10. Generalized linear models

1. 10. Generalized linear models • 10.1 Homogeneous models • Exponential families of distributions, link functions, likelihood estimation • 10.2 Example: Tort filings • 10.3 Marginal models and GEE • 10.4 Random effects models • 10.5 Fixed effects models • Maximum likelihood, conditional likelihood, Poisson data • 10.6 Bayesian Inference • Appendix 10A Exponential families of distributions

2. 10.1 Homogeneous models • Section Outline • 10.1.1 Exponential families of distributions • 10.1.2 Link functions • 10.1.3 Likelihood estimation • In this section, we consider only independent responses. • No serial correlation. • No random effects that would induce serial correlation.

3. Exponential families of distributions • The basic one parameter exponential family is • Here, y is a response and q is the parameter of interest. • The parameter f is a scale parameter that we often will assume is known. • The term b(q) depends only on the parameter q, not the responses. • S(y, f) depends only on the responses and the scale parameter, not the parameter q. • The response y may be discrete or continuous. • Some straightforward calculations show that E y = b¢(q) and Var y = b²(q) f.

4. Special cases of the basic exponential family • Normal • The probability density function is • Take m = q, s2 = f , b(q) = q 2/2 and S(y, f ) = - y2 / (2f) - ln(2 pf))/2 . • Note that E y = b¢(q) = q =m and Var y = b¢¢(m) s2 = s2. • Binomial, n trials and prob p of success • The probability mass function is • Take ln (p/(1-p))= logit (p) = q, 1 = f , b(q) = n ln (1 + eq) and S(y, f ) = ln((n choose y)) . • Note that E y = b¢(q) = neq/(1 + eq) = np and Var y = b¢¢(q) (1) = neq/(1 + eq)2 = np(1-p) , as anticipated.

5. Another special case of the basic exponential family • Poisson • The probability mass function is • Take ln (l) = q, 1 = f , b(q) =eq and S(y, f ) = -ln( y!)) . • Note that E y = b¢(q) = eq = l and • Var y = b¢¢(q) (1) = eq = l , as anticipated.

7. 10.1.2 Link functions • To link up the univariate exponential family with regression problems, we define the systematic component of yit to be hit = xitb . • The idea is to now choose a “link” between the systematic component and the mean of yit , say mit , of the form: hit = g(mit) . • g(.) is the link function. • Linear combinations of explanatory variables, hit = xitb, may vary between negative and positive infinity. • However, means may be restricted to smaller range. For example, Poisson means vary between zero and infinity. • The link function serves to map the domain of the mean function onto the whole real line.

8. Bernoulli illustration of links • Bernoulli means vary between 0 and 1, although linear combinations of explanatory variables may vary between negative and positive infinity. • Here are three important examples of link functions for the Bernoulli distribution: • Logit: h = g(m) = logit(m) = ln (m/(1- m)) . • Probit: h = g(m) = F-1(m) , where F-1 is the inverse of the standard normal distribution function. • Complementary log-log: h = g(m) = ln ( -ln(1- m) ) . • Each function maps the unit interval (0,1) onto the whole real line.

9. Canonical links • As we have seen with the Bernoulli, there are several link functions that may be suitable for a particular distribution. • When the systematic component equals the parameter of interest (h = q ), this is an intuitively appealing case. • That is, the parameter of interest, q , equals a linear combination of explanatory variables, h. • Recall that h = g(m) and m = b¢(q). • Thus, if g-1 = b¢, then h = g(b¢(q)) = q. • The choice of g, such that g-1 = b¢, is called a canonical link. • Examples: Normal: g(q) = q, Binomial: g(q) = logit(q), Poisson: g(q) = ln q.

10. 10.1.3 Estimation • Begin with likelihood estimation for canonical links • Consider responses yit, with mean mit, systematic component hit = g(mit) = xitb and canonical link so that hit = qit. • Assume the responses are independent. • Then, the log-likelihood is

11. MLEs - Canonical links • The log-likelihood is • Taking the partial derivative with respect to b yields the score equations: • because mit = b¢(qit) = b¢(xit¢b ). • Thus, we can solve for the mle’s of b through: 0 = Sitxit (yit - mit). • This is a special case of the method of moments.

12. MLEs - general links • For general links, we no longer assume the relation qit = xit¢b. • We assume that bis related to qit through mit = b¢(qit) and hit = xit¢b = g(mit). • Recall that the log-likelihood is • Further, E yit = mit and Var yit = b¢¢(qit) / f . • The jth element of the score function is • because b ¢(qit) = mit

13. MLEs - more on general links • To eliminate qit, we use the chain rule to get • Thus, • This yields • This is called the generalized estimating equations form.

14. Overdispersion • When fitting models to data with binary or count dependent variables, it is common to observe that the variance exceeds that anticipated by the fit of the mean parameters. • This phenomenon is known as overdispersion. • A probabilistic models may be available to explain this phenomenon. • In many situations, analysts are content to postulate an approximate model through the relation Var yit = 2 b(xitβ) / wit. • The scale parameter  is specified through the choice of the distribution • The scale parameter σ2 allows for extra variability. • When the additional scale parameter σ2 is included, it is customary to estimate it by Pearson’s chi-square statistic divided by the error degrees of freedom. That is,

15. 10.2 Example: Tort filings

17. Offsets • We assume that yit is Poisson distribution with parameter POPit exp(xitβ), • where POPit is the population of the ith state at time t. • In GLM terminology, a variable with a known coefficient equal to 1 is known as an offset. • Using logarithmic population, our Poisson parameter for yit is • An alternative approach is to use the average number of tort filings as the response and assume approximate normality. • Note that in the Poisson model above the expectation of the average response is • whereas the variance is

18. Tort filings • Purpose: to understand ways in which state legal, economic and demographic characteristics affect the number of filings. • Table 10.3 suggests more filings under JSLIAB and PUNITIVE but less under CAPS • Table 10.5 • All variables under the homogenous model are statistically significant • However, estimated scale parameter seems important • Here, only JSLIAB is (positively) statistically significant • Time (categorical) variable seems important

20. 10.3 Marginal models • This approach reduces the reliance on the distributional assumptions by focusing on the first two moments. • We first assume that the variance is a known function of the mean up to a scale parameter, that is, Var yit = v(mit) f . • This is a consequence of the exponential family, although now it is a basic assumption. • That is, in the GLM setting, we have Var yit = b¢¢(qit) f and mit = b¢(qit). • Because b(.) and f are assumed known, Var yitis a known function of mit . • We also assume that the correlation between two observations within the same subject is a known function of their means, up to a vector of parameters t. • That is corr(yir , yis ) = r(mir, mis, t) , for r( .) known.

21. This framework incorporates the linear model nicely; we simply use a GLM with a normal distribution. However, for nonlinear situations, a correlation is not always the best way to capture dependencies among observations. Here is some notation to help see the estimation procedures. Define mi= (mi1,mi2, ..., miTi)´ to be the vector of means for the ith subject. To express the variance-covariance matrix, we define a diagonal matrix of variances Vi = diag(v(mi1),..., v(miTi) ) and the matrix of correlations Ri(t) to be a matrix with r(mir, mis , t) in the rth row and sth column. Thus, Var yi = Vi1/2Ri(t) Vi1/2. Marginal model

22. Generalized estimating equations • These assumptions are suitable for a method of moments estimation procedure called “generalized estimating equations” (GEE) in biostatistics, also known as the generalized method of moments (GMM) in econometrics. • GEE with known correlation parameter • Assuming t is known, the jth row of the GEE is • Here, the matrix • is Tix K*. • For linear models with mit= zit ai + xitb, this is the GLS estimator introduced in Section 3.3.

23. Consistency of GEEs • The solution, bEE, is asymptotically normal with covariance matrix • Because this is a function of the means, mi, it can be consistently estimated.

24. Robust estimation of standard errors • empirical standard errors may be calculated using the following estimator of the asymptotic variance of bEE

25. GEE - correlation parameter estimation • For GEEs with unknown correlation parameters, Prentice (1988) suggests using a second estimating equation of the form: • where • Diggle, Liang and Zeger (1994) suggest using the identity matrix for most discrete data. • However, for binary responses, • they note that the last Ti observations are redundant because yit = yit2 and should be ignored. • they recommend using

26. Tort filings • Assume an independent working correlation • This yields at the same parameter estimators as in Table 10.5, under the homogenous Poisson model with an estimated scale parameter. • JSLIAB is (positively) statistically significant, using both model-based and robust standard errors. • To test the robustness of this model fit, we fit the same model with an AR (1) working correlation. • Again, JSLIAB is (positively) statistically significant. • Interesting that CAPS is now borderline but in the opposite direction suggested by Table 10.3

28. 10.4 Random effects models • The motivation and sampling issues regarding random effects were introduced in Chapter 3. • The model is easiest to introduce and interpret in the following hierarchical fashion: • 1. Subject effects {ai} are a random sample from a distribution that is known up to a vector of parameters t. • 2. Conditional on {ai}, the responses • {yi1,yi2, ... , yiTi } are a random sample from a GLM with systematic component hit = zit ai + xitb .

29. Random effects models • This model is a generalization of: • 1. The linear random effects model in Chapter 3 - use a normal distribution. • 2. The binary dependent variables random effects model of Section 9.2 - using a Bernoulli distribution. (In Section 9.2, we focused on the case zit =1.) • Because we are sampling from a known distribution with a finite/small number of parameters, the maximum likelihood method of estimation is readily available. • We will use this method, assuming normally distributed random effects. • Also available in the literature is the EM (for expectation-maximization) algorithm for estimation - See Diggle, Liang and Zeger (1994).

30. Random effects likelihood • Conditional on ai, the likelihood for the ith subject at the tth observation is • where b¢(qit) = E (yit | ai) and hit = zit ai + xitb= g(E (yit | ai) ). • Conditional on ai, the likelihood for the ith subject is: • We take expectations over ai to get the (unconditional) likelihood. • To see this explicitly, let’s use the canonical link so that qit= hit. The (unconditional) likelihood for the ith subject is • Hence, the total log-likelihood is Si ln li. • The constant SitS(yit , f) is unimportant for determining mle’s. • Although evaluating, and maximizing, the likelihood requires numerical integration, it is easy to do on the computer.

31. Random effects and serial correlation • We saw in Chapter 3 that permitting subject-specific effects, ai, to be random induced serial correlation in the responses yit. • This is because the variance-covariance matrix of yit is no longer diagonal. • This is also true for the nonlinear GLM models. To see this, • let’s use a canonical link and • recall that E (yit | ai) ) = b¢(qit) = b¢(hit ) = b¢(ai + xit b).

32. Covariance calculations • The covariance between two responses, yi1 and yi2 , is Cov(yi1 , yi2 ) = E yi1yi2 - E yi1 E yi2 = E {b¢(ai+xi1b) b¢(ai+xi2b)} - E b¢(ai+xi1b) E b¢(ai+xi2b) • To see this, using the law of iterated expectations, E yi1yi2 = E E (yi1yi2| ai) = E {E (yi1| ai) E(yi2 | ai)} = E {b¢(ai+ xi1 b) b¢(ai+ xi2 b)}

33. More covariance calculations • Normality • For the normal distribution we have b¢(a) = a. • Thus, Cov(yi1 , yi2 ) = E {(ai+ xi1b) (ai + xi2b)} - E (ai + xi1b) E (ai + xi2b) = E ai2 + (xi1b) (xi2b)- (xi1b) (xi2b)= Var ai. • For the Poisson, we have b¢(a) = ea. Thus, E yit = E b¢(ai+ xitb) = E exp(ai+ xitb) = exp(xitb) E exp(ai) and • Cov(yi1 , yi2 ) = E {exp(ai+ xi1b) exp(ai+ xi2b)} - exp((xi1+xi2)b) {E exp(ai)}2 = exp((xi1+xi2)b) {E exp(2a) - (E exp(a))2 } = exp((xi1+xi2)b) Var exp(a) .

34. Random effects likelihood • Recall, from Section 10.2, that the (unconditional) likelihood for the ith subject is • Here, we use zit = 1,f = 1, and g(a) is the density of ai. • For the Poisson, we have b(a) = ea , and S(y, f) = -ln(y!), so the likelihood is • As before, evaluating and maximizing the likelihood requires numerical integration, yet it is easy to do on the computer.

36. 10.5 Fixed effects models • Consider responses yit, with mean mit, systematic component hit = g(mit) = zitai + xitb and canonical link so that hit = qit. • Assume the responses are independent. • Then, the log-likelihood is • Thus, the responses yitdepend on the parameters through only summary statistics. • That is, the statistics Styitzit are sufficient for ai . • The statistics Sityitxit are sufficient for b. • This is a convenient property of the canonical links. It is not available for other choices of links.

37. MLEs - Canonical links • The log-likelihood is • Taking the partial derivative with respect to ai yields: • because mit = b¢(qit) = b¢(zit¢ai + xit¢b ). • Taking the partial derivative with respect to b yields: • Thus, we can solve for the mle’s of ai and b through: 0 = Stzit (yit - mit), and 0 = Sitxit (yit - mit). • This is a special case of the method of moments. • This may produce inconsistent estimates of b , as we have seen in Chapter 9.

39. Conditional likelihood estimation • Assume the canonical link so that qit= hit = zitai + xitb . • Define the likelihood for a single observation to be • Let Si be the random vector representing St zityit and let sumi be the realization of St zityit . • Recall that St zityitare sufficient for ai. • The conditional likelihood of the data set is • This likelihood does not depend on {ai}, only on b. • Maximizing it with respect to b yields root-n consistent estimates. • The distribution of Si is messy and is difficult to compute.

40. Poisson distribution • The Poisson is the most widely used distribution for counted responses. • Examples include the number of migrants from state to state and the number of tort filings within a state. • A feature of the fixed effects version of the model is that the mean equals the variance. • To illustrate the application of Poisson panel data models, let’s use the canonical link and zit = 1, so that ln E (yit | ai) = g(E (yit | ai) ) = qit = hit = ai + xit b . • Through the log function, it links the mean to a linear combination of explanatory variables. It is the basis of the so-called “log-linear” model.

41. Conditional likelihood estimation • We first examine the fixed effects model and thus assume that {ai} are fixed parameters. • Thus, E yit = exp (ai + xit b). • The distribution is • From Section 10.1, St yit is a sufficient statistic for ai. • The distribution of Styit turns out to be Poisson, with mean exp(ai) St exp(xit b) . • Note that the ratio of means, • does not depend on ai.

42. Conditional likelihood details • Thus, as in Section 10.1, the conditional likelihood for the ith subject is

43. Conditional likelihood details • where • This is a multinomial distribution.

44. Multinomial distribution • Thus, the joint distribution of yi1, ..., yiTi given Styit has a multinomial distribution. • The conditional likelihood is: • Taking partial derivatives yields: • where • . • Thus, the conditional MLE, b, is the solution of: