Summary Ecological Methods I

Univariate analysis (exam 1)
Group differences
Introduction
Ecological problems

 Lot of sites/plots
 Many variables
 Lot of natural variation (noise)

 solution: statistics



Sample size (N)

 Larger N  better estimate mean & s, value stabilizes
 Larger N  smaller CI



Don’t learn standard deviation (SD or s) formula

 Standard deviation = s = σ
 Variation = s2



SPSS: levels of measurements

 Nominal => categories
- Habitat, sex, colour
- Prey species killed
 Ordinal => order
- Abundant, frequent, rare
- Droppings
 Scale => ratio scale
- Absolute 0 (weight, length, intake)
- Subtract, add, multiplication
- Fish size & body mass



Distribution type

 Normal => symmetric & continuous
 Lognormal => skewed & continuous
- Exponential growth (biomass)
 Poisson & negative binomial => skewed, non continuous (discrete)
- Counts (quadrants)
 Binomial => 2 outcomes
- Dead/alive, present/absent

,T-tests
 Independent samples (df = N – 2)
1. Unequal variances
2. Equal variances
 Dependent samples (df = N – 1)
3. Paired sample



Hypothesis => testable explanation of observation

 Clear direction: larger/smaller, increase/decrease
 Based on observations + what you already know to be true
 If … then …
 Independent variable (x) = cause
Dependent var (y) = effect
 Example:
- Ho => no difference
- H1 => mean body weight larger in area B than area A
 Standard error of the mean (SE) = standard deviation of X̄ = s/√ N



T-test unequal variances (1)

 Difference between 2 means/SE of that difference




- If X̄ 1- X̄ 2 = 0  t = 0
- If X X̄ 1- X̄ 2 = large or S2/N decreases  t increases
 Threshold: p = 0.05!!
- p ≤ 0.05  reject Ho
- p ≥ 0.05  do not reject Ho



T-test for equal variances (2)

 Difference between 2 means/SE of that difference




 Levene’s test
- Ho: equal variances (p ≥ 0.05)

,Degrees of freedom (df)

 Df < N
- Depends on test & data
 Significant outcome if: calculated t > critical t from table



Reporting statistics

1. Used test
2. Statistical parameter value (e.g. t-value)
3. Df/N
4. P-value



One-sided vs two-sided

 One-tailed test sooner significant
- But: only use if you know that 1 group has higher mean (e.g. more mortality in polluted
area)
 Ecology: lot of noise & uncertainty  often two-tailed
- Doubt  always use two-tailed
- Sometimes one-tailed (F-test)



Type 1 & type 2 error

Type 1 => reject Ho while true

4. Conclude someone is pregnant who is not
 Type 2 => do not reject Ho while false



T-test for paired data (3)

 Example:
- Same animal measured 2x (at two moments in
time)
 z̄ = average value of difference



Power & sample size

 Power => likelihood of test reaching correct conclusion
- Smaller type II error  larger power than visually inspected
 Larger power  implications for:
- Experimental design
- Sample size
- Test results
 Minimum sample size calculation

, Non-parametric & transformation
 Parametric test: assumes data distribution is characterized by mean, SD
- Only use (transformed) normally distributed data (black line)
- Higher power than non-parametric test
 Not normally distributed (red line)  transform data OR use non-parametric test
- Preferably transform, only use non-parametric test as last option
 Test for normality
- Histogram
- Statistical test  Shapiro Wilk test



Data transformation

 Check:
- Variance (s2), mean (X̄ ) & histogram
 Rules of thumb (no hard guidelines)
- S2 > X̄  log(variable) or ln(var)  e.g. growth, biomass
- S2  X̄  √ var  e.g. area, size
- Highly skewed  √ √ var
- Binomial  ln(p/(1-p))  e.g. presence/absence (binomial data)
 0 values
- Use log(x+1) or ln(x+1)
- √ var + 0.5
 Transform whole var  check again!
- Still not normal: use non-parametric test (can also be used for normal data)
- Parametric test has higher power



Non-parametric tests

 Mann Whitney U test
 Wilcoxon matched pairs test
 Kruskal Wallis test



Mann Whitney U test (1)

 Not normal & unmatched pairs
 Ho: 2 medians are equal
 U value for 2 groups
- Compare smallest U value with table
- U < critical value  reject Ho
- Tied data? (same value & rank)  use asymptote p



Wilcoxon matched pairs test (2)