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Quantitative Methods for Lawyers - Class #22 - Regression Analysis - Part 1

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- 1. Quantitative Methods for Lawyers Class #18 Regression Analysis Part 1 @ computational computationallegalstudies.com professor daniel martin katz danielmartinkatz.com lexpredict.com slideshare.net/DanielKatz
- 2. Here is an App that Predicts the Price Per Hour of Various Lawyers City Firm Size Partner Experience Calculate Regression Analysis in Legal Procurement http://tymetrix.com/mobile_apps/
- 3. Here is an App that Predicts the Price Per Hour of Various Lawyers City Firm Size Partner Experience Expected Hourly Rate Regression Analysis in Legal Procurement http://tymetrix.com/mobile_apps/ Our Dependent Variable (i.e. Y) Our Independent Variables (i.e. X1 ... Xn)
- 4. Estimate a lawyer’s rate: Real Rate Report™ Regression model From the CT TyMetrix/Corporate Executive Board 2012 Real Rate Report© $15 1 $16 1 $34 per 10 years$95 +$99 (Finance) -$15 (Litigation) n = 15,353 Lawyers Tier 1 Market Experience Partner Status Practice Area Base + + +/- Source: 2012 Real Rate Report™ 32 $15 Per 100 Lawyers Law Firm Size+ + $161 $151 $15 per 100 lawyers $95 $34 per 10 years -$15 (Litigation) +$99 (Finance)
- 5. Y = βo +/- β1 ( X1 ) +/- β2 ( X2 ) +/- β3 ( X3 ) +/- β4 ( X3 ) +/- β5 ( X3 ) + ε Y = $151 + $15 ( ) + 161 ( ) + 95 ( ) + 34 ( ) +/- β5 ( ) + ε Per 100 Lawyers If Tier 1 Market is True Partner Status is True Per 10 Years Practice Area
- 6. Multiple Regression Example
- 7. Multiple Regression Analysis https://s3.amazonaws.com/KatzCloud/elemapi.dta Load This Data Set from Stata into R
- 8. Multiple Regression Analysis https://s3.amazonaws.com/KatzCloud/elemapi.dta Load This Data Set from Stata into R We Need to Understand these Variables:
- 9. Multiple Regression Analysis Okay Lets Get the Variable Labels from Stata into R
- 10. Here are the measures: academic performance of the school (api00), average class size in kindergarten through 3rd grade (acs_k3) percentage of students receiving free meals (meals) - which is an indicator of poverty percentage of teachers who have full teaching credentials (full) Multiple Regression Analysis regression analysis using the variables api00 as the Y Dependent Variable acs_k3, meals, full X Independent Variable
- 11. Regression Analysis using the variables Y = α +/- β1 ( X1 ) +/- β2 ( X2 ) +/- β3 ( X3 ) + ε api00 = β0 - β1 ( acs_k3 ) - β2 ( meals ) + β3 ( full ) + ε Multiple Regression Analysis
- 12. Regression Analysis using the variables Y = α +/- β1 ( X1 ) +/- β2 ( X2 ) +/- β3 ( X3 ) + ε api00 = β0 - β1 ( acs_k3 ) - β2 ( meals ) + β3 ( full ) + ε Multiple Regression Analysis Some Hypotheses -- We might expect that better academic performance would be associated with ( - ) higher class size ( - ) fewer students receiving free meals ( + ) higher percentage of teachers having full teaching credentials
- 13. api00 = β0 - β1 ( acs_k3 ) - β2 ( meals ) + β3 ( full ) + ε
- 14. api00 = 906.7 - 2.68 ( acs_k3 ) - 3.70 ( meals ) + .108 ( full ) + ε
- 15. the three predictors - are they statistically signiﬁcant and what is the direction of the relationship? The average class size (acs_k3, b=-2.68), is not signiﬁcant (p=0.055), but only just so. The coefﬁcient is negative which would indicate that larger class size is related to lower academic performance -- which is what we would expect.
- 16. Effect of meals (b=-3.70, p=.000) is signiﬁcant and its coefﬁcient is negative indicating that the greater the proportion students receiving free meals, the lower the academic performance. The meals variable is highly related to income level and functions more as a proxy for poverty. Thus, higher levels of poverty are associated with lower academic performance. This result also makes sense.
- 17. Finally, the percentage of teachers with full credentials (full, b=0.11, p=.232) seems to be unrelated to academic performance. This would seem to indicate that the percentage of teachers with full credentials is not an important factor in predicting academic performance -- this result was somewhat unexpected.
- 18. More On Regression Analysis
- 19. “We use regression to estimate the unknown effect of changing one variable over another regression requires making two assumptions: 1) there is a linear relationship between two variables (i.e. X and Y) 2) this relationship is additive (i.e. Y= X1 + X2 + ...+ Xn) (Note: Additivity applies across terms - as within terms there can be a square, log, etc.) Technically, linear regression estimates how much Y changes when X changes one unit.” http://dss.princeton.edu/training/ Regression Analysis
- 20. Example: After controlling by other factors, are SAT scores higher in states that spend more money on education?* Outcome (Y) variable = SAT scores --> variable csat in dataset Predictor (X) variables • Per Pupil Expenditures Primary & Secondary (expense) • % HS of graduates taking SAT (percent) • Median Household Income (income) • % adults with HS Diploma (high) • % adults with College Degree (college) • Region (region) Regression Analysis *Source: search for dataset at http://www.duxbury.com/highered/ Use the ﬁle states.dta (educational data for the U.S.).
- 21. Getting Started Lets Begin by Loading it and Use the Head Command https://s3.amazonaws.com/KatzCloud/states.dta
- 22. Getting Started Use the Summary Command For Additional Information on Each Variable
- 23. Getting Started Lets Start Simple: We Might Hypothesize a Positive Relationship As Expenditures Go Up SAT Performances Also Goes Up Relationship Between Sat Score and Expenditures?
- 24. Getting Started It is Certainly NOT Deﬁnitive But a Scatterplot is a good place to start ...
- 25. Notice the Nature of the Relationship is not what we would naively anticipate Getting Started It is Certainly NOT Deﬁnitive But a Scatterplot is a good place to start ...
- 26. Getting Started It is Certainly NOT Deﬁnitive But a Scatterplot is a good place to start ... It Appears to be a N e g a t i v e Relationship Notice the Nature of the Relationship is not what we would naively anticipate
- 27. Bivariate Regression Notice the -.02228 for expense which is the slope of the regression line shown above w e j u s t ﬁ t t h e regression line to this bivariate relationship
- 28. Bivariate Regression Y = B0 + ( B1 * (X1) ) csat = 1060.7 - (0.022*expense) For each one-point increase in expense, SAT scores decrease by 0.022 points.
- 29. Bivariate Regression Y = B0 + ( B1 * (X1) ) csat = 1060.7 - (0.022*expense) Look at the T Stats, P Values with a Tstat (which is Z when N>30) of Greater than 1.96 we can reject the notion that the coefﬁcient is equal to zero
- 30. A Brief Word about Standard Errors N o t i c e t h a t t h e 9 5 % Conﬁdence Interval is the Beta Coefﬁcient ~ Plus or Minus Two Times the Standard Error The standard error of the estimate tells us the accuracy to expect from our prediction -- The standard error of a correlation coefﬁcient is used to determine the conﬁdence intervals around a true correlation of zero. look at the Standard Error and you can obtain the 95% Conﬁdence Interval 1060.732 + 2(32.7) = ~1126.4 1060.732 - 2(32.7) = ~ 995.0
- 31. Daniel Martin Katz @ computational computationallegalstudies.com lexpredict.com danielmartinkatz.com illinois tech - chicago kent college of law@

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