Finished Theory on Simple Linear Regression Pathologies and Traps in Linear Regression and Correlation Relationships between Categorical Variables Last

Finished Theory on Simple Linear Finished Theory on Simple Linear RegressionRegression

Pathologies and TrapsPathologies and Trapsin Linear Regression and Correlationin Linear Regression and Correlation

Relationships between Relationships between Categorical VariablesCategorical Variables

Last Time:Last Time:

22

22

:Thus rS

Sb

Y

X

Variance ofpredicted Y’s

Variance ofobserved Y’s

Proportion of Varianceof observed Y’s

that is accounted forby the regression

Proportion of Variance explained

2XY How does a Linear RegressionModel approximate (for X=-8,-7,…,7,8)

For these particular datathe regression

model finds

a = 24b = 0

r = 0

Y = X_squared Line Fit Plot

0

50

100

-10 -5 0 5 10

X

Y

Correlation is Zero: No LINEAR Relationship

Is there “no relationship” between X and Y?

There is an extremely strong (nonlinear) relationship here!

ii bXaY ˆ

Regression toward Mediocrity??

60. : and between n Correlatio

:son of IQ

:father of IQ

rYX

Y

XXY ZZ 6.ˆ

2.1)0.2(6.Z :son mediocre morepredict willWe

0.2 Z:fathert intelligenVery

Y

X

2.1)0.2(6.Z :son dumb less apredict willWe

0.2 Z:father dumbVery

Y

X

Predictions are closer to zero (the mean) then the observations!!

Regression toward Mediocrity??

: of Variance XZ

: of Variance YZ

1XZ

S

1YZ

S

Variability of the Z scores is the same!

No slide into mediocrity!!

Implications for Research

It is very risky to study anything based on selection of extreme groups

Test RetestExtremes become less extreme

May look like a treatment effect!

Relationships between Categorical Variables

Baby Held

Right-Handed Mother

Left-Handed Mother

Left 212 25

Right 43 7

237

50

255 32 287

Marginal Distributions


Baby Held

Right-Handed Mother

Left-Handed Mother

Left

Right

.826 (82.6%)

.174 (17.4%)

.889(88.9%)

.111(11.1%)

Marginal Proportions (Percentages)

Vast majority of babies held leftVast majority of mothers right-handed


1 (100%) 1 (100%)

For any given dexterity of the mother,there is an overwhelming tendency to hold the

baby on the left hand side.

Absolute size not taken into account

Baby Held

Right-Handed Mother

Left-Handed Mother

Left .831 .781

Right .169 .219

Segmented BargraphsSegmented Bargraph

0

50

100

150

200

250

300

right-handed left-handed

Dexterity

Fre

qu

ency

right holding

left holding

Simpson’s Paradox

Admit Deny

Male 480 120

Female 180 20

Admit Deny

Male 10 90

Female 100 200

Business School

Law School

Simpson’s Paradox

Admit Deny

Male 490 210

Female 280 220

Admit Deny

Male .70 30

Female .56 .44

Overall:

Overallconditional proportionsper gender

700

500

Men Priviliged!!Gender Discr.!!

Simpson’s Paradox

Admit Deny

Male 480 120

Female 180 20

Admit Deny

Male 10 90

Female 100 200

Admit Deny

Male .80 .20

Female .90 .10

Admit Deny

Male .10 .90

Female .33 .67

600

200

100

300

WomenPriviliged!?!

WomenPriviliged!?!

Simpson’s Paradox

Admit Deny

Male 480 120

Female 180 20

Admit Deny

Male 10 90

Female 100 200

Admit Deny

Male .80 .20

Female .90 .10

Admit Deny

Male .10 .90

Female .33 .67

600

200

100

300

However: Higher admission rate for male dominated discipline

Chapter 3Chapter 3

Producing DataProducing Data

Today:Today:

Where do data come from?

Two Running Examples

Sleazy Politician:

Would like to know voters’ attitudes towards abortion

Wants to say whatever voters

wish to hear

Two Running Examples

Colleague with lower back

problem:

Would like to learn about a treatment

for regrowing lower back ligaments

Has lots of money

%&@!!

Data Collection

Count number of protesters on each side that are picketing

an abortion clinic in town?

Hand out a questionnaireto fathers in the new mothers’

ward at the hospital?

Anecdotal (biased?) information:

Data CollectionAnecdotal (biased?) information:

Call a person who was in bed for 2 yearsbecause of back problems, had the treatment,and is now better? Ask this person whether

the treatment helped?

Heard about this person in the news.

Ask the person for names of other peoplewho recovered dramatically?

Data Collection

Problems with Anecdotal Information:

• Possibly biased (not representative of the overall trend)

• Tends to be dramatized and even inaccurate

• May be confounded with other variables

Problems with Anecdotal Information:(e.g. from Mass Media)




Example from my personal life:






International Newsflash:

‘Environmentalists in Luxembourgbuild tunnel to save frogs from becoming road kill!’










Note:

This story was anecdotal too!

It does not imply that the mass media are always inaccurate or biased

We/they needsystematic statistical designs

to collect data

Sampling: ask a group of people

Depending on how group is chosen (sampled) we can• make statements only about that group

• make statements (inferences) about a larger population

Experiments: Manipulate a variable and

have experimental units go through differential treatments

Depending on how the back treatments are administered we can or cannot• make causal statements about treatment effectiveness

We/they needsystematic statistical designs

to collect data

Goal: Draw Conclusions • about the population of potential/likely voters• about the effectiveness of the back treatment

i.e. make a statistical inference

from a carefully collected set of datato a larger population

andprovide a statement about

how confident we can be in the stated conclusions.

Sampling

Population

(we want to make statements about this)

Sample

HOW??

Simple Random Sample (SRS):

Every sample of size n has the same chance of being drawn

E.g., take list ofregistered

voters and numberthem.

Then draw n numbers using

a random number table.

Sampling

Population

(we want to make statements about this)

Sample

HOW NOT to do that??

Voluntary Response: just look at the opinions stated in the letters sent from constituents?

List of registered voters: pick a voter and the n-1 listed after that voter?

These are not randomly sampled!!

Other Sampling Methods

Stratified Random Sampling:

Example:a% of population are ethnic group A, b% of population are ethnic group B, etc.

Block and draw Simple Random Samples within Blocks.

Example:Draw a Simple Random Sample of size (n)(a)/100 from ethnic group A,

draw a Simple Random Sample of size (n)(b)/100 from ethnic group B, etc.

Other Sampling Methods

Multistage Sampling:(We won’t discuss this.

Too complicated…)

Example:

Randomly select states,from states, randomly select counties,from counties, randomly select schools,from schools, randomly select students.

Problems with Sampling

Nonresponse: Some people in the SRS do not respond

Under-coverage: Some groups left out

Response Biases: e.g., regarding illegal, anti-social behaviore.g., who is asking the questions?

Telescope Effect: People distort timeline of the past

Wording and Order of Questions

Sampling Distributions

Population

Sample Data

Populationcharacterized by parameters

p proportion of population, say, opposing abortion

mean and varianceof a normal distribution

of, say, people’s heights

2,

sample. in the proportion observed theis ˆ where

p, estimate toˆ

estimate to

estimate to

:parameters population theestimate to"statistics sample" use We

22

p

p

S

X

i.e., distributions thatmatter in sampling

Sampling DistributionsThe politician wants to know the proportion

p of voters in the population who favor abortion.

A random sample of n voters is drawn and their opinions recorded.

Suppose in that random sample30% oppose abortion and

70% favor abortion70.ˆ p

The team that collected the random sample reports to the politician that, based on their random sample,

their best guess at the population proportion p is that p is 70%

Sampling DistributionsThe politician is suspicious of statistics.He asks a second team to investigate

the issue.

Another random sample of n voters is drawn and their opinions recorded.

Suppose in that random sample88% oppose abortion and

12% favor abortion12.ˆ p

The team that collected this random sample reports to the politician that, based on their random sample,

their best guess at the population proportion p is that p is 12%


The politician turns crazy, drops his campaign and becomes a statistician.

He spends the rest of his lifedrawing random samples of n voters,

over and over and over again.

Nobody does this in real life. People only collect one single random sample of size n

n. size of sampleth i'in abortion favor that votersof proportion ˆ

...,

n, size of sample secondin abortion favor that votersof proportion ˆ

n, size of samplefirst in abortion favor that votersof proportion ˆ

2

1

ip

p

p

Crazy Politician Turned Statistician

• Realizes that, when his sample size n is “small”, then the sample proportions vary a lot over repeated random samples of size n.

• Because he was trying to save money, his 2 original random samples had only interviewed a few people each.

• When his sample size n is large, then the sample proportions are all very similar, so, he could just have used a single random sample of (large) size n.

• Decides to resume his campaign….


0

5

10

15

20

25

0.1 0.3 0.5 0.7 0.9

Number of results

Politician/Statistician's results

Small n

Large n

50 large samples50 small samples

(nobody would do that)(why not?)


We will study the sampling distribution of sample proportions more precisely in a future class, when we talk about the binomial distribution.

0

5

10

15

20

25

0.1 0.3 0.5 0.7 0.9

Number of results

Politician/Statistician's results

Small n

Large n

The sampling distribution of the sample proportion is the “histogram” that you would obtain

if you generated a new sample (of the given size) infinitely often.

Statistics

• A statistic is something that you calculate from a random sample.

• The value of the statistic varies from one random sample (of size n) to another random sample (also of size n).

• Over infinitely many samples (each of size n) the statistic has a distribution.

• Many statistics have a normal distribution.

Unbiased Statistics(Centered at the true parameter value)

Population Parameter (e.g., true value of p)

Distribution of Sample Statistic (e.g., sample proportion )p

Variability of a Statistic

• Depends only on sample size and true parameter value(s). Larger sample sizes provide smaller variability of the sample statistics. Larger (more expensive) samples provide more accurate assessment of the true population parameter.

• Does not depend on size of the population that we are making inferences about.

Hey, while you are talking about sampling…

My back keeps hurting!!

Let’s design an experiment

to study that back treatment!

Experimental units/subjects/respondents:

Finds 120 people in his hometown who have the back problem, and who

• are considered “hopeless” cases• have no litigation pending• never had back surgery

Collects some information using a pain questionnaire (dependent measure)

How about … giving each of these 120 people the treatment

to see how it works?

• Regression toward the Mean(as a whole, the group might improve even without any treatment)

• Placebo Effect(favorable response on questionnaire even though the treatment

might have no effect,

trust in experimenter and expectation of a cure)

We need a comparison group (control group) against which to compare/assess the results of the treatment group:

Control group has • no treatment or

• a different treatment

Control theinfluence of

outside variables

Who gets into which group?

How about …

• women get the treatment, men don’t?

• people with worst pain get treatment, others don’t?

• people who travel less to get to the hospital get the treatment, others don’t?

• people who get treatment pay, others get paid?

Need Random Assignment: We do not wish results to be biased

We want to conclude that differences between groups are caused by the treatment.

For that, we need random assignment to experimental groups.

Later we will see how toprecisely state how confident we can be

that the differences between experimental groupsare indicative of real differences in the population:

Could the observed differences have arisen by chance?How likely? How unlikely?

If observed differences are unlikely a chance event,then they are called statistically significant.

Why not just one subject/respondent in each group?

Replication:

We want enough respondents in each group,so that the true differences, if they are there,

are visible.

We want the differences between people (that are not due to the treatment)

to “wash out” across members of each group.

How to randomize?

• Assign numbers to all experimental participants.

• Use a random generator / random number table to assign subjects to experimental groups.

Treatment Control

Doctor A

Doctor B

Doctor C

Factorsand Levels of Factors

Here:2 Factors

They mayinteract!

Double Blind Design

Neither doctornor patient

knows who is an experimental subject

who is a control subject

? ?

Ecological Validity

• Would doctors apply treatment the same way outside the experimental study?

• Would patients outside the experiment be comparable to the ones in the study?

• Would the treatment effects found in the study generalize to the “real world”?

• Are the conclusions drawn about the participants of the study generally true for patients who did not participate?

To randomize or to control?

We don’t always want to completely randomize!

Hey! Make up your mind!

To randomize or to control?

Blocking:

A block is a group of subjects similar (matched) in waysthat might relate to the experiment.

Assign subjects at random within each block.

Block becomes another factor.

Examples: Age, Gender, …

Blocking

Treatment Control

Female

Male

By controlling for gender, we may improve our analysis (if treatment effect differs by gender)

BlockingTreatment 1 Treatment 2

Subject 1

Subject 2

…

Subject N

Example:All subjects get both treatments. Treatment order is randomized.

Documents

Finished Theory on Simple Linear Regression Pathologies and Traps in Linear Regression and Correlation Relationships between Categorical Variables Last