Nonlinear Models. What do you do when you don t have a line? What do you do when you don t have a line? A Quadratic Adventure

Transcription

1 What do you do when you don t have a line? Nonlinear Models Spores 0e+00 2e+06 4e+06 6e+06 8e longevity What do you do when you don t have a line? A Quadratic Adventure 1. If nonlinear terms are additive fit with OLS 2. Transform? But think about what it will do to error. 3. Nonlinear Least Squares 4. Generalized Linear Models Spores 0e+00 2e+06 4e+06 6e+06 8e longevity Spores = b0 + b1 Longevity + b2 Longevity 2 + error

2 Putting Nonlinear Terms into an Additive Model Parameters are the Same as Ever summary(fungus.lmsq) fungus.lmsq <- lm(spores longevity + I(longevityˆ2), data=fungus) Call: lm(formula = Spores longevity + I(longevityˆ2), data = fungus) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) longevity e-05 I(longevityˆ2) Residual standard error: on 29 degrees of freedom Multiple R-squared: 0.607, Adjusted R-squared: 0.58 F-statistic: 22.4 on 2 and 29 DF, p-value: 1.3e-06 We Can t use abline Anymore We Can t use abline Anymore plot(spores longevity, data=fungus, pch=19) fungusfun <- function(x) coef(fungus.lmsq)[1] + coef(fungus.lmsq)[2]*x + coef(fungus.lmsq)[3]*xˆ2 curve(fungusfun, add=t, col="red", lwd=2) Spores 0e+00 2e+06 4e+06 6e+06 8e longevity

3 What if It s not a Linear Combination of Terms? Common Transformations bmr.watts log(y) arcsin(sqrt(y)) for bounded data logit for bounded data (more well behaved) Box-Cox Transform May have to add 0.01, 0.5, or 1 to many of these in cases with 0s mass.g You must ask yourself, what do the transformed variables mean? MetabolicRate = a mass b But Where does Error Come In But Where does Error Come In log(bmr.watts) log(metabolicrate) = log(a) + b log(mass) + error implies MetabolicRate = a mass b e error but we often want log(mass.g) MetabolicRate = a mass b + error log(metabolicrate) = log(a) + b log(mass) + error

4 Nonlinear Least Squares Fitting Nonlinear Least Squares Fitting summary(primate.nls) primate.nls <- nls(bmr.watts a*mass.gˆb, data=primates, start=list(a = , b = )) Uses algorithm for fitting. Very flexible. Must specify start values. Formula: bmr.watts a * mass.gˆb Parameters: Estimate Std. Error t value Pr(> t ) a b < 2e-16 Residual standard error: on 15 degrees of freedom Number of iterations to convergence: 4 Achieved convergence tolerance: 7.01e-07 NLS Performs Better Exercise: Kelp! bmr.watts NLS log log Evaluate the Frond Holdfast relationship Fit a model with a log transformation Fit a model with a nls model Compare Check the diagnostics - see anything? mass.g

5 The Kelp Data Envelope Residuals from Log Transform Residuals vs Fitted Normal Q Q Error: object kelp not found are a count variable, cannot be < 0 Residuals Standardized residuals Fitted values Theoretical Quantiles Mild Trumpet even in NLS The Kelp Data residuals(kelp.nls) Maybe the error is wrong fitted(kelp.nls) are a count variable, cannot be < 0

6 Generalized Linear Models Some Common Links Basic Premise: y dist(η, ν) dist is a distribution of the exponential family η is a link function such that η = f(µ) where µ is the mean of a curve φ is a variance function Identity: η = µ - e.g. µ = a + bx Log: η = log(µ) - e.g. µ = e ( a + bx) Logit: η = logit(µ) - e.g. µ = e( a+bx) 1+e ( a+bx) Inverse: η = 1 µ - e.g. µ = (a + bx) 1 For example, if dist is Normal, canonical link is µ, variance is σ 2 Distributions, Canonical Links, and Dispersion Distributions and Other Links Distribution Canonical Link Variance Function Normal identity 1 Poisson log µ Quasipoisson log µθ Binomial logit µ(1 µ) Quasibinomial logit µ(1 µ)θ Negative Binomial log µ + κµ 2 Gamma inverse µ 2 Inverse Normal 1/µ 2 µ 3 Distribution Normal Poisson Quasipoisson Binomial Quasibinomial Negative Binomial Gamma Inverse Normal Links identity, log, inverse log, identity, sqrt log, identity, sqrt logit, probit, cauchit, log, log-log logit, probit, cauchit, log, log-log log, identity, sqrt inverse, identity, log 1/µ 2, inverse, identity, log

7 Deviance and IWLS The Kelp Data Every GLM has a Set of Deviance Function to be Minimized i.e., for a normal distribution DM = (yi ˆµi) 2 Models are Fit using Iteratively Weighted Least Squares algorithm are a count variable, cannot be < 0 Fitting a GLM with a Poisson Error and Log Link How do we Assess Meeting Assumptions? Fronds Poisson( Fronds ˆ ) ˆ F ronds = exp(a + b * holdfast diameter) Deviance Residuals Deviance Residuals kelp.glm <- glm(, data=kelp, family=poisson(link="log")) Fitted Link Fitted Response Response Residuals Response Resdiuals Fitted Link Fitted Response

8 Predicted values Obs. number Theoretical Quantiles Cook's distance Leverage Predicted values Different Types of Residuals How do we Assess Meeting Assumptions? residuals(kelp.glm, type="deviance") residuals(kelp.glm, type="pearson") residuals(kelp.glm, type="response") Residuals Residuals vs Fitted Cook's distance Std. deviance resid Normal Q Q Residuals vs Leverage Std. deviance resid Scale Location Cook's distance Std. Pearson resid GLM Model Coefficients Call: glm(formula =, family = poisson(link = "log"), data = kelp) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) <2e <2e-16 (Dispersion parameter for poisson family taken to be 1) Null deviance: on 107 degrees of freedom Residual deviance: on 106 degrees of freedom Checking Fit cor(fitted(kelp.glm), fitted(kelp.glm) + residuals(kelp.glm, type="response"))ˆ2 [1] summary(kelp.lm)$r.squared [1] 0.277

9 The Fitted Model Prediction Confidence Intervals by Hand upperci <- qpois(0.975, lambda = round(fitted(kelp.glm))) lowerci <- qpois(0.025, lambda = round(fitted(kelp.glm))) HLD <- na.omit(kelp)$ kelp.ggplot + geom_line(mapping=aes(x=hld, y=upperci), lty=2, col="blue") + geom_line(mapping=aes(x=hld, y=lowerci), lty=2, col="blue") Prediction Confidence Intervals by Hand Which Overdispersed Distribution to Use? v(quasipoisson) = µθ v(negative Binomial) = µ + κµ 2 20 see Ver Hoef and Boveng 2007 Ecology 0 Overdispersion?

10 GLM with Negative Binomial Negative Binomial Performs Better anova(kelp.glm, kelp.glm.nb) library(mass) kelp.glm.nb <- glm.nb(, data=kelp) Analysis of Deviance Table Model 1: Model 2: Resid. Df Resid. Dev Df Deviance The Fitted Model Fit with Prediction Error

11 Compare to Quasipoisson with Prediction Error Example: Wolf Inbreeding and Litter Size The Number of Pups is a Count! Fit GLMs with different errors and links Which is the best model? Plot with fit and prediction error