Chapter 4: Generalized Linear Models-II

Transcription

1 : Generalized Linear Models-II Dipankar Bandyopadhyay Department of Biostatistics, Virginia Commonwealth University BIOS 625: Categorical Data & GLM [Acknowledgements to Tim Hanson and Haitao Chu] D. Bandyopadhyay (VCU) 1 / 26

2 4.3 Generalized Linear Models for Counts Overdispersion for Poisson GLMs If data are truly Poisson, then we should have roughly E(Y i ) = var(y i ) = µ i. Data can be grouped into like categories and this can be informally checked. For the horseshoe crab data we have the following (Table 4.4): Width (cm) Sample mean Sample variance < > D. Bandyopadhyay (VCU) 2 / 26

3 4.3 Generalized Linear Models for Counts The sample variance tends to be 2-3 times as much as the mean. This is an example of overdispersion. There is greater variability in the data than we expect under our sampling model. Fixes: Find another sampling model! Include other important, explanatory covariates. Random effects as a proxy to unknown, latent covariates. Quasi-likelihood approach. We ll explore a common approach to the first fix above... D. Bandyopadhyay (VCU) 3 / 26

4 4.3 Generalized Linear Models for Counts Review: PhD 2011 Exam Question on Linear Models, Q1 We say Y NB(α,β), if Y λ Poisson(λ) and λ Gamma(α,β) with density f(λ) = 1 Γ(α)β λ α 1 e λ/β,α,β > 0. α Proof: The probability mass function for the Negative Binomial is h(y) = = = 0 0 g(y λ)f(λ)dλ e λ λ y λ α 1 e λ/β y! Γ(α)β α dλ 1 Γ(y +1)Γ(α)β α 0 λ y+α 1 exp [ λ ( ) ] β 1 dλ β +1 The integrand is the kernel of a Gamma density with shape parameter α+y and scale parameter β/(β +1). D. Bandyopadhyay (VCU) 4 / 26

5 4.3 Generalized Linear Models for Counts To get a Gamma density and make the integral equal 1, we divide the integrand by Γ(α+y) [β/(β +1)] α+y and of course to keep h(y) from changing, we must multiply the factor outside the integral by that quantity. So h(y) = = Γ(α+y)[β/(β +1)]α+y Γ(y +1)Γ(α)β ( α ) Γ(α+y) 1 α ( ) β y. Γ(y +1)Γ(α) 1+β 1+β Recall that the Gamma distribution for λ has mean αβ and variance αβ 2. E(y) = E[E(y λ)] = E[λ] = αβ = µ and Var(Y) = E[Var(Y λ)]+var[e(y λ)] = E[λ]+Var[λ] = αβ +αβ 2 = µ+µβ = µ(1+β) D. Bandyopadhyay (VCU) 5 / 26

6 4.3 Generalized Linear Models for Counts Negative binomial regression Let k = α and µ = αβ, then Y negbin(k,µ) with ( ) Γ(y +k) k k ( p(y) = 1 k ) y for y = 0,1,2,3,... Γ(k)Γ(y +1) µ+k µ+k Then E(Y) = µ and var(y) = µ+µ 2 /k. A sampling model that includes another parameter allows some separation between the mean and variance. The negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number k of failures occurs. The index k 1 is called a dispersion parameter. As k the Poisson distribution is obtained. Here, the variance increases with the mean; is that appropriate for the crab data? Book looks at crab data on p Another modeling approach: adding a random effect for each crab, coming up toward the end of the semester. D. Bandyopadhyay (VCU) 6 / 26

7 4.4 Mean, Variance & Likelihood for GLMs* 4.4 Mean & Variance for GLMs A two parameter exponential family includes a dispersion parameter φ: f(y i θ i,φ) = exp{[y i θ i b(θ i )]/a(φ)+c(y i,φ)}. When φ is known, it simplifies to f(y i θ i ) = a(θ i )b(y i )exp[y i Q(θ i )], the nature one parameter exponential family. This includes binomial, negative binomial, Poisson, normal, and many others. Let L i = logf(y i ;θ i,φ). This is the contribution of the i th observation to the likelihood in terms of θ i and φ. Then L(θ,φ) = N i=1 logf(y i;θ i,φ) = N i=1 L i, where L i = [y i θ i b(θ i )]/a(φ)+c(y i,φ). D. Bandyopadhyay (VCU) 7 / 26

8 4.4 Mean, Variance & Likelihood for GLMs* Then some work gives us µ i = E(Y i ) = b (θ i ) and var(y i ) = b (θ i )a(φ). The model imposes µ i = b (θ i ) = g 1 (x iβ). The N-dimensional µ, or equivalently θ, is reduced to the p-dimensional β (and φ in a 2-parameter family). Then L(β,φ) = N [ yi (b ) 1 (g 1 (x i β)) b((b ) 1 (g 1 (x i β))) i=1 as θ i = (b ) 1 (g 1 (x i β)). a(φ) The MLEs ˆβ and ˆφ are found by taking first derivatives of this, setting equal to zero, and solving (pp ). Things simplify when using the canonical link. ] +c(y i,φ), D. Bandyopadhyay (VCU) 8 / 26

9 4.4 Mean, Variance & Likelihood for GLMs* The asymptotic covariance matrix for ˆβ is the inverse of the fisher information matrix, cov(ˆβ). This is a function of the unknown β and φ, and in practice we just plug in the MLE values ˆβ and ˆφ yielding ĉov(ˆβ). Section shows how Poisson and binomial GLMs fit into the general exponential family form and specifies corresponding b(θ i ), a(φ), and c(y i,φ). Section carries out computations leading to ĉov(ˆβ) in the Poisson regression model with a log link. D. Bandyopadhyay (VCU) 9 / 26

10 4.5 Inference for GLMs Deviance and GOF For now assume we re able to get ˆβ. Anyway, we are able to, in SAS or R! Recall that the saturated model estimates the N µ i s with the N y i s, providing perfect fit. This model does not reduce data, provide a means for prediction for arbitrary covariate values x, allow for meaningful hypotheses to be tested, etc. However, we can use the saturated model to check the fit of a real GLM. If, essentially, the number of distinct covariate vectors remains fixed but N increases then G 2 = 2logL(µ(ˆβ), ˆφ r ;y) logl(y, ˆφ f ;y)] is the LRT statistic for testing H 0 : g(µ i ) = x iβ relative to the alternative that the means µ are unstructured. D. Bandyopadhyay (VCU) 10 / 26

11 4.5 Inference for GLMs In Poisson and binomial regression models a(φ) = 1, i.e. there is no dispersion parameter, and this LRT statistic is equal to the model deviance as described last time for grouped data. When there is a dispersion parameter φ (e.g. normal, negative binomial, or gamma regression models), 2 times the difference in saturated and reduced models log-likelihood is D/φ in most models, called the scaled deviance; see top, p The scaled deviance has an approximate chi-squared distribution when the reduced model holds. D. Bandyopadhyay (VCU) 11 / 26

12 4.5 Inference for GLMs LR model comparison In Poisson and binomial regression, D r = 2[L r L s ] where D is deviance, L r is log-likelihood evaluated at ˆβ for the GLM, and L s is log-likelihood evaluated at ˆµ i = y i under the saturated model. Say we add a few more predictors to the model so the dimension of β goes from p to p +q. Compute the deviance from the smaller model (D r ) and the larger model (D f ). Then D r D f is the likelihood ratio test statistic for testing H 0 : smaller model holds, and is asymptotically χ 2 q when H 0 is true. The larger this difference, the more evidence there is that the new predictors significantly improve model fit (and hence significantly reduce model deviance). Often data are not grouped; in this case it s safer to use L(β;y) directly from the output! D. Bandyopadhyay (VCU) 12 / 26

13 4.5 Inference for GLMs Residuals for GLMs Residuals indicate where model fit is inadequate. The deviance residual d i is defined in such a way that N see p i=1 d2 i = D, The Pearson residual is given by e i = y i ˆµ i. These have variance var(y i ) < 1. The standardized Pearson residuals r i properly standardize the residual to have variance one and in large samples are N(0,1) if the model holds. This means reasonably large n i for binomial data and reasonably large counts for Poisson data. So residuals r i > 3 show rather extreme lack of fit for (x i,y i ) according to the model. Residuals can be plotted versus predictors or against the linear predictor ˆη i = x iˆβ to assess systematic departures from model assumptions. Note: X 2 = N i=1 e2 i χ 2 s p when H 0 : g(µ i ) = x iβ is true. D. Bandyopadhyay (VCU) 13 / 26

14 4.6 Fitting GLMs Newton-Raphson Method in One Dimension Say we want to find where f(x) = 0 for differentiable f(x). Let x 0 be such that f(x 0 ) = 0. Taylor s theorem tells us f(x 0 ) f(x)+f (x)(x 0 x). Plugging in f(x 0 ) = 0 and solving for x 0 we get ˆx 0 = x f(x) f (x). Starting at an x near x 0, ˆx 0 should be closer to x 0 than x was. Let s iterate this idea t times: x (t+1) = x (t) f(x(t) ) f (x (t) ). Eventually, if things go right, x (t) should be close to x 0. D. Bandyopadhyay (VCU) 14 / 26

15 4.6 Fitting GLMs If f(x) : R p R p, the idea works the same, but in vector/matrix terms. Start with an initial guess x (0) and iterate x (t+1) = x (t) [Df(x (t) )] 1 f(x (t) ). If things are done right, then this should converge to x 0 such that f(x 0 ) = 0. We are interested in solving DL(β) = 0 (the score, or likelihood equations!) where DL(β) = L(β) β 1. L(β) β p and D2 L(β) = L(β) β 2 1. L(β) β p β 1 L(β) β 1 β p.... L(β) β 2 p. D. Bandyopadhyay (VCU) 15 / 26

16 4.6 Fitting GLMs So for us, we start with β (0) (maybe through a MOM or least squares estimate) and iterate β (t+1) = β (t) [D 2 L(β (t) )] 1 DL(β (t) ). This is (4.45) on p. 143 disguised. The process is typically stopped when β (t+1) β (t) < ǫ. Newton-Raphson uses D 2 L(β) as is, with the y plugged in. Fisher scoring instead uses E{D 2 L(β)}, with expectation taken over Y, which is not a function of the observed y, but harder to get. The latter approach is harder to implement, but conveniently yields ĉov(ˆβ) [ E{D 2 L(β)}] 1 evaluated at ˆβ when the process is done. D. Bandyopadhyay (VCU) 16 / 26

17 4.7 Quasi-likelihood and Overdispersion The MLE β satisfies: u j (β) = N i=1 (y i µ i )x ij v(µ i ) ( g 1 ) (η i ) = 0, j = 1,...,p, η i where η i = x i β and v(µ i) = var(y i ), a function of µ i. These are the partial derivatives of the log-likelihood function set to zero, also called the score equations. In exponential families, a given µ i = E(Y i ) and v(µ i ) = var(y i ) uniquely determines the distribution. For example, if we say E(Y i ) = µ i and var(y i ) = v(µ i ) = µ i, and that Y i is a distribution in the exponential family, then Y i has to be Poisson. D. Bandyopadhyay (VCU) 17 / 26

18 4.7 Quasi-likelihood and Overdispersion For Poisson data, we know v(µ i ) = µ i ; for Binomial data (E(Y i ) = µ i = n i π i ), we have v(π i ) = n i π i (1 π i ). If we add a dispersion parameter φ and declare that v(µ i ) = φµ i (Poisson) or v(π i ) = φn i π i (1 π i ) (binomial), the resulting family may not be exponential, or not even unique, but the score equations on the previous slide remain the same. So ˆβ does not change. What does change is the estimate ĉov(ˆβ). This estimate is the same as from the original model (where v(µ i ) = µ i or v(π i ) = n i π i (1 π i ) for Poisson or Binomial respectively) except multiplied by φ. Therefore regression effect standard errors are simply multiplied by ˆφ where ˆφ is an estimate of φ. D. Bandyopadhyay (VCU) 18 / 26

19 4.7 Quasi-likelihood and Overdispersion Let X 2 = N i=1 (y i ˆµ i ) 2 /ˆµ i for Poisson and X 2 = N i=1 (y i n iˆπ i ) 2 /[n iˆπ i (1 ˆπ i )] for binomial, the Pearson statistic for assessing (original) model fit. φ is not in the score equations, however, X 2 /φ χ 2 s p (when the dispersion model is true) where s is the number of unique covariate vectors in {x i }. Since E(χ 2 df ) = df, a MOM estimate of φ is ˆφ = X 2 /(s p). When data are grouped, s = N. The adjusted estimate is ĉov a (ˆβ) = ˆφ ĉov(ˆβ). When ˆφ > 1, which happens with overdispersed data, standard errors get properly inflated. This is an easy, ad hoc fix to overdispersion, but commonly done and useful. SAS does everything automatically when you specify SCALE=PEARSON in the MODEL statement of GENMOD. Also: SCALE=DEVIANCE works similarly. D. Bandyopadhyay (VCU) 19 / 26

20 4.7 Quasi-likelihood and Overdispersion To recall, SAS code for crab data without dispersion parameter: proc genmod; model satell = width / dist=poisson link=identity; Output: The GENMOD Procedure Model Informati on Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance Scaled Deviance Pearson Chi Square Scaled Pearson X Log Likelihood Full Log Likelihood AIC ( smaller is better ) AICC ( smaller is better ) BIC ( smaller is better ) Algorithm converged. Analysis Of Maximum Likelihood Parameter Estimates Standard Wald 95% Confidence Wald Parameter DF Estimate Error Li mi ts Chi Square Pr > ChiSq I n t e r c e p t <.0001 width <.0001 Scale D. Bandyopadhyay (VCU) 20 / 26

21 4.7 Quasi-likelihood and Overdispersion SAS code for crab data with dispersion parameter: proc genmod; model satell = width / dist=poisson link=identity scale =pearson; Output: The GENMOD Procedure Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance Scaled Deviance Pearson Chi Square Scaled Pearson X Log Likelihood Full Log Likelihood AIC ( smaller is better ) AICC ( smaller is better ) BIC ( smaller is better ) Algorithm converged. Analysis Of Maximum Likelihood Parameter Estimates Standard Wald 95% Confidence Wald Parameter DF Estimate Error Li mi ts Chi Square Pr > ChiSq I n t e r c e p t <.0001 width <.0001 Scale NOTE: The scale parameter was estimated by the square root of Pearson s Chi Squared/DOF. Note that ˆβ is the same with or without the dispersion parameter. What changes are se(ˆβ j ). D. Bandyopadhyay (VCU) 21 / 26

22 4.7 Quasi-likelihood and Overdispersion This approach to handling overdispersion works well when the mean structure is well modeled. Otherwise, what does ˆφ really estimate? This was a lot of information thrown at you very quickly. Meant to introduce notation and be an overview of things to come. We will slow down and investigate specific models in more detail. Be careful distinguishing s from N! In the saturated model, s is the number of distinct categories that data fall into. However, SAS takes the df for deviance to be the number of records N regardless. Data should be grouped in as few groups as possible when checking for dispersion. See 4.5.3, pp D. Bandyopadhyay (VCU) 22 / 26

23 4.7 Quasi-likelihood and Overdispersion Teratology example Female rats given one of four treatments: placebo, weekly iron supplement, days 7 & 10, days 0 & 7. See p. 152 for the data. The number dead y ij out of litter size n ij was recorded where i = 1,2,3,4 is the treatment group, and j = 1,...,m i is the number of litters in group i (31, 12, 5, 10). Let π i denote the probability of death in group i. The model is simply Y ij bin(n ij,π i ). The sum of two independent binomials with the same probability is also binomial. So according to the model, there really is only four observations: i y i+ n i D. Bandyopadhyay (VCU) 23 / 26

24 4.7 Quasi-likelihood and Overdispersion The idea behind this example is that there is litter-to-litter variability and so the data are really a mixture of binomial distributions and overdispersion might be present. If we do not group the data, then X 2 = 4 m i i=1 j=1 (y ij n ijˆπ i ) 2 n ijˆπ i (1 ˆπ i ). This has an approximate χ distribution when we think of litter sizes n ij. Then ˆφ = 2.86 and there s evidence of overdispersion. According to the model, the groupings are arbitrary; we cannot tell the difference between litters. If we group the data then X 2 = 4 i=1 (y i+ n i+ˆπ i ) 2 n i+ˆπ i (1 ˆπ i ) = 0. D. Bandyopadhyay (VCU) 24 / 26

25 4.7 Quasi-likelihood and Overdispersion The problem is that the latter model being fit is the saturated model. There is no way to check for overdispersion using the grouped data. However, according to the model, the groupings according to data recorded in terms on litters are arbitrary. We know this isn t the case, but the model cannot tell the difference between litters, only treatments. We are using information on litters to assess overdispersion, but not explicitly including this information in a real probability model, but rather through φ. (Better than ignoring the possibility entirely!) A possibly better approach is to include a separate term for each litter! Y ij bin(n ij,µ ij ), logit(µ ij ) = π i +γ ij, where γ ij iid N(0,σ 2 ). This random effects model explicitly includes litter-to-litter heterogeneity in the model. The γ ij serve as a proxy to unmeasured, latent genetic differences among litters. D. Bandyopadhyay (VCU) 25 / 26

26 4.7 Quasi-likelihood and Overdispersion Comments: Which approach is better, estimating φ and inflating the se s for ˆπ i or the random effects model? What assumptions under the random effects model might be violated? What strengths does it have? What assumptions using v(π i ) = φn i π i (1 π i ) might be violated? How does this affect the model? Can you see a potentially bigger problem here in using an estimate ˆφ? How would I analyze these data? With a random effects model, then examine ˆγ ij to check the normality assumption. D. Bandyopadhyay (VCU) 26 / 26