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Scalars
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Scalars
by Dayanand N. Naik, Ravindra Khattree
APPLIED MULTIVARIATE STATISTICS: WITH SAS® SOFTWARE
Copyright
Dedication
Preface
Commonly Used Notation
Multivariate Analysis Concepts
Introduction
Random Vectors, Means, Variances, and Covariances
Multivariate Normal Distribution
Sampling from Multivariate Normal Populations
Some Important Sample Statistics and Their Distributions
Tests for Multivariate Normality
Random Vector and Matrix Generation
Graphical Representation of Multivariate Data
Introduction
Scatter Plots
Profile Plots
Andrews Function Plots
Biplots: Plotting Observations and Variables Together
Q-Q Plots for Assessing Multivariate Normality
Plots for Detection of Multivariate Outliers
Bivariate Normal Distribution
SAS/INSIGHT Software
Concluding Remarks
Multivariate Regression
Introduction
Statistical Background
Least Squares Estimation
ANOVA Partitioning
Testing Hypotheses: Linear Hypotheses
Simultaneous Confidence Intervals
Multiple Response Surface Modeling
General Linear Hypotheses
Variance and Bias Analyses for Calibration Problems
Regression Diagnostics
Concluding Remarks
Multivariate Analysis of Experimental Data
Introduction
Balanced and Unbalanced Data
One-Way Classification
Two-Way Classification
Blocking
Fractional Factorial Experiments
Analysis of Covariance
Concluding Remarks
Analysis of Repeated Measures Data
Introduction
Single Population
k Populations
Factorial Designs
Analysis in the Presence of Covariates
The Growth Curve Models
Crossover Designs
Concluding Remarks
Analysis of Repeated Measures Using Mixed Models
Introduction
The Mixed Effects Linear Model
An Overview of the MIXED Procedure
Statistical Tests for Covariance Structures
Models with Only Fixed Effects
Analysis in the Presence of Covariates
A Random Coefficient Model
Multivariate Repeated Measures Data
Concluding Remarks
References
A Brief Introduction to the IML Procedure
The First SAS Statement
Scalars
Matrices
Printing of Matrices
Algebra of Matrices
Transpose
Inverse
Finding the Number of Rows and Columns
Trace and Determinant
Eigenvalues and Eigenvectors
Square Root of a Symmetric Nonnegative Definite Matrix
Generalized Inverse of a Matrix
Singular Value Decomposition
Symmetric Square Root of a Symmetric Nonnegative Definite Matrix
Kronecker Product
Augmenting Two or More Matrices
Construction of a Design Matrix
Checking the Estimability of a Linear Function p'β
Creating a Matrix from a SAS Data Set
Creating a SAS Data Set from a Matrix
Generation of Normal Random Numbers
Computation of Cumulative Probabilities
Computation of Percentiles and Cut Off Points
Data Sets
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The First SAS Statement
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Matrices
A.2. Scalars
Scalars are specified in the usual way. For example,
c = 7.856;
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