• Today: Quizz 4

• Tomorrow: Lab 3 – SN 4117

• Wed: A3 due

• Friday: Lab 3 due

• Mon Oct 1: Exam I this room, 12 pm

• Mon Oct 1: No grad seminar

Key concepts so far

• Quantity• Measurement scale• Dimensions & Units• Equations• Data Equations

– Sums of squared residuals quantify improvement in fit, compare models

• Quantify uncertainty through frequency distributions– Empirical– Theoretical– 4 forms, 4 uses

Today

Selected examples from:

Read lecture notes

D

Logic of Hypothesis Testing

Reject JUST LUCK Hypothesis

Skill!

Just Luck!!

A B CIzaak Walton

Reject JUST LUCK

• Compared observed outcome to all possible outcomes more tractable to restrict to all possible outcomes given that JUST LUCK hyp is true

Arrangements of 8 fish such that IW catches 7?

Reject JUST LUCK

Arrangements of 8 fish such that IW catches 7?

Assign probabilities to each outcome, assuming that the H0 ‘JUST LUCK’ is true

For each fish, there is a 1 in 5 chance that IW will catch it

IW=8 IW=7 IW=7 IW=7 IW=7

A=1 B=1 C=1 D=1

(1/5)8 (1/5)7 (1/5)7 (1/5)7 (1/5)7

0.00000256 0.0000128 0.0000128 0.0000128 0.0000128

p=0.00005376, i.e. 5 times in 10,000

Hypothesis Testing

• Set of rules for making decisions in the face of uncertainty

• Logic is inductive: from specific to general

• Structure is binary

3 styles of statistical inference

• Likelihood, frequentist and Bayesian inference

• All based on the principle of maximum likelihood

Definition: a model that makes the data more probable (best predicts the observed data) is said to be more likely to have generated the data

Likelihood inference

3 styles of statistical inference

Which model is more likely to have generated the data?

Frequentist inference

Use expected distribution of outcomes to calculate a probability

Bayesian inferenceFind the probability that a hypothesis is true, given the

observed dataContrast to: finding the probability of observing the data I observed

(or more extreme data), assuming that the null hypothesis is true

Integrates prior knowledge we have on the system with new observations to make an informed decision

3 styles of statistical inference

Bayesian inference

3 styles of statistical inference

e.g.: coin flip. Hypothesis: the coin is biased

Observe flips: HTHHHTTHHHH

Frequentist approachNull Hypothesis H0

• H0 just chance

• Research hypothesis (what we really care about) is stated as HA

• So, why work with H0 and not HA?

– Easier to work out probabilities

– Permits yes/no decision

• Working with H0 is not intuitive. Logic is backwards because we want to reject H0, not explain how the world functions through H0

Choice of HA

• Start with research hyp, then challenge it with H0

• HA/H0 defined with respect to population, not sample

• HA/H0 must be defined prior to analysis

• Choice of HA/H0 determines how we calculate p-value

• HA/H0 pair must be exhaustive

• HA/H0 must be mutually exclusive

Choice of HA

How do we choose it?Often HA=effect, H0= no effect

BUT, more informative choices are available:

G: growth rate of plants. c:Control, t: treated with fertilizer

1..

2..

3..

‘tails’ ‘scale’

Type I & Type II error

• Type I (α): reject H0 when it is true ‘false positive’ e.g. in a trial, accused is innocent but goes to jail

H0:

• Type II (β): not rejecting H0 when it is false ‘false negative’ e.g. in a trial, accused is guilty but is set free

H0:

Type I & Type II error

• Type I (α): reject H0 when it is true ‘false positive’

• Type II (β): not rejecting H0 when it is false ‘false negative’

H0 True H0 False

Not rejecting H0

Reject H0

Type I & Type II error

True H0

Reject H0 when it is true

True HA

Type I & Type II error

Draw not rejecting H0 when it is false, i.e. β

Tradeoff between α and β

Draw rejecting H0 when H0 is false, i.e. power

Will present 2 examples (if time allows)

More examples in lecture notes

Selected examples from:

Table 7.1 Generic recipe for decision making with statistics

1. State population, conditions for taking sample2. State the model or measure of pattern……………………………3. State null hypothesis about population…………………………… 4. State alternative hypothesis…………………………………………5. State tolerance for Type I error………………………………………6. State frequency distribution that gives probability of outcomes when

the Null Hypothesis is true. Choices:a) Permutations: distributions of all possible outcomesb) Empirical distribution obtained by random sampling of all possible

outcomes when H0 is truec) Cumulative distribution function (cdf) that applies when H0 is true

State assumptions when using a cdf such as Normal, F, t or chisquare7. Calculate the statistic. This is the observed outcome8. Calculate p-value for observed outcome relative to distribution of

outcomes when H0 is true9. If p less than α then reject H0 in favour of HA

If greater than α then not reject H0

10.Report statistic, p-value, sample sizeDeclare decision

Example: jackal bones

Length of bones from 10 female and 10 male jackals (Manly 1991)

L = length of mandible (L=mm) of Golden jackals

Male Female  

120 110  

107 111  

110 107  

116 108  

114 110  

111 105  

113 107  

117 106  

114 111  

112 111  

113.4 108.6 mean

13.82 5.16 var

   

     

Example: jackal bones

1. Population: All possible measurements on these bonesAll jackals in the world? Need to know if sample representative

2. Measure of pattern: ST = D0 =

3. H0:

4. HA:

5. α=

6. Theoretical dist of D0? UnknownSolution: construct empirical freq dist of D0 when H0 is true by randomization….

Example: jackal bones

2. D0 = mean(Lmale)-mean(Lfem) 3.H0: D0<=0 4.HA:D0>0 5. α=5%

6. Empirical FD. Randomization

a) Assign bones randomly to 2 groups (forget M/F)

b) Compute mean(gr1) and mean(gr2)

c) D0,res= mean(gr1) - mean(gr2)

d) Repeat many times (the more the better, continued later)

e) Assemble random differences into a FD

7. Statistic. Do= 113.4 – 108.6 = 4.8 mm

Example: jackal bones

2. D0 = mean(Lmale)-mean(Lfem) 3.H0: D0<=0 4.HA:D0>0 5. α=5%

8. Compute p-value:100,000 values of D0,res

360 values exceed 4.8p = 360/100000 p = 0.0036

9. p =0.0036< α=0.05 reject H0

in favour of HA (D0>0)

10.D0 = 4.8 mmn = p = male jackal mandible bones significantly longer than those of females

Example: jackal bones

This was laborious

Can be made easier by using theoretical frequency distributions

Trade off: must make assumptions

Example: jackal bones6d) repeat many times

100,000 repetitions

Example: jackal bones6d) repeat many times

10,000 repetitions

Example: jackal bones6d) repeat many times

1,000 repetitions

Example: Oat Yield data

Yield of oats in 2 groups

1. Control

2. Chemical seed treatment

1 common mean1 mean per group

Is the improvement better than random?

Example: Oat Yield data

1. Sample: 8 measurementsPopulation: all possible measurements taken with a stated procedure

2. Measure of pattern: ST = SSmodel

3. H0: E(SSmodel) = 0

4. HA:E(SSmodel) > 0

5. α=5%

6. Theoretical dist of SSmodel? UnknownSolution: construct empirical freq dist of SSmodel when H0 is true by randomization….

Example: Oat Yield data

6. Empirical FD

a) Assign yields to 2 groups (forget treatment/control)

b) Fit common mean model

c) Fit 2 means model

d) Calculate SSmodel

e) Repeat many times (1000)

f) Assemble random differences into a FD

7. Statistic. SSmodel=192.08

Example: Oat Yield data

8. Compute p-value:1,000 values of SSmodel161 values exceed 192.08p = 161/1000 p = 0.161

9. p = 0.161 > 0.05 do not reject H0

The improvement is not better than random

10.SSmodel = 192.08n = 8 p = 0.161we can not reject the JUST LUCK hypothesis

QUIZZ 4

Good luck!