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Power 16

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Power 16

- Post-Midterm
- Cumulative

- Put power point slide show on a high density floppy disk, or e-mail as an attachment, for a WINTEL machine.
- Email [email protected] the slide-show as a PowerPoint attachment

- 1. Project choice
- 2. Data Retrieval
- 3. Statistical Analysis
- 4. PowerPoint Presentation
- 5. Executive Summary
- 6. Technical Appendix
- 7. Graphics

Power_13

- 1. Introduction: Members 1 ,2 , 3
- What
- Why
- How

- 2. Executive Summary: Member 5
- 3. Exploratory Data Analysis: Member 3
- 4. Descriptive Statistics: Member 3
- 5. Statistical Analysis: Member 3
- 6. Conclusions: Members 3 & 5
- 7. Technical Appendix: Table of Contents, Member 6

- Table of Contents
- Spreadsheet of data used and sources or if extensive, a subsample of the data
- Descriptive Statistics and Histograms for the variables in the study
- If time series data, a plot of each variable against time
- If relevant, plot of the dependent Vs. each of the explanatory variables

- Statistical Results, for example regression
- Plot of the actual, fitted and error and other diagnostics
- Brief summary of the conclusions, meanings drawn from the exploratory, descriptive, and statistical analysis.

- Project I: Power 16
- Contingency Table Analysis: Power 14, Lab 8
- ANOVA: Power 15, Lab 9
- Survival Analysis: Power 12, Power 11, Lab 7
- Multi-variate Regression: Power 11 , Lab 6

- Challenger disaster

- Number of O-Rings Failing On Launch i: yi(#) = a + b*tempi + ei
- Biased because of zeros, even if divide equation by 6

- Two Ways to Proceed
- Tobit, non-linear estimation: yi(#) = a + b*tempi + ei
- Bernoulli variable: probability models

- Probability Models: yi(0,1) = a + b*tempi + ei

- Probability Models: yi(0,1) = a + b*tempi + ei
- OLS, Linear Probability Model, linear approximation to the sigmoid
- Probit, non-linear estimate of the sigmoid
- Logit, non-linear estimate of the sigmoid

- Significant Dependence on Temperature
- t-test (or z-test) on slope, H0 : b=0
- F-test
- Wald test

- Plots of Number or Probability Vs Temp.
- Label the axes

- Answer all parts, a-f
- The most frequent sins
- Did not explicitly address significance
- Did not answer b, 660 : all launches at lower temperatures had one or more o-ring failures
- Did not execute c, estimate linear probability model

- The most frequent sins

- Failure of O-rings that sealed grooves on the booster rockets
- Was there any relationship between o-ring failure and temperature?
- Engineers knew that the rubber o-rings hardened and were less flexible at low temperatures
- But was there launch data that showed a problem?

- What: Was there a relationship between launch temperature and o-ring failure prior to the Challenger disaster?
- Why: Should the launch have proceeded?
- How: Analyze the relationship between launch temperature and o-ring failure

- Data
- number of o-rings that failed
- launch temperature

- Launches where there was a problem

Orings temperature

158

157

170

163

170

275

353

.

- All Launches

Plot of failures per observation versus temperature range shows

temperature dependence:

Mean temperature for the 7 launches with o-ring failures was

lower, 63.7, than for the 17 launches without o-ring failures,

72.6. -

Contingency table analysis

Number of O-ring Failures Vs. Temperature

Logit Extrapolated to 31F:

Probit extrapolated to 31F:

Extrapolating OLS to 31F: OLS:

Tobit:

- From extrapolating the probability models to 31 F, Linear Probability, Probit, or Logit, there was a high probability of one or more o-rings failing
- From extrapolating the Number of O-rings failing to 31 F, OLS or Tobit, 3 or more o-rings would fail.
- There had been only one launch out of 24 where as many as 3 o-rings had failed.
- Decision theory argument: expected cost/benefit ratio:

- Decision theory argument: expected cost/benefit ratio:

Difference in mean temperatures for failures and successes

Difference in probability of one or more o-ring failures for high and low temperature ranges

Probabilty models: LPM (OLS), probit, logit

Number of o-ring failure per launch Vs. Temp.

OLS, Tobit

Contingency table analysis

ANOVA

- Challenger example

- Probability one or more o-rings fail
- Low temp: 53-62 degrees
- Medium temp: 63-71 degrees
- High temp: 72-81 degrees

- Average number of o-rings failing per launch
- Low temp: 53-62 degrees
- Medium temp: 63-71 degrees
- High temp: 72-81 degrees

Probability one or more o-rings fails

Number of o-rings failing per launch

- ANOVA and Regression
- (Non-Parametric Statistics)
- (Goodman Log-Linear Model)

- Salesaj = c(1)*convenience+c(2)*quality+c(3)*price+ e
- E[salesaj/(convenience=1, quality=0, price=0)] =c(1) = mean for city(1)
- c(1) = mean for city(1) (convenience)
- c(2) = mean for city(2) (quality)
- c(3) = mean for city(3) (price)
- Test the null hypothesis that the means are equal using a Wald test: c(1) = c(2) = c(3)

One-Way ANOVA and Regression

Regression Coefficients are the City Means; F statistic

- Salesaj = c(1) + c(2)*convenience+c(3)*quality+e
- E[Salesaj/(convenience=0, quality=0)] = c(1) = mean for city(3) (price, the omitted one)
- E[Salesaj/(convenience=1, quality=0)] = c(1) + c(2) = mean for city(1) (convenience)
- c(1) = mean for city(3), the omitted city
- c(2) = mean for city(1) minus mean for city(3)
- Test that the mean for city(1) = mean for city(3)
- Using the t-statistic for c(2)

- Salesaj = c(1) + c(2)*convenience+c(3)*price+e
- E[Salesaj/(convenience=0, price=0)] = c(1) = mean for city(2) (quality, the omitted one)
- E[Salesaj/(convenience=1, price=0)] = c(1) + c(2) = mean for city(1) (convenience)
- c(1) = mean for city(2), the omitted city
- c(2) = mean for city(1) minus mean for city(2)
- Test that the mean for city(1) = mean for city(2)
- Using the t-statistic for c(2)

- Salesaj = c(1) + c(2)*convenience + c(3)* quality + c(4)*television + c(5)*convenience*television + c(6)*quality*television + e, SSR=501,136.7
- Salesaj = c(1) + c(2)*convenience + c(3)* quality + c(4)*television + e, SSR=502,746.3
- Test for interaction effect: F2, 54 = [(502746.3-501136.7)/2]/(501136.7/54) = (1609.6/2)/9280.3 = 0.09

Table of Two-Way ANOVA for Apple Juice Sales

- Salesaj = c(1) + c(2)*convenience + c(3)* quality + e, SSR=515,918.3
- Test for media effect: F1, 54 = [(515918.3-502746.3)/1]/(501136.7/54) = 13172/9280.3 = 1.42
- Salesaj = c(1) +e, SSR = 614757
- Test for strategy effect: F2, 54 = [(614757-515918.3)/2]/(501136.7/54) = (98838.7/2)/(9280.3) = 5.32

- Density, f(t)
- Cumulative distribution function, CDF, F(t)
- Probability you failed up to time t* =F(t*)

- Survivor Function, S(t) = 1-F(t)
- Probability you survived longer than t*, S(t*)
- Kaplan-Meier estimates:
(#at risk- # ending)/# at risk

- Applications
- Testing a new drug

- Current standard for ovarian cancer is taxol and a platinate such as cisplatin
- Previous standard was cyclophosphamide and cisplatin
- Kaplan-Meier Survival curves comparing the two regimens
- Lab 7: ( # at risk- #ending)/# at riak

Taxol ( Bristol-Myers Squibb)

interrupts cell division (mitosis)

It is a cyclical hydrocarbon

Top Panel: European

Canadian and Scottish,

342 at risk for Tc, 292

Survived 1 year

Bottom Panel:

Gynecological Oncology

Group, 196 at risk

For Tc, 168 survived

1 year

- What to do when the sample of observations is not distributed normally?

- Wilcoxon Rank Sum Test for independent samples
- Data Analysis Plus

- Signs Test for Matched Pairs: Rated Data
- Eviews, Descriptive Statistics

- Wilcoxon Signed Rank Sum Test for Matched Pairs: Quantitative Data
- Eviews

- Testing the difference between the means of two populations when they are non-normal
- A New Painkiller Vs. Aspirin, Xm17-02

Ratings

- 30 total ratings for both samples
- 3 ratings of 1
- 5 ratings of 2
- etc

3 15 12

continued

5 30 27

4 19.5 5 27

Rank Sum 276.5 188.5

- E (T )= n1 (n1 + n2 + 1)/2 = 15*31/2 = 232.5
- VAR (T) = n1 * n2 (n1 + n2 + 1)/12
- VAR (T) = 15*31/12 , sT = 24.1
- For sample sizes larger than 10, T is normal
- Z = [T-E(T)]/ sT = (276.5 - 232.5)/24.1 = 1.83
- Null Hypothesis is that the central tendency for the two drugs is the same
- Alternative hypothesis: central tendency for the new drug is greater than for aspirin: 1-tailed test

5%

1.645