Overview

1
Overview

• Why we need multivariate statistics
• Types of designs
• Types of data
• Orthogonality
• Linear composites
• Matrices
• Types of multivariate analyses

2
The following says it all!

• Multivariate thinking is defined as a body of
  thought processes that illuminate interrelations
  between and within sets of variables

3
Why o why do we need multivariate statistics?!

• Theoretical reasons
• Real-world is multidimensional and multicausal
• i.e., multiple IVs (predictors) and DVs
  (outcomes)
• increases precision using multiple measures of a
  construct
• increases completeness
• Statistical reasons
• Examine large data sets in a single analysis
• control for type 1 error rates (omnibus tests)
• generally increasing statistical power (reduces
  error)

4
Experimental and Nonexperimental (Correlational)
Research

• Experimental
• researcher controls levels of IVs
• i.e., systematically controls IV variance
• random assignment is used to reduce third
  variables
• Nonexperimental (correlational)
• research does not control variance of the IV
• causality is more difficult
• IV-DV distinction may be arbitrary

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Multivariate Statistics and Experimental and
Nonexperimental Research

• Nonexperimental (correlational)
• multivariate techniques first developed for this
• goal was to take correlation among measures into
  account
• reduce the number of measured variables
• reduce type I errors
• Experimental
• IVs are typically orthogonal
• however, multiple DVs are typically measured,
  so...
• reduce type I errors

6
Lets get definitional for a moment

• Types of data (i.e., how are variables measured)
• continuous (quantitative)
• scale scores
• discrete (categorical, qualitative)
• can be either ordered (e.g., income, age)
• or not (e.g., ethnicity)
• dichotomous (or binary)
• you tell me

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Types of data continued

• Making a continuous variable discrete or binary
• generally not a good idea loss of information
• median split, tertiary split, quartile split
• Assuming Likert scales are continuous
• individual items are tough (assume continuous?)
• scale scores are typically OK
• Distribution of data, not property of values
  typically more important

8
Orthogonality vs. Obliqueness

• Definition
• Non vs. association between variables (typically
  IVs)
• Old vs. New School

9
Linear Composites

• Typically form a linear combination of variables
• Y W1X1 W2X2 error
• The weights are meaningful (indicate strength)
• Indicates the association between variables
• e.g., pattern matrix, structure matrix
• Algorithms maximize the size of these weights
• Maximum likelihood (ML)
• Expectation-maximization (EM), Full-Information
  ML, Restricted-Information ML

10
Data appropriate for multivariate statistics
Just a look-see at matrices

• N X p Data matrix

Participant Depression Coping Alcohol
Use 1 2 3 4 2 1 1 1 3 4 2 3 4
7 7 7
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Just a look-see at matrices continued

• Correlation matrix (R)
• Square and symmetric
• Metric free

Depression Coping
Alcohol Use Depression 1.0 .30 .15 Coping .3
0 1.0 .50 Alcohol Use .15 .50 1.0
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Just a look-see at matrices continued

• Variance-Covariance (?)
• Nonstandardized values
• Square and symmetric
• Retains metric of the original variables

Depression Coping
Alcohol Use Depression 4.21 1.62 2.52 Coping
1.62 4.02 1.38 Alcohol Use 2.52 1.38 4.34
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Just a look-see at matrices continued

• Determinant
• Single number providing an index of the
  generalized variance in a matrix
• Ranges from 0 to 1
• Tells us how much the variables in a matrix
  differ
• Values close to 0 indicate that variables are
  oblique
• Values close to 1 indicate that variables are
  orthogonal

14
Research Questions and Multivariate Techniques

• Research question, type of data, and number of
  variables determine statistic
• Five (2) big ones
• Degree of relationship among variables
• Significance of group differences
• Prediction of group membership
• Structure
• Time course of events
• Nested data structures
• Profiles of people

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Degree of relationship among variables

• Some form of correlation/regression or chi-square
• Bivariate r
• Multiple r (not multiple regression, which is
  predictive)
• Sequential r (hierarchical multiple regression)
• Canonical r (multiple IVs and DVs)
• Multiway frequency analysis (log-linear analysis)
• logit analysis if we want to predict the DV
• Path analysis (can be predictive)
• temporal relations among observed variables
• figure on next page

16
Path-Analytic Model
Mediator
Predictor (IV)
Criterion (DV)
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Significance of Group Differences

• Youve had some these previously
• Oneway ANOVA
• Oneway ANCOVA
• Factorial AN(C)OVA
• Hotellings T2
• Oneway MAN(C)OVA
• Factorial MAN(C)OVA
• Add Repeated Measures to any of these

18
Prediction of Group Membership

• Predicting group membership (DV) from a set of
  variables (IVs)
• Types
• Discriminant function analysis
• IVs are continuous
• same as MANOVA but the variables have switched
  sides
• Logit analysis
• predictors (IVs) are discrete
• Logistic regression
• -predictors are a mix of continuous and discrete

19
Structure

• What latent variable(s) underlie our observed
  variables?
• Types
• Principal Components Analysis (PCA)
• exploratory analysis for data reduction
• transform correlations of observed variables into
  components
• (Exploratory) Factor Analysis (EFA)
• more theoretical (????????)
• also used for data reduction

20
Structure continued

• Confirmatory Factor Analysis (CFA)
• a priori measurement model is tested
• direct relations between observed and latent
  variables are modeled

21
Structure continued

• Structural Equation Modeling (SEM)
• CFA plus a priori structural model is tested
• direct relations among latent variables are
  modeled
• see figure on next page, too big for here!

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Structure continued SEM Model
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Structure continued

• Multiple Group (Multisample) CFA
• also called testing for invariance
• determines if the measurement model is equivalent
• across groups
• across items, scales, ...
• Multiple Group SEM
• determines if structural model is equivalent
• across groups!

24
Time Course of Events

• Two types
• Survival/Failure Analysis
• How long does it take for something to happen
  (e.g., diagnosis of schizophrenia)?
• Compare groups or determine variables associated
  with time
• Longitudinal Models!!!!!
• Autoregressive, Cross-Lagged Models
• Latent Growth Curve Modeling

25
Nested Data Structures

• Hierarchical linear modeling
• Examples include
• repeated observations nested within individuals
• individuals nested within groups
• groups nested within communities
• communities nested within cultures
• We cannot treat data units at the lowest level
  as independent

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Creating typologies

• Latent class analysis
• Creating groups of individuals based on responses
  to binary variables
• Latent profile analysis
• Creating groups of individuals based on responses
  to continuous variables

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Multivariate statistics are not perfect?

• Garbage in, garbage out!
• there is no substitute for valid, reliable
  measures
• More variables ?More ambiguity, more output,
  more...
• classification indices, factor rotation, where
  does it end?!
• More variables means we need more people
• 10 people when assumptions are met
• 20-50 when assumptions are not met