10. Generalized linear models

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# 10. Generalized linear models - PowerPoint PPT Presentation

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

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

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.
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.
• 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.
• 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.
• 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.
10.1.3 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
• 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.
• 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
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.
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,
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
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
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.
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
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.
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.
Robust estimation of standard errors
• empirical standard errors may be calculated using the following estimator of the asymptotic variance of bEE
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
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
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 .
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).
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.
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).
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)}

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) .

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.
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.
• 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.
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.
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.
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.
Conditional likelihood details
• Thus, as in Section 10.1, the conditional likelihood for the ith subject is
Conditional likelihood details
• where
• This is a multinomial distribution.
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: