Because this estimator is motivated by specification of the first two moments, it can also be viewed as a method-of-moments estimator, a special case ofGMM
whose more general framework is unnecessary here as the number of equations (3.24) equals the number of unknowns. The first-order conditions nest as special cases those for the PoissonMLEandGLM, which replaceω(µi, ˜α) by µi.
3.2.6 Example: Doctor Visits
Consider the following example of the number of doctor visits in the past 2 weeks for a single-adult sample of size 5190 from the Australian Health Survey 1977–78. This and several other measures of health service utilization such as
Table 3.1. Doctor visits: actual frequency distribution
Count 0 1 2 3 4 5 6 7 8 9
Frequency 4141 782 174 30 24 9 12 12 5 1 Relative frequency .798 .151 .033 .006 .005 .002 .002 .002 .001 .000
Table 3.2. Doctor visits: variable definitions and summary statistics
Standard
Variable Definition Mean deviation
DVISITS Number of doctor visits in past 2 weeks .302 .798
SEX Equals 1 if female .521 .500
AGE Age in years divided by 100 .406 .205
AGESQ AGE squared .207 .186
INCOME Annual income in tens of thousands of dollars .583 .369
LEVYPLUS Equals 1 if private health insurance .443 .497
FREEPOOR Equals 1 if free government health insurance due to low income .043 .202 FREEREPA Equals 1 if free government health insurance due to .210 .408
old age, disability or veteran status
ILLNESS Number of illnesses in past 2 weeks 1.432 1.384
ACTDAYS Number of days of reduced activity in past 2 weeks due to .862 2.888 illness or injury
HSCORE General health questionnaire score using Goldberg’s method 1.218 2.124
CHCOND1 Equals 1 if chronic condition not limiting activity .403 .491
CHCOND2 Equals 1 if chronic condition limiting activity .117 .321
days in hospital and number of medicines taken were analyzed in Cameron, Trivedi, Milne, and Piggott (1988) in the light of an economic model of joint determination of health service utilization and health insurance choice. The particular data presented here were also studied by Cameron and Trivedi (1986). The analysis of this example in this chapter (see also sections 3.3, 3.4, 3.5.1, and 3.7.4) is more detailed and covers additional methods.
The dependent variable DVISITS is summarized in Table 3.1. There are few large counts, with 98% of the sample taking values of 0, 1, or 2. The mean number of doctor visits is .302 with variance .637. The raw data are therefore overdispersed, although inclusion of regressors may eliminate the overdisper- sion.
The variables are defined and summary statistics given in Table 3.2. Regres- sors can be grouped into four categories: socioeconomic: SEX, AGE, AGESQ, INCOME; health insurance status indicators: LEVYPLUS, FREEPOOR, and FREEREPA, with LEVY (government Medibank health insurance) the omitted category; recent health status measures: ILLNESS, ACTDAYS; and long-term health status measures: HSCORE, CHCOND1, CHCOND2.
Table 3.3. Doctor visits: PoissonPMLEwith different standard error estimates
Standard errors
Coefficient, t Statistic,
Variable PoissonPMLE MLH MLOP NB1 NB2 RS Boot NB1 ONE −2.224 .190 .144 .219 .207 .254 .265 −10.16 SEX .157 .056 .041 .065 .062 .079 .076 2.42 AGE 1.056 1.001 .750 1.153 1.112 1.364 1.411 .92 AGESQ −.849 1.078 .809 1.242 1.210 1.460 1.547 −.68 INCOME −.205 .088 .062 .102 .096 .129 .130 −2.02 LEVYPLUS .123 .072 .056 .083 .077 .095 .101 1.49 FREEPOOR −.440 .180 .116 .207 .188 .290 .294 −2.12 FREEREPA .080 .092 .070 .106 .102 .126 .133 .75 ILLNESS .187 .018 .014 .021 .021 .024 .025 8.88 ACTDAYS .127 .005 .004 .006 .006 .008 .008 21.87 HSCORE .030 .010 .007 .012 .012 .014 .015 2.59 CHCOND1 .114 .066 .051 .077 .071 .091 .087 1.48 CHCOND2 .141 .083 .059 .096 .092 .122 .121 1.47 −lnL 3355.5
Note: Different standard error estimates due to different specifications ofω, the conditional vari-
ance of y.MLH,ω = µ hessian estimate;MLOP,ω = µ summed outer product of first derivatives estimate;NB1,ω = φµ = (1 + α)µ where here α = .328;NB2,ω = µ + αµ2where hereα = .286;
RS, unspecifiedω robust sandwich estimate;Boot, unspecifiedω bootstrap estimate.
The Poisson maximum likelihood estimates defined by (3.4) are given in the first column of Table 3.3. These estimates are by definition identical to the PoissonPMLestimates. Various estimates of the standard errors are given in the remainder of the table, under different assumptions about the variance of y, where throughout it is assumed that the conditional mean is correctly specified as in (3.2). Standard errors are presented rather than t statistics to allow comparison with the precision of alternative estimators given in later tables.
TheMLHstandard errors are the usual maximum likelihood standard errors using the inverse of the Hessian (3.6). If instead one uses the summed outer product of the first derivatives, the resultingMLOPstandard errors using (3.7) are in this example on average 25% lower thanMLHstandard errors. Comparison of (3.6) and (3.7) shows that this is consistent withE[(yi − µi)2| xi] = φµi
where 1/√φ .75 or α = (φ − 1) .78. More generally for overdispersed data theMLOPstandard errors will be biased downward even more than are the usualMLHstandard errors (3.6).
The columns labeled MLH,NB1, andNB2 specify that the variance of y equals, respectively, the mean, a multiple of the mean, and a quadratic function of the mean. The standard errorsNB1are 1.152 timesMLH standard errors, because ˆφNB1= 1.328 using (3.17), which has square root 1.152. The standard
errorsNB2are obtained using (3.15), where (3.19) yields ˆαNB2= 0.286. These
estimated values of α are not reported in the table, as they are not used in forming an estimate ofβ. They are used only to obtain standard errors of the
PMLEofβ.
Other count applications yield similar results. In the usual case in which data are overdispersed, the MLHandMLOP standard errors are smaller than
NB1andNB2standard errors and should not be used. The differences can be much greater than in this example if data are greatly overdispersed. One should never useMLHorMLOPhere.
The column labeledRSuses the robust sandwich estimates given in (3.20). These are roughly 20 percent larger thanNB1andNB2standard errors. One pos- sibility is that the robust sandwich estimates are biased, due to being influenced by outliers that can lead to large values of (yi− ˆµi)2in (3.20), even in a sample
as large as 5190. One way to assess this is through a bootstrap. The bootstrap standard errors in this situation can be shown to be small-sample–corrected es- timates of the robust sandwich standard errors. The columnBootuses bootstrap estimates with 200 replications. The bootstrap procedure to estimate standard errors, and to conduct hypothesis tests, is detailed in section 5.5.1. The boot- strap standard errors are generally within 5 percent ofRS, indicating little bias in standard error estimation for this example with n= 5190.
Which standard errors should be used? If one is willing to specify that ωi = φµi(orωi = µi+ αµ2i), then one can useNB1(orNB2)standard errors.
If one is unwilling to impose such variance functions, then one can use RS
standard errors in large samples and bootstrap in small samples. In practice
NB1standard errors are very appealing, due to the computational advantage of being a simple rescaling ofMLHstandard errors often reported by maximum likelihood routines. This is also theGLMapproach. It seems to work well in practice and clearly is far superior to using maximum likelihood standard errors, although there appears to be scope for further analysis.
The final column of Table 3.3 gives t statistics based on theNB1standard errors. By far the most statistically significant determinants of doctor visits in the past 2 weeks are recent health status measures – number of illnesses and days of reduced activity in the past 2 weeks – with positive coefficients, confirming that sicker people are more likely to visit a doctor. The long-term health status measure HSCORE and the socioeconomic variable SEX are also statistically significant. Discussion of the impact of these variables on the number of doctor visits is deferred to section 3.5.