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Correlation... beware

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Definition Var(X+Y) = Var(X) + Var(Y) + 2·Cov(X,Y) The correlation between two random variables is a dimensionless number between 1 and -1.

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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

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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.

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Correlation measures the “tightness” of the clustering about a single line. A positive correlation signals that large values of one variable are typically associated with large values of the other.

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A negative correlation signals that large values of one variable are typically associated with small values of the other.

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Independent random variables have a correlation of 0.

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But a correlation of 0 most certainly does not imply independence. Indeed, correlations can completely miss nonlinear relationships.

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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: 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 most- complete model. CostsMileageAgeMake Costs1.0000.7710.023-0.240 Mileage0.7711.000-0.496-0.478 Age0.023-0.4961.0000.164 Make-0.240-0.4780.1641.000

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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.”

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Correlations with Satisfaction leadtime-0.766 ol-tracking-0.242 cost0.097 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: Customers forced to wait are unhappy. Those without access to online order tracking are more satisfied. Those who pay more are somewhat happier. ?????

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Regression: satisfactionconstantleadtimecostol-track coefficient192.7338-6.8856-1.80258.5599 std error of coef16.16430.55350.31374.0729 t-ratio11.9234-12.4391-5.74532.1017 significance0.0000% 4.0092% beta-weight-1.0879-0.45710.1586 standard error of regression13.9292 coefficient of determination75.03% adjusted coef of determination73.70% The Full Regression Customers dislike high cost, and like online order tracking. Why does customer satisfaction vary? Primarily because leadtimes vary; secondarily, because cost varies.

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Reconciliation 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. satisfactionleadtimecostol-tracking satisfaction1.000-0.766-0.097-0.242 leadtime-0.7661.000-0.5430.465 cost-0.097-0.5431.000-0.230 ol-tracking-0.2420.465-0.2301.000

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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 (r- squared). If you give me a correlation, I’ll interpret it by squaring it and looking at it as a coefficient of determination.

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A Pharmaceutical Ad Diagnostic scores from sample of patients receiving psychiatric care So, if your patients have anxiety problems, consider prescribing our antidepressant!

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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.

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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.

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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. ExampleExample 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. ExampleExample

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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 + (b 1 +b 2 Age) 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

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Examples from the Sample Exams Regression: Revenue constantAgeAge 2 SexDirectIndirect Sex Ind coefficient-1224.8462.37502-0.5201-121.8991.9926150.852761.43767 Caligula’s Castle: revenue / $ incentivedirectindirect Men (Sex=0)$1.99$0.85 Women (Sex=1)$1.99$2.29 The Age effect on Revenue is greatest at Age = -(62.37)/(2(-0.5201)) = 59.96 years Give direct incentives (house chips, etc.) to men Give indirect incentives (flowers, meals) to women Revenue pred = -1224.82 + 62.37 Age – 0.5201 Age 2 – 121.9 Sex + 1.99 Direct + (0.8527+1.4377 Sex) Indirect

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Examples from the Sample Exams Regression: CustSat constantWaitWait 2 SizeFranz? Size Franz? coefficient84.4016931-0.8666595-0.0556165-5.6022949-40.0845068.77474654 CustSat pred = 84.40 – 0.8667 Wait – 0.0556 Wait 2 – 5.602 Size + (-40.0845+8.7747 Size) Franz? Hans and 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

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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. ExampleExample 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. ExampleExample

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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 + cX 2 + … (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 = downward- bending) – -b/(2c) indicates where the vertex (either maximum or minimum) of the parabola occurs – Cost: We lose a meaningful interpretation of the beta-weights

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Examples from the Sample Exams Regression: Revenue constantAgeAge 2 SexDirectIndirect Sex Ind coefficient-1224.8462.37502-0.5201-121.8991.9926150.852761.43767 Revenue pred = -1224.82 + 62.37 Age – 0.5201 Age 2 – 121.9 Sex + 1.99 Direct + (0.8527+1.4377 Sex) Indirect Caligula’s Castle: revenue / $ incentivedirectindirect Men (Sex=0)$1.99$0.85 Women (Sex=1)$1.99$2.29 The Age effect on Revenue is greatest at Age = -(62.37)/(2(-0.5201)) = 59.96 years Give direct incentives (house chips, etc.) to men Give indirect incentives (flowers, meals) to women

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Examples from the Sample Exams Regression: CustSat constantWaitWait 2 SizeFranz? Size Franz? coefficient84.4016931-0.8666595-0.0556165-5.6022949-40.0845068.77474654 CustSat pred = 84.40 – 0.8667 Wait – 0.0556 Wait 2 – 5.602 Size + (-40.0845+8.7747 Size) Franz? Hans and 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

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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

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