Math 1710 Class 24pi.math.cornell.edu/~back/m171/slides171/oct23/oct23_v2.pdfMath 1710 Class 24 V2...

Math 1710Class 24

Experimentationas GoldStandard

CausationWithoutExperiment

ExperimentalDesign

Vocabulary

Using aRandomNumber Table

Matched Pairs

ExperimentalDesignDiagrams

Area ofSquares

Math 1710 Class 24

Sampling and ExperimentationDr. Back

Oct. 23, 2009

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Polio Vaccine

NFIP Vaccine Trials

Size Rate (cases/100K)

Grade 2 Vaccine 125K 25Grade 2 No Consent 125K 44Grade 1,3 Control 725K 54

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Polio Vaccine

NFIP Vaccine Trials

PHS Double Blind Vaccine Trials

Treatment 200K 28Control 200K 71

No Consent 350K 46

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Polio Vaccine

NFIP Vaccine Trials

No Consent 350K 46NFIP result confusing, but PHS not.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Polio Vaccine

NFIP Vaccine Trials

No Consent 350K 46NFIP result confusing, but PHS not.Randomized control groups help a lot with unanticipated issues!

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Portacaval Shunt Studies

51 Studies

Enthusiasm:Design Marked Moderate None

No Controls 24 7 1Controls, not randomized 10 3 2Randomized controls 0 1 3

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Gilbert ’75

28 Social and Medical Interventions

++ 21%

– 4%

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Gilbert ’77

36 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%

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Establishing Causation Without Experimentation

(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.

Experimentation is much clearer!

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

(Attempts)

Association strong.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

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 confidencein the scope of your conclusions.

4) Block when possible to increase accuracy/sensitivity andbetter control variability.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Strategies

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Strategies

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Strategies

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Sampling Words

Sample vs. Population

Sample Statistic vs. Population Parameter

Sampling Frame

Voluntary Response Sample

Convenience Sample

Biased Sample

Simple Random Sample (SRS)

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Sampling Words

Census

Strata

Stratified Random Sample

Cluster Sample

Systematic Sampling

Multistage Sample Design

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Sampling Words

Matching in an observational study

Undercoverage

Non-Response Bias

Response Bias

Leading Questions

Sampling Variability

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Stratification

Strata – groups of homogeneous individuals.

Stratified Random Sample – same probability of choicewithin each group.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Stratification

Strata – groups of homogeneous individuals.

Stratified Random Sample – same probability of choicewithin 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.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Types of Bias

Response bias vs. voluntary response bias vs. non-responsebias?

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Types of Bias

Response bias vs. voluntary response bias vs. non-responsebias?Response Bias: problems in the questions or how they areasked.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Types of Bias

Response bias vs. voluntary response bias vs. non-responsebias?Voluntary Response Bias: problems in surveys where onlyvolunteers participate.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Types of Bias

Response bias vs. voluntary response bias vs. non-responsebias?Non-Response Bias: problems associated with which people aremissing in the final results.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Types of Bias

Response bias vs. voluntary response bias vs. non-responsebias?Undercoverage: groups somewhat missing from the samplingframe.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Experiments

Observational Study vs. Experiment

Prospective vs Retrospective Study

Factor in an experiment

Treatment

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Experiments

Control Group

Single-Blind vs. Double-Blind Experiment

One Factor vs. Two Factor Experiment

Placebo

Placebo Effect

Hidden Bias

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Experiments

Completely Randomized Design

Block Design

Matched Pairs Design

Confounding Variables

Statistically Significant Effect

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Factors and Levels

Factors vs. Levels vs. Treatments?

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Factors and Levels

Factor in an Experiment: Variable being manipulated.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Factors and Levels

Levels: Values of a factor.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Factors and Levels

Treatment: What is actively done to the experimental units.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Block Related

Block vs. Completely Randomized Design vs. Block Design vs.Matched Pairs Design

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Block Related

Block vs. Completely Randomized Design vs. Block Design vs.Matched Pairs DesignBlock: homogenous group similar in some important way.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Block Related

Block vs. Completely Randomized Design vs. Block Design vs.Matched Pairs DesignBlock Design: random within each block.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Block Related

Block vs. Completely Randomized Design vs. Block Design vs.Matched Pairs DesignCompletely Randomized Design: Every subject an equal chanceof every treatment.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Block Related

Block vs. Completely Randomized Design vs. Block Design vs.Matched Pairs DesignMatched Pairs Design: block size of 2.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Lurking and Confounding

Lurking: Mostly used in observational situations; a variable(not directly studied) which actually accounts for anassociation.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Lurking and Confounding

Confounding Variables: Mostly used in experiments; variableswhose effects are difficult or impossible to separate from whatyou are observing.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Choose an SRS of size 5 from 30 people

Step 1: Assign ID numbers 01,02,. . .,29,30.all 2-digit!

Step 2: Pick a random place to start in the table.(e.g. row 12, col 3) You might use a wristwatch.

69167 38894 00172 02999 97377 33305

Step 3: Read off pairs of ID digits, selecting those thatcorrespond to real units, rejecting others.

69 16 73 88 94 00 17 20 29 99 97 37 73 33 05

Our SRS is units 16 17 20 29 and 05.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

69167 38894 00172 02999 97377 33305

69 16 73 88 94 00 17 20 29 99 97 37 73 33 05

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

69167 38894 00172 02999 97377 33305

69 16 73 88 94 00 17 20 29 99 97 37 73 33 05

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

69167 38894 00172 02999 97377 33305

69 16 73 88 94 00 17 20 29 99 97 37 73 33 05

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

69167 38894 00172 02999 97377 33305

69 16 73 88 94 00 17 20 29 99 97 37 73 33 05

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

69167 38894 00172 02999 97377 33305

69 16 73 88 94 00 17 20 29 99 97 37 73 33 05

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 1: Have half the people rate pepsi on a 1-7 scale andthe other half rate coke.1 is bad, 7 is good.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 1: Have half the people rate pepsi on a 1-7 scale andthe other half rate coke.1 is bad, 7 is good.Problem: One person’s 6 might be another person’s 5.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 1: Have half the people rate pepsi on a 1-7 scale andthe other half rate coke.1 is bad, 7 is good.Problem: One person’s 6 might be another person’s 5.Would still work if sample size big enough.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 2: Have each person taste both coke and pepsi.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 2: Have each person taste both coke and pepsi.Half try pepsi first, half try coke first.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 2: Have each person taste both coke and pepsi.Half try pepsi first, half try coke first.Look for each person at the difference between their coke andpepsi scores.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 2: Have each person taste both coke and pepsi.Half try pepsi first, half try coke first.Look for each person at the difference between their coke andpepsi scores.Still some problems, e.g. how thirsty someone is when theystart.That’s why we randomize which soda goes first.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Matched Pairs

Coke/Pepsi Taste Test - matched pairs (50 people)Method 2: Have each person taste both coke and pepsi.Half try pepsi first, half try coke first.Look for each person at the difference between their coke andpepsi scores.This is Matched Pairs.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Experimental Design Diagrams

Basic Pattern: Units −→ Treatments −→ Results

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Pattern: Units −→ Treatments −→ ResultsRat experiment - diet to increase weight. (30 rats)

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Pattern: Units −→ Treatments −→ ResultsRat experiment - diet to increase weight. (30 rats)Reduce Electrical Usage - meter/chart/control. (60 houses)

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Pattern: Units −→ Treatments −→ ResultsRat experiment - diet to increase weight. (30 rats)Reduce Electrical Usage - meter/chart/control. (60 houses)Treatment of a Serious Disease - Men vs. Women, 3Alternatives

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Basic Pattern: Units −→ Treatments −→ ResultsRat experiment - diet to increase weight. (30 rats)Reduce Electrical Usage - meter/chart/control. (60 houses)Treatment of a Serious Disease - Men vs. Women, 3AlternativesCoke/Pepsi Taste Test - matched pairs (20 people)

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Eyeball Estimate: Take a 5 second look and make a guess:

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Eyeball Estimate: Take a 5 second look and make a guess:

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Judgement Sample: Each person picks 5 representativerectangles, and computes their average area.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

SRS: Each person uses a table of random numbers to choose 5random rectangles, and computes their average area.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Stratified Random Sample: Call a rectangle big if its area isabove 5; small otherwise.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Stratified Random Sample: Call a rectangle big if its area isabove 5; small otherwise.Have 70% of people find averages of big rectangles and 30%study small ones.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Stratified Random Sample: Call a rectangle big if its area isabove 5; small otherwise.Have 70% of people find averages of big rectangles and 30%study small ones.Our overall estimate will be

.48x̄small + .52x̄large

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Stratified Random Sample: Call a rectangle big if its area isabove 5; small otherwise.Have 70% of people find averages of big rectangles and 30%study small ones.Our overall estimate will be

.48x̄small + .52x̄large

The numbers .48 and .52 come from the fact that there are 48small and 52 big rectangles.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Stratified Random Sample: Call a rectangle big if its area isabove 5; small otherwise.Have 70% of people find averages of big rectangles and 30%study small ones.The fractions 70% and 30% come from the followingcalculation:Type ni σi niσi

small rectangles 48 1.557 74.74big rectangles 52 3.847 200.04

200.04

200.04 + 74.74= .73 ∼ 70%

200.04 + 74.74= .27 ∼ 30%

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

200.04

200.04 + 74.74= .73 ∼ 70%

200.04 + 74.74= .27 ∼ 30%

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

200.04

200.04 + 74.74= .73 ∼ 70%

200.04 + 74.74= .27 ∼ 30%

These proportions give the stratified random sample withsmallest variance for a given sample size.

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Histogram of Actual Areas

Math 1710Class 24

ExperimentalDesign

Vocabulary

Matched Pairs

Area ofSquares

Average Area?

Frequency Breakdown of Actual Areas

Math 1710 Class 24pi.math.cornell.edu/~back/m171/slides171/oct23/oct23_v2.pdfMath 1710 Class 24 V2...

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