1. Choosing and using statistics to test ecological hypotheses
Botany 332 Lab Tutorial
Department of Biological Sciences
University of Alberta
November 2004

2. Patterns in space or time
OBSERVATIONS
Patterns in space or time
MODELS
Explanations or theories
Retain Ho (Null Hypothesis)
Refute hypothesis and model
Reject Ho (Null Hypothesis)
Support hypothesis and model
HYPOTHESIS Predictions based on model
NULL HYPOTHESIS
Logical opposite to hypothesis
EXPERIMENT
Critical test of null hypothesis
INTERPRETATION
Underwood (1997)

3. Ecological experiments
OBSERVE things.
Come up with MODELS (explanations or theories) to explain your observations.
Based on your model, come up with a testable HYPOTHESIS (and a NULL hypothesis).
Design an EXPERIMENT to test your null hypothesis statistically.
Conduct the experiment and collect DATA.
Use STATISTICS with your data to TEST the null hypothesis.
INTERPRET your results. Did you accept or reject the null hypothesis?
Repeat!

4. Testing (null) hypotheses statistically
Recall we can’t prove our hypothesis, so we try to disprove a null hypothesis instead!
Null hypothesis = opposite of our actual hypothesis
H0 = Null Hypothesis
HA = Alternative hypothesis

5. Testing (null) hypotheses statistically
We formally test hypotheses using statistics
Which statistical test to use? Depends on your experimental design, data and your hypotheses
It’s important to understand the basics of statistical hypothesis testing

6. Testing (null) hypotheses statistically
Based on assumptions about the data, statistics tell us the probability that the null hypothesis is true (P-value).
If P is small enough, we can reject the null hypothesis (result is “statistically significant”).
What’s “small enough”?
P < 0.05
Reject null hypothesis (accept our hypothesis)
P > 0.05
Accept null hypothesis (reject our hypothesis)

7. Testing (null) hypotheses statistically
Many statistical methods also tell us the effect size or proportion of variation in the independent variable explained by the dependent variable.
e.g. Regression and correlation
P-values
H0 = No relationship between variables
HA = Relationship between variables
R2 (variation explained)
Can have significant P-values but very small R2

8. Choosing and using statistics
Determine what kinds of data you have
Describe your data
Choose an appropriate statistical test
Perform the test
Report and interpret the results

9. What kinds of data do you have?
Categorical
Fertilizer addition, species identity
Continuous and discrete
Biomass, height, number of bites
Independent and Dependent variables

10. Describe your data Measures of central tendency Measures of dispersion
Mean, median
Measures of dispersion
Variance, standard deviation, standard error, range, quartiles

11. Descriptive Statistics – Visual Aids
Boxplots
- median, upper and lower quartiles, whiskers (fences), outliers
Histograms
- separate, stackbar, or paired
Error Bar Plots

12. Describe your data Normal vs. non-normal distributions
histograms, Q-Q plots, K-S test (significant means non-normal)

13. Data transformation If your data are non-normal
Use non-parametric statistics
Transform your data
square-root transform
log transform
Log-normal Distribution
Many data sets like grain-size of sediments, geochemical concentrations, etc. have a very skewed and long-tailed distribution. In general, such distributions arise when the observed quantities have errors that depend on products rather than sums. It therefore follows that the logarithm of the data may be normally distributed. Hence, taking the logarithm of your data may make the transformed distribution look normal. If this is the case, you can apply standard statistical techniques applicable to normal distributions on your log distribution and convert the results (e.g., mean, standard deviation) back to get the proper units.

14. Choose your statistical test
Choose statistical tests based on your hypothesis, experimental design and the data you have collected
Parametric tests assume data are normal, non-parametric tests do not
Many textbooks have recipes or flowcharts for choosing statistics
Check with your TA’s

15. Common statistical tests
Chi-squared test
t-test (Mann-Whitney U test)
One-way ANOVA (Kruskal-Wallis test)
Two-way ANOVA
ANCOVA
ANOVA with covariate
Correlation and regression

16. Chi-squared test For analysis of tables of counts or frequencies
Good with categorical variables
Non-parametric
# plants
Germinated
Not Germinated
Outcrossed 14 10
Inbred 6

17. t-test
For analysis of categorical independent variable (2 categories) and a continuous dependent variable
Samples may be paired (measurements on same individual) or independent (measurements on two sets of individuals)
Assumes data are parametric
(non-parametric – Mann-Whitney U)

18. ANOVA
Analysis of Variance examines variation within and between groups
For analysis of categorical independent variables (2 or more categories) and a continuous dependent variable
Assumes data are parametric
(non-parametric – Kruskal-Wallis)

19. ANOVA One-way ANOVA Two-way ANOVA
Single independent variable
Main effect
Two-way ANOVA
Two independent variables
Main effects and interaction terms
Significant result means at least one group differed from another
Use post-hoc tests to test for differences among individual treatments

20. ANCOVA Analysis of Covariance
For analysis of categorical independent variables (2 or more categories), a continuous dependent variable, and a covariate
Effects of covariate removed before testing for effect of independent variable(s)

21. Correlation and regression
Tests for relationships between two (or more) continuous variables
Important to consider both significance (P-value) and effect size (R2)

22. Report statistical results
What’s important?
Test used and assumptions tested
Test statistic (t, F, R2, χ2, etc.)
Significance (P-value)
Sample size / degrees of freedom
How to report results?
Text
Figures
Tables

23. Number of flowers per plant
ANOVA, F = 1.8, df = 1,83
P = 0.17
Number of flowers per plant

24. Interpret your results
Remember to relate results/tests to your original hypotheses
Correlation ≠ causation
(P > 0.05) ≠ bad
Recognize trends even when not statistically significant
Talk to your TAs if you have any questions

25. SPSS walkthrough Data entry and transformation Descriptive statistics
Creating figures
Analyses
Chi-square (inbreeding data)
t-test / ANOVA (inbreeding data)
ANCOVA (tomato data)
Correlation and regression (inbreeding data)