1. Randomization Tests
PSU 016
11/2/14

2. Extrasensory Perception
Is there such a thing as extrasensory perception (ESP) or a “sixth sense”?
Do you believe in ESP?

3. Extrasensory Perception
One way to test for ESP is with Zener cards:
Subjects draw a card at random and telepathically communicate this to someone who then guesses the symbol

4. Extrasensory Perception
Let’s do our own study!
Make your own Zener cards:
In groups of two or three
One person will shuffle the cards and attempt to telepathically communicate the symbol (no auditory or visual clues!).
One person will identify (guess) the symbol
Record whether or not correct each time
Do this a total of 15 times
Mark the proportion correct on the board

5. Extrasensory Perception
There are five cards with five different symbols If there is no such thing as ESP, what proportion of guesses should be correct?
Because there are 5 cards, each person has a 1/5 chance of guessing correctly each time, if ESP does not exist.
H0: p = 1/5
Ha: p > 1/5

6. Extrasensory Perception
Statistics vary from sample to sample: even if the population proportion is 1/5, not every sample proportion will be exactly 1/5 How do we determine when a sample proportion is far enough above 1/5 to provide evidence of ESP? More general: How do we determine when a sample statistic is far enough away from H0 to be statistically significant?

7. SIMULATE what would happen if H0 were true!
Key Question
How unusual is it to see a sample statistic as extreme as that observed, if H0 is true?
How do we know how unusual a sample statistic would be if H0 were true?
SIMULATE what would happen if H0 were true!

8. ESP: Simulate!
We just need to set up a rigged system so that the outcome we want happens 1/5 of the time in the long run
Then we use the rigged outcome to simulate outcomes that could happen just by chance.
There are lots of ways to do something like this… what are your ideas?

9. www.lock5stat.com/statkey Lots of simulations!
It’s completely valid to just simulate by hand, but we need LOTS (thousands) of them…
We can use software to do the work for us

10. ESP – Random Chance Are our results statistically significant?
What can we conclude?

11. Randomization Distribution
A randomization distribution is a collection of statistics from samples simulated assuming the null hypothesis is true

12. p-value
The p-value is the chance of obtaining a sample statistic as extreme as (or more extreme than) the observed sample statistic, if the null hypothesis is true

13. Calculating a p-value
What kinds of statistics would we get, just by random chance, if the null hypothesis were true? (randomization distribution)
What proportion of these statistics are as extreme as our original sample statistic?
(p-value)

14. Proportion as extreme as observed statistic
ESP p-value
Distribution of statistics that would be observed, just by random chance, if H0 true
p-value = 0.247
Proportion as extreme as observed statistic
p-value
If you were guessing randomly, the chance of getting a sample proportion as high as is
observed statistic

15. This idea works for any parameter!
Randomization Test
p-values can be calculated by randomization distributions:
Create a randomization distribution by simulating statistics you would see, just by random chance, if H0 were true
Find the p-value as the proportion of simulated statistics as extreme as the observed statistic
This idea works for any parameter!

16. p-value and Ha H0:  = 0 Upper-tail Ha:  > 0 (Right Tail) 𝑥 =2
𝑥 =−1
Lower-tail
(Left Tail)
H0:  = 0
Ha:  ≠ 0
𝑥 =2
Two-tailed

17. Summary p-values can be calculated by randomization distributions:
Create a randomization distribution by simulating statistics you would see, just by random chance, if H0 were true
Find the p-value as the proportion of simulated statistics as extreme as the observed statistic
p-values by randomization distributions will deliver the same results as those estimated using a theoretical reference distribution (Normal, t, etc) as long as the conditions for use are appropriate…

18. homework
Randomization Test Assignment on Canvas (due on Canvas before next class)
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