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Statistical Inference and Regression Analysis: Stat-GB.3302.30, Stat-UB.0015.01

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### Statistical Inference and Regression Analysis: Stat-GB.3302.30, Stat-UB.0015.01

Professor William Greene

Stern School of Business

IOMS Department

Department of Economics

Objectives of Statistical Analysis

- Estimation
- How long do hard drives last?
- What is the median income among the 99%ers?
- Inference – hypothesis testing
- Did minorities pay higher mortgage rates during the housing boom?
- Is there a link between environmental factors and breast cancer on eastern long island?

General Frameworks

- Parametric Tests: features of specific distributions such as the mean of a Bernoulli or normal distribution.
- Specification Tests (Semiparametric)
- Do the data arrive from a Poisson process
- Are the data normally distributed
- Nonparametric Tests: Are two discrete processes independent?

Hypotheses

- Hypotheses - labels
- State 0 of Nature – Null Hypothesis
- State 1 – Alternative Hypothesis
- Exclusive: Prob(H0 ∩ H1) = 0
- Exhaustive: Prob(H0) + Prob(H1) = 1
- Symmetric: Neither is intrinsically “preferred” – the objective of the study is only to support one or the other. (Rare?)

Does the New Drug Work?

- Hypotheses: H0= .50, H1 = .75
- Priors: P0= .40, P1= .60
- Clinical Trial: N = 50, 31 patients “respond’” p = .62
- Likelihoods:
- L0 (31| =.50) = Binomial(50,31,.50) = .0270059
- L1 (31| =.75) = Binomial(50,31,.75) = .0148156
- Posterior odds in favor of H0 = (.4/.6)(.0270059/.0148156) = 1.2152 > 1
- Priors favored H1 1.5 to 1, but the posterior odds favor H0, 1.2152 to 1. The evidence discredits H1even though the ‘data’ seem more consistent with prior P1.

Decision Strategy

- Prefer the hypothesis with the higher posterior odds
- A gap in the theory: How does the investigator do the cost benefit test?
- Starting a new business venture or entering a new market: Priors and market research
- FDA approving a new drug or medical device. Priors and clinical trials
- Statistical Decision Theory adds the costs and benefits of decisions and errors.

An Alternative Strategy

- Recognize the asymmetry of null and alternative hypotheses.
- Eliminate the prior odds (which are rarely formed or available).

http://query.nytimes.com/gst/fullpage.html?res=9C00E4DF113BF935A3575BC0A9649C8B63http://query.nytimes.com/gst/fullpage.html?res=9C00E4DF113BF935A3575BC0A9649C8B63

Classical Hypothesis Testing

- The scientific method applied to statistical hypothesis testing
- Hypothesis: The world works according to my hypothesis
- Testing or supporting the hypothesis
- Data gathering
- Rejection of the hypothesis if the data are inconsistent with it
- Retention and exposure to further investigation if the data are consistent with the hypothesis
- Failure to reject is not equivalent to acceptance.

Asymmetric Hypotheses

- Null Hypothesis: The proposed state of nature
- Alternative hypothesis: The state of nature that is believed to prevail if the null is rejected.

Hypothesis Testing Strategy

- Formulate the null hypothesis
- Gather the evidence
- Question: If my null hypothesis were true, how likely is it that I would have observed this evidence?
- Very unlikely: Reject the hypothesis
- Not unlikely: Do not reject. (Retain the hypothesis for continued scrutiny.)

Some Terms of Art

- Type I error: Incorrectly rejecting a true null
- Type II error: Failure to reject a false null
- Power of a test: Probability a test will correctly reject a false null
- Alpha level: Probability that a test will incorrectly reject a true null. This is sometimes called the size of the test.
- Significance Level: Probability that a test will retain a true null = 1 – alpha.
- Rejection Region: Evidence that will lead to rejection of the null
- Test statistic: Specific sample evidence used to test the hypothesis
- Distribution of the test statistic under the null hypothesis: Probability model used to compute probability of rejecting the null. (Crucial to the testing strategy – how does the analyst assess the evidence?)

Possible Errors in Testing

Hypothesis is Hypothesis is True False

I Do Not Reject the Hypothesis

I Reject the Hypothesis

A Legal Analogy: The Null Hypothesis is INNOCENT

Null Hypothesis Alternative Hypothesis Not Guilty Guilty

Finding: Verdict Not Guilty

Finding: VerdictGuilty

The errors are not symmetric. Most thinkers consider Type I errors to be more serious than Type II in this setting.

(Jerzy) Neyman – (Karl) Pearson Methodology

- “Statistical” testing
- Methodology
- Formulate the “null” hypothesis
- Decide (in advance) what kinds of “evidence” (data) will lead to rejection of the null hypothesis. I.e., define the rejection region
- Gather the data
- Mechanically carry out the test.

Formulating the Null Hypothesis

- Stating the hypothesis: A belief about the “state of nature”
- A parameter takes a particular value
- There is a relationship between variables
- And so on…
- The null vs. the alternative
- By induction: If we wish to find evidence of something, first assume it is not true.
- Look for evidence that leads to rejection of the assumed hypothesis.
- Evidence that rejects the null hypothesis is significant

Example: Credit Scoring Rule

- Investigation: I believe that Fair Isaacs relies on home ownership in deciding whether to “accept” an application.
- Null hypothesis: There is no relationship
- Alternative hypothesis: They do use homeownership data.
- What decision rule should I use?

Hypothesis Test

- Acceptance rate for homeowners = 5030/(5030+1100) = .82055
- Acceptance rate for renters is .74774
- H0: Acceptance rate for renters is not less than for owners.
- H0: p(renters) > .82055
- H1: p(renters) < .82055

The Rejection Region

What is the “rejection region?”

- Data (evidence) that are inconsistent with my hypothesis
- Evidence is divided into two types:
- Data that are inconsistent with my hypothesis (the rejection region)
- Everything else

My Testing Procedure

- I will reject H0 if p(renters) < .815 (chosen arbitrarily)
- Rejection region is sample values of p(renters) < 0.815

Distribution of the Test Statistic Under the Null Hypothesis

- Test statistic p(renters) = 1/N i Accept(=1 or 0)
- Use the central limit theorem:
- Assumed mean = .82055
- Implied standard deviation= sqr(.82055*.17945/7413)=.00459
- Using CLT, normally distributed. (N is very large).
- Use z = (p(renters) - .82055) / .00459

Alpha Level and Rejection Region

- Prob(Reject H0|H0 true) = Prob(p < .815 | H0 is true)= Prob[(p - .82055)/.00459)= Prob[z < -1.209]= .11333
- Probability of a Type I error
- Alpha level for this test

The Test

- The observed proportion is 5469/(5469+1845) = 5469/7314 = .74774
- The null hypothesis is rejected at the 11.333% significance level (by the design of the test)

Application: Breast Cancer On Long Island

- Null Hypothesis: There is no link between the high cancer rate on LI and the use of pesticides and toxic chemicals in dry cleaning, farming, etc.
- Neyman-Pearson Procedure
- Examine the physical and statistical evidence
- If there is convincing covariation, reject the null hypothesis
- What is the rejection region?
- The NCI study:
- Working null hypothesis: There is a link: We will find the evidence.
- How do you reject this hypothesis?

Formulating the Testing Procedure

- Usually: What kind of data will lead me to reject the hypothesis?
- Thinking scientifically: If you want to “prove” a hypothesis is true (or you want to support one) begin by assuming your hypothesis is not true, and look for evidence that contradicts the assumption.

Hypothesis About a Mean

- I believe that the average income of individuals in a population is $30,000.
- H0 : μ = $30,000 (The null)
- H1: μ ≠ $30,000 (The alternative)
- I will draw the sample and examine the data.
- The rejection region is data for which the sample mean is far from $30,000.
- How far is far????? That is the test.

Application

- The mean of a population takes a specific value:
- Null hypothesis: H0: μ = $30,000H1: μ ≠ $30,000
- Test: Sample mean close to hypothesized population mean?
- Rejection region: Sample means that are far from $30,000

Deciding on the Rejection Region

- If the sample mean is far from $30,000, reject the hypothesis.
- Choose, the region, for example,

The probability that the mean falls in the rejection region even though the hypothesis is true (should not be rejected) is the probability of a type 1 error. Even if the true mean really is $30,000, the sample mean could fall in the rejection region.

Rejection

Rejection

29,500 30,000 30,500

Reduce the Probability of a Type I Error by Making the (non)Rejection Region Wider

Reduce the probability of a type I error by moving the boundaries of the rejection region farther out.

Probability outside this interval is large.

28,500 29,500 30,000 30,500 31,500

You can make a type I error impossible by making the rejection region very far from the null. Then you would never make a type I error because you would never reject H0.

Probability outside this interval is much smaller.

Setting the α Level

- “α” is the probability of a type I error
- Choose the width of the interval by choosing the desired probability of a type I error, based on the t or normal distribution. (How confident do I want to be?)
- Multiply the z or t value by the standard error of the mean.

Testing Procedure

- The rejection region will be the range of values greater than μ0 + zσ/√N orless than μ0 - zσ/√N
- Use z = 1.96 for 1 - α = 95%
- Use z = 2.576 for 1 - α = 99%
- Use the t table if small sample, variance is estimated and sampling from a normal distribution.

Deciding on the Rejection Region

- If the sample mean is far from $30,000, reject the hypothesis.
- Choose, the region, say,

Rejection

Rejection

I am 95% certain that I will not commit a type I error (reject the hypothesis in error). (I cannot be 100% certain.)

The Test Procedure

- Choosing z = 1.96 makes the probability of a Type I error 0.05.
- Choosing z = 2.576 would reduce the probability of a Type I error to 0.01.
- Reducing the probability of a Type I error reduces the power of the test because it reduces the probability that the null hypothesis will be rejected.

P Value

- Probability of observing the sample evidence assuming the null hypothesis is true.
- Null hypothesis is rejected if P value <

P value < Prob[p(renter) < .74774] = Prob[z < (.74774 - .82055)/.00459] = (-15.86) = .59946942854362260 * 10-56Impossible

=.11333

Confidence Intervals

- For a two sided test about a parameter, a confidence interval is the complement of the rejection region. (Proof in text, p. 338)

Confidence Interval

- If the sample mean is far from $30,000, reject the hypothesis.
- Choose, the region, say,

Rejection

Confidence

Rejection

I am 95% certain that the confidence interval contains the true mean of the distribution of incomes. (I cannot be 100% certain.)

One Sided Tests

- H0 = 0, H10 Rejection region is sample mean far from 0 in either direction
- H0 = 0, H1>0. Sample means less than 0 cannot be in the rejection region.
- Entire rejection region is above 0.
- Reformulate: H0<0, H1>0.

Carrying Out the LR Test

- In most cases, exact distribution of the statistic is unknown
- Use -2log Chi squared [1]
- For a test about 1 parameter, threshold value is 3.84 (5%) or 6.45 (1%)

Specification Tests

- Generally a test about a distribution where the alternative is “some other distribution.”
- Test is generally based on a feature of the distribution that is true under the null but not true under the alternative.

Poisson Specification Tests

- 3820 observations on doctor visits
- Poisson distribution?

Deviance Test

- Poisson Distribution p(x) = exp(-)x/x!
- H0: Everyone has the same Poisson Distribution
- H1: Everyone has their own Poisson distribution
- Under H0, observations will tend to be near the mean. Under H1, there will be much more variation.
- Likelihood ratio statistic (Text, p. 348)

Dispersion Test

- Poisson Distribution p(x) = exp(-)x/x!
- H0: The distribution is Poisson
- H1: The distribution is something else
- Under H0, the mean will be (almost) the same as the variance
- Approximate Likelihood ratio statistic (Text, p. 348) = N * Variance / Mean
- For the doctor visit data, this is 22,348.6 vs. chi squared with 1 degree of freedom. H0 is rejected.

Specification Test - Normality

- Normal Distribution is symmetric and has kurtosis = 3.
- Compare observed 3rd and 4th moments to what would be expected from a normal distribution.

Kurtosis: t[5] vs. Normal

Kurtosis of normal(0,1) = 3

Kurtosis of t[k] = 3 + 6/(k-4); for t[5] = 3+6/(5-4) = 9.

Testing for a Distribution

- H0: The distribution is assumed
- H1: The assumed distribution is incorrect
- Strategy: Do the features of the sample resemble what we would observe if H0 were correct
- Continuous: CDF of data resemble CDF of the assumed distribution
- Discrete: Sample cell probabilities resemble predictions from the assumed distribution

Chi Squared Test for a Discrete Distribution

- Outcomes = A1, A2,…, AM
- Predicted probabilities based on a theoretical distribution = E1(), E2(),…,EM().
- Sample cell frequencies = O1,…,OM

V2 Rocket Hits

Adapted from Richard Isaac, The Pleasures of Probability, Springer Verlag, 1995, pp. 99-101.

576 0.25Km2 areas of South London in a grid (24 by 24)

535 rockets were fired randomly into the grid = N

P(a rocket hits a particular grid area) = 1/576 = 0.001736 = θ

Expected number of rocket hits in a particular area = 535/576 = 0.92882

How many rockets will hit any particular area? 0,1,2,… could be anything up to 535.

The 0.9288 is the λ for a Poisson distribution:

Poisson Process

- θ = 1/169
- N = 144
- λ = 144 * 1/169 = 0.852
- Probabilities:
- P(X=0) = .4266
- P(X=1) = .3634
- P(X=2) = .1548
- P(X=3) = .0437
- P(X=4) = .0094
- P(X>4) = .0021

λ = 0.852

Probabilities:

P(X=0) = .4266

P(X=1) = .3634

P(X=2) = .1548

P(X=3) = .0437

P(X=4) = .0094

P(X>4) = .0021

There are 169 squares

There are 144 “trials”

Expect .4266*169 = 72.1 to have 0 hits/square

Expect .3634*169 = 61.4 to have 1 hit/square

Etc.

Expect the average number of hits/square to = .852.

Interpreting The ProcessDifference in Means of Two Populations

- Two Independent Normal Populations
- Common known variance
- Common unknown variance
- Different Variances
- One and two sided tests
- Paired Samples
- Means of paired observations
- Treatments and Controls – Diff-in-Diff SAT
- Nonparametric – Mann/Whitney
- Two Bernoulli Populations

Household Incomes, Equal Variances

------------------------------------------------------

t test of equal means INCOME by MARRIED

------------------------------------------------------

MARRIED = 0 Nx = 817 MARRIED = 1 Ny = 3057

t [ 3872] = 3.7238 P value = .0002

------------------------------------------------------

Mean Std.Dev. Std.Error

INCOME ----------------------------------------------

MARRIED = 0 .27982 .12939 .00453

MARRIED = 1 .30145 .15194 .00275

------------------------------------------------------

2 Proportions

- Two Bernoulli Populations:Xi ~ Bernoulli with Prob(xi=1) = xYi ~ Bernoulli with Prob(yi=1) = y
- H0: x = y
- The sample proportions arepx = (1/Nx)ixi and py = (1/Ny)iyi
- Sample variances are px(1-px) and py(1-py).
- Use the Central Limit Theorem to form the test statistic.

z Test for Equality of Proportions

Application: Take up of public health insurance.

------------------------------------------------------

t test of equal means PUBLIC by FEMALE

------------------------------------------------------

FEMALE =0 Nx = 1812 FEMALE =1 Ny = 1565

t [ 3375] = 5.8627 P value = .0000

------------------------------------------------------

Mean Std.Dev. Std.Error

PUBLIC ----------------------------------------------

FEMALE = 0 .84713 .35996 .00846

FEMALE = 1 .91310 .28178 .00712

Paired Sample t and z Test

- Observations are pairs (Xi,Yi), i = 1,…,N
- Hypothesis x = y.
- Both normal distributions. May be correlated.
- Medical Trials: Smoking vs. Nonsmoking (separate individuals, probably independent)
- SAT repeat tests, before and after. (Definitely correlated)
- Test is based on Di = Xi – Yi. Same as earlier with H0:D = 0.

Treatment Effects

- SAT Do Overs
- Experiment: X1, X2, …, XN = first SAT score, Y1, Y2, …, YN = second
- Treatment: T1,…,TN = whether or not the student took a Kaplan (or similar) prep score
- Hypothesis, y > x.
- Placebo: In Medical trials, N1 subjects receive a drug (treatment), N2 receive a placebo.
- Hypothesis: Effect is greater in the treatment group than in the control (placebo) group.

Treatment Effects in Clinical Trials

- Does Phenogyrabluthefentanoel (Zorgrab) work?
- Investigate: Carry out a clinical trial.
- N+0 = “The placebo effect”
- N+T – N+0 = “The treatment effect”
- The hypothesis is that the difference in differences has mean zero.

Placebo Drug Treatment

No Effect N00 N0T

Positive Effect N+0 N+T

A Test of Independence

- In the credit card example, are Own/Rent and Accept/Reject independent?
- Hypothesis: Prob(Ownership) and Prob(Acceptance) are independent
- Formal hypothesis, based only on the laws of probability: Prob(Own,Accept) = Prob(Own)Prob(Accept) (and likewise for the other three possibilities.
- Rejection region: Joint frequencies that do not look like the products of the marginal frequencies.

Contingency Table Analysis

The Data: Frequencies Reject Accept TotalRent 1,845 5,469 7,214Own 1,100 5,030 6,630Total 2,945 10,499 13,444

Step 1: Convert to Actual Proportions Reject Accept TotalRent 0.13724 0.40680 0.54404Own 0.08182 0.37414 0.45596Total 0.21906 0.78094 1.00000

Independence Test

Step 2: Expected proportions assuming independence: If the factors are independent, then the joint proportions should equal the product of the marginal proportions.

[Rent,Reject] 0.54404 x 0.21906 = 0.11918[Rent,Accept] 0.54404 x 0.78094 = 0.42486[Own,Reject] 0.45596 x 0.21906 = 0.09988[Own,Accept] 0.45596 x 0.78094 = 0.35606

When is the Chi Squared Large?

- Critical values from chi squared table
- Degrees of freedom = (R-1)(C-1).

Critical chi squaredD.F. .05 .01 1 3.84 6.63 2 5.99 9.21 3 7.81 11.34 4 9.49 13.28 5 11.07 15.09 6 12.59 16.81 7 14.07 18.48 8 15.51 20.09 9 16.92 21.6710 18.31 23.21

Analyzing Default

- Do renters default more often (at a different rate) than owners?
- To investigate, we study the cardholders (only)

DEFAULT

OWNRENT 0 1 All

0 4854 615 5469

46.23 5.86 52.09

1 4649 381 5030

44.28 3.63 47.91

All 9503 996 10499

90.51 9.49 100.00

Multiple Choices: Travel Mode

- 210 Travelers between Sydney and Melbourne
- 4 available modes, air, train, bus, car
- Among the observed variables is income.
- Does income help to explain mode choice?
- Hypothesis: Mode choice and income are independent.

Travel Mode Choices and Income

+----------------------------------------------------------+

| Travel MODE Data |

+--------+-------------------------------------------------+

|INCOME | AIR TRAIN BUS CAR || Total |

+--------+-------------------------------------++----------+

|LOW | 10 36 9 8 || 63 |

| | 0.04761 0.17143 0.04286 0.03810 || 0.30000 |

|----------------------------------------------++----------+

|MEDIUM | 19 20 13 24 || 76 |

| | 0.09048 0.09524 0.06190 0.11429 || 0.36190 |

|----------------------------------------------++----------+

|HIGH | 29 7 8 27 || 71 |

| | 0.13810 0.03333 0.03810 0.12857 || 0.33810 |

|==============================================++==========+

|Total | 58 63 30 59 || 210 |

| | 0.27619 0.30000 0.14286 0.28095 || 1.00000 |

+--------+-------------------------------------+-----------+

Contingency Table

+----------------------------------------------------------+

| Travel MODE Data |

+--------+-------------------------------------------------+

|INCOME | AIR TRAIN BUS CAR || Total |

+--------+-------------------------------------++----------+

| | 10 36 9 8 || 63 |

|LOW | 0.04761 0.17143 0.04286 0.03810 || 0.30000 || | 0.08286 0.09000 0.04286 0.08429 ||

|----------------------------------------------++----------+

| | 19 20 13 24 || 76 |

|MEDIUM | 0.09048 0.09524 0.06190 0.11429 || 0.36190 || | 0.09995 0.10857 0.05170 0.10168 ||

|----------------------------------------------++----------+

| | 29 7 8 27 || 71 |

|HIGH | 0.13810 0.03333 0.03810 0.12857 || 0.33810 || | 0.09338 0.10143 0.04830 0.09499 ||

|==============================================++==========+

|Total | 58 63 30 59 || 210 |

| | 0.27619 0.30000 0.14286 0.28095 || 1.00000 |

+--------+-------------------------------------+-----------+

Assuming independence, P(Income,Mode) = P(Income) x P(Mode).

Computing Chi Squared

For our transport mode problem, R = 3, C = 4, so DF = 2x3 = 6. The critical value is 12.59. The hypothesis of independence is rejected.

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