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Correlation ... beware Definition Var(X+Y) = Var(X) + Var(Y) + 2·Cov(X,Y) Corr ( X , Y ) Cov ( X , Y ) StdDev ( X ) StdDev ( Y ) The correlation between two random variables is a dimensionless number between 1 and -1. Interpretation Correlation measures the strength of the linear relationship between two variables. • Strength – not the slope • Linear – misses nonlinearities completely • Two – shows only “shadows” of multidimensional relationships A correlation of +1 would arise only if all of the points lined up perfectly. Stretching the diagram horizontally or vertically would change the perceived slope, but not the correlation. A positive correlation signals that large values of one variable are typically associated with large values of the other. Correlation measures the “tightness” of the clustering about a single line. A negative correlation signals that large values of one variable are typically associated with small values of the other. Independent random variables have a correlation of 0. But a correlation of 0 most certainly does not imply independence. Indeed, correlations can completely miss nonlinear relationships. Correlations Show (only) Two-Dimensional Shadows In the motorpool case, the correlations between Age and Cost, and between Make and Cost, show precisely what the manager’s two-dimensional tables showed: Costs Mileage Age Make Costs 1.000 0.771 0.023 -0.240 Mileage 0.771 1.000 -0.496 -0.478 Age 0.023 -0.496 1.000 0.164 Make -0.240 -0.478 0.164 1.000 There’s little linkage directly between Age and Cost. Fords had higher average costs than did Hondas. But each of these facts is due to the confounding effect of Mileage! The pure effect of each variable on its own is only revealed in the mostcomplete model. Potential for Misuse (received via email from a former student, all employer references removed) “One of the pieces of the research is to identify key attributes that drive customers to choose a vendor for buying office products. “The market research guy that we have hired (he is an MBA/PhD from Wharton) says the following: “‘I can determine the relative importance of various attributes that drive overall satisfaction by running a correlation of each one of them against overall satisfaction score and then ranking them based on the (correlation) coefficient scores.’ “I am not really certain if we can do that. I would tend to think we should run a regression to get relative weightage.” Customer Satisfaction • Consider overall customer satisfaction (on a 100-point scale) with a Web-based provider of customized software as the order leadtime (in days), product acquisition cost, and availability of online order-tracking (0 = not available, 1 = available) vary. • Here are the correlations: Correlations with Satisfaction leadtime -0.766 ol-tracking -0.242 cost 0.097 Customers forced to wait are unhappy. Those without access to online order tracking are more satisfied. Those who pay more are somewhat happier. ????? The Full Regression Regression: satisfaction coefficient std error of coef t-ratio significance beta-weight constant leadtime 192.7338 -6.8856 16.1643 0.5535 11.9234 -12.4391 0.0000% 0.0000% -1.0879 standard error of regression coefficient of determination adjusted coef of determination cost ol-track -1.8025 8.5599 0.3137 4.0729 -5.7453 2.1017 0.0000% 4.0092% -0.4571 0.1586 13.9292 75.03% 73.70% Customers dislike high cost, and like online order tracking. Why does customer satisfaction vary? Primarily because leadtimes vary; secondarily, because cost varies. Reconciliation satisfaction leadtime cost ol-tracking satisfaction 1.000 -0.766 -0.097 -0.242 leadtime -0.766 1.000 -0.543 0.465 cost -0.097 -0.543 1.000 -0.230 ol-tracking -0.242 0.465 -0.230 1.000 • Customers can pay extra for expedited service (shorter leadtime at moderate extra cost), or for express service (shortest leadtime at highest cost) – Those who chose to save money and wait longer ended up (slightly) regretting their choice. • Most customers who chose rapid service weren’t given access to order tracking. – They didn’t need it, and were still happy with their fast deliveries. Finally … • The correlations between the explanatory variables can help flesh out the “story.” • In a “simple” (i.e., one explanatory variable) regression: – The (meaningless) beta-weight is the correlation between the two variables. – The square of the correlation is the unadjusted coefficient of determination (rsquared). Regression: Costs If you give me a correlation, I’ll interpret it by squaring it and looking at it as a coefficient of determination. coefficient std error of coef t-ratio significance beta-weight constant Mileage 364.476942 19.812076 76.8173302 4.54471998 4.7447 4.3594 0.0383% 0.0774% 0.7706 standard error of regression coefficient of determination adjusted coef of determination 73.8638412 59.38% 56.26% A Pharmaceutical Ad Correlation(anxiety,depression) = 0.7 100 90 80 70 anxiety 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 depression Diagnostic scores from sample of patients receiving psychiatric care So, if your patients have anxiety problems, consider prescribing our antidepressant! Evaluation • At most 49% of the variability in patients’ anxiety levels can potentially be explained by variability in depression levels. – “potentially” = might actually be explained by something else which covaries with both. • The regression provides no evidence that changing a patient’s depression level will cause a change in their anxiety level. Association vs. Causality • Polio and Ice Cream • Regression (and correlation) deal only with association – Example: Greater values for annual mileage are typically associated with higher annual maintenance costs. – No matter how “good” the regression statistics look, they will not make the case that greater mileage causes greater costs. – If you believe that driving more during the year causes higher costs, then it’s fine to use regression to estimate the size of the causal effect. • Evidence supporting causality comes only from controlled experimentation. – This is why macroeconomists continue to argue about which aspects of public policy are the key drivers of economic growth. – It’s also why the cigarette companies won all the lawsuits filed against them for several decades. Structural Variations • Interactions – When the effect of one explanatory variable on the dependent variable depends on the value of another explanatory variable • The “trick”: Introduce the product of the two as a new artificial explanatory variable. Example • Nonlinearities – When the impact of an explanatory variable on the dependent variable “bends” • The “trick”: Introduce the square of that variable as a new artificial explanatory variable. Example Interactions: Summary • When the effect (i.e., the coefficient) of one explanatory variable on the dependent variable depends on the value of another explanatory variable – Signaled only by judgment – The “trick”: Introduce the product of the two as a new artificial explanatory variable. After the regression, interpret in the original “conceptual” model. – For example, Cost = a + (b1+b2Age)Mileage + … (rest of model) – The latter explanatory variable (in the example, Age) might or might not remain in the model – Cost: We lose a meaningful interpretation of the beta-weights Examples from the Sample Exams Caligula’s Castle: Regression: Revenue coefficient constant Age -1224.84 62.37502 Age2 Sex Direct Indirect SexInd -0.5201 -121.899 1.992615 0.85276 1.43767 Revenuepred = -1224.82 + 62.37Age – 0.5201Age2 – 121.9Sex + 1.99Direct + (0.8527+1.4377Sex)Indirect revenue / $ incentive direct indirect Men (Sex=0) $1.99 $0.85 Women (Sex=1) $1.99 $2.29 Give direct incentives (house chips, etc.) to men Give indirect incentives (flowers, meals) to women The Age effect on Revenue is greatest at Age = -(62.37)/(2(-0.5201)) = 59.96 years Examples from the Sample Exams Hans and Franz: Regression: CustSat coefficient constant Wait Wait2 Size Franz? SizeFranz? 84.4016931 -0.8666595 -0.0556165 -5.6022949 -40.084506 8.77474654 CustSatpred = 84.40 – 0.8667Wait – 0.0556Wait2 – 5.602Size + (-40.0845+8.7747Size)Franz? Set Franz? = 0 (assign Hans) when the party size is < 40.0845/8.7747 = 4.568 Customers’ anger grows more quickly the longer they wait Structural Variations • Interactions – When the effect of one explanatory variable on the dependent variable depends on the value of another explanatory variable • The “trick”: Introduce the product of the two as a new artificial explanatory variable. Example • Nonlinearities – When the impact of an explanatory variable on the dependent variable “bends” • The “trick”: Introduce the square of that variable as a new artificial explanatory variable. Example Nonlinearity: Summary • When the direct relationship between an explanatory variable and the dependent variable “bends” – Signaled by a “U” in a plot of the residuals against an explanatory variable – The “trick”: Introduce the square of that variable as a new artificial explanatory variable: Y = a + bX + cX2 + … (rest of model) – One trick can capture 6 different nonlinear “shapes” – Always keep the original variable (the linear term, with coefficient “b”, allows the parabola to take any horizontal position) – c (positive = upward-bending parabola, negative = downwardbending) – -b/(2c) indicates where the vertex (either maximum or minimum) of the parabola occurs – Cost: We lose a meaningful interpretation of the beta-weights Examples from the Sample Exams Caligula’s Castle: Regression: Revenue coefficient constant Age -1224.84 62.37502 Age2 Sex Direct Indirect SexInd -0.5201 -121.899 1.992615 0.85276 1.43767 Revenuepred = -1224.82 + 62.37Age – 0.5201Age2 – 121.9Sex + 1.99Direct + (0.8527+1.4377Sex)Indirect revenue / $ incentive direct indirect Men (Sex=0) $1.99 $0.85 Women (Sex=1) $1.99 $2.29 Give direct incentives (house chips, etc.) to men Give indirect incentives (flowers, meals) to women The Age effect on Revenue is greatest at Age = -(62.37)/(2(-0.5201)) = 59.96 years Examples from the Sample Exams Hans and Franz: Regression: CustSat coefficient constant Wait Wait2 Size Franz? SizeFranz? 84.4016931 -0.8666595 -0.0556165 -5.6022949 -40.084506 8.77474654 CustSatpred = 84.40 – 0.8667Wait – 0.0556Wait2 – 5.602Size + (-40.0845+8.7747Size)Franz? Set Franz? = 0 (assign Hans) when the party size is < 40.0845/8.7747 = 4.568 Customers’ anger grows more quickly the longer they wait Sample Datasets • Four datasets continuing to review material from Session 2, with some added modeling issues. • Two very thorough sample exams. – One based on Harrah’s success in understanding its patrons – One based on a restaurateur comparing maitres d’hotel, with a 90-minute prerecorded Webex tutorial