1. Correct statistics in ecological research
Han Y. H. Chen
Lakehead University
Canada

2. Selecting the appropriate inferential statistics
Generalized Linear Model (GLM) as a dominant method in statistics

3. lm includes all except path analysis
install.packages(glm2)
library("glm2")
install.packages(lme4)
library(lme4)
install.packages("ggplot2")
library(ggplot2)

4. Read file into R mydata <-read.csv(file.choose()) attach(mydata)
summary (mydata)
edit (mydata)

5. What are the assumptions for general linear model (lm)
Both for regression and ANOVA
Independence of observations (can not be met in almost all situations!)
Normality –the distributions of the residuals are normal
Equality (homogeneity) of variances – the variance of data in groups (or along the x gradients) is the same
Unit-treatment additivity –cannot be directly falsified
Additional for regression
Linearity
Why this assumption is not applicable in ANOVA?

6. What if some assumptions can not be met?
Use Generalized Linear Models (GLM)
If normality and homogenous variance can not be met
Use Generalized Non-linear Models (GNM)
If linearity can not be met

7. 6 Steps to verify assumptions
Correct specification of distribution
Family
Variance
Link
Normal
Gaussian
identity
Binomial
logit, probit or cloglog
Poisson
log, identity or sqrt
Gamma
inverse, identity or log
inverse.gaussian
1/mu^2
Quasi
user-defined
What is the distribution for species richness?

8. Types of distributions
Raw data is almost never as well behaved as we would like it to be
Generalized extreme value distribution

9. A good starting place to learn R graphics

10. Normality

11. Practical example to check distributions
#Verfiy normal distribution
hist(AGBw, 50)
qqnorm(AGBw)
shapiro.test(AGBw)
#is log-transformation makes it better?
logAGB<-log(AGBw+15)
hist(logAGB, 50)
qqnorm(mydata$logAGB)
shapiro.test(mydata$logAGB)
#root square transformation?
rAGB<-(AGBw+15)^0.5

12. 2. Correct form for the explanatory variable x?
Categorical or numerical
Linear or nonlinear?

13. Non-linearity

14. Check non-linearity and homogenous variances
a <- ggplot(mydata, aes(y=AGBw, x=SA)) +ylab(expression(paste(Aboveground~biomass~(Mg~ha^-1~year^-1))))+xlab(expression(paste(ln~(Forest~age~(years)))))
a + stat_smooth(method = "lm", size = 1.5, alpha = 0.9, colour=2)+geom_point(shape=1, size=1)+ theme_bw()+ theme(panel.grid.minor=element_blank(), panel.grid.major=element_blank())+ stat_smooth(size = 1, alpha = 0.5)
c = ggplot(mydata, aes(y=AGBw, x=log(SA))) +ylab(expression(paste(Aboveground~biomass~(Mg~ha^-1~year^-1))))+xlab(expression(paste(ln~(Forest~age~(years)))))
c + stat_smooth(method = "lm", size = 1.5, alpha = 0.9, colour=2)+geom_point(shape=1, size=1)+ theme_bw()+ theme(panel.grid.minor=element_blank(), panel.grid.major=element_blank())+ stat_smooth(size = 1, alpha = 0.5)
Transform x to make relationships linear, start with polynomials
Still does not work, go with gnm

15. 3. Statistical independence of the n observations
All default models assume “completely randomized experimental or sampling design”, but reality is that this assumption can hardly be met in ecological data.
This is why we learn Fundamentals of Experimental Design
Is there a hierarchical structure in your data?
Example 1
Influences of stand type (S) and soil layer (L) on nitrogen concentration?
A typical example of nested design: Yijk = Si + Lj(i) + ek(ij)
What if you have repeated measures over the growing season?
Example
Random plot effect in repeated measures
m1<-lmer(AGBw~log(SA)+(1|Plot), data=mydata)
m11<-lmer(AGBw~log(SA)+(log(SA)|Plot), data=mydata)

16. 3. Statistical independence of the n observations –clustering
Is your sampling/experiment pseudo-replicated?
Is there spatial auto-correlation?
Suitable tests are Mantel and Moran I
Note: Our Spruce Forest plots are spatially auto-correlated, subject to criticism of pseudo-replication! But this is dilemma for studies of natural experiments
Nevertheless, researchers shall be ready to defend this

17. 4. Correct specification of the variance function v
lm –Ordinary Least Square
GLM – Maximum likelihoods
If you have properly specified the distribution, this will take care of homogenous variance assumption!

18. 5. Overdispersion and underdispersion
Overdispersion: The presence of greater variability (statistical dispersion) in a data set than would be expected based on a given simple statistical model
Causes: model is too simple
Underdispersion means that there was less variation in the data than predicted
Model is too complex; data is overfitted
Practical solution—AIC or BIC to select the “best” model

19. 6. Lack of undue influence of individual observations on the fit
Deletion diagnostics: the influence of individual observations on a GLM fit is measured by the change in estimated coefficient/parameter (mean, intercept, or slope)
It can be done by following this
You can model how individual observations affect your model
m2<-lm(AGBw~SA*ForestType, data=mydata)
rstandard(m2, infl = lm.influence(m2, do.coef = FALSE), sd = sqrt(deviance(m2)/df.residual(m2)))

20. Take home message
Simple statistical analysis is the best if it meets all assumption, but ecological reality is more complex
Simple questions have been studied for so long, no niche for novelty/discovery
Statistics without verifying assumptions are not reliable and do not serve the purpose