Principal Components Analysisand Factor Analysis

by

Dr. Winai Bodhisuwan

Principal Component Analysis

Principal components analysis transforms the original set of variables into a smaller set of linear combinations that account for most of the variance in the original set. The purpose of PCA is to determine factors (i.e., principal components) in order to explain as much of the total variation in the data as possible.

Principal Component Analysis

The are 2 bases of analysis.

Based on Covariance Matrices

Based on Correlation Matrices

Radiotherapy Data StudyThe data consist of average ratings over the course of treatment for The data consist of average ratings over the course of treatment for

patients undergoing radiotherapy. Variables measured include patients undergoing radiotherapy. Variables measured include xx11 (number of symptoms, such as sore throat or nausea); (number of symptoms, such as sore throat or nausea);

xx22 (amount of activity, on a 1-5 scale);(amount of activity, on a 1-5 scale);

xx33 (amount of sleep, on a 1-5 scale);(amount of sleep, on a 1-5 scale);

xx44 (amount of food consumed, on a 1-5 scale); (amount of food consumed, on a 1-5 scale);

xx55 (appetite, on a 1-5 scale); and(appetite, on a 1-5 scale); and

xx66 (skin reaction, on a 0-3 scale) (skin reaction, on a 0-3 scale)

*Refer to the data set file, radiotherapy.MTW

Mineral Contents in BonesAt the start of a study to determine whether exercise or dietary

supplements would slow bone loss in older women, an investigator measured the mineral content of bones by photon absorptiometry. Measurements were recorded for three bones on the dominant and nondominant sides.

*Refer to the data set file, mineralcontents.MTW

Air PollutionsThe data set file are 42 measurements on air-pollution variables

recorded at 12:00 noon in the Los Angeles area on different days.

*Refer to the data set file, airpollution.MTW

Principal Component AnalysisMinitab Command:

Using the menu: Stat >> Multivariate >> Principal Components

Principal Component AnalysisMinitab Command:

Click Graphs and Storage to produce score plot and store the resulted score.

Factor Analysis

Factor analysis is a multivariate tool that is very similar to PCA. Factor analysis is also used to condense a set of observed variables into smaller of transformed variables called factors.

Rotated Factor Matrix

Correlation Matrix

or

Covariance Matrix

Data Collection

MLE

Factor Model

PCA

Unrotated Factor Matrix

Factor AnalysisMinitab Command:

Using the menu: Stat >> Multivariate >> Factor Analysis

Factor AnalysisMinitab Command:

Using the menu: Stat >> Multivariate >> Factor Analysis

Factor AnalysisMinitab Command:

Using the menu: Stat >> Multivariate >> Factor Analysis