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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

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10 generalized linear models
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
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
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
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
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
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.
bernoulli illustration of links
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
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
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
mles canonical links
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
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
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
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
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
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
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.
marginal model
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
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
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
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
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 filings1
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
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
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
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
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
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
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 likelihood1
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
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.
mles canonical links1
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.
conditional likelihood estimation
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
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 estimation1
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
Conditional likelihood details
  • Thus, as in Section 10.1, the conditional likelihood for the ith subject is
conditional likelihood details1
Conditional likelihood details
  • where
  • This is a multinomial distribution.
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:
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