BSAD 6314

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BSAD 6314 Multivariate Statistics
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• Major Goals
• Expand your repertoire of analytical options.
• Choose appropriate techniques
• Conduct analyses using available software.
• Interpret findings
• Know where to go for additional help.

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What are Multivariate Stats?

• Correlation

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• Canonical Correlation

• t test

• One-way ANOVA

• One way MANOVA

IV

• Multiple Regression

• Canonical Correlation

gt1

• Factorial ANOVA

• Factorial MANOVA

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DV
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Hair Taxonomy
Structure/ Independence
Dependence
Type of relationship
variables?
Structure
Single Rel. Mult DVs
One DV, Mult IVs
Structure among cases/ people
Mult Rel. IVs DVs
Structure among variables
IV Cat.
DV Cont.
IV Cont.
DV Cat.
Factor Analysis
Cluster Analysis
SEM
MANOVA
Canon. Corr.
Discrim Log Regr
Mult. Regr.
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Matrices

• The math behind the statistics!
• Two dimensional math represented in an array
• People x Variables (OB/HR).
• Organization x Variables (Strategy/OT).

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The variables (V) can be continuous measures,
categories represented by numbers, and
transformations, products or combinations of
other variables.
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Nearly all statistical proceduresunivariate and
multivariateare based on linear combinations.
Understanding that basic fact has far-reaching
implications for using statistical procedures to
their fullest advantage. A linear combination
(LC) is nothing more than a weighted sum of
variables LC W1V1 W2V2 . . . WKVK
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LC W1V1 W2V2 . . . WKVK A very simple
example is the total score on a questionnaire.
The individual items on the questionnaire are the
variables V1, V2, V3, etc. The weights are all
set to a value of 1 (i.e., W1 W2 . . . Wk
1). We call this unit weighting.
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The items combined in a linear combination need
not be variables. The items combined are often
cases (e.g., people, groups, organizations). LC
W1P1 W2P2 . . . WKPN A good example is the
sample mean. In this case the weights are set to
the reciprocal of the sample size (i.e., W1 W2
. . . Wk 1/N).
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Different statistical procedures derive the
weights (W) in a linear combination to either
maximize some desirable property (e.g., a
correlation) or to minimize some undesirable
property (e.g., error). The weights are sometimes
assigned rather than derived (e.g., dummy,
effect, and contrast codes) to produce linear
combinations of particular interest.
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Choice of Statistical Techniques

• Research Purpose
• Prediction, Explanation, Data Reduction
• Number of IVs
• Number of DVs
• Measurement Scale
• Continuous, categorical

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Dependence/Prediction

• Correlation
• Regression
• Multiple Regression
• With nominal variables Polynomials
• Logistic regression/Dicriminant analysis
• Canonical Correlation
• MANOVA

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The simplest possible inferential statistic-the
bivariate correlation-involves just two variables.
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Continuous
Continuous
In its usual form, the correlation is calculated
on variables that are continuous.
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Continuous
Categorical
When one of the variables is categorical, the
calculation produces a point-biserial correlation.
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Categorical
Categorical
When both variables are categorical, the
calculation produces a phi coefficient.
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Regression All forms of these correlations,
however, can be recast as a linear combination
V2 predicted A BV1
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OLS (ordinary least squares) Regression B and A
can be chosen so that the sum of the squared
deviations between V2 predicted and V2 are
minimized. Solving for B and A using this rule
also produces the maximum possible correlation
between V1 and V2.
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Least Squared Error
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The problem can be easily expanded to include
more than one predictor. This is multiple
regression V4predicted B1V1 B2V2 B3V3 A
Continuous
Continuous
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The values for B1, B2, B3, and A are found by the
least squares rule
Continuous
Continuous
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V1, V2, and V3 could be categorical contrast
variables, perhaps coding the two main effects
and the interaction from an experimental design.
In that case, the multiple regression produces an
analysis of variance.
Continuous
Categorical
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Or V2 might be the square of V1 and V3 might be
the cube of V1. Then the multiple regression
examines the polynomial trends.
Continuous
Continuous
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If the outcome variable is categorical, the
basic nature of the analysis does not change. We
still seek an optimal linear combination of V1,
V2, and V3.
Categorical
R
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Set A The basic multiple regression problem can
be generalized to situations that involve more
than one outcome variable.
Set B
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Set A LCA W1V1 W2V2 W3V3 W4V4
W5V5 W6V6
Set B LCB W7V7 W8V8 W9V9
W10V10 W11V11 W12V12
RAB
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Structure of Data

• Goal is typically data reduction
• How do the data hang together?
• Techniques include
• Factor analysis (the items)
• Cluster analysis (the cases, people,
  organizations)
• MDS (the objects)

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Sometimes we are not interested in relations
between sets of variables but instead focus on a
single set and seek a linear combination that has
desirable properties.
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Or we might wonder how many dimensions underlie
the 12 variables. These multiple dimensions also
would be represented by linear combinations,
perhaps constrained to be uncorrelated. These
questions are addressed in principal components
analysis and factor analysis.
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A
B
C
D
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Sometimes we might shift the status of people
and variables in our analysis. Our interest
might be in whether a smaller number of
dimensions or clusters might underlie the larger
collection of people.
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Approaches such as multidimensional scaling and
cluster analysis can address such questions.
These are conceptually similar to principal
components analysis, but on a transposed matrix.
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Structural Equation Modeling

• Examines both structure of the data and
  predictive relationships simultaneously
• Measurement model
• Structural model

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Structural equation models examine the relations
among latent variables. Two kinds of linear
combinations are needed.
A1
A
A2
D3
B1
D
B
D2
B2
D1
C1
C
C2
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The key idea is that the original data matrix can
be transformed using linear combinations to
provide useful ways to summarize the data and to
test hypotheses about how the data are
structured. Sometimes the linear combinations are
of variables and sometimes they are of people.
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Once a linear combination is created, we also
need to know something about its variability.
Everything we need to know about the variability
of a linear combination is contained in the
variance-covariance matrix of the original
variables (S). The weights that are applied to
create a linear combination can also be applied
to S to get the variance of that LC.
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