Statistics

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Statisticsrevisited
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Standard statistics revisited
Bolker
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Standard statistics revisitedSimple Variance
Structures
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Standard statistics revisited
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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.

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Linear regression
Standard R code gtlm.reglt-lm(YX) gtsummary(lm) gta
nova(lm.reg)
Likelihood R code gtlmfunlt-function(a, b,
sigma) Y.predlt-abx -sum(dnorm(Y,
meanY.pred, sdsigma, logTRUE))
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Analysis of variance (ANOVA)
Standard R code gtlm.onewayaovlt-lm(Yf1) gtsummary
(lm.aov) gtanova(lm.aov) will give you an ANOVA
table
Likelihood R code gtaovfunlt-function(a11, a12,
sigma) Y.predlt-c(a11,a12) -sum(dnorm(DBH,
meanY.pred, sdsigma, logTRUE))
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Analysis of covariance (ANCOVA)
Standard R code gtlm.anclt-lm(YfX) gtsummary(lm.a
nc) gtstr(summary(lm.anc))
Likelihood R code gtancfunlt-function(a11, a12,
slope1, slope2, sigma) Y.predlt-c(a11,a12)f
c(slope1, slope2)fX -sum(dnorm(Y,
meanY.pred, sdsigma, logTRUE))
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Standard statistics revisited
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Nonlinearlity Non-linear least squares
Uses numerical methods similar to those use in
likelihood
Standard R code gtnls(yaxb,
startlist(a1,b1) gtsummary(nls) gtstr(summary(lns
))
Likelihood R code gtnlsfunlt-function(a, b,
sigma) Y.predlt-axb -sum(dnorm(Y,
meanY.pred, sdsigma, logTRUE))
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Standard statistics revisited
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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!

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GML Poisson regression
Standard R code gtglm.poislt-glm(YX,
familypoisson) gtsummary(gml.pois)
Likelihood R code gtpoisregfunfunction(a,b) Y.
predlt-exp(abX) -sum(dpois(Y, lambdaY.pred,
logTRUE))
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GML Logistic regression
Standard R code gtglm2lt-glm(yx,
familybinomial) gtsummary(gml2)
Likelihood R code gtlogregfunfunction(a,b,N) p
.predlt-exp(a bX))/(1exp(a
bX)) -sum(dbinom(Y, sizeN, probp.pred,
logTRUE))
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Standard statistics revisited
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Generalized (non)linear least-squares
modelsVariance changes with a covariate or
among groups
Standard R code gtglslt-gls(y1,weightsvarIdent(f
orm1f) gtsummary(gls)
Likelihood R code gtvardifffunfunction(a,
sd1,sd2) sdvallt-c(sd1,sd2)f -sum(dbinom(Y,
meana, sdsdval, logTRUE)
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Standard statistics revisited Complex Variance
Structures
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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
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Complex error structures
(x
x
Increasing variance
General case
Independent
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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

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Complex error structures An exampleSpatially-cor
related errors
R code gtrho0.5 gtmmatrix(nrow5,
ncol5) gtmlt-rho(abs(row(m)-col(m))
OR gtmabs(row(m)-col(m))1rho mvliklt-functi
on(a,b, rho) pred.radabdbh nlength(radius
) mdiag(n) generates diag matrix of n rows, n
columns mabs(row(m)-col(m))1rho
-dmvnorm(radius, pred.rad, Sigmam, logTRUE)
mle(mvlik, startlist(a0.5, b3,rho0.5),
method"L-BFGS-B", lower0.001)
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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.
• GLMMs are generalized linear models with random
  effects

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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
• ..

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Bolker