Applied Multivariate Quantitative Methods Summarization of Multivariate Data

Applied Multivariate Quantitative Methods Summarization 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 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 1

Summarization of Multivariate Data n Limitations of Graphical Presentation n n Two-dimensional only At one time, only three variables Not possible for one variable plotting another three at the same time No simple way to describe the relationship among more than four or more variables 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 2

Summarization of Multivariate Data n n Plotting index variables Reduction of the number of variables to a few index variables n Principal components n n Principal component 2 against principal component 1 Principal component 3 versus other two Key difference between the objects may be lost in the reduction Index variables may not have any meaningful interpretation 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 3

1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 4

1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 5

Summarization of Multivariate Data n The draftman’s plot n n A plots of the variables for each variable against variable for each of other variables The individual graphs are small enough so that they can be viewed at the same time Only need two-dimensional plots Can not depict aspects of the data that would only be apparent when three or more variables are considered together 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 6

1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 7

Summarization of Multivariate Data n The Representation of Individual Data Points n n n Each of the objects for which variables are measured by a symbol Different characteristics of this symbol varying according to different variables: faces (Chernoff, 1973) or stars (Welsch, 1976) Advantage: displaying all variables simultaneously Disadvantage: depending strongly on the order in which objects are displayed and the order in which variables are assigned to different aspects of the symbols Few statistical packages to produce such graphs 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 8

Summarization of Multivariate Data n Example: n Prehistoric Thai dog: faces n n n 1/23/2022 Mandible breadth eye size Mandible height nose size Length of first molar brow size Breadth of first molar ear size Length from first to third molar mouth size Length from first to fourth premolars the amount of smile Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 9

1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 10

Summarization of Multivariate Data n Stars n Six variables were assigned to rays according to the order of n n n 1/23/2022 Mandible breadth (six o’clock) Mandible height (clockwise) Length of first molar Breadth of first molar Length from first to third molar Length from first to fourth premolars Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 11

Summarization of Multivariate Data n Profiles of variables n Plotting the values for variables n n Plotting the bars n n Horizontal axis: variables Vertical axis: values of measurements Values of objects are connected by line Horizontal axis: objects Vertical axis: values of measurements Values of variables for each object are grouped together Only for variables with the same units 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 12

1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 13

1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 14

Summarization of Multivariate Data n Summary n n n Difficulty for plotting results of all variables simultaneously Index variables Draftman’s plots Use of symbols for individual data points Profiles of variables 1/23/2022 Copyright by Jen-pei Liu, Ph. D and Wei-Chu Chie, MD, Ph. D 15