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2. Significance Testing
- What is a p-value?
- How to build a Null Hypothesis
3. How about the Alternative Hypothesis?
- False Alarms and Power
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2. HYPOTHESIS TESTING
1. Science and Falsification
2. Significance Testing
2.1. What is a p-value?
2.2. How to build a Null Hypothesis
3. How about the Alternative Hypothesis?
3.1. False Alarms and Power
3. FALSIFICATIONISM
● Denying the consequent (modum tollens)
((P → Q) ^ ¬Q) → ¬P
(model → data) ^ ¬data) → ¬model
● Models can only be disproven
● Not explicitly probabilistic
4. STATISTICAL FALSIFICATIONISM
● Data is a consequence of the true model
● That consequence is probabilistic
Likelihood (Model) = P(Data | Model)
● Model unlikely if data obtained would have low
probability under that model.
6. TEST INGREDIENTS
● A hypothesis to reject: the null model (H0
)
● Some data
● A summary of the data: a statistic
● A way to calculate the probability
distribution of the statistic given H0
NULL HYPOTHESIS SIGNIFICANCE TESTING
8. NHST
Null model (H0
)
Parameter
Data (X)
Statistic
Probability
Null model (H0
) Lady can’t tell the difference
Parameter Probability of mistake (pm
= 0.5)
Data (X) 1 mistake out of 10 (n = 10)
Statistic Proportion of mistakes (Pm
=0.1)
Probability What is the probability of 1/10 mistakes
if H0
is true and pm
= 0.5?
11. pr
(1-p)n-rn!
r! (n-r)!P(r, n | p ) =
What is the probability of making
r mistakes out of n trials given p?
p = probability of mistake r = number of mistakes
n = number of trials
12. The BINOMIAL TEST
Probability of 1 or less
mistakes= 0.0107
IF H0
(pm
= 0.5)
p-value
0.0009
0.0098
0.0439
0.1171
0.2051
0.2461
0.1171
0.2051
0.0439
0.0098
0.0009
14. P-vALUES
● NOT the probability of the null hypothesis being true
● NOT applicable to all distributions
15. P-vALUES
● NOT the probability of the null hypothesis being true
● NOT applicable to all distributions
● NOT a measure of effect size or importance
16. NHST
Null model (H0
) Lady can’t tell the difference
Parameter Probability of mistake (pm
= 0.5)
Data (X) 450 mistakes out of 1000 (n = 1000)
Statistic Proportion of mistakes (Pm
= 0.45)
P-value What is the probability of making
≤ 450/1000 mistakes if pm
= 0.5?
18. P-vALUES
● NOT the probability of the null hypothesis being true
● NOT applicable to all distributions
● NOT a measure of effect size or importance
● ANY effect will be significant given enough data
19. SIGNIFICANCE VS. MAGNITUDE
10% mistakes
p = 0.0008p = 0.0107
45% mistakes
How good is 0.45?
What would be the a
better null hypothesis?
20. NHST
Null model (H0
) One in three mistakes
Parameter Probability of mistake (pm
= 0.33)
Data (X) 1 mistake out of 10 (n = 10)
Statistic Proportion of mistakes (Pm
=0.1)
P-value What is the probability of making
≤1/10 mistakes if pm
= 0.33?
25. RANDOMIZATION
1. Draw a random sample
2. Calculate statistic
3. Do it many times
re
= 40.2 re
= 47.2 re
= 58.7 re
= 56.1 re
= 44.4 re
= 51.4
26. RANDOMIZATION
1. Draw a random sample
2. Calculate statistic
3. Do it many times
re
= 40.2 re
= 47.2 re
= 58.7 re
= 56.1 re
= 44.4 re
= 51.4
27. RANDOMIZATION
1. Draw a random sample
2. Calculate statistic
3. Do it many times
4. Distribution of those statistics
Distribution of the test statistic (re
)
under the NULL HYPOTHESIS
30. P-vALUES
● NOT the probability of the null hypothesis being true
● NOT applicable to all distributions
● NOT a measure of effect size or importance
● NOT appropriate for modeling rare events
31. What is the probability of a Scientist winning a (science)
Nobel Prize?
P(Nobel | Scientist) = 0.00001
Marie Curie won 2 Nobel Prizes
p-value = P(≥2 Nobel | Scientist) ≅ 0.00000000001
Therefore, Marie Curie is unlikely to be a Scientist?
WHY IS THIS WRONG?
32. CONSIDERING the ALTERNATIVE
Probability of a Scientist winning a Nobel Prize?
P(2 Nobel | Scientist) = 0.000012
P(2 Nobel |¬ Scientist) ≅ 0.000000000012
Likelihood of
being a scientist
Likelihood of
NOT being a scientist
LIKELIHOOD RATIO = = 1012
0.000000000012
0.000012
33. NULL vs. ALTERNATIVE
H0
is true Ha
is true
Accept H0 ✓ Type II error
Accept Ha
Type I error ✓
● Consider both a NULL and an ALTERNATIVE hypothesis
34. NULL vs. ALTERNATIVE
H0
is true Ha
is true
Accept H0
1 − α 1 − β
Accept Ha
α β
False Positives
You’re
pregnant
POWER
37. Null model (H0
) Even sex ratio (1:1)
Parameter Probability of producing a male ( = 0.5)
Data (X) Number of males and females (n = 15)
Statistic Proportion of males
Probability
inspired by
Hamilton 1967.
Science
38. Null model (H0
) Even sex ratio (1:1)
Parameter Probability of producing a male ( = 0.5)
Data (X) Number of males and females (n = 15)
Statistic Proportion of males
Alternative (HA
)
Model
inspired by
Hamilton 1967.
Science
39. Null model (H0
) Even sex ratio (1:1)
Parameter Probability of producing a male ( = 0.5)
Data (X) Number of males and females (n = 15)
Statistic Proportion of males
Alternative (HA
) Biased sex ratio (1:2)
Model Probability of producing a male ( = 0.33)
inspired by
Hamilton 1967.
Science
41. NULL vs. ALTERNATIVE
α = 0.05
P(data|H0
)
critical
value
NULL MODEL
( = 0.5)
Number of males
Accept HA
Accept H0
42. NULL vs. ALTERNATIVE
α = 0.05
P(data|H0
)
NULL MODEL
( = 0.5)
power = 0.41
Number of males
ALTERNATIVE MODEL
( = 0.33)
Accept HA
Accept H0
P(data|HA
)
48. Type I errorp-value
● Refers to single test
● Data-based random value
● Property of the data
● Inductive evidence
● Refers to multiple tests
● Fixed quantity set a priori
● Property of the test
● Deductive assessment
FISHER’s APPROACH NEYMAN-PEARSON APPROACH
52. HOMEWORK
● Work in groups
● Read one of the assigned articles applying a
nonparametric test in biological research
● Try to understand the test in question
● Present your findings to the class
30.01.17
10:00h
53. QUESTIONS
1. What is the question? What is the purpose of the test?
2. What is the statistic and what does it measure?
3. How is the null hypothesis built and what does it assume?
4. How does the test answer the question?
5. Find out how to implement the test in R
6. What other applications could this test have?
55. EvALUATION
● Questions: Relate lecture concepts to the new test
● Delivery: Quality and clarity of the presentation
● Discussion: Both asking and answering questions
● Group-level AND individual-level
Group work ≠ Dividing tasks
(except in presenting)