Quantitative Methods for Lawyers - Class #13 - Professor Daniel Martin Katz - MSU College of Law

T Statistic, F-Statistic,ANOVA and More STATA Quantitative Methods for Lawyers Class #13 Daniel Martin Katz Michigan State University - College of Law

A Quick ReviewFrom Last Time

Students “T”Distribution

Students “T” DistributionStudent “T” Distribution is preferred statisticfor dealing with continuous datasample sizes are sometimes small, and often wedo not know the standard deviation of thepopulation.When either of these problems occur,statisticians rely on “t” distribution

Students “T” DistributionThe t distribution should NOT be used withsmall samples from populations that areNOT approximately normalThe particular form of the t distributionis determined by its degrees of freedom

Students “T”Distribution http://www.nku.edu/~longa/ stats/taryk/TDist.html

Students “T”Distribution Different Forms Single Sample T Test Problem Comparing the Means of Two Samples

Students “T” DistributionAcme Corporation manufactures light bulbs.The CEO claims that an average Acme lightbulb lasts 300 days. A researcher randomlyselects 15 bulbs for testing. The sampledbulbs last an average of 290 days, with astandard deviation of 50 days.If the CEO’s claim were true, what is theprobability that 15 randomly selected bulbswould have an average life of no more than290 days?

Students “T” DistributionAcme Corporation manufactures light bulbs.The CEO claims that an average Acme lightbulb lasts 300 days. A researcher randomlyselects 15 bulbs for testing. The sampledbulbs last an average of 290 days, with astandard deviation of 50 days.If the CEO’s claim were true, what is theprobability that 15 randomly selected bulbswould have an average life of no more than290 days? This is Single Sample T Test Problem

Students “T”Distribution

Students “T”Distribution P Value

Students “T”Distribution http://stattrek.com/Tables/T.aspx

Getting Started in Stata

NoticeDamageAwards.xlsx Loaded Variables are Loaded

Variables are Case Sensitive -Must be Typed as Capitals if Capitals

T Stat = -1.0724 Pooled SDP Value

Lets Plot a Histogram

Quick Aside about Hive Knowledge VersusLocal Knowledge

Do Not Need to KnowHow To Do Everything ... Need to Learn How toLearn the Method to dowhatever you need to do

Start a Log File / Do FileLog File =Retains a Copy of YourCommands and theOutputHas the .scml FileExtensionDo File =Script of CommandsReally Creating aProgramYou Can Run the Scriptto Quickly recreate youroutput

YouTube Stata Ecosystem Professor Huber can teach you about “Log Files” “Do Files” and lots of other things in Statahttp://www.youtube.com/ user/ProfessorHuber

Some AdditionalThoughts on the “T” Statistic

Assumptions Associated with the “T” StatisticEqual Variances (~ Roughly EqualVariances)Normality in the Underlying Data BeingTestedIndependent Samples(as opposed to paired samples)

Equal Variances(Roughly Equal Variances)How Do We Know Whether the VariancesAre Equal or Equal Enough?Bartlett’s or Levene’s Test for Equalityof VariancesConducted like other statistical testwith the typical pvalue > .05 thanreject criteria

Normality in the Underlying Data Being Tested Wilcoxon Rank Sum Test This is a Non-Parametric Test (like Chi Squared - Does not scale with magnitude of the observation) Conducted By Ranking the Data and Comparing those ranks from each group

Normality in the Underlying Data Being TestedHere is Howyou do it inStata

note:could alsotype thecommand atthe commandline The Wilcoxon Test is Better given the skewing in the underlying data Our Prior Result Still Holds - Fail to Reject given the P Value Above

Independent Samples (as opposed to paired samples)T Test Typically Assumes IndependenceIf Not True - than used the PairedSamples Version of the T-Test

ANOVA and the F-Statistic

Northern Western Southern EasternDistrict District District District2500 3500 1325 15001500 900 1500 10001500 995 1799 15001300 1299 2780 17992000 1900 1800 16001500 995 1399 16001500 771 2225 20002000 1250 1700 25002000 900 3800 12001500 749 2299 25001400 1200 1800 20001500 950 1450 15002000 1200 800 9953000 13001500 16002000 160120001700 1000 1371 Attorneys Fees in25002299 2400 1500 Chapter 7 BK’s in19002050 Texas Districts2101 (From Lawless, et al)116021011300

Limitation of “T Test” in this ContextPairwise Comparison Might Not MakeSense If We Are Interested in AnsweringQuestions Such as is there a statisticallysignificant difference across all fourjudicial districtsNote: Given a 5% Threshold and a Totalof Six Comparisons{(W,N)(S,N)(E,N)(S,W)(E,W)(E,S)}~26% Chance of Generating StatSignificance in at least 1 Comparison

ANOVA and F statComparing Multiple Means at Once

ANOVAAnalysis of Variance = ANOVAConceptually ANOVA Relies on a Ratio oftwo different measures (1) Between Group Difference Weighted Difference between the mean of each group and the overall mean of all groups (Squared to eliminate Negative Signs) called the between group sum of squares

ANOVAAnalysis of Variance = ANOVAConceptually ANOVA Relies on a Ratio oftwo different measures (2) Within Group Difference Difference between observations and the overall mean (Squared to eliminate Negative Signs) called the Within group sum of squares

ANOVABetween group sum of squares (SSb) (number of observations in each group) x (mean of each group - overall mean)2Within group sum of squares (SSw) (each observation - group mean)2

ANOVAMean group sum of squares (MSb) = (SSb)_______ degrees of freedomMean Within group sum of squares (MSw) = (SSw)_______ degrees of freedom F= (MSb) (MSw)

ANOVA and F stathttp://www.educator.com/mathematics/statistics/son/

Great OnlineLearning Platform (~$35 per month) http://www.educator.com/

http://www.psych.utah.edu/stat/introstats/anovaflash.html

Take a look at this page for amoment as it mayhelp build your intuition abouthow the Fstatistic is calculated

Can Use Stata but this is easier for our purposes http://www.physics.csbsju.edu/stats/anova.html

http://www.physics.csbsju.edu/stats/anova.html

Other Uses for F Testshttp://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm

A Quick WordAbout Where WeAre Now Heading in the Class

Quantitative Methods for Lawyers - Class #13 - Professor Daniel Martin Katz - MSU College of Law

by Daniel Katz on Feb 22, 2012

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Quantitative Methods for Lawyers - Class #13 - Professor Daniel Martin Katz - MSU College of Law — Presentation Transcript