Statistics: Unlocking the Power of Data Lock 5 Randomization Tests Dr. Kari Lock Morgan PSU 016...

Statistics: Unlocking the Power of Data Lock5

Randomization Tests

Dr. Kari Lock Morgan

PSU 016

11/5/14


Extrasensory PerceptionIs there such a thing as extrasensory

perception (ESP) or a “sixth sense”?Do you believe in ESP?


Extrasensory PerceptionOne 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


Extrasensory PerceptionLet’s do our own study!Make your own Zener cards:Randomly choose a symbolFind a partner, telepathically communicate

your symbol (no auditory or visual clues!), and have them guess your symbol.

Switch roles.Did you guess correctly?


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 = 1/5Ha: p > 1/5


Extrasensory PerceptionStatistics 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?


Key Question

How do we know how unusual a sample statistic would be if H0 were true?

How unusual is it to see a sample statistic as extreme as that observed, if H0 is true?

SIMULATE what would happen if H0 were true!


ESP: Simulate!• How could we simulate what would happen,

just by random chance, if the null hypotheses were true for the ESP experiment?

Randomly choose a symbol.

• Return it to the rest, shuffle, and choose again for the (random) guess.

• Did you (randomly) get the correct symbol?


Lots of simulations!

• We need many more simulations!

www.lock5stat.com/statkey

http://www.lock5stat.com/statkey


ESP – Random Chance

Are our results statistically significant?

What can we conclude?


Randomization Distribution

A randomization distribution is a collection of statistics from samples

simulated assuming the null hypothesis is true


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


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


ESP p-value

p-value = 0.247

If you were all just guessing randomly, the chance of us getting a sample proportion as high as 0.294 is 0.247.

p-value

observed statistic

Proportion as extreme as observed statistic

Distribution of statistics that would be observed, just by random chance, if H0 true


• 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

• Is Desipramine better than Lithium at treating cocaine addiction?

Cocaine Addiction

pD, pL: proportion of cocaine addicts who relapse after taking Desipramine or Lithium, respectively

H0: pD = pL

Ha: pD < pL


R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R

R R R R R R

R R R R R R

R R R R R R

R R R R

R R R R R R

R R R R R R

R R R R R R

Desipramine Lithium

1. Randomly assign units to treatment groups


R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R R R R N N

N N N N N N

RR

N N N N N N

R = RelapseN = No Relapse

R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R R R R RR

R R N N N N

RR

N N N N N N

2. Conduct experiment

3. Observe relapse counts in each group

LithiumDesipramine

10 relapse, 14 no relapse 18 relapse, 6 no relapse

1. Randomly assign units to treatment groups

10 18

24

ˆ ˆ

24.333

D Lp p


To see if a statistic provides evidence against H0, we need to

see what kind of sample statistics we would observe,

just by random chance, if H0 were true

Measuring Evidence against H0


• “by random chance” means by the random assignment to the two treatment groups

• “if H0 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 H0 were true…

Cocaine Addiction


R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R R R R N N

N N N N N N

RR

N N N N N N



R R R R R R

R R R R N N

N N N N N N

N N N N N N

R R R R R R

R R R R R R

R R R R R R

N N N N N N

R N R N

R R R R R R

R N R R R N

R N N N R R

N N N R

N R R N N N

N R N R R N

R N R R R R

Simulate another randomization

Desipramine Lithium


ˆ ˆ16 12

24 240.167

LDp p


R R R R

R R R R R R

R R R R R R

N N N N N N

RRR R R R

R N R R N N

R R N R N R

RR

R N R N R R

Simulate another randomization

Desipramine Lithium


ˆ ˆ17 11

24 240.250

D Lp p


• Shuffle your cards and deal them into two piles. What is your sample difference in proportions?

• Why did you re-deal your cards?

• Why did you leave the outcomes (relapse or no relapse) unchanged on each card?

Cocaine Addiction

You want to know what would happen

• by random chance

• if the null hypothesis is true


Lots of simulations!

• We need many more simulations!





p-valueProportion as extreme as observed statistic

observed statistic

If the two drugs are equal regarding cocaine relapse rates, we have a 1.3% chance of seeing a difference in proportions as extreme as that observed.

Distribution of statistics that would be observed, just by random chance, if H0 true



Randomization Testp-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!


Your Turn! Correlation

3.0 3.5 4.0 4.5 5.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Malevolence Rating of Uniform

z-sc

ore

for

Pen

alty

Yar

ds

r = 0.43

NFL Teams • Do NFL teams with more malevolent uniforms get more penalty yards?


p-value and Ha

H0: = 0Ha: > 0

Upper-tail(Right Tail)

H0: = 0Ha: < 0

Lower-tail(Left Tail)

H0: = 0Ha: ≠ 0

Two-tailed


Summaryp-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

Statistics: Unlocking the Power of Data Lock 5 Randomization Tests Dr. Kari Lock Morgan PSU 016...

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