Transcript Statistics review - University of British Columbia

Statistics review 1
Basic concepts:
• Variability
measures
• Distributions
• Hypotheses
• Types of error
Common analyses
• T-tests
• One-way ANOVA
• Two-way ANOVA
• Randomized
block
Variance
Ecological rule # 1: Everything varies
…but how much does it vary?
Variance
s2 
2
(x

x
)
 i
n 1
x

3cm
Sum-of-square
cake

Urchin size
15cm
x
3cm
Urchin size
15cm
Sum-of-square
cake

x
3cm
Urchin size
15cm
Variance
s
2
(x


i
 x)
2
n 1
What is the mean and variance of 4, 3, 3, 2 ?

Mean = 3, Variance = 0.67
What are the units?
Variance variants
1. Standard deviation (s, or SD)
= Square root (variance)
Advantage: units
Variance variants
2. Standard error (S.E.)
s
s.e. 
n
Advantage: indicates precision

How to report
Tourist boats observed 29.7 (+ 5.3)
shark attacks on seals (mean + S.E.)
A mean (+ SD) of 29.7 (+ 7.4) shark
attacks were seen per month
+ 1SE or SD
- 1SE or SD
Distributions
Normal
• Quantitative data
Poisson
• Count
(frequency) data
Normal distribution
16
67% of data
within 1 SD of
mean
14
12
10
8
6
4
2
0
mean
95% of data
within 2 SD of
mean
Poisson distribution
18
16
14
12
10
8
6
4
2
0
mean
Mostly, nothing happens (lots of zeros)
Poisson distribution
• Frequency data
• Lots of zero (or minimum value)
data
• Variance increases with the mean
What do you do with
Poisson data?
1. Correct for correlation between mean and
variance by log-transforming y (but log (0)
is undefined!!)
2. Use non-parametric statistics (but low
power)
3. Use a “generalized linear model”
specifying a Poisson distribution
Hypotheses
• Null (Ho): no effect of our
experimental treatment, “status quo”
• Alternative (Ha): there is an effect
Whose null hypothesis?
Conditions very strict for rejecting Ho,
whereas accepting Ho is easy (just a
matter of not finding grounds to
reject it).
Preliminary study?
A criminal trial?
Chance of a disease epidemic?
Hypotheses
Null (Ho) and alternative (Ha):
always mutually exclusive
So if Ha is treatment>control…
Types of error
Reject Ho
Ho true
Ho false
Accept Ho
Type 1 error
Type 2 error
Types of error
• Usually ensure only 5% chance of
type 1 error (ie. Alpha =0.05)
• Ability to minimize type 2 error:
called power