Other Analytic DesignsOther Analytic DesignsPsy 420AinsworthLatin Square DesignsLatin Square Designs In a basic latin square (LS) design a researcher has a single variable of interest in a design and you want to control for other nuisance variables.other nuisance variables. To analyze the variable of interest while controlling for the other variables in a fully crossed ANOVA would be prohibitively large.Latin Square DesignsLatin Square Designs In these designs only the main effects are of interest The interaction(s) are confounded with the tests of the main effects and the error the tests of the main effects and the error termLatin Square DesignsLatin Square Designs Typically applied to repeated measures designs (to control for carryover, testing, etc.)Can be used with between groups Can be used with between groups variables (typically used to control for unavoidable nuisance variables like time of day, different instruments, etc.)Latin Square DesignsLatin Square Designs Basic latin square designb1b2b3b4a1c1c2c3c4a2c2c4c1c3a2c2c4c1c3a3c3c1c4c2a4c4c3c2c1Latin Square DesignsLatin Square Designs Limited view into the interactionsc1c2b1b2b3b4b1b2b3b4a1Xa1Xa2Xa2Xa3Xa3Xa4Xa4Xc3c4b1b2b3b4b1b2b3b4a1Xa1Xa2Xa2Xa3Xa3Xa4Xa4XLatin Square DesignsLatin Square Designs Other Types◦ Latin Square with replications◦ Crossover Designs – special name for a 2 level LS designlevel LS design◦ Greco-latin square design Another nuisance variable is incorporated 2 latin square designs superimposed◦ Incomplete Block Designs When a complete latin square is not possible Time constraints, limited number of subjects, etc.Screening/Incomplete DesignsScreening/Incomplete Designs Screening designs are analytical models that allow researchers to test for the effects of many variables while only using a few subjectsa few subjects Applicable to pilot-testing and limited subject pools (e.g. expense, time, etc.)Screening/Incomplete DesignsScreening/Incomplete Designs Resolution in Screening/Incomplete designs◦ Resolution refers to what aspects (factors) of a screening design are testable◦ Low resolution refers to designs in which only main effects can be tested◦ High resolution refers to designs in which main effects and two-way interactions can be testedScreening/Incomplete DesignsScreening/Incomplete DesignsResolution Description When To UseIII Main effects are independent of each other but are aliased with interationsYou are willing to assume that all interactions are negligible and will base future research only on screened main effectsIVMain effects are independent of You are willing to assume that all third- IVMain effects are independent of each other and independent of two-way interactions, but some two-way interactions are aliased with othersYou are willing to assume that all third- and higher-order effects are negligible. You are unable to determine ahead of time which two-way interactions are negligible and which are worth testing.V Main effects and two-way interactions are independent of each otherYou are willing to assume that all third- and higher-order effects are negligible. You want to test all main effects and two-way interactions.Screening/Incomplete DesignsScreening/Incomplete Designs 2-level Fractional Factorial Designs◦ Typically indicated by a 2 raised to the number of IVs (e.g. 25with five IVs)◦Fractional Factorial designs can be tested with Fractional Factorial designs can be tested with less “runs” by simply reducing the number of cells tested in the design and reducing the number of replications◦ Reduced fractional models are indicated by subtracting some value, q, from the factorial (e.g. 2k-q)Screening/Incomplete DesignsScreening/Incomplete Designs A 25fractional factorial with replicationb1b2b1b2s1s17s33s49s2s18s34s50s3s19s35s51s4s20s36s52a1a2cd1e1e2s4s20s36s52s5s21s37s53s6s22s38s54s7s23s39s55s8s24s40s56s9s25s41s57s10s26s42s58s11s27s43s59s12s28s44s60s13s29s45s61s14s30s46s62s15s31s47s63s16s32s48s64c2d1e1e2d2e1e2c1d2e1e2Screening/Incomplete DesignsScreening/Incomplete Designs A 25-1half fractional factorial with replicationb1b2b1b2s1s3s2s4s5s7s6s8ca1a2d1e1e2s6s8s9s11s10s12s13s15s14s16s17s19s18s20s21s23s22s24s25s27s26s28s29s31s30s32c1c2e1e2d2d1d2e1e2e1e2Screening/Incomplete DesignsScreening/Incomplete Designs A 25-2quarter fractional factorial without replicationb1b2b1b2e1s8e2s5e1s6e2s7a1a2c1d1d2e2s7e1s4s1e2e1s3e2s2c2d1d2Screening/Incomplete DesignsScreening/Incomplete Designs Other designs◦ Plackett-Burman (resolution III) – created to maximize main effects with limited subjects◦Taguchi –created for quality control and Taguchi –created for quality control and focus on combination of variables (not sig testing) and test signal to noise ratios converted to dB scaleScreening/Incomplete DesignsScreening/Incomplete Designs Other designs◦ Response-surface methodology Box-Behnken – used with 3 three-level quantitative IVs. Tests for trend on the main effects with minimum subjectsminimum subjects Central-Composite – Same as Box-Behnken but each IV has 5 levels instead of 3 Mixture/Lattice models – used when you are testing variables that are blends or mixtures of quantitative IVs where the sum for each IV is a fixed amount (e.g. 100%)Random Effects ANOVARandom Effects ANOVA So far, everything has assumed that the IVs were Fixed Fixed effects means that we as researchers pick the levels of the IV(s) and are not typically interested in generalizing beyond the levels we chosechose It’s also assumed that the levels are without error (no variability) But what if we do want to generalize beyond or if we feel that there is some variability inherent in the IV levels?Random Effects ANOVARandom Effects ANOVA So far we have had one effect that we have considered random; subjects What makes them random?Why do we want random subjects?Why do we want random subjects? It’s the same reason(s) we want random levels of an IVRandom Effects ANOVARandom Effects ANOVA With random effects we create a “population” of possible levels (e.g. quantitative levels) for an IV and randomly select from itselect from it So we have a random “sample” of IV levels that will vary from study to study The goal is to increase the generalizabilityof the results beyond just the levels that were selectedRandom Effects ANOVARandom Effects ANOVA Generalizability is increased