1. Statistics review Basic concepts: Variability measures Distributions
Hypotheses
Types of error
Common analyses
T-tests
One-way ANOVA
Two-way ANOVA
Regression

2. Variance Ecological rule # 1: Everything varies
…but how much does it vary?

3. Variance
S2= Σ (xi – x )2
n-1 x

4. Variance
S2= Σ (xi – x )2
n-1 x

5. Variance
S2= Σ (xi – x )2
n-1
What is the variance of 4, 3, 3, 2 ?
2/3

6. 1. Standard deviation (s, or SD) = Square root (variance)
Variance variants
1. Standard deviation (s, or SD)
= Square root (variance)
Advantage: units

7. Advantage: indicates reliability
Variance variants
2. Standard error (S.E.)
= s n
Advantage: indicates reliability

8. How to report
We observed 29.7 (+ 5.3) grizzly bears per month (mean + S.E.).
A mean (+ SD)of 29.7 (+ 7.4) grizzly bears were seen per month
+ 1SE or SD
- 1SE or SD

9. Distributions
Normal
Quantitative data
Poisson
Count (frequency) data

10. Normal distribution 67% of data within 1 SD of mean

11. Poisson distribution
mean
Mostly, nothing happens (lots of zeros)

12. Poisson distribution Frequency data
Lots of zero (or minimum value) data
Variance increases with the mean

13. What do you do with Poisson data?
Correct for correlation between mean and variance by log-transforming y (but log (0) is undefined!!)
Use non-parametric statistics (but low power)
Use a “general linear model” specifying a Poisson distribution

14. Hypotheses
Null (Ho): no effect of our experimental treatment, “status quo”
Alternative (Ha): there is an effect

15. Whose null hypothesis?
Conditions very strict for rejecting Ho, whereas accepting Ho is easy (just a matter of not finding grounds to reject it).
A criminal trial?
Environmental protection?
Industrial viability?
Exotic plant species?
WTO?

16. Hypotheses Null (Ho) and alternative (Ha): always mutually exclusive
So if Ha is treatment>control…

17. Types of error Type 1 error Reject Ho Accept Ho Ho true Type 2 error
Ho false

18. Types of error
Usually ensure only 5% chance of type 1 error (ie. Alpha =0.05)
Ability to minimize type 2 error: called power

19. Statistics review Basic concepts: Variability measures Distributions
Hypotheses
Types of error
Common analyses
T-tests
One-way ANOVA
Two-way ANOVA
Regression

20. The t-test Asks: do two samples come from different populations? YESNO
DATA Ho A B

21. The t-test
Depends on whether the difference between samples is much greater than difference within sample. A B
Between >> within… A B

22. The t-test
Depends on whether the difference between samples is much greater than difference within sample. A B
Between < within… A B

23. Difference between means Standard error within each sample
The t-test
T-statistic=
Difference between means
Standard error within each sample
sp2 + sp2
n1 n2

24. How many degrees of freedom?
The t-test
How many degrees of freedom?
(n1-1) + (n2-1)
sp2 + sp2
n1 n2

25. T-tables v
0.10
0.05
0.025 1
3.078
6.314
12.706 2
1.886
2.920
4.303 3
1.638
2.353
3.182 4
1.533
2.132
2.776
infinity
1.282
1.645
1.960
Two samples, each n=3, with t-statistic of 2.50: significantly different?

26. T-tables v
0.10
0.05
0.025 1
3.078
6.314
12.706 2
1.886
2.920
4.303 3
1.638
2.353
3.182 4
1.533
2.132
2.776
infinity
1.282
1.645
1.960
Two samples, each n=3, with t-statistic of 2.50: significantly different? No!

27. If you have two samples with similar n and S. E
If you have two samples with similar n and S.E., why do you know instantly that they are not significantly different if their error bars overlap? v
0.10
0.05
0.025 1
3.078
6.314
12.706 2
1.886
2.920
4.303 3
1.638
2.353
3.182 4
1.533
2.132
2.776
infinity
1.282
1.645
1.960

28. If you have two samples with similar n and S. E
If you have two samples with similar n and S.E., why do you know instantly that they are not significantly different if their error bars overlap? v
0.10
0.05
0.025 1
3.078
6.314
12.706 2
1.886
2.920
4.303 3
1.638
2.353
3.182 4
1.533
2.132
2.776
infinity
1.282
1.645
1.960 }
Careful! Doesn’t work the other way around!!
the difference in means < 2 x S.E., i.e. t-statistic < 2
and, for any df, t must be > 1.96 to be significant!

29. General form of the t-test, can have more than 2 samples
One-way ANOVA
General form of the t-test, can have more than 2 samples
Ho:
All samples the same…
Ha:
At least one sample different

30. General form of the t-test, can have more than 2 samples
One-way ANOVA
General form of the t-test, can have more than 2 samples A B C
DATA A B C Ho Ha A B C A C B

31. One-way ANOVA
Just like t-test, compares differences between samples to differences within samples A B C
Difference between means
Standard error within sample
T-test statistic (t)
MS between groups
MS within group
ANOVA statistic (F)

32. MS= Sum of squares df
Mean squares:
Analogous to
variance

33. Variance:
S2= Σ (xi – x )2
n-1
Sum of squared differences

34. } } ANOVA tables df SS MS F p Treatment (between groups) df (X) SSX
MSX
MSE
Look
up !
Error
(within groups)
df (E)
SSE
Total
df (T)
SST } }

35. Do three species of palms differ in growth rate
Do three species of palms differ in growth rate? We have 5 observations per species. Complete the table! df SS MS F p
Treatment
(between groups) 69
Error
(within groups)
k(n-1)
Total
104

36. Hint: For the total df, remember that we calculate total SS as if there are no groups…MS F p
Treatment
(between groups) 69
Error
(within groups)
k(n-1)
Total
104

37. At alpha = 0.05, F2,12 = 3.89 df SS MS F p
Treatment
(between groups) 2 69
34.5
11.8 ?
Error
(within groups) 12 35
2.92
Total 14
104

38. Two-way ANOVA
Just like one-way ANOVA, except subdivides the treatment SS into:
Treatment 1
Treatment 2
Interaction 1&2

39. Two-way ANOVA
Suppose we wanted to know if moss grows thicker on north or south side of trees, and we look at 10 aspen and 10 fir trees:
Aspect (2 levels, so 1 df)
Tree species (2 levels, so 1 df)
Aspect x species interaction (1df x 1df = 1df)
Error?
k(n-1) = 4 (10-1) = 36

40. v df SS MS F
Aspect 1
SS(Aspect)
MS(Aspect)
MS(As)
MSE
Species
SS(Species)
MS(Species)
MS(Sp)
Aspect x Species
SS(Int)
MS(Int)
Error
(within groups) 36
SSE
Total 39
SST

41. Interactions Combination of treatments gives non-additive effect
North
South
Alder
Fir

42. Interactions Combination of treatments gives non-additive effect
Anything not parallel!
North
South
North
South

43. Careful!
If you log-transformed your variables, the absence of interaction is a multiplicative effect:
log (a) + log (b) = log (ab) y
Log (y)
North
South
North
South

44. Regression
Problem: to draw a straight line through the points that best explains the variance

45. Regression
Problem: to draw a straight line through the points that best explains the variance

46. Regression
Problem: to draw a straight line through the points that best explains the variance

47. Regression Test with F, just like ANOVA:
Variance explained by x-variable / df
Variance still unexplained / df
Variance
explained
(change in
line lengths2)
Variance
unexplained
(residual
line lengths2)

48. Regression Test with F, just like ANOVA:
Variance explained by x-variable / df
Variance still unexplained / df
In regression, each x-variable will normally have 1 df

49. Regression Test with F, just like ANOVA:
Variance explained by x-variable / df
Variance still unexplained / df
Essentially a cost: benefit analysis –
Is the benefit in variance explained worth the cost in using up degrees of freedom?

50. Regression example Total variance for 32 data points is 300 units.
An x-variable is then regressed against the data, accounting for 150 units of variance.
What is the R2?
What is the F ratio?

51. Regression example Total variance for 32 data points is 300 units.
An x-variable is then regressed against the data, accounting for 150 units of variance.
What is the R2?
What is the F ratio?
R2 = 150/300 = 0.5
F 1,30 = 150/1 = 15
300/30
Why is df error = 30?

52. ANCOVA In regression, x-variables can be continuous or categorical.
Eg. To convert a treatment (size) with two levels (small, large) into a regression variable, we could code small=0, large= 1.
Y= size*B + constant
Mean value for small
Difference between large and small
Test significance of “size” in regression

53. ANCOVA
In an Analysis of Covariance, we look at the effect of a treatment (categorical) while accounting for a covariate (continuous)
Fertilized N+P
Fertilized N
Growth rate (g/day)
Plant height (cm)

54. ANCOVA Fit full model (categorical treatment, covariate, interaction)
Test for interaction (if significant- stop!)
Test for differences in intercept
Fertilized N+P
Fertilized N
Growth rate (g/day)
No interaction
Intercepts differ
Plant height (cm)