PSY400 Research Methods and Analysis 4 from University of the Sunshine Coast

PSY400 Research Methods and Analysis 4

UniSC
PSY400 Research Methods
and Analysis 4
Dr Joshua Adie
University of the Sunshine Coast | CRICOS Provider Number: 01595DUniSC
Week 6
Multivariate Predictors with Multivariate
Outcome Designs
University of the Sunshine Coast | CRICOS Provider Number: 01595DReading - covers undertaking these techniques in SPSS

• Field, A. (2018) Discovering statistics using IBS SPSS statistics
• Chapters 17.9-17.12 (Discriminant analysis)

Workshop content

1. Multiple Discriminant Analysis (MDA)
2. Why use MDA?
3. Research designs for MDA
4. Assumptions of MDA

University of the Sunshine Coast | CRICOS Provider Number: 01595D
UniSCOverview of Multivariate Methods

What type of
relationship is
being examined?

Dependence

Interdependence

Multivariate
technique

How many
variables are
being
predicted?

Is the structure
of the
relationships
among:

Multiple relationships of
dependent and independent
variables

Several dependent
variables in a single
relationship

One dependent variable
in a single relationship

Variables

Cases/respondents

Objects

• Structural
  Equation
  Modelling

What is the
measurement
scale of the
DV?

What is the
measurement
scale of the
DV?

Factor analysis
Confirmatory
factor analysis
Cluster analysis

How are the
attributes
measured?

Metric
Nonmetric
Metric
+
Nonmetric
+
Metric
Nonmetric
7

What is the
measurement
of the
predictor
variable?

Canonical
correlation
analysis with
dummy variables
Conjoint analysis
Linear
Probability
Models
Metric
Nonmetric
+
Canonical
correlational
analysis
MANOVA
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UniSC
Nonmetric
Multiple
Regression
Multiple
Discriminant
Analysis
Multidimensional
scaling
Correspondence
analysis
Decision
point

Multiple Discriminant Analysis (MDA)

MDA Overview

aka Discriminant Function Analysis, Profile Analysis
. The goal of discriminant analysis is to predict group membership from a set of predictors.
. Multiple Discriminant Analysis (MDA) is effectively an inverse of MANOVA - and is also an extension from logistic regression but
enables prediction of group membership where there are more than 2 groups.
· Applying and interpreting an MDA is similar to a regression analysis; in MDA the discriminant function is a linear
combination (variate) of metric measurements for two or more IVs and is used to describe or predict a single DV.
· key difference is that MDA is appropriate for research problems in which the DV is categorical (nominal or nonmetric), whereas
regression is utilized when the DV is metric.
· MDA is also comparable to "reversing" multivariate analysis of variance (MANOVA). In MDA, the single DV is
categorical, and the IVs are metric. The opposite is true of MANOVA, which involves metric DVs and categorical IV(s).
University of the Sunshine Coast | CRICOS Provider Number: 01595D
UniSCSimple MANOVA model

Level 1
DV1
Level 2
Level 3
IV1
Variate
DV2
.
.
DVK
Level n
e.g., group differences on variate of DV1-DVK
Basic structure of a logistic regression

IV1
a
IV1 unique variance
Group 1
(DV level 1)
Variate
b
IV3 unique variance
IV3
IVn unique variance
Group 2
(DV level 2)
IVn
Multiple Discriminant Analysis model

IV1
Group 1
Group 2
Group 3
IV2
Variate
(discriminant
function)
.
DV
(group)
IVk
Group n
Zjk = a + W1X1k + W2X2k+ ... + WnXnk
Where:
Zjk = discriminant Z score of discriminant function j for object k
a = intercept
W; = discriminant weight for independent variable i
Xik = independent variable i for object
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M UniSC
IV2
IV2 unique variance· MDA - appropriate statistical technique for testing the hypothesis that the group means of a set of IVs for two or more
groups are equal.
· MDA multiplies each IV by its corresponding weight and adds these products together. The result is a single composite
discriminant Z score for each individual in the analysis (similar to a regression linear equation).
· By averaging the discriminant scores for all the individuals within a group, we arrive at the group mean. This group
mean is referred to as a centroid.
. When the analysis involves 2 groups, there are 2 centroids; with 3 groups, there are 3 centroids; and so forth. A centroid
indicates the average location of any individual from a particular group, and a comparison of the group centroids shows how far
apart the groups are along the dimension being tested.
· test of the statistical significance of the discriminant function is a generalized measure of the distance between the group
centroids. It is computed by comparing the distributions of the discriminant scores for the groups. If the overlap in the
distributions is small, the discriminant function separates the groups well. If the overlap is large, the function is a poor
discriminator between the groups.

Discriminant Function Examples

Good discriminator

Group A
Group B
z
Discriminant function
Poor discriminator

Group A
Group B
Z
Discriminant function
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UniSC· MDA is unique:
· if >2 groups in the DV, MDA will calculate >1 discriminant function.
· A MDA will calculate NG - 1 discriminant functions (NG = number of
groups.
· In a 3-group DV, each group will have a score for discriminant
functions 1 and 2, allowing the groups to be plotted in two
dimensions, with each dimension representing a discriminant
function.
· MDA is not limited to a single variate (as in multiple
regression) but creates multiple variates representing
dimensions of discrimination among the groups.

Multiple Discriminant Analysis Decision Diagram

Research problem
Selective objective(s):

• Evaluate group differences on a multivariate profile
• Classify observations into groups
• Identify dimensions of discrimination between groups

Research design issues
Selection of IVs
Sample size considerations
Creation of analysis and holdout samples
Assumptions

• Normality of IVs
• Linearity of relationships
• Lack of multicollinearity among IVs
• Equal dispersion matrices

Estimation of the Discriminant Function(s)

• Simultaneous or stepwise estimation
• Significance of the discriminant function
  Assess predictive accuracy with classification matrices
• Determining optimal cutting score
• Specify criteria for assessing hit ratio
• Statistical significance of predictive accuracy

ONE
Interpretation of the
Discriminant Function(s)
How many DFs will be
interpreted?
2 or
more
1
I
Evaluation of single function
Evaluation of separate functions

• Discriminant weights
• Discriminant weights
• Discriminant loadings
• Discriminant loadings
• Partial F values
• Partial F values

1
Evaluation of combined functions

• Rotation of functions
• Potency index
  . Graphical displays of group centroids
  . Graphical display of loadings
  Validation of Discriminant results
  Split-sample or cross-validation
  Profiling group differences
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  UniSC

Objectives of MDA

· MDA can address any of the following research objectives:

1. Determining whether statistically significant differences exist between the average score profiles on a set of variables for two
   (or more) apriori defined groups.
2. Determining which of the IVs account the most for the differences in the average score profiles of the two or more groups
3. Establishing procedures for classifying objects into groups on the basis of their scores on a set of IVs.
4. Establishing the number and composition of the dimensions of discrimination between groups formed from the set of IVs.

· MDA is useful when the researcher is interested either in understanding group differences or in correctly classifying
objects into groups or classes. MDA is most appropriate where there is a single categorical DV and several metrically
scaled IVs.
. MDA provides an objective assessment of differences between groups on a set of IVs - so is similar to MANOVA
. The use of sequential estimation methods allows for identifying subsets of variables with the greatest discriminatory
power.
. For classification purposes, MDA provides a basis for classifying not only the sample used to estimate the discriminant
function but also any other observations that can have values for all the IVs. In this way, the MDA can be used to
classify other observations into the defined groups.
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UniSC

Research Design for MDA

Selecting IVs and DVs for MDA

3.1 Selecting IVs and DVs:
. The DV is categorical. The number of DV groups (categories) can be 2 or more, but these groups must be mutually
exclusive and exhaustive (each observation can be placed into only one group). In some cases, the DV may involve 2
groups (dichotomous), in other cases, the DV may involve several groups (multichotomous).
· the IV(s) are metric.
After a decision has been made on the DV, then need to decide which IVs to include in the analysis. IVs are usually
selected in two ways:

1. identifying variables either from previous research or from the theoretical model that is the underlying basis of the research
   question.
2. Intuition: utilising the researcher's knowledge and intuitively selecting variables for which no previous research or theory exists
   but that logically might be related to predicting the groups for the DV.

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Sample Size Considerations for MDA

3.2 Sample Size:
. Many studies suggest a ratio of 20 observations for each predictor variable.
. The minimum size recommended is 5 observations per IV (applies to all variables considered in the analysis, even if all of the
variables considered are not entered into the MDA such as in stepwise estimation).
· At a minimum, the smallest group size must exceed the number of IVs. As a practical guideline, each group should
have at least 20 observations.
. But must also consider the relative sizes of the groups. If the groups vary widely in size, this may impact the estimation of the
MDA and the classification of observations.
. If the group sizes vary markedly - randomly sample from the larger group(s), thereby reducing the size of the largest group(s) to a
level comparable to the smaller group(s).
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UniSC