Applied Multivariate Quantitative Methods Introduction and Examples of

Applied Multivariate Quantitative Methods Introduction and Examples of Multivariate Data By Jen-pei Liu, Ph. D Division of Biometry, Department of Agronomy, National Taiwan University and Wei-Chie, MD, Ph. D Department of Public Health National Taiwan University 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 1

Introduction and Examples of Multivariate Data n n n 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 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 2

Introduction and Examples of Multivariate Data n Example n Clinical trials: n Efficacy n n Safety n n n 1/15/2022 Stroke n NIH stroke scale n Bartel score n Modified Rankin score Adverse events Laboratory evaluations Vital signs Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 3

Introduction and Examples of Multivariate Data n Example n Quality of Life n n n Physical Mental Social Well-being Microarray n 1/15/2022 Intensity of tens of thousands of transcripts Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 4

Introduction and Examples of Multivariate Data n n Continuous Endpoints Numerical discrete data n n n Heart beats per minutes Total NINSS Total Hamilton Rating Scale for Depression Total Alzheimer’s Disease Assessment Scale Numerical continuous data Age n Weight n ALT n Peak flow rate (liters per minute) Copyright by Jen-pei Liu, Ph. D and n FEV 1 (% of predicted value) 1/15/2022 Wei-chu Chie, MD, Ph. D n 5

Introduction and Examples of Multivariate Data n n Categorical Endpoints Nominal scale data Classification of patients according to their attributes n Gender n Race n Occurrence of a particular adverse reaction n Occurrence of ALT>3 times upper normal limit n Ordinal scale data A certain order among different categories 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 6

Introduction and Examples of Multivariate Data n Ordered categorical data Symptom score 0 = no symptom, 1 = mild, 2 = moderate, 3 = severe n Score of questionnaires (Likert Scale) Not at all Slightly Moderately Quite a bit Greatly 0 1 2 3 4 n n Censored Endpoints n n 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 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 7

Introduction and Examples of Multivariate Data Cross-sectional vs. longitudinal data n Cross-sectional data Clinical data are collected and evaluated at a particular time point during the trial n Longitudinal data (repeated measurements Clinical data collected and evaluated over a series of time points during the trial 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 8

Introduction and Examples of Multivariate Data n 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 ? ? ? n 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 9

Introduction and Examples of Multivariate Data n Dataset: n Egyptian skulls Four measurements of male skulls early pre-dynastic period (4000 B. C. ) late pre-dynastic period (3300 B. C. ) the 12 th and 13 th dynasties (1850 B. C. ) Ptolemaic period (200 B. C. ) 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 10

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Introduction and Examples of Multivariate Data n Questions n n 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? 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 12

Introduction and Examples of Multivariate Data n Dataset: n n Distribution of a butterfly Uphydryas editha over 16 counties in California and Oregon Data n Four environmental variables n n Six genetic variables n 1/15/2022 Altitude, annual precipitation, minimum and maximum temperature Percent frequencies of different Pgi genes determined by electrophoresis Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 13

1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 14

Introduction and Examples of Multivariate Data n Questions: n n 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? 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 15

Introduction and Examples of Multivariate Data n Prehistoric Dogs from Thailand n n 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, cuon, dingo? 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 16

Introduction and Examples of Multivariate Data n Dataset: n n 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 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 17

Introduction and Examples of Multivariate Data n Questions: n n n 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 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 18

Introduction and Examples of Multivariate Data n Quality of Life in Treatment of Cancer Patients n Adjuvant Breast Cancer Trial n n n 1/15/2022 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 (HRQo. L) of CAF is hypothesized to be superior to dose-intensive therapy Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 19

Introduction and Examples of Multivariate Data n Quality of Life in Treatment of Cancer Patients n Advanced non-small Cell lung cancer (NSCLC) n n n 1/15/2022 ECOG compared two new paclitaxel-cisplatin regimens with traditional etoposide-cisplatin regimen in treatment of stage IIIB-IV NSCLC To compare the HRQo. L among three regimens and correlated quality of life to toxicity (Common Toxicity Criteria, CTC) Both HRQo. L and CTC are multivariate in nature Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 20

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 n 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 21

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Introduction and Examples of Multivariate Data n Example n n 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) 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 23

Introduction and Examples of Multivariate Data n n Course and Instructor Evaluation Courses n n Accounting, Decision Science, Finance, Management, and Marketing Instructors n From 17 for accounting to 27 in Finance 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 24

Introduction and Examples of Multivariate Data n n Course and Instructor Evaluation Items (0 -5) n n n n n 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 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 25

Introduction and Examples of Multivariate Data n Measurements of Iris species n Species n n Iris setosa Iris versicolor Iris virginica Measurements n n 1/15/2022 Sepal length Sepal width Petal length Petal width Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 26

Introduction and Examples of Multivariate Data n Example: Microarray Chip The Roche Amli. Chip 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 P 450 genes which are active in the liver to metabolize drugs and other compounds such as grape fruit. 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 27

Introduction and Examples of Multivariate Data n n Cytochrome (CYP) P 450 is a family of genes in all living creatures CYP 450 plays a primary role in metabolism CYP 450 genes have been in existence for more than 3. 5 billion years In humans, enzymes encoded by the CYP 450 are found primarily in liver, where they metabolize drugs, toxins, and other foreign substances that enter the body 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 28

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Introduction and Examples of Multivariate Data n Phenotypes of CYP 2 D 6 can be classified n n n poor (no enzyme activity) intermediate (reduced enzyme activity) extensive (”normal” enzyme activity) Ultra-rapid (higher than normal enzyme activity) Phenotypes of CYP 2 C 19 can be classified n n Poor (no enzyme activity) extensive (”normal” enzyme activity) 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 30

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1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 32

Introduction and Examples of Multivariate Data n n n Unequal genetic variations (polymorphism) of CYP 2 C 19 and CYP 2 D 6 has more than 80 distinct allelic variants Poor metabolizers of CYP 2 D 6 n n n Caucasians: 7% African-American: 2 -4% Asian: 1 -2% n n 1/15/2022 CYP 2 D 6*10 allele (50% allele frequency) CYP 2 D 6*17 and CYP 2 D 6*29 alleles (~30%) Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 33

Introduction and Examples of Multivariate Data From Carco Y. NEJM 2005 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 34

Introduction and Examples of Multivariate Data n CYP 2 D 6 Gene duplications (ultra rapid metabolizers) n n n Ethiopians: 29% Southern Europeans: 10% Northern European: 1 -2% 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 35

Introduction and Examples of Multivariate Data n n n CYP 2 C 19*2 and CYP 2 C 19*3 alleles for poor metabolizer Null alleles are caused by a SNP that either causes a splice site or a stop codon Frequency n n n Asian: 13 -23% Caucasian: 3 -5% African-American: 3 -5% 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 36

Introduction and Examples of Multivariate Data n n 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 CYP 2 D 6 and CYP 2 C 19 and predictive phenotype of associated enzyme activities Prevention of harmful drug interactions and assurance of the use of the optimal dose 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 37

Introduction and Examples of Multivariate Data n Characteristics of Ampli. Chip CYP 450 Microarray n n 15, 129 probes with ~107 of the specific oligonucleitide probe each Length of probe sequence: 16 -22 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 042259 (US FDA, 2005) 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 38

Introduction and Examples of Multivariate Data n Characteristics of Ampli. Chip CYP 450 Microarray n n n 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 CYP 2 D 6 including gene duplications and gene deletion To identify 27 distinct alleles including 7 CYP 2 D 6 gene duplication genes Source: Decision summary of k 042259 (US FDA, 2005) 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 39

Introduction and Examples of Multivariate Data n Class Discovery n n Class Comparison n n Finding new classes (unsupervised) Identification of differentially expressed genes Class Prediction n Selection of features and summary measure for prediction of a new member 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 40

Introduction and Examples of Multivariate Data n Class Discovery n Multidimensional Scaling n Cluster Analysis n Factor Analysis n Pattern Recognition n Self Organizing Maps 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 41

Introduction and Examples of Multivariate Data n Class Comparison n Unpaired t-test n Analysis of Variance n Permutation Tests n Paired Comparisons n 1/15/2022 Control of Experimentwise Error Rate and False Discovery Rate Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 42

Introduction and Examples of Multivariate Data n Class Prediction n Discriminant Analysis n Nearest Neighbor Classification n Classification Tree and Logistic Regression n Multinomial Regression Analysis n Support Vector Machines 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 43

Introduction and Examples of Multivariate Data n Evaluation of Classification Error Rates n Cross-validation n Training set and test set n Leave-one-out Method n Jackknife Method n Bootstrap Method 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 44

Introduction and Examples of Multivariate Data n All variables are n n n 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 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 45

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Summarization and comparison of mean characteristics n n n 1/15/2022 Health Related Quality of Life questionnaires Identification of differentially expressed genes effectiveness in teaching methods Bumpus’s female sparrows Egyptian skulls Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 46

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Reduction of number of variables n Principal components analysis Female sparrows The sum of 5 variables general size of the birds Y 1 = X 1 + X 2 + X 3 + X 4 + X 5 The difference between the sum of the first three variables and sum of the last two Y 2 = X 1 + X 2 +X 3 – X 4 –X 5 n 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 47

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Reduction of number of variables n Factor Analysis – to account for the variation of original variables using a smaller number of index variables or factors n 1/15/2022 Female sparrows – a two-factor model n X 1 = a 11 F 1 + a 12 F 2 + e 1 n X 2 = a 21 F 1 + a 22 F 2 + e 2 n X 3 = a 31 F 1 + a 32 F 2 + e 4 n X 4 = a 41 F 1 + a 42 F 2 + e 4 n X 5 = a 51 F 1 + a 52 F 2 + e 5 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 48

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Reduction of number of variables n n n 1/15/2022 Factor Analysis F 1 and F 2 are factors ei represents the variation in Xi independent of the variation in the other X variables aij are constant F 1: size aij all positive F 2: shape aij some + and some Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 49

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Classification n Discrimination analysis n n n 1/15/2022 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 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 50

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Classification n Cluster analysis n n n 1/15/2022 Identification of groups of similar characteristics The number of classes may or may not be prespecified Unsupervised analysis Exploratory Grouping subjects based on their gene expression data Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 51

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Canonical correlations n n n 1/15/2022 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 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 52

Introduction and Examples of Multivariate Data n Preview of Multivariate Methods n Multidimensional scaling n n 1/15/2022 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, cuon, dingo A map of European countries based on their employment patterns Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 53

Introduction and Examples of Multivariate Data n The multivariate Normal Distribution n Computer Software n n Most of multivariate were derived under the assumption of multivariate normal distribution – mean vector and covariance matrix Transformation SAS, SPSS, MINITAB, STATA, STATISTICA Graphic Methods n A picture worth thousands of words n 1/15/2022 2 and 3 -dimensional graphs Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 54

Introduction and Examples of Multivariate Data n Summary n n n Examples Applications Datasets Preview of Methods Multivariate Normal Distribution Graphical Methods 1/15/2022 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 55