A graduate-level illustrated introduction to and tutorial for Pearson correlation, Spearman's rank correlation (rho), Kendall's rank correlation (tau-b). polyserial correlation, biserial correlation, polychoric correlation, tetrachoric correlation, phi, point-biserial correlation, rank-biserial correlation (Somers' D), and more.
Why we think it's important: Factor analysis, structural equation modeling, and other procedures accept correlation matrix input. No existing software package automatically creates matrices using forms of correlation designed for all types of variable pairings. Using the appropriate types of correlation can make a difference in substantive conclusions.
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Below is the unformatted table of contents.
CORRELATION Table of Contents Overview 6 Key Concepts and Terms 8 Deviation 8 Covariance 8 Standardization 8 Use of correlation matrices 8 Data example 9 Pearson correlation (for interval data) 11 Overview 11 Coefficient of determination, r2 11 Attenuation of correlation 12 SPSS 17 Input example 17 The apples.sav example 19 SAS 21 Input example 21 The apples.sas7bdat example 24 Stata 25 Spearman's rho (for ordinal data) 27 Overview 27 SPSS 27 SAS 28 Stata 29 Kendall's tau-b (for ordinal data) 30 Overview 30 SPSS 31 SAS 31 Stata 32 Polyserial correlation (for a continuous with an ordinal or binary variable) 33 Overview 33 SPSS 34 SAS 34 Stata 35 Polychoric correlation (for ordinal and binary variables) 35 Overview 35 SPSS 36 SAS 36 Stata 37 Other software for polychoric correlation 41 Phi (for two binary variables) 41 Overview 41 SPSS 41 SAS 42 Stata 42 Other types of correlation 42 Point-biserial correlation 42 Converting point-biserial to biserial correlation 43 Rank-biserial correlation (Somers' D) 43 SPSS 43 SAS 44 Stata 45 Correlation ratio, eta 47 Coefficient of intraclass correlation (ICC) 47 Assumptions 47 Data level 47 Linear relationships 48 Homoscedasticity 48 No outliers 48 Minimal measurement error 48 Unrestricted variance 49 Similar underlying distributions 49 Common underlying normal distributions 49 Normally distributed error terms 50 Frequently Asked Questions 50 Do I want one-tailed or two-tailed significance? 50 How many correlations will there be among k variables? 50 What rules exist for determining the appropriate significance level for testing correlation coefficients? 50 How do I convert correlations into z scores? 51 Z-Score Conversions of Pearson's r 51 How is the significance of a Pearson correlation coefficient computed? 53 Significance of r 53 Significance of the difference between two correlations from two independent samples 53 Significance of the difference between two dependent correlations from the same sample 54 How do I set confidence limits on my correlation coefficients? 54 I have ordinal variables and thus used Spearman's rho. How do I use these ordinal correlations in SPSS for partial correlation, regression, and other procedures? 55 Are polyserial and biserial correlations simply Spearman's rho applied to appropriate variables? 55 What is the relation of correlation to ANOVA? 55 What is the relation of correlation to validity? 56 What is the SPSS syntax for correlation? 56 Bibliography 57 Acknowledgment 59 Pagecount: 61