Dr. Allen Back. Sep. 30, 2016

Transcription

1 Dr. Allen Back Sep. 30, 2016

2 Extrapolation is Dangerous

3 Extrapolation is Dangerous And watch out for confounding variables. e.g.: A strong association between numbers of firemen and amount of damge at a fire does not mean firemen cause

4 Extrapolation is Dangerous High Leverage Point: A data point (x i, y i ) with x i far from x.. Consequently the point might (depending on the actual value of y i ) have a large impact on the line of regression.

5 Was it Fair? The first draft lottery during the Vietnam War: 366 balls labeled by dates. Mixed up and pulled out in a random order.

6 Was it Fair? Scatterplot

7 Was it Fair? Boxplots for each month

8 Was it Fair? Scatterplot with Line

9 Was it Fair? Correlation Display

10 Was it Fair? Correlation Display Around 1 in a thousand chance of a correlation coefficient this far from 0 if the lottery was fair.

11 Was it Fair? Around 1 in a thousand chance of a correlation coefficient this far from 0 if the lottery was fair. The balls were probably not mixed well enough.

12 How Many Rooms Can x Clean? x crews working for a building contractor go out each night and clean y rooms. Understand the relationship?

13 How Many Rooms Can x Clean? Scatterplot

14 How Many Rooms Can x Clean? Num summary

15 How Many Rooms Can x Clean? RoomsCleaned Summary

16 How Many Rooms Can x Clean? Scatterplot with Line

17 How Many Rooms Can x Clean? Display

18 How Many Rooms Can x Clean? Display

19 How Many Rooms Can x Clean? Display RoomsCleaned = 3.70 Num

20 How Many Rooms Can x Clean? Residual Plot

21 How Many Rooms Can x Clean? There are important deviations from the the assumptions of an ideal linear regression model here.

22 Highlight of and Distance The slope b 1 of Fare = b 1 Distance + b 0 is the average increase in fare per extra mile. Fare = Distance and Distance = Fare are different lines! (Note ) If you want to compute r on a TI-83/84, the place to look is stat calc linreg. And ONCE, you need to set DiagnosticsOn in the Catalog.

23 Highlight of and Distance Phrase about the regression of y on x: The proportion of the variance of y explained by the regression is r 2.

24 Highlight of and Distance Phrase about the regression of y on x: The proportion of the variance of y explained by the regression is r 2. My view: Right psychologically but unclear at first glance what it means.

25 Highlight of and Distance Phrase about the regression of y on x: The proportion of the variance of y explained by the regression is r 2. What it actually means is Var(ŷ i ) Var(y i ) = r 2 where the variances refer to the 1 variable data sets {y i } and {ŷ i }.

26 Highlight of and Distance Phrase about the regression of y on x: The proportion of the variance of y explained by the regression is r 2. My view: Right psychologically but unclear at first glance what it means. My view: The companion statement Var(Residuals) Var(y i ) = 1 r 2 does really explain why r 2 near 1 says something important about the quality of the approximation offered by the regression model.

27 by Locality (rm outliers?, transform?) vs Housing Prices in 1996 Crime Rate is Crimes Per 1000 Housing Prices in Dollars

28 by Locality (rm outliers?, transform?) scatterplot

29 by Locality (rm outliers?, transform?) with regression line ĤP = 577 CR + 177K r 2 =.06 (SMALL)

30 by Locality (rm outliers?, transform?) regression display

31 by Locality (rm outliers?, transform?) Residuals

32 by Locality (rm outliers?, transform?) Now analyze without the Center City Outlier

33 by Locality (rm outliers?, transform?) scatterplot

34 by Locality (rm outliers?, transform?) with regression line ĤP = 2290 CR + 225K r 2 =.18 (vs..06 before)

35 by Locality (rm outliers?, transform?) regression display

36 by Locality (rm outliers?, transform?) Residuals

37 by Locality (rm outliers?, transform?) Now transform from CR to 1 CR.

38 by Locality (rm outliers?, transform?) scatterplot

39 by Locality (rm outliers?, transform?) with regression line ĤP = 1.3M But Center City included. 1 CR K r 2 =.17

40 by Locality (rm outliers?, transform?) regression display

41 by Locality (rm outliers?, transform?) Residuals

42 For both men and women: 1 IQ s average about SD about 15

43 A large study showed: 1 For men with IQ of 140, average wife s IQ was For women with IQ of 120, average husband s s IQ was Note the Z score of 140 is twice the Z score of The above kind of comparison is typical because of the two regression lines.

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45 e.g. if r =.5, 1 Ẑ w = rz m, Z m = Ẑ w = Ẑ m = rz w, Z w = Ẑ m =.667.

46 Polio Vaccine NFIP Vaccine Trials Size Rate (cases/100k) Grade 2 Vaccine 125K 25 Grade 2 No Consent 125K 44 Grade 1,3 Control 725K 54

47 Polio Vaccine NFIP Vaccine Trials Size Rate (cases/100k) Grade 2 Vaccine 125K 25 Grade 2 No Consent 125K 44 Grade 1,3 Control 725K 54 PHS Double Blind Vaccine Trials Size Rate (cases/100k) Treatment 200K 28 Control 200K 71 No Consent 350K 46

48 Polio Vaccine NFIP Vaccine Trials Size Rate (cases/100k) Grade 2 Vaccine 125K 25 Grade 2 No Consent 125K 44 Grade 1,3 Control 725K 54 PHS Double Blind Vaccine Trials Size Rate (cases/100k) Treatment 200K 28 Control 200K 71 No Consent 350K 46 NFIP result confusing, but PHS not.

49 Polio Vaccine NFIP Vaccine Trials Size Rate (cases/100k) Grade 2 Vaccine 125K 25 Grade 2 No Consent 125K 44 Grade 1,3 Control 725K 54 PHS Double Blind Vaccine Trials Size Rate (cases/100k) Treatment 200K 28 Control 200K 71 No Consent 350K 46 NFIP result confusing, but PHS not. Randomized control groups help a lot with unanticipated issues!

50 Portacaval Shunt Studies 51 Studies Enthusiasm: Design Marked Moderate None No Controls Controls, not randomized Randomized controls 0 1 3

51 Gilbert Social and Medical Interventions ++ 21% + 21% 0 46% - 7% 4%

52 Gilbert Surgical and Anaesthetic Innovations innovation highly preferred 14% innovation preferred 19% innovation a success but not much better 11% innovation a disappointment but not much worse 28% standard preferred 6% standard highly preferred 11%

53 Establishing Association strong. (Attempts)

54 Establishing Association strong. Association consistent. (Attempts)

55 Establishing (Attempts) Association strong. Association consistent. Higher doses give stronger responses.

56 Establishing (Attempts) Association strong. Association consistent. Higher doses give stronger responses. Alleged cause precedes effect.

57 Establishing (Attempts) Association strong. Association consistent. Higher doses give stronger responses. Alleged cause precedes effect. Alleged cause is plausible.

58 Establishing (Attempts) Association strong. Association consistent. Higher doses give stronger responses. Alleged cause precedes effect. Alleged cause is plausible. Rule out other plausible explanations.

59 Establishing (Attempts) Association strong. Association consistent. Higher doses give stronger responses. Alleged cause precedes effect. Alleged cause is plausible. Rule out other plausible explanations. This is hard to do reliably.

60 Establishing (Attempts) Association strong. Association consistent. Higher doses give stronger responses. Alleged cause precedes effect. Alleged cause is plausible. Rule out other plausible explanations. This is hard to do reliably. is much clearer!

61 Basic Strategies 1) Control extraneous sources of variation.

62 Basic Strategies 1) Control extraneous sources of variation. 2) Randomize to deal with uncontrollable sources of variation.

63 Basic Strategies 1) Control extraneous sources of variation. 2) Randomize to deal with uncontrollable sources of variation. 3) Replicate to increase accuracy and gain greater confidence in the scope of your conclusions.

64 Basic Strategies 1) Control extraneous sources of variation. 2) Randomize to deal with uncontrollable sources of variation. 3) Replicate to increase accuracy and gain greater confidence in the scope of your conclusions. 4) Block when possible to increase accuracy/sensitivity and better control variability.

65 Sampling Words Sample vs. Population Sample Statistic vs. Population Parameter Sampling Frame (not in your text?) Voluntary Response Sample (not in your text?) Convenience Sample Biased Sample Simple Random Sample (SRS)

66 Sampling Words Census Strata Stratified Random Sample Cluster Sample Multistage Sample Design

67 Sampling Words Matching in an observational study cohort Undercoverage (not in your text?) Non-Response Bias Response Bias (not in your text?) Leading Questions Sampling Variability

68 Stratification Strata groups of homogeneous individuals. Stratified Random Sample same probability of choice within each group.

69 Stratification Strata groups of homogeneous individuals. Stratified Random Sample same probability of choice within each group. Advantages include: Every stratum well represented. Can be more accurate for a given sample size. Strata with greater variability should be better represented.

70 Types of Bias Response bias vs. voluntary response bias vs. non-response bias?

71 Types of Bias Response bias vs. voluntary response bias vs. non-response bias? Response Bias: problems in the questions or how they are asked.

72 Types of Bias Response bias vs. voluntary response bias vs. non-response bias? Voluntary Response Bias: problems in surveys where only volunteers participate.

73 Types of Bias Response bias vs. voluntary response bias vs. non-response bias? Non-Response Bias: problems associated with which people are missing in the final results.

74 Types of Bias Response bias vs. voluntary response bias vs. non-response bias? Undercoverage: groups somewhat missing from the sampling frame.

75 s Observational Study vs. Prospective vs Retrospective Study Factor in an experiment Level Treatment

76 s Control Group Single-Blind vs. Double-Blind One Factor vs. Two Factor Placebo Placebo Effect

77 s Block Block Design Matched Pairs Design Confounding Variables Statistically Significant Effect

78 Factors and Levels Factors vs. Levels vs. Treatments?

79 Factors and Levels Factors vs. Levels vs. Treatments? Factor in an : Variable being manipulated.

80 Factors and Levels Factors vs. Levels vs. Treatments? Levels: Values of a factor.

81 Factors and Levels Factors vs. Levels vs. Treatments? Treatment: What is actively done to the experimental units.

82 Block Related Block vs. Block Design vs. Matched Pairs Design

83 Block Related Block vs. Block Design vs. Matched Pairs Design Block: homogenous group similar in some important way.

84 Block Related Block vs. Block Design vs. Matched Pairs Design Block Design: random within each block.

85 Block Related Block vs. Block Design vs. Matched Pairs Design Matched Pairs Design: block size of 2.