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## Multivariate Statistical Data Analysis with Its Applications

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### Multivariate Statistical Data Analysis with Its Applications Hua-Kai Chiou Ph.D., Assistant Professor Department of Statistics, NDMC hkchiou_at_rs590.ndmc.edu.tw – PowerPoint PPT presentation

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Title: Multivariate Statistical Data Analysis with Its Applications

1
Multivariate Statistical Data Analysis with Its
Applications
• Hua-Kai Chiou
• Ph.D., Assistant Professor
• Department of Statistics, NDMC
• hkchiou_at_rs590.ndmc.edu.tw

September, 2005
2
Agenda
1. Introduction
3. Sampling Estimation
4. Hypothesis Testing
5. Multiple Regression Analysis
6. Logistic Regression
7. Multivariate Analysis of Variance
8. Principal Components Analysis

3
1. Factor Analysis
2. Cluster Analysis
3. Discriminant Analysis
4. Multidimensional Scaling
5. Canonical Correlation Analysis
6. Conjoint Analysis
7. Structural Equation Modeling

4
1
• Introduction

5
Some Basic Concept of MVA
• What is Multivariate Analysis (MVA)?
• Impact of the Computer Revolution
• Multivariate Analysis Defined
• Measurement Scales
• Type of Multivariate Techniques

6
• Dependence technique the objective is
prediction of the dependent variable(s) by the
independent variable(s), e.g., regression
analysis.
• Dependent variable presumed effect of, or
response to, a change in the independent
variable(s).
• Dummy variable nometrically measured variable
transformed into a metric variable by assigning 1
or 0 to a subject, depending on whether it
possesses a particular characteristic.
• Effect size estimate of the degree to which the
phenomenon being studied (e.g., correlation or
difference in means) exists in population.

7
• Indicator single variable used in conjunction
with one or more other variables to form a
composite measure.
• Interdependence technique classification of
statistical techniques in which the variables are
not divided into dependent and independent sets
(e.g., factor analysis).
• Metric data also called quantitative data,
interval data, or ratio data, these measurements
identify or describe subjects (or objects) not
only on the possession of an attribute but also
by the amount or degree to which the subject may
be characterized by attribute. For example, a
persons age and weight are metric data.

8
• Multicollinearity extent to which a variable
can be explained by the other variables in the
analysis. As multicollinearity increases, it
complicates the interpretation of the variate as
it is more difficult to ascertain the effect of
any single variable, owing to their
interrelationships.
• Nonmetric data also called qualitative data.
• Power probability of correctly rejecting the
null hypothesis when it is false, that is,
correctly finding a hypothesized relationship
when it exists. Determined as a function of
(1)the statistical significance level (a) set by
the researcher for a Type I error, (2) the sample
size used in the analysis, and (3) the effect
size being examined.

9
• Practical significance means of assessing
multivariate analysis results based on their
substantive findings rather than their
statistical significance. Whereas statistical
significance determines whether the result is
attributable to chance, practical significance
assesses whether the result is useful.
• Reliability extent to which a variable or set
of variables is consistent in what it is intended
to measure. Reliability relates to the
consistency of the measure(s).
• Validity extent to which a measure or set of
measures correctly represents the concept of
study. Validity is concerned with how well the
concept is defined by the measure(s).

10
• Type I error probability of incorrectly
rejecting the null hypothesis.
• Type II error - probability of incorrectly
failing to reject the null hypothesis, it meaning
the chance of not finding a correlation or mean
difference when it does exist.
• Variate linear combination of variables formed
in the multivariate technique by deriving
empirical weights applied to a set of variables
specified by the researcher.

11
• The Relationship between Multivariate Dependence
Methods
• Analysis of Variance (ANOVA)
• (metric) (nometric)
• Multivariate Analysis of Variance (MANOVA)
• (metric)
(nometric)
• Canonical Correlation
• (metric, nometric)
(metric, nometric)

12
• Discriminant Analysis
• (nometric) (metric)
• Multiple Regression Analysis
• (metric) (metric,
nometric)
• Conjoint Analysis
• (metric, nometric)
(nometric)

13
• Structural Equation Modeling
• (metric) (metric,
nometric)

14
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15
A Structured Approach to Multivariate Model
Building
• Stage 1 Define the research problem, objectives,
and multivariate technique to be used
• Stage 2 Develop the analysis plan
• Stage 3 Evaluate the assumptions underlying the
multivariate technique
• Stage 4 Estimate the multivariate model and
assess overall model fit
• Stage 5 Interpret the variate(s)
• Stage 6 Validate the multivariate model

16
2

17
• HATCO Case
• Primary Database
case from existing customers of HATCO.
• The primary database consists 100 observations on
14 separate variables.
• Three types of information were collected
• The perceptions of HATCO, 7 attributes (X1 X7)
• The actual purchase outcomes, 2 specific measures
(X9,X10)
• The characteristics of the purchasing companies,
5 characteristics (X8, X11-X14).

18
Table 2.1 Description of Database Variables
(Hair et al., 1998)
19
Fig 2.1 Scatter Plot Matrix of Metric Variables
(Hair et al., 1998)
20
Fig 2.2 Examples of Multivariate Graphical
Displays (Hair et al., 1998)
21
Missing Data
• A missing data process is any systematic event
external to the respondent (e.g. data entry
errors or data collection problems) or action on
the part of the respondent (such as refusal to
• The impact of missing data is detrimental not
only through its potential hidden biases of the
results but also in its practical impact on the
sample size available for analysis.

22
• Understanding the missing data
• Ignorable missing data
• Remediable missing data
• Examining the pattern of missing data

23
Table 2.2 Summary Statistics of Pretest Data
(Hair et al., 1998)
24
Table 2.3 Assessing the Randomness of Missing
Data through Group Comparisons of Observations
with Missing versus Valid Data (Hair et al., 1998)
25
Table 2.4 Assessing the Randomness of Missing
Data through Dichotomized Variable Correlations
and the Multivariate Test for Missing Completely
at Random (MCAR) (Hair et al., 1998)
26
Table 2.5 Comparison of Correlations Obtained
with All-Available (Pairwise), Complete Case
(Listwise), and Mean Substitution Approaches
(Hair et al., 1998)
27
Table 2.6 Results of the Regression and EM
Imputation Methods (Hair et al., 1998)
28
Outliers
• Four classes of outliers
• Procedural error
• Extraordinary event can be explained
• Extraordinary observations has no explanation
• Observations fall within the ordinary range of
values on each of the variables but are unique in
their combination of values across the variables.
• Detecting outliers
• Univariate detection
• Bivariate detection
• Multivariate detection

29
Outliers detection
• Univariate detection threshold
• For small samples, within 2.5 standardized
variable values
• For larger samples, within 3 or 4 standardized
variable values
• Bivariate detection threshold
• Varying between 50 and 90 percent of the ellipse
representing normal distribution.
• Multivariate detection
• The Mahalanobis distance D2

30
Table 2.7 Identification of Univariate and
Bivariate Outliers (Hair et al., 1998)
31
Fig 2.3 Graphical Identification of Bivariate
Outliers (Hair et al., 1998)
32
Table 2.8 Identification of Multivariate
Outliers (Hair et al., 1998)
33
Testing the Assumptions of Multivariate Analysis
• Graphical analyses of normality
• Kurtosis refers to the peakedness or flatness of
the distribution compared with the normal
distribution.
• Skewness indicates the arc, either above or below
the diagonal.
• Statistical tests of normality

34
Fig 2.4 Normal Probability Plots and
Corresponding Univariate Distribution (Hair et
al., 1998)
35
Homoscedasticity vs. Heteroscedasticity
• Homoscedasticity is an assumption related
primarily to dependence relationships between
variables.
• Although the dependent variables must be metric,
this concept of an equal spread of variance
across independent variables can be applied
either metric or nonmetric.

36
Fig 2.5 Scatter Plots of Homoscedastic and
Heteroscedastic Relationships (Hair et al., 1998)
37
Fig 2.6 Normal Probability Plots of Metric
Variables (Hair et al., 1998)
38
Table 2.9 Distributional Characteristics,
Testing for Normality, and Possible Remedies
(Hair et al., 1998)
39
Fig 2.7 Transformation of X2 (Price Level) to
Achieve Normality (Hair et al., 1998)
40
Table 2.10 Testing for Homoscedasticity (Hair et
al., 1998)
41
3
• Sampling Distribution

42
Understanding sampling distributions
• A histogram is constructed from a frequency
table. The intervals are shown on the X-axis and
the number of scores in each interval is
represented by the height of a rectangle located
above the interval.

43
• A bar graph is much like a histogram, differring
in that the columns are separated from each other
by a small distance. Bar graphs are commonly used
for qualitative variables.

44
What is a normal distribution?
• Normal distributions are a family of
distributions that have the same general shape.
They are symmetric with scores more concentrated
in the middle than in the tails. Normal
distributions are sometimes described as bell
shaped. The height of a normal distribution can
be specified mathematically in terms of two
parameters the mean (m) and the standard
deviation (s).

45
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