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10. Generalized linear 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
- 10.1.2 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.

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.

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 = xitb .

- 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 = xitb, 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 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.

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.

10.1.3 Estimation

- Begin with likelihood estimation for canonical links
- Consider responses yit, with mean mit, systematic component hit = g(mit) = xitb and canonical link so that hit = qit.
- Assume the responses are independent.

- Then, the log-likelihood is

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.

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

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(xitβ) / 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(xitβ),

- 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 modelGeneralized estimating equations simply use a

- 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 + xitb, this is the GLS estimator introduced in Section 3.3.

Consistency of GEEs simply use a

- 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 simply use a

- empirical standard errors may be calculated using the following estimator of the asymptotic variance of bEE

GEE - correlation parameter estimation simply use a

- 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 simply use a

- 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 simply use a

- 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 + xitb .

Random effects models simply use a

- 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 simply use a

- Conditional on ai, the likelihood for the ith subject at the tth observation is
- where b¢(qit) = E (yit | ai) and hit = zit ai + xitb= 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 simply use a

- 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 simply use a

- The covariance between two responses, yi1 and yi2 , is
Cov(yi1 , yi2 ) = E yi1yi2 - E yi1 E yi2

= E {b¢(ai+xi1b) b¢(ai+xi2b)}

- E b¢(ai+xi1b) E b¢(ai+xi2b)

- 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 simply use a

- Normality
- For the normal distribution we have b¢(a) = a.
- Thus, Cov(yi1 , yi2 )
= E {(ai+ xi1b) (ai + xi2b)} - E (ai + xi1b) E (ai + xi2b)

= E ai2 + (xi1b) (xi2b)- (xi1b) (xi2b)= Var ai.

- For the Poisson, we have b¢(a) = ea. Thus,
E yit = E b¢(ai+ xitb) = E exp(ai+ xitb)

= exp(xitb) E exp(ai) and

- Cov(yi1 , yi2 )
= E {exp(ai+ xi1b) exp(ai+ xi2b)} - 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 simply use a

- 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 simply use a

- Consider responses yit, with mean mit, systematic component hit = g(mit) = zitai + xitb 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.

MLEs - Canonical links simply use a

- 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 simply use a

- Assume the canonical link so that qit= hit = zitai + xitb .
- 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 simply use a

- 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 simply use a

- 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 simply use a

- Thus, as in Section 10.1, the conditional likelihood for the ith subject is

Conditional likelihood details simply use a

- where
- This is a multinomial distribution.

Multinomial distribution simply use a

- 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:

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