1

Maximum Likelihood

• L[hypothesis | data] = Pr[data | hypothesis]

• Allows you to solve problems and test

hypotheses that would be extremely difficult

in any other way

2

Example

• What proportion of this class has shoplifted

an item worth more than $10?

• Flip a coin

• Don’t tell ANYONE the result

– If “heads,” answer “heads”

– If “tails,” answer “heads” if you’ve shoplifted

something, “tails” otherwise

3

Pseudoreplication

• The error that occurs when samples are not

independent, but are treated as though they

are

4

Example: “The transylvania effect”

A study of 130,000 calls for police assistancein 1980 found that they were more likely thanchance to occur during a full moon.

5

Example: “The transylvania effect”

A study of 130,000 calls for police assistancein 1980 found that they were more likely thanchance to occur during a full moon.

Problem: There may have been 130,000calls in the data set, but there were only 13full moons in 1980. These data are notindependent.

6

Pseudoreplication

• We are making a false claim about the

number of independent samples in our data

• Very common mistake in biology

• Easiest solution: use the average of all the

pseudoreplicates

7

Very small samples and

assumptions

• Question from the class:“Say there's a test which you desire to carry out which is expensive and therefore you canafford only 2 treatments, each with two replicates. How would we go about analysing anydifference, because are sample would be so small that we wouldn't be able to know if ourdata followed a normal distribution, right? and would these tests be worth carrying out sincethey would have pretty low power?”

• Answer: most scientists will justproceed with the test

• Interpret the results as “if ourassumptions are true (and we have noidea), then…”

8

Very small samples and

assumptions

• Example: does the Earth have more speciesof living things than other planets in thesolar system?

• Data: Earth=10,000,000-100,000,000

• Mercury, Venus, Mars, Jupiter, Saturn,Uranus, Neptune=0 (as far as we know)

9

Hypothesis testing

• Null hypothesis are usually very simple, and

often known beforehand to be false

• You will eventually reject them if you have

a big enough sample size

10

Example

• Study on logging

• Ho: The density of large trees is greater in

unlogged versus logged areas

11

Fewer trees

12

Statistical significance !

Biological importance

• “Statistically significant” means P < 0.05

• But it does not necessarily mean important!

• Likewise, nonsignificant results can be

biologically important

• It’s always useful to estimate a parameter or

effect size, with a confidence interval

13

Examples

• Some studies of thousands of children havefound statistically significant associations ofIQ with birth order

• These differences are on the order of 1-2 IQpoints

• Such differences are not biologicallyimportant for individuals, and can’t explainwhy your sister is smarter than you!

14

Examples

• Large study of hormone replacement

therapy showed no significant benefit of

HRT to post-menopausal women

• Confidence interval for the effect size

suggested that any possible undetected

effect is likely to be extremely small

15

Correlation does not require

causation

16

Correlation and Causation

Hot weather

Ice cream

Violent

crime

17

Data for many countries:

18

Confounding variables

• Variables that mask or distort the

association between measured variables in a

study

• Two approaches:

– Try to measure them all

– Do an experiment

19

Make a Plan

• Develop a clear statement of the question

• List possible outcomes

• Develop an experimental plan

• Keep the design as simple as possible

• Check for common design problems

• Is sample size big enough?

• Discuss with other people!

20

The importance of controls

• Placebo effect - an improvement in a

medical condition that results from the

psychological effects of medical treatment

– Most people get better over time

– Humans like to please others, including their

doctors

– Benefits of doctors beyond drugs

– Direct psychological effects on health

21

The importance of controls

• Well-documented for pain relief

• Up to 40% of people report improvement in

pain when given sugar pills

• Drugs and treatments must be analyzed in

this context

22

Head On = stick of wax

23

“I’m addicted to placebos. I

could quit but it wouldn’t

matter.”

Steven Wright

24

Mistakes

• Two types of mistakes:

– Experimental mistakes

– Statistical mistakes (“Type III error”)

25

Mistakes

• Two types of mistakes:

– Experimental mistakes

– Statistical mistakes (“Type III error”)

26

Experimental Mistakes

27

Mistakes

• Two types of mistakes:

– Experimental mistakes

– Statistical mistakes (“Type III error”)

28

Statistical Mistakes

• 1/3 to 1/2 of scientific papers that use

statistics make at lease minor mistakes

• ~ 8% major mistakes - enough to alter the

conclusions of the paper

• Be careful when reading papers

• Be careful with your own work!

29

Data dredging

• The process of carrying out statistical tests

on your data until you come up with a

statistically significant result.

30

P = 0.05

+ second digit

31 32

Beware multiple comparisons

Probability of a Type I error in N tests = 1-(1-!)N

For 20 tests, the

probability of at least

one Type I error is

~65%.

33

Example - ESP

34

Six or more correct answers: you have ESP!

35

Bonferroni correction

!

"*=

"

number of tests

Anyone in the class have 8 or more correct?

36

Garbage-in, garbage-out

• Small P-values do not rescue a poor

measurement

• Example: IQ test bias

37

Aboriginal-based IQ Test

1.What number comes next in thesequence, one, two, three,__________?

MANY

38

Aboriginal-based IQ Test

2. As wallaby is to animal socigarette is to __________

TREE

39

Aboriginal-based IQ Test

3. Three of the following items maybe classified with salt-watercrocodile. Which are they?

marine turtle brolga

frilled lizard black snake

40

Fraud happens

Original Haeckel's copy

(echidna embryos)

41

Recent Fraud Example

• Woo Sek Hwang, human cloning

• Much of the data suspected to be fabricated

42

Regression to the mean

• When repeated measurements are taken

over time…

• Individuals with extreme values for the first

measurement tend to be nearer to the mean

for the second measurement

43

Regression to the Mean

44

Regression to the Mean

The “sophomore slump”

45

Publication bias

Papers are more likely to be published if P<0.05

This causes a bias in the science reported in the literature.

46

Meta-analysis

• Compiles all known scientific studies

testing the same null hypothesis and

quantitavely combines them to give an

overall estimate of the effect and its

statistical properties

• This is a GREAT honours project…