Introduction to Statistical Models and Factoring dependent and independent models “traditional” and possible applications ofindependent models 3-way sampling problem basic steps in factor analysesDependent multivariate modelsDependent models are used when we divide our variables into “criterion” and “predictor” variablesthe value of the criterion(ia) is “dependent” on the value of the predictor(s) -- statistically / causallysimple reg y’ = bx + amultiple reg y’ = bx + bx + bx + acanonical reg a + by + by = bx + bx + bx + ax & y vars are quantitative, binary, coded, or interaction termsDependent multivariate models, cont.Dependent models …these are the General Linear Models from “multivariate class”research questions/hypotheses are about which predictors (with what weightings) are useful for estimating what criteriaIndependent multivariate modelsIndependent models are used when there is no “predictor vs. criterion” distinction among our variables.The independent models we will examine are …Factor Analysis Cluster AnalysisMultidimensional ScalingResearch questions are about the number and identity(interpretation) of groupings among the “things” being analyzedIndependent multivariate models: “Traditional” UsesFactor variables to find the number and identity of the different kinds of information Factor the 25 questions from client’s intake interviewsCluster people to find the number and identity of the different kinds of characteristic profiles Cluster 250 students to took a standardized test Scale stimuli to find the “rules of stimulus similarity and dissimilarity” MDscale 24 shape stimuliBefore we get into the “alternative” uses of these models ...3-way sampling problem from a statistical or research design perspective “sampling”usually refers to the selection of some set of people from whichdata will be collected, for the purposes of representing what the results would be if data were collected from the entire population of people in which the researcher is interested from a psychometric perspective “sampling” is a broader issue, with three dimensionssampling respondents to represent the desired population of individualssampling attributes to represent some desired domain of characteristicssampling stimuli (things or people) to represent the desired category(ies) ofobjects3-way samplingExamples 20 patients each rate the complexity, meaningfulness and pleasantness of the 10 Rorschach cards 3 co-managers judge the efficiency, effectiveness, efficacy and elegance of the 15 workers they share 10 psychologists rate each of 30 clients on their amenability to treatment, dangerousness and treatment progress200 respondents complete a 50 item self-report personality measureStimuliAttributesPeopleLet’s look at how “people”, “attributes”and “stimuli” are used...From the examples...Example#1#2#3#4People Stimuli Attributes20 patients 10 cards cmp, ples. mng3 co-man 15 workers e, e, e & e10 psychists 30 clients amen, dang, tp200 responds 1 -- “self” 50 itemsSo, why is it called the 3-way sampling “problem” ??? one “problem” is that most data analysis models (both dependent and independent) start from a 2-way data set, most commonly… So, the 3-way data must be “prepared” for analysis, by either ...limited collection (only collect 2-way data-- only one person, one stimulus, or one attribute involved)selection (only use one 2-way “layer”from the 3-way sample) aggregation (combine across one “way”of the 3-way sample to get a 2-way layer)“Variables”“Cases”Let’s look at an example of each ...Examples of data prep...limited collectiononly collect 2-way data-- only one person, one stimulus, or one attribute involved Example -- 200 respondents complete a 50 item self-report personality measureonly one stimulus (“self” or “I”) -- so only a 2-way sampling (people x attributes) 2-way data table would look like200 Respondents50 ItemsExamples of data prep...selectiononly use one 2-way layer from the 3-way sample Example -- 20 patients each rate the complexity, meaningfulness and pleasantness of the 10 Rorschach cardshere’s what the 3-way data array would look like  Imagine the researcher were interested in only the meaningfulness dataonly those data would be selectedcomp mean plesnt20 Patients10 CardsExample of selection, cont.The resulting 2-way table would look like ...20 Patients10 cardsAll data are mean-ingfullness ratingsExamples of data prep...aggregationonly use one 2-way layer from the 3-way sample Example -- 3 co-managers judge the efficiency, effectiveness, efficacy and elegance of the 15 workers they sharehere’s what the 3-way data array would look like  Imagine the researcher was interested in how the workers differed in terms of the attributesef ef ef el3 co-mangrs15 workersExample of aggregation, cont. In this case, the co-manger ratings would be considered “replications”of each other -- existing primarily to get more stable data (than one manager’s rating) So, we would aggregate (take the mean) across the three co-managers for each attribute of each worker The resulting 2-way table would look like...15 workersAll data are average ratingsef ef ef elA second example of aggregation Imagine the researcher was interested in how the workers differed in the ratings given by the three co-mangers  In this case, the attributes would be considered “replications” of each other --existing primarily to get more stable data (than using one attribute) So, we would aggregate (take the mean) across the four attributes ratings from each co-manager, for each worker The resulting 2-way table would look like...15 workersAll data are average ratingsco#1 co#2 co#3Different ways of treating data for the different modelsThe 2-way data table we have been discussing is often labeled the “X” matrixstarting with “X”, different things are done to prepare the data for different modelLet’s look at these ..Remember, we’ll start with the “traditional” uses of the different models, and then look at the different ways they can be usedFactor variables to find the number and identity of the