Non-Gaussian Response Variables

Transcription

1 Non-Gaussian Response Variables

2 What is the Generalized Model Doing? The fixed effects are like the factors in a traditional analysis of variance or linear model The random effects are different A generalized linear mixed model (GLMM) includes fixed and random factors The random effects are modeled as coming from a specified distribution (i.e., each female has a different baseline offspring size, and this baseline size has a normal distribution with an estimated variance) The model basically searches through parameter values to find the set of slopes and intercepts that maximize the probability of the observed data By including a random effect, you can reduce that variable s impact on the fixed effect analysis you usually will not care too much about the random effect (but you might)

3 A Flexible Modeling Framework GLMMs (and related models) allow a lot of modeling flexibility Explicit modeling of heterogeneity Including a spatial or temporal component Response variables that are not normally distributed Include fixed and random factors Naturally allows nested designs and repeated measures

4 Exponential Family of Distributions Normal: symmetric, continuous Poisson: Asymmetric, discrete Rare events, like number of robberies per week in College Station The mean equals the variance Binomial: Asymmetric, discrete Number of occurrences The mean is larger than the variance Negative binomial: Asymmetric, discrete Like the binomial, except the variance is larger than the mean Gamma: asymmetric, continuous Can have a variety of shapes, but all observations are positive

5 Parts of a GLM The distribution of the response variable Usually we assume it s normal Specification of the systematic component in terms of explanatory variables The fixed effects If we had random factors, it would be a GLMM The link between the systematic part and the response variables In our usual models, the link is the identity link, where the expected value of the response value is directly estimated (like from the equation for a line: y = mx+b)

6 Implementing a Poisson GLM Now the response variable has a Poisson distribution We specify the systematic part of the model in the usual way (same goes for random parts if we want those) The link is logarithmic, which ensures that the predicted values are always non-negative (a Poisson distribution doesn t allow negative values)

7 Example: Amphibian Roadkills Dataset: Roadkills of amphibians at 52 sites of varying distance from a natural park Number of roadkills is not normally distributed Amphibian getting run over by car might be a rare, random event, so you might expect it to have a Poisson distribution (at each distance)

8 Total Roadkills Plot of Roadkills on Distance Distance from Park

9 Fitting a Poisson GLM > M1 <- glm(tot.n ~ D.PARK, family=poisson, data=roadkills) > summary(m1)

10 Poisson GLM Output Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) 4.316e e <2e-16 *** D.PARK e e <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for poisson family taken to be 1) Null deviance: on 51 degrees of freedom Residual deviance: on 50 degrees of freedom AIC: Number of Fisher Scoring iterations: 4

11 Meaning of Deviance Null and residual deviances are kind of like maximum likelihood equivalents of the total and residual sums of squares An R 2 like term can be obtained from: null 100 deviance - residual null deviance deviance Applying this relationship to the previous model, we find that it explains 63.5% of the variation

12 Total Roadkills Fitting a Line for the Model Distance from Park

13 Code for the Lines MyData <- data.frame(d.park = seq(from = 0, to = 25000, by=1000)) G <- predict(m1, newdata=mydata, type="link", se=true) F <- exp(g$fit) FSEUP <- exp(g$fit+1.96*g$se.fit) FSELOW <- exp(g$fit-1.96*g$se.fit) lines(mydata$d.park, F, lty=1, lwd=3) lines(mydata$d.park, FSEUP, lty=2, lwd=3) lines(mydata$d.park, FSELOW, lty=2, lwd=3)

14 Model Selection in a Poisson GLM Option 1: Drop terms sequentially and test full and reduced models Option 2: Use the drop1 command to drop each explanatory variable in turn Option 3: Use the anova command to sequentially remove each term and compare the resulting models to the original full model

15 The drop1 command Example: Still roadkills, but with nine explanatory variables > M2 <- glm(tot.n ~ OPEN.L + MONT.S + SQ.POLIC + D.PARK + SQ.SHRUB + SQ.WATRES + L.WAT.C + SQ.LPROAD + SQ.DWATCOUR, family=poisson, data=rk) > summary(m2) > drop1(m2, test= Chi )

16 Results of drop1() Single term deletions Model: TOT.N ~ OPEN.L + MONT.S + SQ.POLIC + D.PARK + SQ.SHRUB + SQ.WATRES + L.WAT.C + SQ.LPROAD + SQ.DWATCOUR Df Deviance AIC LRT Pr(>Chi) <none> OPEN.L MONT.S e-09 *** SQ.POLIC e-05 *** D.PARK < 2.2e-16 *** SQ.SHRUB e-07 *** SQ.WATRES ** L.WAT.C e-16 *** SQ.LPROAD *** SQ.DWATCOUR Signif. codes: 0 *** ** 0.01 *

17 Overdispersion Recall that the Poisson distribution assumes the variance is equal to the mean If the variance is greater than the mean, then a Poisson will not accurately describe the data This problem is called overdispersion

18 Detecting Overdispersion Calculate: ˆ D is the residual deviance of the model [It was in model M1 a few slides ago]. n p represents the degrees of freedom for the residual deviance [also reported by the summary() function in this case 50] If this value is around 1, then overdispersion should not be a problem If it is greater than 1, then overdispersion is a problem /50 = 7.8, so overdispersion is a problem in this dataset. n D p

19 Overdispersion in a Poisson GLM One approach is to use a quasi-poisson GLM This model includes a dispersion parameter to better model the variance relative to the mean If the dispersion parameter (φ) is large then it might be better to use a different model

20 Fitting a Quasipoisson > M4 <- glm(tot.n ~ D.PARK, family=quasipoisson, data=rk) > summary(m4)

21 Results Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 4.316e e < 2e-16 *** D.PARK e e e-11 *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for quasipoisson family taken to be ) Null deviance: on 51 degrees of freedom Residual deviance: on 50 degrees of freedom AIC: NA Number of Fisher Scoring iterations: 4

22 Model Selection in Quasipoisson AIC is not defined for a quasipoisson model, so you can t use AIC It s possible to compare models using F-tests drop1(m5, test= F )

23 Model Validation in Poisson GLM Pearson residuals: scaled by the expected mean for a given value of the explanatory variable (because the variance of the poisson changes with the mean) Deviance residuals: the contribution of each observation to the residual deviance. In other words, a measure of how badly that point fits. The default is to use the deviance residuals for model validation, and they will usually be the best choice.

24 What to Plot Deviance residuals versus: The fitted values Each explanatory variable in the model Each explanatory variable dropped from the model Against time (if it s available) Against any spatial aspect of the data We don t expect normality, but we are looking for patterns and fit

25 Std. deviance resid Std. Pearson resid Residuals Std. deviance resid Model Validation Plots Residuals vs Fitted Normal Q-Q Predicted values Theoretical Quantiles Scale-Location Residuals vs Leverage Cook's distance Predicted values Leverage

26 EP ED E EP Model Validation Plots Response residuals Pearson residuals mu mu Pearson residuals scaled Deviance residuals mu mu

27 Code for the Validation Plots #Model validation example M5 <- glm(tot.n ~ D.PARK, family = quasipoisson, data=rk) plot(m5) EP <- resid(m5, type="pearson") ED <- resid(m5, type="deviance") mu <- predict(m5, type="response") E <- RK$TOT.N - mu EP2 <- E/sqrt( *mu) op <- par(mfrow = c(2,2)) plot(x = mu, y = E, main="response residuals") plot(x = mu, y = EP, main="pearson residuals") plot(x = mu, y = EP2, main="pearson residuals scaled") plot(x = mu, y = ED, main="deviance residuals") par(op)

28 Interpretation This model has a couple of problems First, the residuals have a clear pattern, where they are above the predicted line at some distances and below it at others Second, some outliers are strongly influencing the results

29 Negative Binomial GLM Assumes: The distribution of the response variable is negative binomial for any value of X. Recall that the variance is larger than the mean for a negative binomial distribution The link function is logarithmic, which ensures that the fitted values are always non-negative

30 Fitting a Negative Binomial GLM > library(mass) > M6 <- glm.nb(tot.n ~ OPEN.L + MONT.S + SQ.POLIC + D.PARK + SQ.SHRUB + SQ.WATRES + L.WAT.C + SQ.LPROAD + SQ.DWATCOUR, link="log", data=rk) > summary(m6, cor=false)

31 Some output Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) 3.951e e <2e-16 *** OPEN.L e e ** MONT.S 5.846e e SQ.POLIC e e D.PARK e e <2e-16 *** SQ.SHRUB e e SQ.WATRES 1.631e e L.WAT.C 2.076e e * SQ.LPROAD 5.944e e SQ.DWATCOUR e e Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for Negative Binomial(5.5178) family taken to be 1) Null deviance: on 51 degrees of freedom Residual deviance: on 42 degrees of freedom AIC:

32 Tools for Model Selection The z-statistic from the summary (previous slide) Analysis of deviance table from anova(m6, test= Chi ) does sequential testing Drop each term in turn using drop1(m6, test= Chi ) Manually specify a nested model and compare them using anova(m6, M7, test= Chi )

33 Results Model after model selection procedure: > M8 <- glm.nb(tot.n ~ OPEN.L + D.PARK, link = "log", data=rk) > summary(m8) > plot(m8)

34 Std. deviance resid Std. Pearson resid Residuals Std. deviance resid Residuals vs Fitted Normal Q-Q Negative Binomial Plots Predicted values Theoretical Quantiles Scale-Location Residuals vs Leverage Cook's distance Predicted values Leverage

35 Std. deviance resid Std. Pearson resid Residuals Std. deviance resid Residuals vs Fitted Normal Q-Q Poisson Plots Which is better? Predicted values Theoretical Quantiles Scale-Location Residuals vs Leverage Cook's distance Predicted values Leverage

36 Adding Random Effects in a GLMM What if you have a non-gaussian response variable AND want to include random effects in your model? The answer is a GLMM Several packages are available in R, but we will use glmer from the lme4 package

37 Example: Deer Parasites Data consist of whether or not each deer has parasites Deer differ by sex, size and farm of origin Which factors seem like they should be fixed and which are random? Because the response variable is binary, a binomial distribution is appropriate

38 Implementing the GLMM > library(lme4) > DE.lme4 <- glmer(ec01 ~ CLength * fsex + (1 ffarm), family=binomial, data=deer) > summary(de.lme4)

39 Results Part I Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) [glmermod] Family: binomial ( logit ) Formula: Ec01 ~ CLength * fsex + (1 ffarm) Data: deer AIC BIC loglik deviance df.resid Scaled residuals: Min 1Q Median 3Q Max Random effects: Groups Name Variance Std.Dev. ffarm (Intercept) Number of obs: 826, groups: ffarm, 24

40 Results Part II Fixed effects: Estimate Std. Error z value Pr(> z ) (Intercept) ** CLength e-08 *** fsex ** CLength:fSex ** --- Signif. codes: 0 *** ** 0.01 * Correlation of Fixed Effects: (Intr) CLngth fsex2 CLength fsex CLngth:fSx

41 Summary Generalized Linear Models can accommodate non-gaussian response variables It s possible to include fixed and random effects, and then the model is called a Generalized Linear Mixed Model The syntax for the random effects depends upon the package that s being used for the analysis, so be careful

42 Summary Other features can be modeled as well, and you should consult Zuur et al. and the literature if your data include: Temporal autocorrelation Spatial autocorrelation An excess or deficit of individuals in the zero category compared to the expectations of the exponential family of distributions