3 Completely randomized The typical situation with only between-subjects factors (IVs)Participants are randomly assigned to treatment groupsDon’t trust yourself to do this, peeps is bad at randomizingExample in Rx=c(1,2,3,4) groups 1-4y=c(.25,.25,.25,.25) long run probability of occurrencesample(x, 80, replace=TRUE, prob = y)
4 Randomized block design Suppose an effect is present that may influence results, but for which you may not be interested inNuisance variableWe can extend the randomization process to control for a variety of factorsRandomized Block DesignAn extension of the completely randomized design in which a single extraneous variable that might affect test units’ response to the treatment has been identified and the effects of this variable are isolated by blocking out its effects.Examples:Researchers collecting the dataProbably a relationship between who is collecting and outcomeExperimenter/participant expectationsAgeCreate blocks of similarly aged individuals
5 Randomized block design In such a situation we want to randomly assign to treatment groups within each blocking levelExample: Treatment AExperimenter 1: condition 1 condition 2 condition 3Experimenter 2: condition 1 condition 2 condition 3Experimenter 3: condition 1 condition 2 condition 3
7 Randomized block design Partitioning the sums of squaresF-statistic is now MSb/t/MSresidualThis will be a more powerful test, although less so as SSresdiual approaches SSw/ini.e. less effect due to blocking variable
8 Randomized block design Conditions:Correlation between blocking variable and DVTreatment and blocking levels > 2Random assignment within each blocking unitGoal is to see homogeneous groups w/in each blocking level and treatment levels will look similarly with respect to one anotherMethods of implementationSubject matching (age example)Repeated measures (everyone gets all conditions, subjects serve as their own block)Other matched pair random assignment (e.g. twins, husband-wife)Latin-squares and Graeco-Latin squares extend the blocking to 2 and 3 nuisance variables respectivelyConsult Winer or Kirk texts
9 Latin Square DesignA balanced, two-way classification scheme that attempts to control or block out the effect of two or more extraneous factors by restricting randomization with respect to the row and column effectsRandomly assign treatment levels according to levels of the nuisance variable such that each level of the treatment variable is assigned to the levels of the nuisance variablesCan help with sequence effects with RM variablesThis special sort of balancing means that the systematic variation between rows, or similarity between columns, does not affect the comparison of treatments.
10 Previous therapy (none, yes but no drugs, yes w/ drugs) Latin SquareExample controlling for previous therapy experience and therapist (A is the treatment, e.g. type of therapy)A1 A2 A3 2 A2 A3 A A3 A1 A2Previous therapy (none, yes but no drugs, yes w/ drugs)Example: 3 groups receive Therapy A1Group1- therapist1, no previousGroup2- therapist2, drugsGroup3- therapist3, no drugsTherapist
11 LS design has limited utility in behavioral sciences Not too many cases with factors all with the same number of levelsKirk has suggested that with less than 5 levels it’s not very practical due to few degrees of freedom for error term (except for when several levels of treatment are assigned to each cell e.g. in a mixed design situation)Nevertheless, we see we have various ways to control for a number of effects that might be present in our data but not of interest to the study
12 Repeated measures design Subjects participate in multiple levels of a factorSimplest is the paired t-test situation, however ‘repeated measures design’ often implies 3 or more applications of some treatmentExample: pre and post-test
13 Mixed design Between subjects factors with repeated measures Simple but common example is control-treatment, pre-postSubjects are randomly assigned to control or treatment groups and tested twicePrePostControlTreatment
14 Mixed designThis example may appear a straightforward method for dealing with dataHowever we could analyze it in two additional waysANCOVATest for differences at time 2 controlling (adjusting) for differences seen at time 1A simple t-test on the gain scores from pre to postSame result as mixed but w/ different (simpler) outputOne method may speak more to the exact nature of your research question (e.g. only interested in time 2 or the rate of improvement from time 1 to 2)
15 Mixed designConsider also the possibility of an effect in posttest scores (or effect of treatment) due solely to exposure to the pretestE.g. with no prior training GRE scores would naturally be assumed to increase the second time around more often than notSolomon four-group designSplit-plot factorial controls for a nuisance variable in mixed designsWe’ll take a more in depth look at repeated measures and mixed designs later
16 Random effects modelsIn most ANOVA design we are dealing with fixed effectsRandom effectsIf the levels included in our analysis represent a random sample of all the possible levels, we have a random-effects ANOVA.With fixed effects the same levels of the effect would be used again if the experiment was repeated.With random effects different levels of the effect would be used.The conclusion of the random-effect ANOVA applies to all the levels (not only those studied).
17 Random effects models Questions to ask: Were the individual levels of the factor selected due to a particular interest, or were they chosen completely at random?Will the conclusions be specific to the chosen levels, or will they be applied to a larger population?If the experiment were repeated, would the same levels be studied again, or would new levels be drawn from the larger population of possible levels?
18 Random effects models Some examples: Persuasiveness of commercials Effect of worker on machine outputVehicle emissions impact on the environment
19 Random effects modelsDetermination of fixed vs. random effects affects the nature of the tests involved, how the F-statistic will be calculated (sometimes), and the conclusions to be drawnSome designs will incorporate both fixed and random effectsMixed effects designsDon’t confuse with previous mixed design factorial discussed previously