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

September, 2005
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Agenda

1. Introduction
2. Examining Your Data
3. Sampling Estimation
4. Hypothesis Testing
5. Multiple Regression Analysis
6. Logistic Regression
7. Multivariate Analysis of Variance
8. Principal Components Analysis

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

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1

• Introduction

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Some Basic Concept of MVA

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

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• 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.

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• 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.

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• 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.

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• 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).

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• 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.

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• 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)

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

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

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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

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2

• Examining Your Data

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• HATCO Case
• Primary Database
• This example investigates a business-to-business
  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).

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Table 2.1 Description of Database Variables
(Hair et al., 1998)
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Fig 2.1 Scatter Plot Matrix of Metric Variables
(Hair et al., 1998)
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Fig 2.2 Examples of Multivariate Graphical
Displays (Hair et al., 1998)
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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
  answer) that leads to missing values.
• 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.

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

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Table 2.2 Summary Statistics of Pretest Data
(Hair et al., 1998)
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Table 2.3 Assessing the Randomness of Missing
Data through Group Comparisons of Observations
with Missing versus Valid Data (Hair et al., 1998)
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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)
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Table 2.5 Comparison of Correlations Obtained
with All-Available (Pairwise), Complete Case
(Listwise), and Mean Substitution Approaches
(Hair et al., 1998)
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Table 2.6 Results of the Regression and EM
Imputation Methods (Hair et al., 1998)
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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

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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

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Table 2.7 Identification of Univariate and
Bivariate Outliers (Hair et al., 1998)
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Fig 2.3 Graphical Identification of Bivariate
Outliers (Hair et al., 1998)
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Table 2.8 Identification of Multivariate
Outliers (Hair et al., 1998)
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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

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Fig 2.4 Normal Probability Plots and
Corresponding Univariate Distribution (Hair et
al., 1998)
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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.

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

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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.

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• 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.

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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).

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