1. Statistics……revisited

2. Standard statistics revisited
What we have learned a method to fit parameters to almost anything. Most models you will encounter in traditional stats are specific cases of what you have learned in this class.
Bolker

3. Standard statistics revisited: Simple Variance Structures

4. Standard statistics revisited

5. General linear models
Predictions are a linear function of a set of parameters.
Includes:
Linear models
ANOVA
ANCOVA
Assumptions:
Normally distributed, independent errors
Constant variance
Not to be confused with generalized linear models!
Distinction between factors and covariates.

6. Linear regression Standard R code: >lm.reg<-lm(Y~X)
>summary(lm)
>anova(lm.reg)
Likelihood R code:
>lmfun<-function(a, b, sigma) {
Y.pred<-a+b*x
-sum(dnorm(Y, mean=Y.pred, sd=sigma, log=TRUE)) }

7. Analysis of variance (ANOVA)
Standard R code:
>lm.onewayaov<-lm(Y~f1)
>summary(lm.aov)
>anova(lm.aov) # will give you an ANOVA table
Likelihood R code:
>aovfun<-function(a11, a12, sigma) {
Y.pred<-c(a11,a12)
-sum(dnorm(DBH, mean=Y.pred, sd=sigma, log=TRUE)) }

8. Analysis of covariance (ANCOVA)
Standard R code:
>lm.anc<-lm(Y~f*X)
>summary(lm.anc)
>str(summary(lm.anc))
Likelihood R code:
>ancfun<-function(a11, a12, slope1, slope2, sigma) {
Y.pred<-c(a11,a12)[f] + c(slope1, slope2)[f]*X
-sum(dnorm(Y, mean=Y.pred, sd=sigma, log=TRUE)) }

9. Standard statistics revisited

10. Nonlinearlity: Non-linear least squares
Uses numerical methods similar to those use in likelihood
Standard R code:
>nls(y~a*x^b, start=list(a=1,b=1)
>summary(nls)
>str(summary(lns))
Likelihood R code:
>nlsfun<-function(a, b, sigma) {
Y.pred<-a*x^b
-sum(dnorm(Y, mean=Y.pred, sd=sigma, log=TRUE)) }

11. Standard statistics revisited

12. Generalized linear models
Assumptions:
Non-normal distributed errors ( but still independent and only certain kinds of non-normality)
Non-linear relationships are allowed but only if they have a linearizing transformation (the link function).
Linearizing transformations:
Non-normal distributed errors ( but still independent and only certain kinds of non-normality). These include the exponential family and are typically used with a specific linearizing function.
Poisson: loglink
Binomial: logit transfomation
Gamma: inverse Gaussian
Fit by iteratively reweighed least square methods: estimate variance associated with each point for each estimate of parameter(s).
Not to be confused with general linear models!

13. GML: Poisson regression
Standard R code:
>glm.pois<-glm(Y~X, family=poisson)
>summary(gml.pois)
Likelihood R code:
>poisregfun=function(a,b)
{Y.pred<-exp(a+b*X)
-sum(dpois(Y, lambda=Y.pred, log=TRUE))}

14. GML: Logistic regression
Standard R code:
>glm2<-glm(y~x, family=binomial)
>summary(gml2)
Likelihood R code:
>logregfun=function(a,b,N)
{p.pred<-exp(a + b*X))/(1+exp(a + b*X))
-sum(dbinom(Y, size=N, prob=p.pred, log=TRUE))}

15. Standard statistics revisited

16. Generalized (non)linear least-squares models: Variance changes with a covariate or among groups
Standard R code:
>gls<-gls(y~1,weights=varIdent(form=~1|f)
>summary(gls)
Likelihood R code:
>vardifffun=function(a, sd1,sd2)
{sdval<-c(sd1,sd2)[f]
-sum(dbinom(Y, mean=a, sd=sdval, log=TRUE)}

17. Standard statistics revisited: Complex Variance Structures

18. Complex error structures
Error structures are not independent
Complex likelihood functions
Includes:
Time series analysis
Spatial correlation
Repeated measures analysis
Variance-covariance matrix x x
Vector of
means (pred)
Vector of
data

19. Complex error structures
Independent
Increasing variance
General case

20. Complex error structures
Variance/covariance matrix is symmetric so we need to specify at most n(n-1)/2 parameters.
V/C matrix must also be positive definite (logical), this translates to having a positive eigenvalue or positive diagonal values.
Select elements of matrix that define the error structure and ensure positive definite.
In this example, correlation drops off with the number ofd steps between sites.c

21. Complex error structures: An example Spatially-correlated errors
R code:
>rho=0.5
>m=matrix(nrow=5, ncol=5)
>m<-rho^(abs(row(m)-col(m)) #OR#
>m[abs(row(m)-col(m))==1]=rho
mvlik<-function(a,b, rho) {
pred.rad=a+b*dbh
n=length(radius)
m=diag(n) #generates diag matrix of n rows, n columns
m[abs(row(m)-col(m))==1]=rho
-dmvnorm(radius, pred.rad, Sigma=m, log=TRUE) }
mle(mvlik, start=list(a=0.5, b=3,rho=0.5), method="L-BFGS-B", lower=0.001)

22. Mixed models & Generalized linear mixed models (GLMM)
Samples within a group (block, site) are equally correlated with each other.
Fixed effects: effects of covariates
Random effects: block, site etc.
GLMM’s are generalized linear models with random effects
First, independent case.
Second, sums of squares weighted by their variance
General case

23. Complex variance structures
So how do you incorporate all potential sources of variance?
Block effects
Individual effects (repeated measures includes both individual and temporal correlation)
Measurement vs. process error
…..

24. Bolker