Pvalues

1
P-values

• Second approach to statistical inference
• Hypothesis testing
• H0 null hypothesis
• Ha alternative hypothesis
• Two different possibilities for a parameter such
  as p in the binomial

2
P-values

• The two hypotheses are treated differently
• H0
• No proof
• No action
• Status quo
• Equals

3
P-values

• Ha
• Often what we are seeking to prove
• The treatment improves the chances of survival.
• One-sided pgt1/2
• Two-sided p not equal 1/2

4
P-values

• Must read and understand the scientific setting
  to formulate H0 and Ha
• These should be formulated BEFORE data is
  collected
• They are properties of the PROBLEM, not the DATA

5
P-values

• After we collect data, we can assess how
  plausible H0 is
• P-value Prob(obs or something more extreme when
  H0 is true)
• We need to include more extreme because in many
  situations, every observation is unlikely on its
  own
• Eg, N very large in binomial

6
P-values

• If PV is small, then what we observed is not very
  likely when H0 is true
• So H0 is not very plausible
• Casts doubt on H0
• NOTE PV is a probability of our DATA, not a
  probability of H0
• H0 is not random

7
P-values

• Suppose you are playing a game of chance with a
  friend
• Prob(you win) ½ (you think)
• But after 10 rounds, youve only won once
• This is very unlikely with p1/2
• Casts doubt on H0
• You would tend to think your friend is cheating

8
P-values

• Note that PV measures how sure you are that H0 is
  not precisely true
• It does not tell you how different your situation
  is from H0
• Use confidence for that
• Each approach has its advantages and
  disadvantages

9
P-values

• Lets calculate PV for only winning 1 in 10 when
  p1/2
• H0 p1/2
• Ha plt1/2 if he is cheating
• Observe 1 success in 10 trials
• But 0 is even more extreme
• gtgt pvbprob(10,1/2,0,1)
• 0.0107
• If he wasnt cheating, it would be very unlikely
  that I would win only 1 time in 10

10
P-values

• Exercises (Same H0, Ha)
• 1. If I win 2 times in 10, find the PV for
  cheating
• 2. If I win 4 times in 20
• 3. If I win 4 times in 15

11
P-values

• We could define a 90 confidence bound as the
  value of p so that PV0.10

12
Sign Test

• Suppose we have data from some a continuous RV
• H0 MedianM
• Ha MedianltM
• (Recall Prob(Xltmedian) Prob(Xgtmedian) ½)

13
Sign Test

• We can count the of observations that are ltM
• If this is a lot, then it looks like the median
  is smaller
• Use BPROB to calculate PV

14
Sign Test

• Example
• Test H0 Median25 vs Ha Medianlt25
• Data 1.6, 2.3, 5.9, 18.3, 21.2, 28, 37
• 5/7 are lt25
• gtgt bprob(7,.5,5,7)
• 0.2266
• PV not real small, so Median might be 25