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Unit 11: Chi-Squared Tests

Unit 11: Chi-Squared Tests. Testing %’s of 2 populations: equal?. Take a sample from each population; their  x’s (or %’s) won’t be equal. Are they different enough to say populations’ μ ’s (or %’s) are unequal?

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Unit 11: Chi-Squared Tests

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  1. Unit 11: Chi-Squared Tests

  2. Testing %’s of 2 populations: equal? • Take a sample from each population; theirx’s (or %’s) won’t be equal. Are they different enough to say populations’μ’s (or %’s) are unequal? • Fact: SE of difference in sample avgs from two populations is √[(SE1)2 + (SE2)2] • So for CI of diff in %: • SE of diff = √[p1(1-p1)/n1 + p2(1-p2)/n2] where • p’s are samples’ %’s & n’s are sample sizes • (… assuming n’s are big enough for z-test) • For sig test of diff in %, H0 : %’s are = ; bootstrap for common population % with “pooled estimate” p = (n1p1 + n2p2)/(n1+n2), : • SE = √[p(1-p)(1/n1 + 1/n2)]

  3. Example: NYRI • In Morrisville, a sample of 100 people shows 65% against NYRI. In Hamilton, a sample of 150 is 60% against. Is NYRI more unpopular in Morrisville than Hamilton? • SE for CI: √[.65(.35)/100+.6(.4)/150] ≈ 6.22% • so 95% CI is 65% ± 2(6.22%) • which includes 60% • SE for z-test: “pooled” p = (.65(100)+.6(150)) / (100+150) = .62 • so SE is √[.62(.38)(1/100+1/150)] ≈ 6.26% • so z = (5%-0%)/6.26% ≈ .80, not significant diff

  4. Example: prenatal AZT • Does giving the drug AZT to HIV-positive pregnant women save their babies from HIV? (Newsweek, March 7, 1994, p. 53) Out of 163 babies born to mothers treated with AZT, only 13 were HIV-positive, while out of 161 born to mothers treated with a placebo, 40 were HIV-positive.

  5. Testing μ’s (not %’s) of 2 populations: equal? • Difference ofx’s of two samples approximates difference of populations’ μ’s • The formula for SE of sample difference is now √[(s12/n1) + (s22/n2)] (bootstrapping) • If samples are small enough to need a t-test, the value for df is (a mess, but) ≥ min(n1-1,n2-1) (and usually near n1 + n2 – 2)

  6. Example: better at math? • A standard math test is given to 25 randomly chosen Colgate students and 25 Hamilton College students. The Colgate students averaged 155 of a possible 200, with an SD of 15; the Hamilton students averaged 150, with an SD of 20. Are Colgate students better at math?

  7. Example: OAEs • Is there a prenatal basis for homosexuality? (McFadden & Pasanen, Proc. Natl. Acad. Sci. USA, 95 (1998), 2709-2713) If a quiet click sound is made outside a person’s ear, the inner ear responds with “otoacoustic emissions” (OAEs), measured by a mic in ear canal. The inner ears of women usually generate stronger OAEs than men for same volume click (maybe because of androgens in womb). This study: Right ears of 57 heterosexual women produced OAEs of avg amplitude 18.2 dB SPL, with SE of 0.8 dB (in response to click of 75 dB); for 37 homosexual women, avg 16.0 and SE 0.7. Is the difference significant?

  8. Example: Music lessons for math? Does music help a child learn math? (Inspired by Newsweek, July 24, 2000, pp. 50-52): One group of 26 second-graders gets piano instruction plus practice with a math video game; another group of 29 gets extra English lessons plus the math game. After four months, the first group gets an average score of 69 on a test of ratios and fractions, with an SD of 10, while the second averages 60, with an SD of 15.

  9. χ2-tests for counts (I) • Given a list of frequencies (counts), are they distributed (significantly) differently from an expected list? • Or, given a table of frequencies, are the counts distributed (significantly) differently in different rows [or columns]? • I.e., does the choice of category represented by the row affect the distribution of the frequencies in the categories represented by the columns? • Or are the column categories “independent” of the row categories, and v.v.?

  10. Ex of χ2-test: Admittance bias by HS? A certain college claims it does not use an applicant's high school as a factor in the decision to admit him/her. Does the following data support that claim?

  11. χ2-tests for counts (II) • [Toy example: Frequencies are too small for true “asymptotic” test; all cells should have ≥ 5] • H0: % admitted would be same for all HSs • so expected = (22/75)(# applied) • χ2 = Σ (observed–expected)2/expected • If observed = expected, term would be 0 • Always ≥ 0 • Must be counts, not %’s or fractions (would change X2 value) • Here, = 2.69 • Useχ2-table with df = #cols – 1 = 6 • [ times (# rows – 1), which is 1 here ]

  12. Ex of χ2-test: Family sizes & geography • A survey gives the following information on the numbers of children that couples have in different regions of the country. Are the differences between the regions due to chance?

  13. Family size & geography, ctnd. • H0: distribution of family sizes is same all over. • expected = (row sum)(column sum)/(total) • χ2 = Σ(obs - exp)2/exp = 9.244 • df = (4-1)(3-1) = 6

  14. So you’ve done a significance test and gotten significance. Now what? • It may still be just chance. • It may reflect bias in the experiment or study. • *It must reflect a box model with chance errors, even though the arithmetic doesn’t refer to it (as in a χ2 test) • *It may not be important, even if the use of a large sample makes a small significance level.

  15. Type I and type II errors • A “type I” error in a sig test is to reject H0 when it is true. • Many “discoveries” are type I errors • By def, α-value is P(type I error | H0 is true) • A “type II” error is to fail to reject H0 when it is false. • We never “accept H0” – no science is exactly right. (E.g., Einstein corrected Newton.)

  16. Example: α-level and type I or II Flip a coin, get 8 heads. With an H0 of a fair coin, P(count ≥ 8) = [C(10,8) + C(10,9) + C(10,10)] / 210≈ 5.5% So with α = 10%, we reject null, while with α = 5%, we do not reject null. • Therefore, if the coin is fair, 10% makes a type I error, and 5% yields the correct answer, • while if the coin is unfair, 10% yields the correct answer, and 5% makes a type II error.

  17. The “power” of a test is P(no type II error | H0 is false) = 1 - P(type II | H0 false), but ... • ... we can’t compute the power because it depends on how false H0 is, and usually we don’t even know whether it is false. • Ex: Test a coin for fairness (H0 : P(head) = 0.5) with 20 flips. With α = 5%, test says coin unfair if #heads <= 5 or >= 15.Sample (binomial) distributions whenP(head) = 0.6 and 0.7: P(type II error) = 87% and 58% respectively.

  18. Another example: An oracle speaks • Valesky vs. Brown for State Senate, interviewing 400 people. • H0: p = 50% of voters for Val; Ha: p > 50% • EV of (sample) % = .5, SE of % = √[.5(.5)/400] = .025 • For sig, need z = ((#/400)-.5)/.025 ≥ 1.96, i.e., # ≥ 220 • Oracle reveals true p = 52%: EV = .52, SE = √(.52(.48)/400) = .025 : P(Type II error) = P(z ≤ ((220/400)-.52)/.025 = 1.2) = 89% • Oracle reveals true p = 55%: EV = .55, SE = √(.55(.45)/400) = .025 : P(Type II error) = P(z ≤ ((220/400)-.55)/.025 = 0) = 50%

  19. Still, … • As P(Type I error) [rejecting H0 when you shouldn’t] goes up -- maybe by picking a larger α or using a larger sample -- P(Type II error) [failing to reject H0 when you should] goes down, and v.v.

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