Chapter 9 Measures of Strength of a Relationship Learning Objectives Understand the strength of association between two variables Explain an association from a table of joint frequencies Understand a proportional reduction of errors Understand the difference between symmetric and asymmetric measures Learning Objectives Calculate the strength of association for nominal, ordinal, interval, and ratio level data Interpret the strength of association for using lamba, Somers d, Spearman s rho, and Pearson s r Explain and interpret a correlation matrix Introduction Measure of association Important to know the strength of association Descriptive statistical procedures, not inferential ones What is Association? Improve prediction of a value based on X Predict category Y and are told category X Proportional error reduction Calculating how much better the prediction of DV with the knowledge of some IV over knowledge of DV alone.
Nominal Level Data Nominal Level Data Calculating Lambda Number of errors made predicting Y with no information Minus number of errors made predicting Y with information from X Divide by number of errors made predicting Y with no information Nominal Level Data Interpreting Lambda Range values = 0 to 1 0 = no strength 1 = perfect association Between 0 and 1 shows error predicting Y mode X and Y should be either: Ordinal Dichotomized nominal Categorized interval Can tell you both strength and direction of association Goal: Predict Rank order score pairs rather than modal value
1 if rank case knowledge of one perfectly predicts the other 0 if rank case knowledge of one cannot predict the other Nothing gained by including IV Five possibilities of pairing: Concordant Pairs: X and Y ranked same order Discordant Pairs: Ranked oppositely Pairs tied on Y but not X Pairs tied on X but not Y Pairs tied on X and Y Values for ordinal level variables: Range = ± 1 to 0 Positive values = positive relationship Negative values = negative relationship Increased # concordant and discordant = MOA closer to 1 If same #, MOA = 0 Options for analyzing ordinal variables: Partially ordered and ordinal: Tau b and c Somers d
Fully ordered: Spearman s Rho All ordinal MOA have same numerator Tau Three measures for ordinal variables: Tau a = simplest MOA [squared tables] Tau b = square root of two Somers d s [squared tables] % reduction in errors Tau c [for unequal tables] Tau a, b and c range = 1 to 0 to +1 Positive value: more concordant than discordant Negative value: more discordant than concordant Compares the number of times variables are the same with the number times they are different PRE measure only for untied pairs If ties exist, Gamma better than Tau If no ties, Gamma and Tau same If many ties, Gamma gives conservative assessment Can be used with any size table If all other pairs tied, value of 1 based one pair Symmetric measure of association
Doesn t specify which variable is the IV Calculating Gamma Ratio of concordant minus discordant pairs, divided by concordant pairs plus discordant pairs Interpreting Gamma Interpreted essentially same as Tau except with no ties Ratio of Gamma determines sign of Gamma Higher values = stronger positive or negative association Range from 1 to 0 to +1 Somers d Asymmetric counterpart to Gamma Used on any size table Interpreted essentially same as Tau and Gamma Range = 1 to 0 to +1 +1 when all pairs concordant 1 when all pairs discordant Spearman s Rho Measures differences in ranks of individual cases Measures strength, significance, and direction Not a PRE measure Pearson s r applied to ranks of data points Spearman s Rho
+1 if ranks match perfectly 1 if ranked inversely 0 if no association Limitation influenced by a large number of ties of the ranks Interval Level Data Advanced, interval analyses possible if both variables are interval or ratio Supports conclusion X might be cause of Y Proportion of change in one variable explained by change in another Requires linearity Interval Level Data Goals and methods Better prediction (than mean) for exact score of Y Use straight line Best straight line summarizes linear relationship Least squares method Establishes line where sum of squares is smallest Must be straight line Pearson s r It does the following: Measures strength Examines existence Determines direction
Least squares line (LSL) Measures spread around LSL and line slope Spread amount shows association strength Pearson s r Value range = ±1 +1 = perfect positive relationship 1 = perfect negative relationship 0 usually = no relationship If relationship is not curvilinear r 2 also called coefficient of determination Pearson s r Limitations May show low strength though definite patterns Influenced by sample size Weak correlations may be significant when sample size is large Unstable correlations when sample is small Conclusions: Selecting the Most Appropriate Measure of Strength Determine if the relationship has any ability to predict values of the dependent variable Measures of significance do not provide all needed information Measure of strength determines ability to predict mode, median, or mean Conclusions: Selecting the Most Appropriate Measure of Strength Choose measure that represents the data Advisable to examine the direction and nature of the data