Presentation on theme: "MANOVA (and DISCRIMINANT ANALYSIS) Alan Garnham, Spring 2005"— Presentation transcript:
1MANOVA (and DISCRIMINANT ANALYSIS) Alan Garnham, Spring 2005 STATISTICAL ANALYSISMANOVA(and DISCRIMINANT ANALYSIS)Alan Garnham, Spring 2005
2What is MANOVA?Like ANOVA, applied to regimented experimental designs.But in cases where there is more than one DEPENDENT variableExample: text comprehension experiment with three dependent variablesclause reading timequestion answering timequestion answering accuracyusually analysed in separate ANOVAs, but could do MANOVA).
3Carreiras et al. 1996 Stereotyping Experiment The electrician examined the light fitting.He needed a special attachment to fix it.ORShe needed a special attachment to fix it.Was the electrician mending a stereo?Design: 2 (male/female stereotype) x 2 (pronoun matches or mismatches stereotype)
4Carreiras et al. 1996 Stereotyping Experiment In the paper we actually analysed the data using multiple univariate ANOVAsWe could have used MANOVAThis tells you something about typical practice in the field of psycholinguistics
5MANOVA - further examples Questionnaire data with subtest scores (the DVs) and respondents classified as e.g. male vs female, old vs young etc.Any other type of study with multiple tests (e.g. reading, writing, maths) and participants of different kinds (boys / girls; 6 year olds / 8 year olds etc.)
6What is MANOVA?Like ANOVA, MANOVA is a special case of the General Linear Model.y = Xb + eWhere y is a vector of criterion variables (DVs), X is a matrix of predictors (IVs, reflecting the study’s design), b is a vector of regression coefficients (weightings), and e is a vector of error terms.So, in SPSS: Analyse, GLM, Multivariate
7What is MANOVA?Looks to see if there are differences between groups on a linear combination of standardised DVsWhich is effectively a single new DVThis new DV is the linear combination of DVs which maximises group differencesDifferent combinations of DVs are selected for each main effect or interaction in the design
8Statistical Reasons for MANOVA Fragmented univariate ANOVAs lead to type 1 errorsseeing effects that aren’t really there.Because MANOVA effectively uses a single DV it protects against type 1 errors arising by chance from performing multiple testsUnivariate ANOVAs throw away info - correlation among dependent variables.
9Statistical Reasons for MANOVA Can get differences on a "combined" MANOVA measure, when none of the differences on the individual ANOVA measures are significant (so avoiding type 2 errors)in particular if treatments have different effects on the dependent variables, but the dependent variables are strongly correlated within any particular treatments (giving a small multivariate error term).(Extension of above) can avoid cancelling out effectsHowever, in practice this advantage is rarely realised
10MANOVA - Disadvantages More complexAdditional assumptionsOutcome can be ambiguousUsually lower power than ANOVA
11Null hypothesis in MANOVA Groups (experimental conditions) have the same mean for all the dependent (criterion) variables
12MANOVA - RestrictionCannot have too many DVs (fewer than cases)
13MANOVA: When and HowMay not be a good idea to put all dependent variables in one MANOVA. Better to put those that there is a good rationale for including in the main MANOVA and perhaps doing another on speculative variables.Reason: if there are no effects on the speculative dependent variables, they will just add noise to the analysis.
14Assumptions of MANOVAIndependence of observations (as in univariate ANOVA)Multivariate normality - all dependent variables and linear combinations of them are distributed normallyEquality of covariance matrices (cf homogeneity of variance in univariate). (Box's test to check, but set alpha to .001).
15Assumptions of MANOVASecond and third assumptions are more stringent than corresponding univariate assumptions in univariate ANOVA.
16MANOVA StatsGeneralisation of Student's t (replaces scalars by vectors/matrices) leads to Hotelling's T2 - only for 2 group case, though.For the multigroup case, no single agreed statistic. Best known is Wilk's lambda.
17MANOVA StatsSignificance means: there is a linear combination of the dependent variables (the discriminant function) that distinguishes the groups.Need post hoc tests to find out which dependent variables make significant contributions to discriminant function.For the multigroup case it is possible to use Hotelling's T2 tests for post hoc pairwise multivariate analyses.Hotelling's T2 can be followed up in this and the simple 2 group multivariate case by univariate t's.
18MANOVA STATISTICS Pillai-Bartlett Trace Hotelling's Trace Wilk's LambdaRoy's Greatest RootALL 4 are reported by SPSS
19MANOVA STATISTICS Each will have an F value associated with it These Fs are typically different (for the different tests) in the case of a "within" factor and any interaction including a within factor.
20MANOVA AND REPEATED MEASURES Repeated measures on a single individual, usually treated as a “within” factor in a univariate ANOVA can be thought of as measures on multiple dependent variables.So, repeated measures designs can be alternatively analysed using MANOVA.Recent versions of SPSS report MANOVA statistics for repeated measures designs.
21MANOVA AND REPEATED MEASURES Advantage: Avoids assumptions about equality of covariances required in repeated measures ANOVA.Violation of this assumption may be particularly problematic for specific comparisons.Problem: MANOVA may have less power.
22Discriminant Analysis As we have seen, MANOVA produces discriminant functionsLinear combinations of DVs that best separate the levels of an IV (or an interaction of IVs)Discriminant Analysis can be regarded as the inverse of (one-way) MANOVA
23Discriminant Analysis and MANOVA In discriminant analysis we ask if group membership can be predicted by a set of variablesE.g. Can party voted for at General Election be predicted from age, income, social class etc.
24Discriminant Analysis and MANOVA So, the IVs in MANOVA (specifically the levels of the single factor in one-way MANOVA) become the groups to which an individual might belong (Labour voter, Conservative voter etc.)And the DVs in the MANOVA become the predictorsWhether one thinks of a study as requiring MANOVA or discriminant analysis depends on extra-statistical considerations.
25Discriminant Analysis and MANOVA The mathematics is equivalent, just as ANOVA and multiple regression are equivalent, and all of them (ANOVA, MANOVA, MR, Discriminant Analysis) are special cases of the GLM.
26Discriminant Analysis and Logistic Regression Logistic Regression can also be used to predict group membership from a set of other variables.It has a different set of assumptions from Discriminant Analysis and is preferred by many authorities.In particular it unproblematically allows binary (in particular, and discontinuous, in general) predictors (as well as continuous ones).
27MANOVA - SummaryAn apparently attractive extension of ANOVA to the case of multiple dependent variables - included in a single analysisIt has more complex assumptions and less is known about robustness in relation to violations of assumptionsIn practice, its advantages are rarely realised
28Discriminant Analysis -Summary MANOVA produces discriminant functionsLooked at in a different way, one can ask whether the “DVs” in a MANOVA can predict “group membership” of the levels of the IV in the MANOVALogistic Regression, an alternative approach to such prediction, has advantages over discriminant analysis