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Inference: Fisher's Exact P-‐‑Values 2/4/14 Quiz 1 Grades Inference: Fisher’s Exact p-values STA 320 Design and Analysis of Causal Studies Dr. Kari Lock Morgan and Dr. Fan Li Department of Statistical Science Duke University > summary(Quiz1) Min. 1st Qu. Median Mean 3rd Qu. Max. 8.00 15.00 16.00 15.33 17.00 19.00 Review of Quiz Review of Last Class • Potential outcomes: what would spring GPA be if student tents AND what would it be if • Randomizing units to treatments creates student doesn’t tent? balanced treatment groups • Covariates must be pre-treatment • Placebos and blinding are important • Assumptions: o SUTVA no interference: Yi not affected by Wj • Four types of classical randomized o individualistic: Wi not affected by Xj or Yj experiments: unconfounded: W not affected by Y or Y o i i j, o Bernoulli randomized experiment conditional on X o Completely randomized experiment • Wi can depend on Wj o Stratified randomized experiment • Context: don’t just recite definitions o Paired randomized experiment Diet Cola and Calcium Diet Cola and Calcium Drink Calcium Excreted • Does drinking diet cola leach calcium Diet cola 50 from the body? Diet cola 62 Diet cola 48 Diet cola 55 • 16 healthy women aged 18-40 were Diet cola 58 randomly assigned to drink 24 ounces of Diet cola 61 Diet cola 58 either diet cola or water Diet cola 56 Water 48 • Their urine was collected for 3 hours, and Water 46 Water 54 calcium excreted was measured (in mg) Water 45 Water 53 • Is there a significant difference? Water 46 Water 53 Water 48 1 2/4/14 Test Statistic Diet Cola and Calcium • A test statistic, T, can be any function of: • Difference in sample means between obs o the observed outcomes, Y treatment group (diet cola drinkers) and o the treatment assignment vector, W o the covariates, X control group (water drinkers) • The test statistic must be a scalar (one T obs ≡ Y obs −Y obs number) T C n n • Examples: WiYi (1) (1−Wi )Yi (0) Difference in means ∑ ∑ o i=1 i=1 o Regression coefficients = − o Rank statistics NT NC o See chapter 5 for a discussion of test statistics = 6.875 mg Key Question p-value • Is a difference of 6.875 mg more extreme • T: A random variable than we would have observed, just by • Tobs: the observed test statistic computed in random chance, if there were no the actual experiment difference between diet cola and water regarding calcium excretion? • The p-value is the probability that T is as extreme as Tobs, if the null is true • What types of statistics would we see, • GOAL: Compare Tobs to the distribution of T just by the random assignment to under the null hypothesis, to see how treatment groups? extreme Tobs is • SO: Need distribution of Tobs under the null Sir R.A. Fisher Randomness • In Fisher’s framework, the only randomness is the treatment assignment: W • The potential outcomes are considered fixed, it is only random which is observed • The distribution of T arises from the different possibilities for W • For a completely randomized experiment, N choose NT possibilities for W 2 2/4/14 Sharp Null Hypothesis Diet Cola and Calcium • Fisher’s sharp null hypothesis is there is no • There is NO EFFECT of drinking diet cola treatment effect: (as compared to water) regarding calcium excretion • H0: Yi(0) = Yi(1) for all i • So, for each person in the study, their • Note: this null is stronger than the typical amount of calcium excreted would be hypothesis of equality of the means the same, whether they drank diet cola or water • Advantage of Fisher’s sharp null: under the null, all potential outcomes “known”! Sharp Null Hypothesis Randomization Distribution • Key point: under the sharp null, the • The randomization distribution is the vector Yobs does not change with distribution of the test statistic, T, assuming different W the null is true, over all possible assignment vectors, W • Therefore we can compute T exactly • For each possible assignment vector, under the null for each different W! compute T (keeping Yobs fixed, because we are assuming the null) • Assignment mechanism completely determines the distribution of T under • The randomization distribution gives us the null exactly the distribution of T, assuming the sharp null hypothesis is true • (why is this not true without sharp null?) Diet Cola and Calcium Diet Cola and Calcium • 16 choose 8 = 12,870 different possible assignment vectors • For each of these, calculate T, the difference in sample means, keeping the values for calcium excretion fixed 3 2/4/14 Exact p-value Diet Cola and Calcium • From the randomization distribution, computing the p-value is straightforward: • The exact p-value is the proportion of p-value test statistics in the randomization = 0.005 distribution that are as extreme as Tobs • This is exact because there are no distributional assumptions – we are using the exact distribution of T Diet Cola and Calcium Notes • If there were no difference between • This approach is completely diet cola and water regarding calcium nonparametric – no model specified in excretion, only 5/1000 of all terms of a set of unknown parameters randomizations would lead to a difference as extreme as 6.875 mg (the • We don’t model the distribution of observed difference) potential outcomes (they are considered fixed) • Drinking diet cola probably does leach calcium from your body! • No modeling assumptions or assumptions about the distribution of the potential outcomes Approximate p-value Approximate p-value • For larger samples, the number of • Repeatedly randomize units to treatments, possible assignment vectors (N choose and calculate test statistic keeping Yobs fixed N ) gets very large T • If the number of simulations is large enough, • Enumerating every possible assignment this randomization distribution will look very vector becomes computationally much like the exact distribution of T difficult • Note: estimated p-values will differ slightly • It’s often easier to simulate many from simulation to simulation. This is okay! (10,000? 100,000?) random assignments • The more simulations, the closer this approximate p-value will be to the exact p-value 4 2/4/14 Diet Cola and Calcium StatKey www.lock5stat.com/statkey p-value ≈ 0.004 To Do • Read Ch 5 • Bring laptops to class Wednesday • HW 2 due next Monday 5 .
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