A Tale of Three Numbers

1. A Tale of Three Numbers Statistical Significance, Effect Size, and Sample Size

2. Brief review

3. Causation vs. Correlation When two variables A and B are correlated, there are four possibilities: • A causes B • B causes A • A common cause C causes both A and B • The correlation is accidental

4. So, discovering that countries with democratic elections get in fewer wars, we might conclude: • Democracy causes peace. • Peace causes democracy. • Christianity causes both democracy and peace. • Democracy and peace are only accidentally correlated.

5. Observational Studies Importantly, if we just observe the facts and collect data on how things are, we cannot tell which hypothesis is true. Observational studies find correlations, not the causal structure of the world. (This is what HW4 was about.)

6. The Best Evidence So far, we’ve learned that a good experiment or clinical trial is: • Randomized • Double-blind • Controlled This is often abbreviated ‘RCT’: Randomized Controlled Trial.

9. Controls An experiment with no controls is useless. It tells us what happens when we do X, but not what happens when we don’t do X (control). Maybe the same results would happen from not doing X. Maybe X does nothing. Or a lot. Or a little. With no controls, it is impossible to tell.

10. Randomization An experiment or trial is randomized when each person who is participating in the experiment/ trial has a fair and equal chance of ending up either in the control group or the experimental group.

11. Benefits of Randomization Proper randomization: Minimizes experimenter bias– the experimenter can’t bias who goes into which group. Minimizes allocation bias– lowers the chance that the control group and experimental group differ in important ways.

12. Selection Bias Randomization cannot get rid of all selection bias. For example, many psychology experiments are just performed on American undergraduates by their professors. This means both groups over-represent young Westerners. (“Sampling bias”)

13. Allocation Bias Randomization also guards against allocation bias, where the control group and experimental group are different in important ways. For example, if you assign the first 20 people to enroll in the experiment to the control and the next 20 to the experimental group, there may be allocation bias: the first to enroll may be more eager to take part, because they are sicker.

14. The Importance of Randomization Previously we saw that improper randomization procedures on average exaggerated effects by 41%. This is an average result, so improper randomization often leads to exaggerations that are even larger than 41%.

15. Why RCTs? The importance of the experimental method (as opposed to scientific observation) is that it allows us to discern the causal structure of the world.

16. Causal Structure If we find a correlation between our experimental treatment T and our desired outcome O, we can rule out: • O caused T in the experiment. • A common cause C caused both O and T in the experiment.

17. Causal Structure But can we determine whether the correlation between T and O is real in the first place and not accidental? Yes!

18. Statistical significance

19. Statistical Significance We say that an experimental correlation is statistically significant if it’s unlikely to be accidental. How can we tell when it’s unlikely to be accidental?

20. Null Hypothesis We give a name to the claim that there is no causal connection between the variables being studied. It is called the null hypothesis. Our goal is to reject the null hypothesis when it is false, and to accept it when it is true.

22. Rejecting the Null Hypothesis All experimental data is consistent with the null hypothesis. Any correlation can always be due entirely to chance. But sometimes the null hypothesis doesn’t fit the data very well. When the null hypothesis suggests that our actual observations are very unlikely, we reject the null hypothesis.

23. P-Values One way to characterize the significance of an observed correlation is with a p-value. The p-value is the probability that we would observe our data on the assumption that the null hypothesis is true. p = P(observations/ null hypothesis = true)

24. P-Values Obviously lower p-values are better, that means your observed correlation is more likely to be true. In science we have an arbitrary cut-off point, 5%. We say that an experimental result with p < .05 is statistically significant.

25. Statistical Significance What does p < .05 mean? It means that the probability that our experimental results would happen if the null hypothesis is true is less than 5%. According to the null hypothesis, there is less than a 1 in 20 chance that we would obtain these results.

26. Note Importantly, p-values are not measures of how likely the null hypothesis is, given the data. They are measures of how likely the data is, given the null hypothesis. p = P(data/ null hypothesis = true) ≠ P(null hypothesis = true/ data)

27. Example Suppose I have a coin, and I hypothesize that the coin is biased toward heads. The null hypothesis might be “this is a fair coin, it is equally likely to land heads or tails”. Suppose I then flip it 5 times and it lands HHHHH– heads 5 times in a row.

28. Example We know that the probability of this happening if the coin is fair is 1/25= 1/32 = 0.03125 or about 3%. P(HHHHH/ the coin is fair) = P(HHHHH/ null hypothesis = true) = p = 3%

29. Example So p = .03 < .05, and we can reject the null hypothesis. The bias toward heads is statistically significant.

30. Importance Just because the results of an experiment (or observational study) are “statistically significant” does not mean the revealed correlations are important. The effect size also matters, that is the strength of the correlation.

31. Effect sizes

33. Effect Size One NAEP analysis of 100,000 American students found that science test scores for men were higher than the test scores for women, and this effect was statistically significant These results are unlikely if the null hypothesis, that gender plays no role in science scores, were true.

34. Effect Size However, the average difference between men and women on the test was just 4 points out of 300, or 1.3% of the total score. Yes, there was a real (statistically significant) difference. It was just a very, very small difference.

35. Effect Size One way to put the point might be: “p-values tell you when to reject the null hypothesis. But they do not tell you when to care about the results.”

36. Measures of Effect Size There are lots of measures of effect size: Pearson’s r, Cohen’s f, Cohen’s d, Hedges’ g, Cramér’s V,… Here we will just talk about two measures that are commonly reported: odds ratios and relative risks.

37. Odds Ratio First, let’s introduce the idea of a binary variable. A binary variable is a variable that can have only two values. “height” is not a binary variable, because there are more than two heights people can have. “got an A” is a binary variable, because either you got an A or you didn’t.

38. Odds Whenever you have a binary variable, you can ask about the odds of that variable– what are the odds of getting an A? If 10 students got A’s out of 50 students, then 10 students passed and 40 failed. The odds of getting an A are 10:40 or 1:4 or 25%.

39. Odds vs. Probabilities Odds are not probabilities. There are 50 students and 10 of them got A’s. The probability of getting an A: 10/50 = 20% The odds of getting an A: 10/40 = 25%

40. Odds Ratios Suppose I have another binary variable “studied”– students either studied for the exam or they didn’t. I can ask about the odds that a student who studied got an A, and the odds that a student who didn’t study got an A.

41. In Table Format

42. Odds Ratio So the odds of getting an A among studiers are 6:15 or 40%. And the odds of getting an A among non-studiers are 4/25 or 16%.

43. Odds Ratio The odds ratio is the ratio of these odds, or 40%:16% ≈ 2.5 This means that (in our example) studying raises the odds that someone will get an A by 150%. Alternatively: a student who studies has two and a half times better odds of getting an A.

44. Relative Risk While odds ratios are appropriate when we have two correlated binary variables in an observational study (as when I observe the effects of studying on getting an A), the effect sizes in RCTs are usually reported by relative risks, which are also called risk ratios.

45. Relative Risk Relative risks are just like odds ratios except they compare probabilities and not odds. The odds that a studying student passes are 6:15 = 40% The probability is 6/(6 + 15) = 6/21 ≈ 29%

46. Example The odds that a non-studying student passes are 4:25 = 16%. The probability is 4/(4 + 25) = 4/29 ≈ 14%.

47. Example Whereas the odds ratio was 40:16 = 250%, we get a relative risk of: 29%:14% = 29:14 = 2.07 = 207% These numbers are similar, but obviously not the same. The risk ratio tells you that a student who studies is twice as likely to get an A.

48. Relation As the probabilities of events get smaller the odds approach the probabilities, and odds ratios and relative risks are similar. However, as the probabilities of the events get higher, the odds and risk ratios get very different.

49. Here’s our table again…

50. Odds Ratio for High Probability Events The probability of not getting an A is much higher than the probability of getting an A: 40/50 >> 10/50. The odds of study = no, A = no: 25/4 = 6.25 The odds of study = yes, A = no: 15/6 = 2.5 Odds ratio: 6.25/2.5 = 250%. Not studying increases odds of A = no by one and a half times.