Experimental design
Based on Chapter 2 of D. Heath (1995). An Introduction to Experimental Design and
Statistics for Biology. CRC Press.
Four critical features of experimental designHurlbert 1984
• Controls
• Randomization
• Replication
• Interspersion
Possible explanations?Research hypothesis
(or hypotheses)
The design of a experiment
• Factor: humidity
• Variable: direction
Removing other possible effects
• Dealing with bias
Other design issues
• Number of woodlice
• Which woodlice
• They must be representative of the population of reference
Confounding factors
Independent observations
Analysis
• Null hypothesis:
• Alternative hypothesis:
Probability of damp turn = 0.5
Probability of damp turn = 0.5
Expected frequencies for four trails
dry
dry damp
dry damp
dry damp
dry damp
dry damp
dry damp
dry damp
damp
dry damp
dry damp
dry damp
dry damp
dry damp
dry damp
dry damp
Example
• Damp*Damp*Damp*Damp• If order does not matter there is only one way to
obtain four damp turns and the combined probability (under the assumption of independence) is 0.5*0.5*0.5*0.5= 0.0625
• Calculate the probability of the other possible outcomes under the null hypothesis
Exercise • There are four ways to obtain three damp turns:
Damp*Damp*Damp*DryDamp*Damp*Dry*DampDamp*Dry*Damp*DampDry*Damp*Damp*Damp
• and the combined probability (under the assumption of independence) is 0.5*0.5*0.5*0.5= 0.0625 four times = 0.25
• Calculate the probability of the other possible outcomes under the null hypothesis
Binomial distribution (4 trials)Under the null hypothesis
0.000.050.100.150.200.250.300.350.40
0 1 2 3 4
Number of damp turns
Expected frequency
Distribution under the null hypothesis(17 trials)
0.00
0.05
0.10
0.15
0.20
0:17 2:15 4:13 6:11 8:09 10:07 12:05 14:03 16:01
Number of damp:dry turns
Expected frequency
What do you conclude if we observed 14 damp turns out of 17 ?
Binomial distribution
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0 2 4 6 8 10 12 14 16
Number of damp turns
Rejection region Rejection region
0.0000+0.0001+0.0010+0.0052+0.0182=
2.45%
0.0182+0.0052+0.0010+0.0001+0.0000=
2.45%
likelyunlikely unlikely
Why we start with the null hypothesis?
The main points
• Use a mathematical model to produce a sampling distribution of all possible values of the test statistic assuming that the null hypothesis is true
• Find the probability associated with a a particular value occurring in a particular experiment
• Use the probability to make a decision about whether a particular result is likely or unlikely
The Binomial DistributionOverview
• However, if order is not important, then
where is the number of ways to obtain X successes
in n trials, and n! = n (n – 1) (n – 2) … 2 1
n!
X!(n – X)! ppXX q qn – n –
XX
PP((XX) =) =
n!
X!(n – X)!