Multivariate StatisticsStat Review 1IV vs. DVIV vs. DVSlide 5Extraneous vs. Confounding VariablesUnivariate, Bivariate, MultivariateExperimental vs. Non-ExperimentalWhy multivariate statistics?Slide 10Why multivariate?When is MV analysis not usefulStat Review 2Continuous, Discrete and Dichotomous dataSlide 15Slide 16Slide 17Slide 18Slide 19Normal Probability FunctionSlide 21Slide 22OrthogonalityOrthogonalitySlide 25Standard vs. Sequential AnalysesStandard vs. Sequential AnalysesSlide 28MatricesSlide 30Slide 31Slide 32Slide 33Multivariate Statistics Multivariate Statistics Psy 524Psy 524Andrew AinsworthAndrew AinsworthStat Review 1Stat Review 1IV vs. DVIV vs. DV Independent Variable (IV)Independent Variable (IV)–Controlled by the experimenterControlled by the experimenter–an d/oran d/or hypothesized influence hypothesized influence–an d/oran d/or represent different groups represent different groupsIV vs. DVIV vs. DVDependent variablesDependent variables–the response or outcome variablethe response or outcome variableIV and DV - “input/output”, “stimulus/response”, etc.IV and DV - “input/output”, “stimulus/response”, etc.IV vs. DVIV vs. DVUsually represent sides of an equationUsually represent sides of an equationx yb a+ =x yb a= +x y�x y z� �Extraneous vs. Confounding Extraneous vs. Confounding VariablesVariablesExtraneous Extraneous –left out (intentionally or forgotten)left out (intentionally or forgotten)–Important (e.g. regression)Important (e.g. regression)Confounding – Confounding – –Extraneous variables that offer Extraneous variables that offer alternative explanation alternative explanation –Another variable that changes along Another variable that changes along with IVwith IVUnivariate, Bivariate, MultivariateUnivariate, Bivariate, MultivariateUnivariateUnivariate–only one DV, can have multiple IVsonly one DV, can have multiple IVsBivariateBivariate–two variables no specification as to IV or two variables no specification as to IV or DV (r or DV (r or 2)2)MultivariateMultivariate–multiple DVs, regardless of number of IVsmultiple DVs, regardless of number of IVsExperimental vs. Non-ExperimentalExperimental vs. Non-ExperimentalExperimentalExperimental–high level of researcher control, direct high level of researcher control, direct manipulation of IV, true IV to DV causal flowmanipulation of IV, true IV to DV causal flowNon-experimentalNon-experimental–low or no researcher control, pre-existing groups low or no researcher control, pre-existing groups (gender, etc.), IV and DV ambiguous(gender, etc.), IV and DV ambiguousExperiments = internal validityExperiments = internal validityNon-experiments = external validityNon-experiments = external validityWhy multivariate Why multivariate statistics?statistics?Why multivariate statistics?Why multivariate statistics?RealityReality–Univariate stats only go so far when Univariate stats only go so far when applicableapplicable–““Real” data usually contains more than Real” data usually contains more than one DVone DV–Multivariate analyses are much more Multivariate analyses are much more realistic and feasiblerealistic and feasibleWhy multivariate?Why multivariate?““Minimal” Increase in ComplexityMinimal” Increase in ComplexityMore control and less restrictive More control and less restrictive assumptionsassumptionsUsing the right tool at the right timeUsing the right tool at the right timeRememberRemember–Fancy stats do not make up for poor planningFancy stats do not make up for poor planning–Design is more important than analysisDesign is more important than analysisWhen is MV analysis not usefulWhen is MV analysis not usefulHypothesis is univariate use a Hypothesis is univariate use a univariate statisticunivariate statistic–Test individual hypotheses univariately Test individual hypotheses univariately first and use MV stats to explorefirst and use MV stats to explore–The Simpler the analyses the more The Simpler the analyses the more powerfulpowerfulStat Review 2Stat Review 2Continuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataContinuous data Continuous data –smooth transition no steps smooth transition no steps –any value in a given range any value in a given range –the number of given values the number of given values restricted only by instrument restricted only by instrument precisionprecisionContinuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataDiscreteDiscrete–CategoricalCategorical–Limited amount of values and always Limited amount of values and always whole values whole values DichotomousDichotomous–discrete variable with only two categoriesdiscrete variable with only two categories–Binomial distributionBinomial distributionContinuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataContinuous to discreteContinuous to discrete–Dichotomizing, Trichotomizing, etc.Dichotomizing, Trichotomizing, etc.–ANOVA obsession or limited to one analysesANOVA obsession or limited to one analyses–Power reduction and limited interpretation Power reduction and limited interpretation –Reinforce use of the appropriate stat at the Reinforce use of the appropriate stat at the right timeright timeContinuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataX1 dichotomized at median >=11 and x2 at median >=10Continuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataCorrelation of X1 and X2 = .922Correlation of X1 and X2 = .922Correlation of X1di and X2di = .570Correlation of X1di and X2di = .570Continuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataDiscrete to continuousDiscrete to continuous–cannot be done literally (not enough info cannot be done literally (not enough info in discrete variables)in discrete variables)–often dichotomous data treated as often dichotomous data treated as having underlying continuous scalehaving underlying continuous scaleNormal Probability FunctionNormal Probability FunctionContinuous, Discrete and Continuous, Discrete and Dichotomous dataDichotomous dataCorrelation of X1 and X2 when Correlation of X1 and X2 when continuous scale assumed = .895continuous scale assumed = .895(called Tetrachoric correlation)(called Tetrachoric correlation)Not perfect, but closer to