1. THE SCIENTIFIC METHOD: It’s the method you use to study a question scientifically.

2. The five stages of the scientific method: 1)Identification of a the problem (a question): what is it that we wish to know more about? 2)Form a hypothesis (often, but not necessarily a Null- hypothesis): preferably a deduction from known theory a tentative answer to the question, a testable assertion 3)Design an experiment or observation that tests the hypothesis: the experiment or observation is dictated by the hypothesis 4)Collect and analyze the data: formally compare observation (results) with expectation under the hypothesis 5)Derive conclusions and speculate: reject the hypothesis or not? What do the results mean in terms of the original question? Is there more to do to completely address the question? How do results compare to results of similar experimentation or observation?

3. Experimental Design Considerations: Observe or manipulate? What to measure? Measure everything or a subset? How to chose the subset? How many replicates? What type of data?

4. Types of data: Counts (e.g. number of individuals, number of events) Continuous variables (e.g. biomass, height, length of time) Sets of dependent data (e.g. the height and biomass of individuals) The data type determines what analysis one must use!

5. Sampling: Must be random! with respect to the factor of interest Must be representative! of all other factors that might be important but are not explicitly considered This is easier said than done, it helps to formulate sampling rules ahead of time. Non-random, non-representative sampling creates bias, i.e. the sampling mean can be expected to differ from the true mean.

6. Uncertainty, variability and error Sources of variability: observation error Interference by factors other than the one considered in the experiment or observation Uncertainty are mathematically expressed in measures of variation around a mean: Mean: Variance: Standard Deviation:

7. Uncertainty, variability and error The greater the uncertainty surrounding a measurement, the more samples must be taken to detect a pattern. TRUE MEAN:3.05.03.05.03.05.0 2.9894.9592.9265.0236.4052.754 3.0345.0273.2215.232-1.1352.314 2.9874.982.945.497-0.8354.749 3.0174.9832.5564.7492.8119.216 2.9954.9832.5314.9565.1662.878 3.0285.043.3595.387-1.2030.562 3.0444.9513.195.2972.0552.02 2.9794.9683.0854.980.9978.293 3.04652.5714.5497.5990.379 2.9845.0442.6624.94.4593.984 Mean3.014.9932.9045.0572.6323.715

8. The Normal Distribution Continuous data often follow the normal distribution. T-tests and ANOVAs can be applied, if we can assume that the data are approximally normally distributed. mean, variance

9. Uncertainty, variability and error The greater the uncertainty surrounding a measurement, the more samples must be taken to detect a pattern. TRUE MEAN:3.05.03.05.03.05.0 2.9894.9592.9265.0236.4052.754 3.0345.0273.2215.232-1.1352.314 2.9874.982.945.497-0.8354.749 3.0174.9832.5564.7492.8119.216 2.9954.9832.5314.9565.1662.878 3.0285.043.3595.387-1.2030.562 3.0444.9513.195.2972.0552.02 2.9794.9683.0854.980.9978.293 3.04652.5714.5497.5990.379 2.9845.0442.6624.94.4593.984 Mean3.014.9932.9045.0572.6323.715 Variance 0.000690.00110.09480.087310.2918.8768

10. Uncertainty, variability and error The t-test can be used for simple comparisons of two sample means, or of one sample mean with a fixed number: 353535 2.9894.9592.9265.0236.4052.754 3.0345.0273.2215.232-1.1352.314 2.9874.982.945.497-0.8354.749 3.0174.9832.5564.7492.8119.216 2.9954.9832.5314.9565.1662.878 3.0285.043.3595.387-1.2030.562 3.0444.9513.195.2972.0552.02 2.9794.9683.0854.980.9978.293 3.04652.5714.5497.5990.379 2.9845.0442.6624.94.4593.984 Mean3.014.9932.9045.0572.6323.715 Variance 0.000690.00110.09480.087310.2918.8768 t-value147.315.950.782 degrees of freedom18 tcrit1.734 Degrees of freedom: n-2 Significant difference Significant difference No difference

11. Statistical methods do not preclude wrong conclusions Type I error: a test leads to the rejection of a true hypothesis (also: false positive). Type II error: a test fails to reject a false hypothesis (also: false negative).

12. Types of Analyses: - Counts - Counts must be analyzed in contingency tables (non- parametric). Example: comparing individual vigilance between two groups of different size: 100 animals in groups of 10 100 animals in groups of 20 animals with heads up3520 animals with heads down6580 total100

13. Types of Analyses: - Counts - Counts must be analyzed in contingency tables (non- parametric). Example: comparing individual vigilance between two groups of different size: 100 animals in groups of 10 100 animals in groups of 20Jointly expected for both groups, if vigilance same animals with heads up35205527.5 animals with heads down658014572.5 total100 200100

14. Types of Analyses: - Counts - The G-statistic for counts: 100 animals in groups of 10 100 animals in groups of 20 expected if same obs*ln(obs/ exp) animals with heads up352027.5 35*ln(35/27.5) = 8.44 animals with heads down658072.5 65*ln(65/72.5) = -7.10 total100 1.34 G = 2*1.34 = 2.68 Degrees of freedom = no of observation categories (rows) -1 =1 Gcrit (df=1,  =0.05) = 3.841 We fail to reject the Null Hypothesis that individual vigilance is different in groups of 10 and 20!

15. Types of Analyses: - Dependent data - Dependent data can be analyzed by regression. Example: the height and basal circumference of individual trees in a dense and open forest. thinned forest dense forest tree Circum- ferenceheight Circum- ferenceheight 10.662.28 4.0223.74 22.006.54 0.834.51 33.859.53 1.7710.28 42.666.43 0.271.86 54.7514.10 6.4027.49 66.3815.74 2.7914.10 75.5812.65 6.3030.25 88.1223.77 0.252.06 90.692.40 9.5146.91 109.1622.37 3.6117.15

16. Types of Analyses: - Dependent data - Is the slope different? s 1 = 4.68, var 1 = 0.21 s 2 = 2.51, var 2 = 0.18 Degrees of freedom: n-4 t=2.17/0.147=14.76 t crit (df= 16,  = 0.05)=1.764 The slopes are significantly different: trees in the dense forest are taller for their height, compared to trees in the open forest!

17. Summary: 1.An ecological experiment is motivated by an observation evaluated in the light of ecological theory or existing evidence. 2.The experiment or systematic observation is based on a clearly stated hypothesis that is derived from theory and anticipates the experimental design. It must be testable. It must relate to ecological principles. 3.Experiments have to be designed with careful consideration of (unbiased) random sampling and adequate sample size. The larger the random variation or smaller the expected differences between groups, the larger the sample size should be. 4.The statistical analysis chosen must fit the type of data collected. Counts should never be analyzed by t-test or ANOVA! The selection of the critical value depends on the correct determination of the degrees of freedom.