1. Multivariate Data Analysis
Chao-Min Chiu
National Sun Yat-sen University
2010
Lecture 1

2. The Instructor BA: National Taiwan Institute of Technology
Industrial Management
MS: National Taiwan Institute of Technology
Information Management
PhD: Rutgers University (The State University of New
Jersey)
Computers & Information Systems
Phone: #4733
2010
Lecture 1

3. What Is It?
Referring to all statistical methods that simultaneously analyze multiple measurements on each individual or object under investigation
Loosely: any simultaneous analysis of more than two variables
Truly Multivariate: All variables must be random and interrelated in the such ways that their different effects cannot meaningfully be interpreted separately
Purpose: To measure, explain and predict the degree of relationship among variates (multiple combinations of variables)
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4. Basic Concepts X1 = income X2 = education The Variate
A linear combination of variables with empirically determined weights
A variable of n weighted variables (X1 to Xn)
Variate Value = w1X1+w2X2+ … + wnXn
Each respondent has a variate value (Y’).
The Y’ value is a linear combination of the entire set of variables. It is the dependent variable.
Potential Independent Variables:
X1 = income X2 = education
X3 = family size X4 = ??
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5. Measurement Scales The key: measurement
Important in accurately representing the concept of interest
Instrumental in the selection of the appropriate multivariate method of analysis
Nonmetric (qualitative)
Metric (quantitative)
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6. Types of Data and Measurement Scales
Nonmetric or
Qualitative
Metric or
Quantitative
Nominal
Scale
Ordinal
Scale
Interval
Scale
Ratio
Scale
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7. Measurement Scales
Nominal Scales: Assigns numbers as a way to label or identify subjects or objects.
E.g., Gender (0: female; 1:male)
Demographic attributes: gender、religion、occupation etc.
Ordinal Scales: Variables can be ordered or ranked in relation to the amount of the attribute possessed.
Satisfaction with products: (1: product A >
2: product B > 3: product C)
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8. Measurement Scales
Interval Scales: permitting nearly any mathematical operation to be performed.
Difference between any two adjacent points on any part of the scale are equal.
use an arbitrary zero point, e.g., temperature.
Ratio Scales: All mathematical operations are permissible.
use an absolute zero point, e.g., height.
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9. Measurement Error
Measurement Error: The degree to which the observed values are not representative of the “true” values
Must address validity and reliability
Validity is the degree to which a measure accurately represents what it is supposed to.
Reliability is the degree to which the observed variable measures the true value and is error free; thus, it is the opposite of measurement error.
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10. Measurement Error Multivariate measurements (summated scales)
Several variables are joined to in a composite measure to represent a concept
Using several variables as indicators
Satisfaction (S1, S2, S3, and S4)
Perceived Usefulness (PU1, …… PU6)
The impact of measurement error cannot be directly seen, requiring efforts to reduce it.
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11. Statistical Significance & Statistical Power
Rely on statistical inference, except for cluster analysis & multidimensional scaling
Require specifying the acceptable levels of statistical error
Most common: Type I error, or Alpha (α)
The prob. of rejecting the null hypothesis when actually true
Showing statistical significance when it actually is not present – a “false positive”
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12. When specifying Type I error, you also determine Type II error, or beta
Failing to reject the null hypothesis when it is actually false
An more interesting prob. is 1-ß, the power of statistical inference: the prob of correctly rejecting the null hypothesis when it should be rejected; i.e., statistical inference will be indicated if it is present
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13. Hypothesis testing Reality H0: no difference (H0 : true)
Ha: difference
(H0 : false) H0
(Accept H0)
Beta
Type II Error
1 - Alpha
Statistical
Decision
Alpha
Type I Error Ha
(Reject H0)
1 - Beta
Power
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14. Power is determined by three factors
Effect size: the actual magnitude of the effect of interest in the population, e.g., the SD of mean difference, and the actual correlation between the variables
Alpha:
As alpha becomes more restrictive (smaller), power decreases
Reducing the chance of finding an incorrect significant effect, the probability of correctly finding an effect also decreases
Sample size: Given an alpha level, increased sample size always produce greater power
The danger of producing “too much” power
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17. The relationships among alpha, sample size, effect size and power are very complicated
General guideline of Cohen: alpha level at least .05 with power levels of 80%
Power become acceptable at sample sizes of 100 or more in situations with a moderate effect size (0.5).
If anticipating the effects to be small -> larger sample sizes and/or less restrictive alpha levels
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18. A Classification
Dependence Techniques: having a dependent variable to be predicted
Multiple regression
Interdependence Techniques: simultaneous analysis of a set of variables
Factor analysis
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19. Dependence Techniques
Two characteristics:
the # of dependent variables
Single vs. multiple
the type of measurement scale employed by the variables
Metric (quantitative/numerical) vs. nonmetric (qualitative/categorical)
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20. Dependence Methods Canonical correlation MANOVA ANOVA
Y1 + Y Yn (metric, nonmetric) = X1 + X Xn (metric, nonmetric)
MANOVA
Y1 + Y Yn (metric) = X1 + X Xn (nonmetric)
ANOVA
Y1 (metric) = X1 + X Xn (nonmetric)
Multiple Discriminant Analysis
Y1 (nonmetric) = X1 + X Xn (metric)
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21. Multiple Regression Analysis Conjoint Analysis
Y1 (metric) = X1 + X Xn (metric, nonmetric)
Conjoint Analysis
Y1 (metric, nonmetric) = X1 + X Xn (nonmetric)
Structural Equation Modeling
Y1 (metric) = X11 + X X1n (metric, nonmetric)
Y2 (metric) = X21 + X X2n (metric, nonmetric)
Ym (metric) = Xm1 + Xm Xmn (metric, nonmetric)
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22. TAM PU1 PU2 PU3 Perceived Usefulness Intention Usage to Use Behavior
Ease of Use
Perceived Usefulness
PU1
0.82
PU2
0.88
PU3
0.90
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23. Problems A substitute for the necessary conceptual development
Complexity in the results and interpretation
No single “answer” exists
Some guidelines as a philosophy of multivariate analysis
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24. Guidelines and Interpretation
Practical & statistical significance
Focusing solely on the achieved significance of the results without understanding their interpretations, good or bad
Look at practical significance: “So what?”
Substantive & theoretical implications
E.g., repurchase intention, predicted as 50% at the .05 significance level, yet could vary as much as ±20%
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25. Sample Size Affects All Results
Too little statistical power for the test to identify significant results
Too easily an “overfitting” of the data such that the results are artificially good (no generalizability)
Large sample size: overly sensitive
Multiple groups with unequal sample sizes
Always assess the results in light of the sample used in the analysis
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26. Know Your Data Complex relationships require careful examination
More rigorous examine the data because the influence of outliers, violations of assumptions, and missing data
“Know where to look” for alternative formulations of the original model, such nonlinear and interactive relationships
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27. Strive for Model Parsimony
Conceptual model development first
Specification error: omitting a critical predictor variable; inserting variables indiscriminately – let the technique sort out
Increasing fit at the expense of overfit
Mask the true effect because of multicollinearity
Avoiding conceptually irrelevant variables
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28. Look at Your Error
Rarely do we achieve the best prediction in the first analysis, so modification?
But, “where does one go from here?”
Look at the errors in prediction
Residuals, Misclassification, outliers
Errors: not a measure of failure, nor merely something to eliminate
As a starting point for diagnosing the validity and an indication of the unexplained relationships
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29. Validate Your Results
Complex interrelationships means specific only to the sample and not generalizable
Sufficient observations per estimate: overfitting
Split the sample, holdout & test
A bootstrapping technique
Gathering a separate sample
The objective is not to find the “best” fit, but develop a model that best describe the population as a whole.
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30. Model Building
A series of guidelines that emphasizes a model-building approach
Focus on analyzing a well-defined research plan, starting with a conceptual model detailing the relationships to be examined
Then empirical issues can be addressed
Interpretation results
Diagnosis of generalizability
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31. Define the Research Problem, Objectives, and Multivariate Technique to Be Used
Specifying in conceptual terms
Primary importance of conceptual model, or theory
Defining the concepts and identifying the fundamental relationships to be investigated
A conceptual model need not be complex and detailed
A concept, rather than a variable, is defined
Identify the ideas or topics of interest
Dependency vs. Interdependency
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32. Develop the Analysis Plan
Turn to the implementation issues
With a techniques chosen, develop a analysis plan that addresses the set of issues particular to its purpose and design
Minimum or desired sample sizes
Allowable or required types of variables
Estimation methods
Resolving specific details and finalize the model formulation and requirements for data collection
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33. Evaluate the Assumptions Underlying the Multivariate Technique
First: to evaluate the underlying assumptions, conceptual or statistical
For dependency techniques:
Multivariate normality, linearity, independence of error terms, equality of variance
Conceptual assumptions:
Model formulation and types of relationship represented
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34. Estimate the Multivariate Model and Assess Overall Model Fit
Estimate the model and assess overall model fit
Choose among options to meet specific characteristics of the data or maximize the fit to the data
Level of significance; proposed relationships; practical significance
Before proceeding, obtain an acceptable model
Results unduly affected by any single or small set of observations?
Results are robust and stable
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35. Interpret the Variate(s)
Interpreting the variate(s) reveals the nature of the multivariate relationship
Individual variables: estimated coefficients
Multiple variates: underlying dimensions of comparison or association
May lead to additional respecification of variables and/or model formulation
Identify empirical evidence of multivariate relationships in the sample data that can be generalized to the total population
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36. Validate the Multivariate Model
Before accepting the results, subject to one final set of diagnostic analyses that assess the degree of generalizability of the results by the available validation methods
Adding little to the results, but can be viewed as “insurance” that the results are
The most descriptive of the data
Generalize to the population
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37. Bootstrapping, a computational nonparametric technique for " re-sampling, " enables researchers to draw a conclusion about the characteristics of a population strictly from the existing sample
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