Statistical Tests Using Experimental Data

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1 Statistical Tests Using Experimental Data Alec Brandon July 15, 2015

2 Alternative title So you ve worked your tail off and have some experimental data. Now what?

3 Why are we even talking about statistics? 1. It s fun? Often this isn t true 2. It s useful? Often this is true

4 Options 1. Classical inference: Upside: People will understand what you re talking about. E.g., statistical significance, p-values, etc. Downside: Bordering on illogical 2. Bayesian inference: Upside: Logical Downside: No one will understand what you re talking about We ll focus on classical inference. (Hope you like double negatives!)

5 Classical inference Recipe to specify your research question in the context of classical inference: 1. Pick a null hypothesis. Typically something along the lines of the status quo. 2. Pick an alternative hypothesis. Typically your favorite explanation. Examples? Pitfalls?

6 When conducting inference we try to minimize two types of mistakes 1. Rejecting the null when it s true (aka Type I Error) 2. Not rejecting the null when it s false (aka Type II Error)

7 Null hypothesis: Is person pregnant Other examples?

8 In other words We might be skeptical that anything works so classical inference places the burden of proof on the alternative hypothesis by minimizing Type I error for a fixed level of Type II error.

9 What tools do we have? All of statistics in one slide 1. Averages are pretty good estimates of averages (Law of Large Numbers) 2. On average averages are distributed normally (Central Limit Theorem) Furthermore, LLN and CLT hold when continuous functions applied. (Not important for this presentation but called Continuous Mapping Theorem.)

10 Law of Large Numbers Tells us that we might want to focus on comparing treatment and control average. Why?

11 Central Limit Theorem 10 : X : X : X Tells us how to estimate quantities like statistical significance. Why?

12 How do these tools add up to anything useful? z = ( X trt X ctrl ) ˆ sigma/ n Where: 1. Xtrt = Average of treatment. Calculate it with the mean or average functions in Excel or whatever. 2. ˆσ = Standard deviation of data. Calculate it with stdev in Excel. 3. n = Number of observations. Which puts us only 1 step away from the quantity of interest... the p value!

13 Go ahead and dust off that ole stats textbook

14 Or use a computer E.g., in Excel: =1-NORMDIST(z, 0, 1, TRUE) Intuitively p value is the smallest significance level at which you can reject the null hypothesis

15 Cult of 5 % What s the largest p value that we feel comfortable rounding down to 0?

16 Work through a simple example Anyone have a favorite (simple!) experiment?

17 Review Recipe so far: 1. Pick a null and alternative hypothesis 2. Estimate the p value from your statistical test 3. If p < 0.05 then reject the null and congratulate yourself for having taught the world something Hopefully this makes sense because next we re going to see how this framework can lead us astray

18 Common pitfalls of classical inference Roadmap: 1. Data dependence 2. Multiple outcomes 3. Multiple subgroups 4. Multiple treatments Examples?

19 Data dependence When is your data truly independent?

20 Failing to account for data dependence can lead to incredibly misleading results Bertrand, Duflo, and Mullainathan (2004 QJE): 1. Download 20 years worth of female wage data across the United States. 2. Create a new variable called fakelaw for a randomly selected state. 3. Estimate the effect of fakelaw. Repeat. 4. Find that fakelaw changed female wages at the 5% level 45% of the time. Yikes!

21 Solutions to data dependence Anyone recall issue with sample size and clustering? How did we adjust?

22 Solutions to data dependence Cluster the standard errors! Few notes: 1. What to cluster on? 2. In practice, just type, vce(cluster grp) into Stata after your regression. 3. Beware small standard errors if your number of clusters is small! Works for many different types of clustering: Time, spatial, social, etc.

23 One note on data dependence Note for the practitioners: If dealing with panel data and t > n then parametric (e.g., panel equivalent of Newey-West standard errors) solutions might be more appropriate.

24 Multiple outcomes/subgroups/treatment arms When is this an issue? 1. Well specified experiments 2. Kitchen sink experiments

25 Multiple outcomes (Bennet et al., 2009) What s your null?

26 Statistically significant differences with fish (Bennet et al., 2009)

27 Estimating effect for many subgroups leads to similar problem To prove this to yourself: 1. Download years and years of NBA gambling data. 2. Find a subsample (e.g., games on TV on the weekends that feature a team from LA that s the underdog) where betting is profitable. 3. Travel to Las Vegas and place bets.

28 Similar problem arises with many treatments: Run 20 trials on effect of different colored jelly beans and acne xkcd.com/882/

29 What s going wrong when we enter a multiple outcome/treatment/subgroup world? Recall the thought experiment for statistical significance...

30 Correct p values for multiple hypothesis testing Many efforts to factor in these concerns. One that gives a p value and relies on AB testing framework: Based on List, Shaikh, and Xu (201? WP): 1. Estimate test statistic for each outcome/subgroup. 2. Bootstrap B test statistics for each outcome/subgroup. 3. For each bootstrap save the largest test statistic for all the outcomes. 4. Calculate the proportion of times that the test statistic from #1 is greater than the test statistic for # that proportion is your new p value. To factor in many treatment comparisons do this for every comparison of treatments.

31 Wrapping up Questions?

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