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Measurement Math DeShon - 2006 Univariate Descriptives Mean Variance, standard deviation Var ( x) S 2 X i X X i X n 1 S X i X X i X n 1 Skew & Kurtosis If normal distribution, mean and SD are sufficient statistics Normal Distribution Univariate Probability Functions Bivariate Descriptives Mean and SD of each variable and the correlation (ρ) between them are sufficient statistics for a bivariate normal distribution Distributions are abstractions or models Used to simplify Useful to the extent the assumptions of the model are met 2D – Ellipse or Scatterplot Galton’s Original Graph 3D Probability Density Covariance Covariance is the extent to which two variables co-vary from their respective means X Cov( x, y ) i X Yi Y n 1 Case X Y x=X-3 y =Y-4 xy 1 1 2 -2 -2 4 2 2 3 -1 -1 1 3 3 3 0 -1 0 4 6 8 3 4 12 Sum 17 Cov(X,Y) = 17/(4-1) = 5.667 Covariance Covariance ranges from negative to positive infinity Variance - Covariance matrix Variance is the covariance of a variable with itself Var ( X ) Cov( XY ) Cov( XZ ) Cov( XY ) Var (Y ) Cov(YZ ) Cov( XZ ) Cov(YZ Var ( Z ) Correlation Covariance is an unbounded statistic Standardize the covariance with the standard deviations -1 ≤ r ≤ 1 rp Cov( X , Y ) Var ( X )Var (Y ) Correlation Matrix Table 1. Descriptive Statistics for the Variables Correlations Variables Mean s.d 1 2 3 4 5 6 7 8 9 Self-rated cog ability 4.89 .86 .81 Self-enhancement 4.03 .85 .34 .79 Individualism 4.92 .89 .40 .41 .78 Horiz individualism 5.19 1.05 .41 .25 .82 .80 Vert individualism 4.65 1.11 .25 .42 .84 .37 .72 Collectivism 5.05 .74 .21 .11 .08 .06 .06 .72 21.00 1.70 .12 .01 .17 .13 .16 .01 -- Gender 1.63 .49 -.16 -.06 -.11 .07 -.11 -.02 -.01 Academic seniority 2.17 1.01 .17 .07 .22 .23 .14 .06 .45 .12 -- 10.71 1.60 .17 -.02 .08 .11 .03 .07 -.02 -.07 .12 Age Actual cog ability 10 -- -- Notes: N = 608; gender was coded 1 for male and 2 for female. Reliabilities (Coefficient alpha) are on the diagonal. Coefficient of Determination r2 = percentage of variance in Y accounted for by X Ranges from 0 to 1 (positive only) This number is a meaningful proportion Other measures of association Point Biserial Correlation Biserial Correlation Tetrachoric Correlation Polychoric Correlation binary variables ordinal variables Odds Ratio binary variables Point Biserial Correlation Used when one variable is a natural (real) dichotomy (two categories) and the other variable is interval or continuous Just a normal correlation between a continuous and a dichotomous variable Biserial Correlation When one variable is an artificial dichotomy (two categories) and the criterion variable is interval or continuous Tetrachoric Correlation Estimates what the correlation between two binary variables would be if you could measure variables on a continuous scale. Example: difficulty walking up 10 steps and difficulty lifting 10 lbs. Difficulty Walking Up 10 Steps no d d if iff fi L e ve Tetrachoric Correlation Assumes that both “traits” are normally distributed Correlation, r, measures how narrow the ellipse is. a, b, c, d are the proportions in each quadrant d c a b Tetrachoric Correlation For α = ad/bc, Approximation 1: 1 Q 1 Approximation 2 (Digby): 1 Q 34 1 34 Tetrachoric Correlation Example: Tetrachoric correlation = 0.61 Pearson correlation = 0.41 o o Assumes threshold is the same across people Strong assumption that underlying quantity of interest is truly continuous Difficulty Walking Up 10 Steps Difficulty Lifting 10 lb. No Yes No 40 10 50 Yes 20 30 50 60 40 100 Odds Ratio Measure of association between two binary variables Risk associated with x given y. Example: odds of difficulty walking up 10 steps to the odds of difficulty lifting 10 lb: OR p1 /(1 p1 ) p2 /(1 p2 ) ad bc ( 40 )( 30 ) ( 20 )(10 ) 6 Pros and Cons Tetrachoric correlation Odds Ratio same interpretation as Spearman and Pearson correlations “difficult” to calculate exactly Makes assumptions easy to understand, but no “perfect” association that is manageable (i.e. {∞, -∞}) easy to calculate not comparable to correlations May give you different results/inference! Dichotomized Data: A Bad Habit of Psychologists Sometimes perfectly good quantitative data is made binary because it seems easier to talk about "High" vs. "Low" The worst habit is median split Usually the High and Low groups are mixtures of the continua Rarely is the median interpreted rationally See references Cohen, J. (1983) The cost of dichotomization. Applied Psychological Measurement, 7, 249-253. McCallum, R.C., Zhang, S., Preacher, K.J., Rucker, D.D. (2002) On the practice of dichotomization of quantitative variables. Psychological Methods, 7, 19-40. Simple Regression The simple linear regression MODEL is: y = b 0 + b 1 x +e x y e describes how y is related to x b0 and b1 are called parameters of the model. e is a random variable called the error term. E ( y) yˆ b 0 b1 x Simple Regression E ( y) b 0 b1 x Graph of the regression equation is a straight line. β0 is the population y-intercept of the regression line. β1 is the population slope of the regression line. E(y) is the expected value of y for a given x value Simple Regression E(y) Regression line Intercept b0 Slope b1 is positive x Simple Regression E(y) Regression line Intercept b0 Slope b1 is 0 x Estimated Simple Regression The estimated simple linear regression equation is: ŷ b0 b1 x The graph is called the estimated regression line. b0 is the y intercept of the line. b1 is the slope of the line. ŷ is the estimated/predicted value of y for a given x value. Estimation process Regression Model y = b0 + b1x +e Regression Equation E(y) = b0 + b1x Unknown Parameters b0, b1 b0 and b1 provide estimates of b0 and b1 Sample Data: x y x1 y1 . . . . xn yn Estimated Regression Equation ŷ b0 b1 x Sample Statistics b0, b1 Least Squares Estimation Least Squares Criterion where: min (y i y i ) 2 yi = observed value of the dependent variable for the ith observation ^ yi = predicted/estimated value of the dependent variable for the ith observation Least Squares Estimation Estimated Slope b1 x y ( x y ) / n Cov( X , Y ) Var ( X ) x ( x ) / n i i i 2 i Estimated y-Intercept b0 y bx i 2 i Model Assumptions 1. X is measured without error. 2. X and e are independent 3. The error e is a random variable with mean of zero. 4. The variance of e , denoted by 2, is the same for all values of the independent variable (homogeneity of error variance). 5. The values of e are independent. 6. The error e is a normally distributed random variable. Example: Consumer Warfare Number of Ads (X) 1 3 2 1 3 Purchases (Y) 14 24 18 17 27 Example Slope for the Estimated Regression Equation b1 = 220 - (10)(100)/5 = 5 24 - (10)2/5 y-Intercept for the Estimated Regression Equation b0 = 20 - 5(2) = 10 Estimated Regression Equation y^ = 10 + 5x Example Scatter plot with regression line 30 25 Purchases 20 ^ y = 10 + 5x 15 10 5 0 0 1 2 # of Ads 3 4 Evaluating Fit Coefficient of Determination SST = SSR + SSE 2 2 ^ )2 ( y i y ) ( y^i y ) ( y i y i where: r2 = SSR/SST SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error Evaluating Fit Coefficient of Determination r2 = SSR/SST = 100/114 = .8772 The regression relationship is very strong because 88% of the variation in number of purchases can be explained by the linear relationship with the between the number of TV ads Mean Square Error An Estimate of 2 The mean square error (MSE) provides the estimate of 2, S2 = MSE = SSE/(n-2) where: SSE (yi yˆi ) 2 ( yi b0 b1 xi ) 2 Standard Error of Estimate An Estimate of S To estimate we take the square root of 2. The resulting S is called the standard error of the estimate. Also called “Root Mean Squared Error” SSE s MSE n2 Linear Composites Linear composites are fundamental to behavioral measurement Prediction & Multiple Regression Principle Component Analysis Factor Analysis Confirmatory Factor Analysis Scale Development Ex: Unit-weighting of items in a test Test = 1*X1 + 1*X2 + 1*X3 + … + 1*Xn Linear Composites Sum Scale Unit-weighted linear composite ScaleA = X1 + X2 + X3 + … + Xn ScaleA = 1*X1 + 1*X2 + 1*X3 + … + 1*Xn Weighted linear composite ScaleA = b1X1 + b2X2 + b3X3 + … + bnXn Variance of a weighted Composite X Y Y Var(X) Cov(XY) Y Cov(XY) Var(Y) Var ( X Y ) w12 *Var ( X 1 ) w22 *Var ( X 2 ) 2w1w2 * Cov( X , Y ) Effective vs. Nominal Weights Nominal weights The desired weight assigned to each component Effective weights the actual contribution of each component to the composite function of the desired weights, standard deviations, and covariances of the components Principles of Composite Formation Standardize before combining!!!!! Weighting doesn’t matter much when the correlations among the components are moderate to large As the number of components increases, the importance of weighting decreases Differential weights are difficult to replicate/cross-validate Decision Accuracy Truth Yes False Negative True Positive True Negative False Positive No Fail Pass Decision Signal Detection Theory Polygraph Example Sensitivity, etc…