level is yijfor j = 1, . . . , ni.The model for the observed data isyij= µi+ eij, eij∼ iidN(0, σ2e),where µiis the population mean for the ith treatment group,j = 1, . . . , ni, i = 1, . . . , k.An alternative expression of the model isyij= µ + αi+ eij, eij∼ iidN(0, σ2e),where µ is the grand population mean µ =1kPki=1µiand αi= µi− µis the difference between the ith trt mean and the grand mean.Note thatPki=1αi= 0.This parameterization is not the default in R.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 4 / 11CRD One-Way ANOVA Balanced DesignsBalanced DesignsIn a balanced design, sample sizes are equal in each treatment group(ni= n for all i).Equations asso ciated with ANOVA F tests are simpler in balanceddesigns.The hypothesis of equal treatment means is equivalent to thehypothesis that all treatment effects are zero.H0: µ1= µ2= · · · = µkversus Ha: not all µi’s are equalH0: αi= 0 for all i versus Ha: not all αi= 0Sums of squares have these formula:SSTot =Pki=1Pnj=1(yij− ¯y··)2on df = kn − 1.SSTrt =Pki=1n(¯yi·− ¯y··)2on df = k − 1.SSErr =Pki=1Pnj=1(yij− ¯yi·)2on df = k(n − 1).where ¯y··is the grand mean and ¯yi·is the ith group mean.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 5 / 11CRD One-Way ANOVA Balanced DesignsANOVA for Balanced DesignsThe ANOVA table for a balanced design is:Source df SS MS FTrt k − 1 SSTrt MSTrt MSTrt/MSErrError k(n − 1) SSErr MSErr –Total kn − 1 SSTot – –Under model assumptions, the F statistic has an F distribution withk − 1 and k(n − 1) degrees of freedom.The estimated variance is ˆσ2e= MSErr.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 6 / 11CRD One-Way ANOVA Balanced DesignsExpected Mean Square (EMS)Both MSTrt and MSErr are random quantities.Each has a distribution and an expected value.Facts for balanced designs:E (MSErr) = σ2eE (MSTrt) = σ2e+nPki=1α2ik − 1.It follows for balanced designs thatE (MSTrt)E (MSErr)= 1 + 1σ2e×nPki=1α2ik − 1!.For unbalanced designs, there is a messier formula, but the samebasic principle holds.When treatment effects (αi) are not all zero, the ratio of expectedvalues is greater than one.The F -test is significant when the ratio is large enough.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 7 / 11Randomization Why randomize?RandomizationRandomization provides a way to control potential selection bias.If unknown factors affect the response variable, with randomizationthese factors are likely to be fairly balanced by the treatmentallocation.If known factors affect the response, these factors should be measuredand included in the mo del or in the design (with blocking).Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 8 / 11Randomization Randomization in RRandomization in RThe sample function in R can be used to allocate individuals totreatment groups at random.Here is an example to allocate 20 individuals into treatment groups A,B, C, and D equally.The function rep repeats the first argument some number of times.The function sample with only one argument places the elements ofthe argument in a random order.> trt = rep(c("A", "B", "C", "D"), each = 5)> trt[1] "A" "A" "A" "A" "A" "B" "B" "B" "B" "B" "C" "C" "C" "C" "C" "D" "D" "D" "D"[20] "D"> trt = sample(trt)> trt[1] "C" "C" "A" "B" "B" "A" "B" "A" "D" "C" "A" "C" "D" "D" "B" "C" "D" "D" "B"[20] "A"Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 9 / 11Random Effects MotivationMotivating ExampleExample 1:Compare the average reading skill of students in 8 specific secondgrade classes in Madison.Take a random sample of 10 students from each class and measuretheir reading skills.Example 2:Of interest is the reading skill among all second grade classes inMadison.Take a random sample of 8 classes.Within each class, take a random sample of 10 students and measuretheir reading skills.Note the difference in the objectives.In Example 1, we want to compare 8 specific classes and hence theclass effect is fixed.In Example 2, we want to assess the variabilities among classes and usea random sample of the classes to make inference about all the classesin Madison. Hence the class effect is random.Statistics 572 (Spring 2007) CRD and Random Effects March 8, 2007 10 / 11Random Effects ModelsModelsFixed effect model:yij= µ + αi+ eij, eij∼ iidN(0, σ2e)wherePki=1αi= 0.Random effect model:yij= µ + ai+ eij, j = 1, . . . , ni, i = 1, . . . , k,where aiis