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Statistics: Unlocking the Power of Data Lock 5 Hypothesis Testing: p-value STAT 250 Dr. Kari Lock Morgan SECTION 4.2 Randomization distribution p-value

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Statistics: Unlocking the Power of Data Lock 5 Two Plausible Explanations If the sample data support the alternative, there are two plausible explanations: 1. The alternative hypothesis (H a ) is true 2. The null hypothesis (H 0 ) is true, and the sample results were just due to random chance Do the data provide enough evidence to rule out #2?

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Statistics: Unlocking the Power of Data Lock 5 Key Question If it is very unusual, we have statistically significant evidence against the null hypothesis Today’s Question: How do we measure how unusual a sample statistic is, if H 0 is true? How unusual is it to see a sample statistic as extreme as that observed, if H 0 is true? SIMULATE what would happen if H 0 were true!

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Statistics: Unlocking the Power of Data Lock 5 In a randomized experiment on treating cocaine addiction, 48 people were randomly assigned to take either Desipramine (a new drug), or Lithium (an existing drug), and then followed to see who relapsed Question of interest: Is Desipramine better than Lithium at treating cocaine addiction? Cocaine Addiction

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Statistics: Unlocking the Power of Data Lock 5 Cocaine Addiction Is Desipramine better than Lithium at treating cocaine addiction (preventing relapses)? What parameter(s) are we interested in? a) Proportion b) Mean c) Difference in proportions d) Difference in means e) Correlation

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Statistics: Unlocking the Power of Data Lock 5 Cocaine Addiction Is Desipramine better than Lithium at treating cocaine addiction (preventing relapses)? p D : proportion of cocaine addicts who relapse after Desipramine p L : proportion of cocaine addicts who relapse after Lithium What are the relevant hypotheses? a) H 0 : p D = p L, H a : p D ≠ p L b) H 0 : p D = p L, H a : p D < p L c) H 0 : p D = p L, H a : p D > p L d) H 0 : p D < p L, H a : p D = p L e) H 0 : p D > p L, H a : p D = p L

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Statistics: Unlocking the Power of Data Lock 5 RRRRRR RRRRRR RRRRRR RRRRRR RRRRRR RRRRRR RRRRRR RRRRRR RRRR RRRRRR RRRRRR RRRRRR RRRR RRRRRR RRRRRR RRRRRR Desipramine Lithium 1. Randomly assign units to treatment groups

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Statistics: Unlocking the Power of Data Lock 5 RRRR RRRRRR RRRRRR NNNNNN RRRRRR RRRRNN NNNNNN RR NNNNNN R = Relapse N = No Relapse RRRR RRRRRR RRRRRR NNNNNN RRRRRR RRRRRR RRNNNN RR NNNNNN 2. Conduct experiment 3. Observe relapse counts in each group Lithium Desipramine 10 relapse, 14 no relapse18 relapse, 6 no relapse 1. Randomly assign units to treatment groups

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Statistics: Unlocking the Power of Data Lock 5 To see if a statistic provides evidence against H 0, we need to see what kind of sample statistics we would observe, just by random chance, if H 0 were true Measuring Evidence against H 0

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Statistics: Unlocking the Power of Data Lock 5 “by random chance” means by the random assignment to the two treatment groups “if H 0 were true” means if the two drugs were equally effective at preventing relapses (equivalently: whether a person relapses or not does not depend on which drug is taken) Simulate what would happen just by random chance, if H 0 were true… Cocaine Addiction

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Statistics: Unlocking the Power of Data Lock 5 RRRR RRRRRR RRRRRR NNNNNN RR RRRR RRRRNN NNNNNN RR NNNNNN 10 relapse, 14 no relapse18 relapse, 6 no relapse

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Statistics: Unlocking the Power of Data Lock 5 RRRRRR RRRRNN NNNNNN NNNNNN RRRRRR RRRRRR RRRRRR NNNNNN RNRN RRRRRR RNRRRN RNNNRR NNNR NRRNNN NRNRRN RNRRRR Simulate another randomization Desipramine Lithium 16 relapse, 8 no relapse12 relapse, 12 no relapse

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Statistics: Unlocking the Power of Data Lock 5 RRRR RRRRRR RRRRRR NNNNNN RR RRRR RNRRNN RRNRNR RR RNRNRR Simulate another randomization Desipramine Lithium 17 relapse, 7 no relapse11 relapse, 13 no relapse

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Statistics: Unlocking the Power of Data Lock 5 Distribution of Statistic under H 0

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution A randomization distribution is a collection of statistics from samples simulated assuming the null hypothesis is true The randomization distribution shows what types of statistics would be observed, just by random chance, if the null hypothesis were true

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Statistics: Unlocking the Power of Data Lock 5 Key Question A randomization distribution allows us to assess this! How unusual would it be to see a sample statistic as extreme as that observed, if H 0 were true?

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution a) 10.2 b) 12 c) 45 d) 1.8

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution Center A randomization distribution is centered at the value of the parameter given in the null hypothesis. A randomization distribution simulates samples assuming the null hypothesis is true, so

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution a) How extreme 10.2 is b) How extreme 12 is c) How extreme 45 is d) What the standard error is e) How many randomization samples we collected

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution a) 0 b) 1 c) 21 d) 26 e) 5

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Statistics: Unlocking the Power of Data Lock 5 Randomization Distribution a) The standard error b) The center point c) How extreme 26 is d) How extreme 21 is e) How extreme 5 is

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Statistics: Unlocking the Power of Data Lock 5 Remember: Statistical Significance When results as extreme as the observed sample statistic are unlikely to occur by random chance alone (assuming the null hypothesis is true), we say the sample results are statistically significant

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Statistics: Unlocking the Power of Data Lock 5 Significant? Do you think the results from the cocaine addiction experiment are significant? a) Yes b) No

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Statistics: Unlocking the Power of Data Lock 5 Quantifying Evidence We need a way to quantify evidence against the null…

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Statistics: Unlocking the Power of Data Lock 5 p-value The p-value is the chance of obtaining a sample statistic as extreme (or more extreme) than the observed sample statistic, if the null hypothesis is true The p-value can be calculated as the proportion of statistics in a randomization distribution that are as extreme (or more extreme) than the observed sample statistic

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Statistics: Unlocking the Power of Data Lock 5 1. What kinds of statistics would we get, just by random chance, if the null hypothesis were true? (randomization distribution) 2. What proportion of these statistics are as extreme as our original sample statistic? (p-value) Calculating a p-value

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Statistics: Unlocking the Power of Data Lock 5 p-value Proportion as extreme as observed statistic observed statistic If the two drugs are equal regarding cocaine relapse rates, we have a 2% chance of seeing a difference in proportions as extreme as Cocaine Addiction Distribution of statistic if H 0 true

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Statistics: Unlocking the Power of Data Lock 5 p-value p-values can be calculated by randomization distributions: simulate samples, assuming H 0 is true calculate the statistic of interest for each sample find the p-value as the proportion of simulated statistics as extreme as the observed statistic

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Statistics: Unlocking the Power of Data Lock 5 Cocaine Addiction In the cocaine addiction experiment, people were actually randomized to one of three groups: Desipramine, Lithium, or Placebo Does Desipramine do better than just a placebo at preventing relapses? Does Lithium do better than just a placebo at preventing relapses?

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Statistics: Unlocking the Power of Data Lock 5 Cocaine Addiction Is Desipramine better than a placebo at preventing relapses? p D : proportion of cocaine addicts who relapse after Desipramine p L : proportion of cocaine addicts who relapse after placebo What are the relevant hypotheses? a) H 0 : p D = p P, H a : p D ≠ p P b) H 0 : p D = p P, H a : p D < p P c) H 0 : p D = p P, H a : p D > p P d) H 0 : p D < p P, H a : p D = p P e) H 0 : p D > p P, H a : p D = p P

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Statistics: Unlocking the Power of Data Lock 5 Desipramine vs Placebo p-value observed statistic Distribution of statistic if H 0 true If there were no difference between Desipramine and placebo regarding cocaine relapses, we would only see a difference as extreme as that observed 1 out of 1000 times.

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Statistics: Unlocking the Power of Data Lock 5 Significant? Do you think the Despiramine vs placebo results from the cocaine addiction experiment are significant? a) Yes b) No

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Statistics: Unlocking the Power of Data Lock 5 Lithium vs Placebo p-value observed statistic Distribution of statistic if H 0 true If there were no difference between Lithium and placebo regarding cocaine relapses, we would see a difference as extreme as the one observed about 34% of the time.

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Statistics: Unlocking the Power of Data Lock 5 Significant? Do you think the Lithium vs placebo results from the cocaine addiction experiment are significant? a) Yes b) No

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Statistics: Unlocking the Power of Data Lock 5 A one-sided alternative contains either > or < A two-sided alternative contains ≠ The p-value is the proportion in the tail in the direction specified by H a For a two-sided alternative, the p-value is twice the proportion in the smallest tail Alternative Hypothesis

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Statistics: Unlocking the Power of Data Lock 5 p-value and H a Upper-tail (Right Tail) Lower-tail (Left Tail) Two-tailed

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Statistics: Unlocking the Power of Data Lock 5 Sleep versus Caffeine Recall the sleep versus caffeine experiment from last class s and c are the mean number of words recalled after sleeping and after caffeine. H 0 : s = c H a : s ≠ c Let’s find the p-value! Two-tailed alternative

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Statistics: Unlocking the Power of Data Lock 5 Sleep or Caffeine for Memory? WordsGroup 9sleep 11sleep 13sleep 14sleep 14sleep 15sleep 16sleep 17sleep 17sleep 18sleep 18sleep 21sleep WordsGroup 6caffeine 7 10caffeine 10caffeine 12caffeine 12caffeine 13caffeine 14caffeine 14caffeine 15caffeine 16caffeine 18caffeine sleep mean = caffeine mean = Is sleep actually better than caffeine for memory, or did the people with better memory just happen to get randomly assigned to sleep? sleep mean – caffeine mean = 3

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Statistics: Unlocking the Power of Data Lock 5 WordsGroup 9sleep 11sleep 13sleep 14sleep 14sleep 15sleep 16sleep 17sleep 17sleep 18sleep 18sleep 21sleep WordsGroup 6caffeine 7 10caffeine 10caffeine 12caffeine 12caffeine 13caffeine 14caffeine 14caffeine 15caffeine 16caffeine 18caffeine mean = 15.25mean = What kinds of results would you see, just by random chance, if there were no difference between sleep and caffeine? Sleep or Caffeine for Memory?

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Statistics: Unlocking the Power of Data Lock 5 WordsGroup WordsGroup sleep mean = sleep caffeine sleep caffeine sleep caffeine caffeine mean = sleep mean – caffeine mean = 1 What kinds of results would you see, just by random chance, if there were no difference between sleep and caffeine? Sleep or Caffeine for Memory?

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Statistics: Unlocking the Power of Data Lock 5 Sleep or Caffeine for Memory? p-value = 2 × = 0.044

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Statistics: Unlocking the Power of Data Lock 5 p-value and H 0 If the p-value is small, then a statistic as extreme as that observed would be unlikely if the null hypothesis were true, providing significant evidence against H 0 The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative

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Statistics: Unlocking the Power of Data Lock 5 The smaller the p-value, the stronger the evidence against H o. p-value and H 0

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Statistics: Unlocking the Power of Data Lock 5 Summary The randomization distribution shows what types of statistics would be observed, just by random chance, if the null hypothesis were true A p-value is the chance of getting a statistic as extreme as that observed, if H 0 is true A p-value can be calculated as the proportion of statistics in the randomization distribution as extreme as the observed sample statistic The smaller the p-value, the greater the evidence against H 0

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Statistics: Unlocking the Power of Data Lock 5 To Do Read Section 4.2 HW 4.1, 4.2 due Friday 3/6

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