1. Applied Multivariate Quantitative Methods
Introduction and Examples of
Multivariate Data
By Jen-pei Liu, PhD
Division of Biometry, Department of Agronomy,
National Taiwan University
and
Wei-Chie Chie, MD, PhD
Department of Public Health
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

2. Introduction and Examples of Multivariate Data
Most of statistical methods only focus on analysis of one random variable at a time – Univariate analysis
Ordinary regression analysis investigates the empirical relationship between a random variable (dependent variable) and a set of explanatory variables (nonrandom independent variables) and is still a univariate analysis
However, univariate analysis can not provide a comprehensive and overall description for most of phenomena we are interested in
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

3. Introduction and Examples of Multivariate Data
Clinical trials:
Efficacy
Stroke
NIH stroke scale
Bartel score
Modified Rankin score
Safety
Adverse events
Laboratory evaluations
Vital signs
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

4. Introduction and Examples of Multivariate Data
Quality of Life
Physical
Mental
Social
Well-being
Microarray
Intensity of tens of thousands of transcripts
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

5. Introduction and Examples of Multivariate Data
Continuous Endpoints
Numerical discrete data
Heart beats per minutes
Total NINSS
Total Hamilton Rating Scale for Depression
Total Alzheimer’s Disease Assessment Scale
Numerical continuous data
Age
Weight
ALT
Peak flow rate (liters per minute)
FEV1 (% of predicted value)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

6. Introduction and Examples of Multivariate Data
Categorical Endpoints
Nominal scale data
Classification of patients according to their attributes
Gender
Race
Occurrence of a particular adverse reaction
Occurrence of ALT>3 times upper normal limit
Ordinal scale data
A certain order among different categories
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

7. Introduction and Examples of Multivariate Data
Ordered categorical data
Symptom score
0 = no symptom, 1 = mild, 2 = moderate, 3 = severe
Score of questionnaires (Likert Scale)
Not at all Slightly Moderately Quite a bit Greatly
Censored Endpoints
Time to the occurrence of a pre-defined event.
The occurrence of the event may not observed for some patients. Then the time to the occurrence of the event for these subjects is censored
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

8. Introduction and Examples of Multivariate Data
Cross-sectional vs. longitudinal data
Cross-sectional data
Clinical data are collected and evaluated at a particular time point during the trial
Longitudinal data (repeated measurements
Clinical data collected and evaluated over a series of time points during the trial
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

9. Introduction and Examples of Multivariate Data
Dataset:
Storm survival of sparrows
After a sever storm on Feb. 1, 1898, 49 moribund sparrow were taken to Brown university.
Professor Bumpus made morphological measurements and weight of each of 21 surviving female birds and 38 died female birds
Relationship between survival and morphology of female birds: shorter, light, longer wing bones, longer legs, greater brain capacity 
natural selection ???
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

10. Introduction and Examples of Multivariate Data
Dataset:
Egyptian skulls
Four measurements of male skulls
early pre-dynastic period (4000 B.C.)
late pre-dynastic period (3300 B.C.)
the 12th and 13th dynasties (1850 B.C.)
Ptolemaic period (200 B.C.)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

11. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

12. Introduction and Examples of Multivariate Data
Questions
How are the four measurements related?
Any statistical significance differences in the sample means, and if so, gradual changes over time in the shape and size of skulls
Any statistical significance differences in the sample standard deviations, and if so, gradual changes over time in the shape and size of skulls in the amount of variation
Is possible to construct a function of the four variables that describes the changes over time?
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

13. Introduction and Examples of Multivariate Data
Dataset:
Distribution of a butterfly Uphydryas editha over 16 counties in California and Oregon
Data
Four environmental variables
Altitude, annual precipitation, minimum and maximum temperature
Six genetic variables
Percent frequencies of different Pgi genes determined by electrophoresis
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

14. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

15. Introduction and Examples of Multivariate Data
Questions:
Are the Pgi frequencies similar for colonies that are close in space?
To what extent, if any, are the Pgi fequencies related to environmental variables?
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

16. Introduction and Examples of Multivariate Data
Prehistoric Dogs from Thailand
A collection of dog bones covering a period from about 3500 B.C. to present in excavations of prehistoric sites in northern Thailand
Mandible (lower jaw) measurements to clarify ancestor of the prehistoric dog – golden jackal, Chinese wolf, Indian wolf, Indian wolf, cuon, dingo?
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

17. Introduction and Examples of Multivariate Data
Dataset:
Employment in European Countries
Percentages of the labor force in nine different types of industry for 30 European countries
Nine types of industry: agriculture, mining and quarrying, manufacturing, power and water supplies, construction, finance, services, social and personal services, transport and communications
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

18. Introduction and Examples of Multivariate Data
Questions:
Similar patterns of employments?
Understanding the relationships between countries
Differences between countries that related to political grouping: European Union, European Free Trade Area, Eastern European countries, and others
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

19. Introduction and Examples of Multivariate Data
Quality of Life in Treatment of Cancer Patients
Adjuvant Breast Cancer Trial
Eastern Cooperative Oncology Group (ECOG) compared a 16-week dose-intensive therapy to a conventional 24-week CAF therapy
Dose-intensive therapy: increasing physical symptom and impact on psychosocial aspect of patient’s lives caused by inconvenience and fatigue.
Health-related quality of life (HRQoL) of CAF is hypothesized to be superior to dose-intensive therapy
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

20. Introduction and Examples of Multivariate Data
Quality of Life in Treatment of Cancer Patients
Advanced non-small Cell lung cancer (NSCLC)
ECOG compared two new paclitaxel-cisplatin regimens with traditional etoposide-cisplatin regimen in treatment of stage IIIB-IV NSCLC
To compare the HRQoL among three regimens and correlated quality of life to toxicity (Common Toxicity Criteria, CTC)
Both HRQoL and CTC are multivariate in nature
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

21. Introduction and Examples of Multivariate Data
Functional Assessment of Cancer Therapy (FACT) -Lung (v 2)
Items Not A little Some Quite Very
at all bit what a bit much
34. I have been short of breath
35. I am losing weight
37. My thinking is clear
38. I have been bothered by hair loss
39. I have a good appetite
40. I feel tightness in my chess
41. Breathing is easy for me
If you ever smoking answer 42
42. I regret my smoking
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

22. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

23. Introduction and Examples of Multivariate Data
Four different teaching methods on reading proficiency and analytical ability of pupils in primary schools
Student evaluations for the fall and spring semester over five academic years on Instructor teaching ability and Course overall quality from 1 (poor) to 5 (excellent)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

24. Introduction and Examples of Multivariate Data
Course and Instructor Evaluation
Courses
Accounting, Decision Science, Finance, Management, and Marketing
Instructors
From 17 for accounting to 27 in Finance
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

25. Introduction and Examples of Multivariate Data
Course and Instructor Evaluation
Evaluation Items (0-5)
Ability to present course material
Ability to stimulate student interest
Knowledge of subject matter
Apparent interest in students
Objectivity and fairness
Overall evaluation of the instructor
Course contribution to technical or analytical skills
Course contribution to understanding of complex phenomena
Value of reading and assigned work
Overall evaluation of the course
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

26. Introduction and Examples of Multivariate Data
Measurements of Iris species
Species
Iris setosa
Iris versicolor
Iris virginica
Measurements
Sepal length
Sepal width
Petal length
Petal width
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

27. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
Introduction and Examples of Multivariate Data
Example: Microarray Chip
The Roche AmliChip CYP 450 Test was the first DNA microarray diagnostic test to be approved by the US FDA on Dec. 23, 2004 to analyze one of the genes from a family of genes called cytochrome P450 genes which are active in the liver to metabolize drugs and other compounds such as grape fruit.
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

28. Introduction and Examples of Multivariate Data
Cytochrome (CYP) P450 is a family of genes in all living creatures
CYP450 plays a primary role in metabolism
CYP450 genes have been in existence for more than 3.5 billion years
In humans, enzymes encoded by the CYP450 are found primarily in liver, where they metabolize drugs, toxins, and other foreign substances that enter the body
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

29. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

30. Introduction and Examples of Multivariate Data
Phenotypes of CYP2D6 can be classified
poor (no enzyme activity)
intermediate (reduced enzyme activity)
extensive (”normal” enzyme activity)
Ultra-rapid (higher than normal enzyme activity)
Phenotypes of CYP2C19 can be classified
Poor (no enzyme activity)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

31. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

32. Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

33. Introduction and Examples of Multivariate Data
Unequal genetic variations (polymorphism) of CYP2C19 and CYP2D6
CYP2D6 has more than 80 distinct allelic variants
Poor metabolizers of CYP2D6
Caucasians: 7%
African-American: 2-4%
Asian: 1-2%
CYP2D6*10 allele (50% allele frequency)
CYP2D6*17 and CYP2D6*29 alleles (~30%)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

34. Introduction and Examples of Multivariate Data
From Carco Y. NEJM 2005
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

35. Introduction and Examples of Multivariate Data
CYP2D6 Gene duplications (ultra rapid metabolizers)
Ethiopians: 29%
Southern Europeans: 10%
Northern European: 1-2%
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

36. Introduction and Examples of Multivariate Data
CYP2C19*2 and CYP2C19*3 alleles for poor metabolizer
Null alleles are caused by a SNP that either causes a splice site or a stop codon
Frequency
Asian: 13-23%
Caucasian: 3-5%
African-American: 3-5%
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

37. Introduction and Examples of Multivariate Data
Poor or immediate metabolizer: toxic adverse events (side effects) even at lower dose
Ultra rapid metabolizer: no efficacy even at higher dose
Necessity of the microarray-based pharmacogenomic tests to provide genotyping of the CYP2D6 and CYP2C19 and predictive phenotype of associated enzyme activities
Prevention of harmful drug interactions and assurance of the use of the optimal dose
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

38. Introduction and Examples of Multivariate Data
Characteristics of AmpliChip CYP450 Microarray
15,129 probes with ~107 of the specific oligonucleitide probe each
Length of probe sequence: bases
A single Probe Set consists of 4 Probes (or Features) which have a fixed target except for at the substitution position wherean A, C, G, and T are included to generate four unique probes: one Perfect Match (PM) and three Mismatch (MM)
Data are many variables ( 10000) and very few cases (10)
Source: Decision summary of k (US FDA, 2005)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

39. Introduction and Examples of Multivariate Data
Characteristics of AmpliChip CYP450 Microarray
A Probe Set Pair consists of a Wild-type Probe Set (for Wild-type allele) and a Mutant Probe Set (for a known polymorphism).
To distinguish 29 polymorphisms in CYP2D6 including gene duplications and gene deletion
To identify 27 distinct alleles including 7 CYP2D6 gene duplication genes
Source: Decision summary of k (US FDA, 2005)
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

40. Introduction and Examples of Multivariate Data
Class Discovery
Finding new classes (unsupervised)
Class Comparison
Identification of differentially expressed genes
Class Prediction
Selection of features and summary measure for prediction of a new member
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

41. Introduction and Examples of Multivariate Data
Class Discovery
Multidimensional Scaling
Cluster Analysis
Factor Analysis
Pattern Recognition
Self Organizing Maps
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

42. Introduction and Examples of Multivariate Data
Class Comparison
Unpaired t-test
Analysis of Variance
Permutation Tests
Paired Comparisons
Control of Experimentwise Error Rate and False Discovery Rate
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

43. Introduction and Examples of Multivariate Data
Class Prediction
Discriminant Analysis
Nearest Neighbor Classification
Classification Tree and Logistic Regression
Multinomial Regression Analysis
Support Vector Machines
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

44. Introduction and Examples of Multivariate Data
Evaluation of Classification Error Rates
Cross-validation
Training set and test set
Leave-one-out Method
Jackknife Method
Bootstrap Method
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

45. Introduction and Examples of Multivariate Data
All variables are
continuous and in the same unit
continuous and in the different unit
All categorical data with the same scoring system
All categorical data with different number of categories and different scoring system
Combination of all mentioned above
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

46. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Summarization and comparison of mean characteristics
Health Related Quality of Life questionnaires
Identification of differentially expressed genes
effectiveness in teaching methods
Bumpus’s female sparrows
Egyptian skulls
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

47. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Reduction of number of variables
Principal components analysis
Female sparrows
The sum of 5 variables  general size of the birds
Y1 = X1 + X2 + X3 + X4 + X5
The difference between the sum of the first three variables and sum of the last two
Y2 = X1 + X2 +X3 – X4 –X5
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

48. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Reduction of number of variables
Factor Analysis – to account for the variation of original variables using a smaller number of index variables or factors
Female sparrows – a two-factor model
X1 = a11F1 + a12F2 + e1
X2 = a21F1 + a22F2 + e2
X3 = a31F1 + a32F2 + e4
X4 = a41F1 + a42F2 + e4
X5 = a51F1 + a52F2 + e5
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

49. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Reduction of number of variables
Factor Analysis
F1 and F2 are factors
ei represents the variation in Xi independent of the variation in the other X variables
aij are constant
F1: size  aij all positive
F2: shape  aij some + and some -
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

50. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Classification
Discrimination analysis
Classification of different groups on the available measurements
Surviving vs. non-surviving sparrows
Prediction of new specimens
Supervised analysis
The number of classes are pre-specified
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

51. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Classification
Cluster analysis
Identification of groups of similar characteristics
The number of classes may or may not be pre-specified
Unsupervised analysis
Exploratory
Grouping subjects based on their gene expression data
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

52. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Canonical correlations
Variables are divided in groups
Interest centers on the relationship between these groups
The relationship between genetic variables and 4 environmental variables at the different colonies of a butterfly Uphydryas editha
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

53. Introduction and Examples of Multivariate Data
Preview of Multivariate Methods
Multidimensional scaling
Distances apart of a number of subjects
Map on how the objects are related
A map of relationship among prehistoric dogs with golden jackal, Chinese wolf, Indian wolf, Indian wolf, cuon, dingo
A map of European countries based on their employment patterns
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

54. Introduction and Examples of Multivariate Data
The multivariate Normal Distribution
Most of multivariate were derived under the assumption of multivariate normal distribution – mean vector and covariance matrix
Transformation
Computer Software
SAS, SPSS, MINITAB, STATA, STATISTICA
Graphic Methods
A picture worth thousands of words
2 and 3-dimensional graphs
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD

55. Introduction and Examples of Multivariate Data
Summary
Examples
Applications
Datasets
Preview of Methods
Multivariate Normal Distribution
Graphical Methods
2018/9/22
Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD