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

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  1. 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

  2. Part 5 – Hypothesis Testing

  3. 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?

  4. 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?

  5. 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?)

  6. Testing Strategy

  7. Posterior (to the Evidence) Odds

  8. 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.

  9. 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.

  10. An Alternative Strategy • Recognize the asymmetry of null and alternative hypotheses. • Eliminate the prior odds (which are rarely formed or available).

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

  12. 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.

  13. 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.

  14. 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.)

  15. 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?)

  16. Possible Errors in Testing Hypothesis is Hypothesis is True False I Do Not Reject the Hypothesis I Reject the Hypothesis

  17. 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.

  18. (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.

  19. 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

  20. 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?

  21. Some Evidence = Homeowners 5469 5030 1845 1100

  22. 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

  23. 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

  24. My Testing Procedure • I will reject H0 if p(renters) < .815 (chosen arbitrarily) • Rejection region is sample values of p(renters) < 0.815

  25. 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

  26. 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

  27. Distribution of the Test Statistic and the Rejection Region Area=.11333

  28. 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)

  29. Power of the test

  30. Power Function for the Test(Power = size when alternative = the null.)

  31. 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?

  32. 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.

  33. 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.

  34. 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

  35. 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

  36. 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.

  37. 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.

  38. 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.

  39. 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.)

  40. The Testing Procedure (For a Mean)

  41. 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.

  42. P Value • Probability of observing the sample evidence assuming the null hypothesis is true. • Null hypothesis is rejected if P value < 

  43. P value < Prob[p(renter) < .74774] = Prob[z < (.74774 - .82055)/.00459] = (-15.86) = .59946942854362260 * 10-56Impossible =.11333

  44. 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)

  45. 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.)

  46. One Sided Tests • H0 = 0, H10 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.

  47. Likelihood Ratio Tests

  48. 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%)

  49. Poisson Likelihood Ratio Test

  50. Generalities About LR Test