Medical Care Demand and Health Systems Portuguese Case Isabel Pereira Departament d’Economia i d’Història Econòmica Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona), Spain
March 2004
Abstract In the Portuguese health system, both public and private components co-exist, allowing a double bene…t for many citizens and, therefore, a selective choice of both services. The main purpose of this empirical study is to search in which way the medical demand behavior of those who bene…t from sub-systems di¤ers from those who only have access to the National Health Service (NHS). A complementary question is related with possible evidences that distinct demand may be induced by sub-systems. Econometric procedures used are: Logit model, Probit model and Linear Probability model (LPM). JEL Classi…cation: I10 Keywords: Sub-systems, NHS, Logit, Probit, LPM.
1
Introduction
Inspired in the British National Health Service, the Portuguese National Health Service (NHS) was created in 1976 by the Portuguese Constitution, being universal, general and free. In 1979, it was enacted the NHS Law, anticipating that in cases where it would be impossible to render medical assistance by public means, those treatments could be provided by external entities through the establishment of contracts or reimbursement of the users. Accordingly, by the end of 70s, in Portugal, there was a public healthcare system that theoretically reached 100% of population, together with several entities named sub-systems, usually associated with labor activities. These did not only take upon themselves health expenditures of their bene…ciaries and relatives, but sometimes also provide medical care.1 As far as the present study is concerned, we use the sub-system expression to refer to all Portuguese health systems rather than NHS. Although we recognize that this can be sometimes an abusive application, the data used doesn’t allow us another option and we are quite certain that the incongruities that might exist can be taken as insigni…cant. I would like to thank the useful comments on earlier versions from Carlos Gouveia Pinto, Miguel Gouveia, Leonor Modesto, Joao Caravana Santos Silva, Patricia Cruz, Jean-Pierre Gomes and seminar participants at IDEA-UAB. I am especially grateful for the suggestions from my supervisor Pedro Pita Barros. All errors are, of course, mine. This study was developed when I was at the Master Program of Faculdade de Economia da Universidade Nova de Lisboa. 1 To have more precise information about sub-systems, please see Annex, section 5.1..
In a health system where both public and private components co-exist, we can see that there has been a double bene…t for many citizens, allowing them a selective choice of both services (Reis, 1999). Besides this, is almost consensual among public opinion that sub-systems "favour a better health system and more satisfaction, with lower per capita cost than NHS and with control of the relation cost/quality; in summary, better management and more satisfaction" (Leal, 1999). Having the modest belief that this is not a close issue, our purpose is to study in which way the demand behavior of those who bene…t from sub-systems medical care di¤ers from those who only have access to the NHS. Di¤erent reasonings can lead us to expect opposite medical demand behavior of sub-systems bene…ciaries. On the one hand, accepting the connection between medical behavior and health state, the argument that sub-systems favour a better health state can lead us to forecast a smaller demand for medical services, and if we assume that the worse someone feels, the greater the propensity to go to a specialist, then this will translate in an even smaller demand for specialized treatments. On the other hand, the existence of an insurance easily accessible and with restricted participation of the consumer in the direct payment of medical care, may encourage the demand for those treatments. This lead us to defend sub-systems’high accessibility and motivates us to anticipate a higher propensity to demand medical services among those who bene…t from sub-systems, when comparing with whom only have access to NHS. Therefore, we apparently have two di¤erent e¤ects in‡uencing medical care demand of sub-systems. To throw light on which demand e¤ect is dominant for the Portuguese case is essentially an empirical question and is the main purpose of the present study. To pursue this goal, we turn to the data collected by the National Health Survey in Portugal during the years of 1995 and 1996.2 A primary but rudimentary approach to the potential resolution of our question can be done by analyzing the person’s behavior towards medical appointments, in the last three months, crosstabled with the mostly used health system:3 Type of Demand Specialist General Practitioner Didn’t go to doctor Total
Sub-systems 780 (26.6%) 936 (32%) 1.212 (41.4%) 2.928 (100%)
NHS 1.783 (13.3%) 5.801 (43.1%) 5.871 (43.6%) 13.455 (100%)
Total 2.563 (15.6%) 6.737 (41.1%) 7.083 (43.2%) 16.383 (100%)
Source: INE(1997)
At a glance, our impetus may be to conclude in favour of the second hypothetical behavior introduced, by which sub-systems bene…ciaries have a greater tendency to go to a medical appointment, comparing to those who are only within NHS’protection. However, this can be an inconsiderate conclusion for two essential reasons: …rst, the percentages shown don’t allow us to make such a clear distinction (among those who have demanded a specialized consultation, the proportion of sub-systems’ bene…ciaries is larger, but NHS seems to be associated with a higher propensity to consult a general practitioner); second and mainly, because the characteristics of the citizens who use sub-systems services can be di¤erent from those who have only access to the NHS. Therefore, we need to control those features and to isolate the health system e¤ect. This is done in the next section using Logit Model. 2 Although
the Survey has 49 718 cases, only 16 383 are manageable for the econometric treatment of this study. to data’s limitations, we are restricted to consider the type of the last medical consultation occurred in the previous three months. 3 Due
1
Alternatives models are: Linear Probability Model (LPM) and Probit Model. If LPM has the simplicity advantage, the other two are unanimously considered more adequate tools to solve some of LPM limitations when analyzing a categorical variable as we have here, namely the Type of Medical Demand. Between Logit and Probit techniques, "the di¤erences that are apparent, rarely seem to matter much in practice" (Davidson and Mackinnon, 1993). So the choice for Logit model is justi…ed mainly by calculus simplicity. Consistency of results among the di¤erent models estimated, reinforces the presentation of only one model. Finding evidence of di¤erent medical treatment demand, another interesting question is then possible: can this distinct demand be induced by sub-systems themselves? If this is so, we shall expect subsystems’ bene…ciaries to demand medical services with a di¤erent health state than the one felt by NHS bene…ciaries, whenever they require medical assistance. We try to clear up this supply e¤ect, as a complementary purpose of the present study. The rest of this paper is organized as follows: in section 2, we formalize the analysis of di¤erent demand behavior regarding individual characteristics and health system; in section 3, we try to develop a theoretical way of detecting induced behavior; in section 4, we conclude.
2
Di¤erent Demand Behavior ?
As it was previously clari…ed, our main goal is to explain how the individual’s medical care demand changes with his own characteristics and with the chosen health system. For this purpose, we may consider that each individual t has a health state index Ht , which depends on K personal characteristics: Ht =
0
+
1 X1t
+ ::: +
K XKt
where ut is a random variable with mean zero and variance
+ ut 2
(1)
and t = 1; :::; T .
However, in the real world what we observe is a particular behavior of medical demand, Yt . Thus, we can consider Yt as an ordinal indicator of individual’s health state. For simplicity, we start by considering this indicator as: 8 0, in the last three months the person inquired went at least > > < to one medical appointment, if Ht Yt = > 1, in the last three months the person inquired didn’t go to any > : medical appointment, if Ht > where
is a parameter that is normalized to 0 (zero), following the usual econometric procedures.
The most commonly used approaches to estimate this type of model are: 1. the Linear Probability Model (LPM), 2. the Binary Logit Model, 3. the Binary Probit Model. Despite the simplicity of LPM, the other two models have in their behalf the advantage of more adequate modelization of our problem, since we consider a discrete and categorical dependent variable (Yt ). Going through three di¤erent approaches may also give us more con…dence in results. However, simplicity and similarity of results allow us to focus only in one of those three models, Logit.
2
2.1 2.1.1
Binary Logit The Model
The attractiveness of using LPM is directly connected with the simplicity of estimating it by OLS. However, since the 1970’s there has been increasing literature on qualitative dependent variable models, which stands up for more appropriate econometric approaches.4 The main argument is based on the fact that LPM doesn’t take into account the discrete and ordinal nature of the dependent variable, whose codes just re‡ect a ranking with meaningless di¤erences between categories. Furthermore, Gujarati (1995) reveals that: "(LPM) assumes that the probability of Yt increases linearly with Xt , that is, the marginal or incremental e¤ect of Xt remains constant throughout (...) but what we need is a probability model which approaches zero at slower rates as Xt gets small and approaches one at slower and slower rates as Xt gets very large." Therefore, to explain how the probability of each of the categories of the medical demand behavior changes with the characteristics of the individual, it becomes clear that a cumulative distribution function of a random variable is more appropriate:5 p0
= P r(Yt = 0) = P r(Ht 6 0) = P r(ut 6 =
p1
P r(ut 6
1
0
+ Xt ) = 1
F(
0
0
+ Xt )
= P r(Yt = 1) = P r(Ht > 0) = P r(ut > = F(
0
Xt )
0
+ Xt )
(2) Xt ) (3)
The most popular alternatives considered in the literature for this cumulative function F (:) are the logistic and the normal distributions, given rise to Binary Logit and Binary Probit models, respectively. In the Binary Logit model, it is assumed that the cumulative distribution of ut is logistic, with var(ut )= 2 ; which means that expressions (2) and (3) become:
p0
p1
where
= P r(Yt = 0) = 1 ( 0+ exp( 0 + Xt ) = 1 1 + exp( 0 + Xt ) = P r(Yt = 1) = ( 0 + exp( 0 + Xt ) = 1 + exp( 0 + Xt )
Xt ) (4)
Xt ) (5)
is the standard logistic distribution function.
As far as the sample of this study is concerned, we assume it is i.i.d., so the method of maximum likelihood can be used to estimate the model. Therefore, the likelihood function and its logaritm come 4 Greene(1993)
refers that earliest works on Probit modeling were applications of grouped data in laboratory experiments, made by Finney(1971) and Cox(1970). The introduction of the Ordered Probit and Logit models was due to Zavoina & McElvey(1975). 5 For simplicity of notation, from now on we refer to as the K column vector of coe¢ cients and Xt as the K row vector of personal characteristics for individual t.
3
as: L =
T Y 1 Y
P r(yt = jjxt ) =
t=1 j=0
=
T Y
[1
T Y
P r(yt = 0jxt ):P r(yt = 1jxt )
t=1
(
0
+
xt )]It0 :[ (
0
xt )]It1
+
(6)
t=1
l(
)
= ln L =
T X t=1
fIt0 : ln[1
+It1 : ln[ ( where: Itj =
1 0
if yt = j otherwise
and
0
+
j = 0; 1
(
0
+
xt )]g
, and
xt )]+ (7)
=
:
In the Binary Probit, the development of the model is similar to the previous one, but assuming that the residual ut s N (0; 2 ) or, equivalently, zt = ut s N (0; 1): The maximum likelihood estimates of to the unknown coe¢ cients, iteratively.
can be obtained, maximizing expression (7) with respect
As Crawford et al.(1998) mentioned, "a feature of qualitative choice models is the absence of an easy interpretation of how the outcome probability varies in response to changes in the explanatory variables". Thus, when interpreting the estimates achieved, we must have in attention that: 1. There is a normalization done to the parameters, which means we only have an estimation for the ratios = . To overcome this problem partially, it is usually assumed that for the Binary Logit model = p3 and for the Binary Probit model = 1. Nevertheless, as mentioned by Becker and Kennedy(1992): "to the absolute magnitude of the coe¢ cient cannot be given any meaning". 2. The interpretation of regressors’coe¢ cients di¤er most as the explanatory variables are continuous or dummies. In the present study we use both methods, regarding the treatment of explanatory variables. At …rst, all variables are converted in dummies variables, which means that in interpreting the estimates of k (k = 1; :::; K), we must have in mind that they are just an indicator of how a speci…c category of a qualitative characteristic di¤erentiates itself from the class chosen to be the basis for that qualitative feature. In a second step, we have the opportunity to take some regressors as "continuous" variables.6 In that case, the estimates of k obtained after the normalization by , don’t give us directly the impact of an unitary change of the explanatory variable on the probability of the dependent variable. This is so, because from expressions (2) and (3) there comes: @p0 = f ( 0 + X): k @Xk @p1 = f ( 0 + X): k @Xk
(8) (9)
where f (:) denotes the probability density function of the logistic distribution, if we consider the Binary Logit model, or the probability density function of the normal distribution, if we consider the Binary Probit model. 6 The
term "continuous" is used in an abusive way, since in fact we joint the information of several dummies into discrete variables.
4
As a consequence, we realize that the e¤ect of a continuous regressor on the probability of the dependent variable depends not only on the magnitude of the slope coe¢ cient, but also on its sign and on the pattern of the probability density function considered. This complexity is probably one reason for the extensive use of LPM. Which model is preferable, Logit or Probit, it is a question with no straight answer. Cumulative logistic and the normal distributions are very similar to each other, as they both have probability density functions which are symmetric around the mean. The main di¤erence comes at the tails, where logistic distribution is ‡atter than the normal function, which means the normal curve approaches the axes more quickly. In spite of the similarities between the two models, to compare estimations of the coe¢ cients k achieved from each of the models, it is necessary to pay attention that the value of the variance assumed for p each one is di¤erent. In fact, only if we multiplied the k estimations obtained in Binary Logit by 3 ; it would probably approximate the results of Binary Probit. Davidson and Mackinnon(1993) also reinforce this idea of di¤erent results for the coe¢ cients estimates, when they said that because of "the 2 way the elements of are scaled, the variance of the distribution for the logistic function is 3 , while for the normal is unitary. The Logit estimates therefore all tend to be larger than the Probit estimates". Nevertheless, many authors, such as Gujarati(1995) concludes that "the choice between the two is one of mathematical convenience and ready availability of computer programs". As far as this study is concerned, this argument of calculus simplicity favours the use of Logit model, specially when speci…cation tests are considered. 2.1.2
Speci…cation Tests for Binary Logit
Greene(1993) suggests starting with the usual way of testing individual signi…cance of the coe¢ cients’ estimates, based on t tests. Then, he pursues with other familiar methods for more involved restrictions: Wald test and Likelihood Ratio test.7 In the classical regression model, the existence of omitted variables and heteroscedasticity result in biased estimates for the coe¢ cients in the …rst problem and in ine¢ cient but consistent estimates in the latter case. However, Yatchew and Griliches(1984) analyze the consequences of those drawbacks in the Logit and Probit models and …nd out that in both cases, the e¤ects are even worse. In fact, they detect that if there are omitted variables, the coe¢ cients of the included ones become inconsistent and if the residual of the underlying regression presents heteroscedasticity, the maximum likelihood estimators are not only ine¢ cient but also inconsistent. As Greene(1993) notices "the second result is particularly troubling in view of the fact that the Probit (Logit) model is most often used with microeconomic data, which is frequently heteroscedastic." Speci…cation tests referring to heteroscedasticity problem, follow the methodology proposed by Godfrey(1988). According to this author, the null hypothesis of homoscedasticity of the residuals can be tested against heteroscedasticity of the form 2t = h(Zt0 ); where Zt is a q-dimensional vector of exogenous variables and is a q-dimensional vector of unknown parameters, using as test statistic the explained sum of squares (here labelled as ESS) for the OLS regression of the variable
on the (k + q) variables 7 For
h
yt
1
Ft (^)
i h = Ft (^): 1
further details, please see Annex, section 5.2.1..
5
Ft (^)
i0:5
h
i h ^ Xt :Zt ; Xt :ft (^)= Ft (^): 1
i0:5
Ft (^)
h i0:5 h 1 ^ ^ where Ft (^) = 1+exp( Ft (^) = Ft (^): 1 ^ Xt ) and ft ( )= Ft ( ): 1 model, and Ft (^) = 1 ( ^ Xt ) and ft (^) = ( ^ Xt ) for the Probit model.
Ft (^)
Comparing the ESS values thus obtained with upper-tail critical values of not the existence of equal variance for the residuals.
2.2 2.2.1
2
i0:5
for the Logit
(q), we may reject or
Ordered Logit The Model
With a small increase in sophistication, we can further consider a model with three categories for the indicator variable: 8 > 0, the last medical appointment was to a specialist, if Ht 1 > > > > < 1, the last medical appointment was to a general practitioner (GP), Yt = if 1 < Ht 2 > > > 2, in the last three months the person inquired didn’t go to any > > : medical appointment, if Ht > 2
The econometric procedures normalize 1 to be 0 (zero) and estimate vector (please refer to expression (1)) along with .
2
= (> 0), as usually, and try to
The econometrics models used are now: 1. the Linear Probability Model (LPM),8 2. the Ordered Logit Model, 3. the Ordered Probit Model. Using the same arguments as in the binary context, we give more evidence to Ordered Logit Model. In the case of a dependent variable with three categories, the probability of each one can be expressed as:
p0
p1
= P r(Yt = 0) = P r(Ht =
1
=
1
0) = P r(ut
( 0 + Xt ) = 1 ( exp( 0 + Xt ) 1 + exp( 0 + Xt )
0
0
=
Xt )
=
( 0 exp( 1 + exp(
0 0
Xt ) ( Xt ) Xt )
8A
Xt )
Xt ) (10)
= P r(Yt = 1) = P r(0 < Ht = P r(ut
+
0
) P r(ut 0
Xt ) exp( 0 1 + exp(
0
Xt )
0
Xt ) Xt )
(11)
more developed theoretical analysis of LPM, for the case of an indicator variable Yt with three categories, can be seen in Annex, section 5.2.3..
6
p2
= P r(Yt = 2) = P r(Ht > ) = P r(ut =
1
( 0 exp( 1 + exp(
= 1 where
Xt )
0
Xt ) Xt ) Xt )
0 0
(12)
is the standard logistic distribution function and we consider
=
and
=
.
The likelihood function and its logaritm change accordingly:
T Y 2 Y
L =
P r(yt = jjxt )
t=1 j=0 T Y
=
P r(yt = 0jxt ):P r(yt = 1jxt ):P r(yt = 2jxt )
t=1 T Y
=
[1
(
0
+
xt )]It0 :[ (
xt )
0
(
xt )]It1 :
0
t=1
:[1
l(
(
;
)
0
=
ln L =
xt )]It2 T X t=1
(13)
fIt0 : ln[1
+It1 : ln[ ( +It2 : ln[1 where: Itj =
2.2.2
1 0
if yt = j otherwise
and
(
xt )]+ (
xt )]+
0
xt )]g
0
j = 0; 1; 2
+
xt )
0
(
0
, and
(14) ,
=
=
:
Speci…cation Tests for Ordered Logit
Based on a Lagrange Multiplier (LM) test and on results derived by Davidson and Mackinnon, Murphy(1996) deduces mis-speci…cation tests for Ordered Logit model in such convenient and practical way, which become the turning-point in favour of Ordered Logit, against Ordered Probit. These LM tests for omitted variables, neglected heteroscedasticity and asymmetry are calculated through an arti…cial regression. Let us see how it works. In a discrete choice model with T individuals, denoted by t = 1; :::; T , and three ordered alternatives,9 we must …rst de…ne:
G0t
= G( =
G1t
(1 + exp(
= G( =
0
0
(1 + exp(
Xt ) = P r(ut > Xt ))
0
1
Xt ) = P r(ut > Xt ))
0
9 Murphy(1996)
Xt )
0
1
(15) 0
Xt ) (16)
refers generically to J + 1 alternatives, but in this study J = 2, so it will be used this particular value in this brief theoretic exposition.
7
P0t P1t
= 1
G0t
= G0t P2t
(17)
G1t
(18)
= G1t
(19)
Then, we know that: If there are I omitted variables Mit (i = 1; :::; I), the previous de…ned G0t (15) and G1t (16) are transformed in, respectively: L0t
= L(
Xt
0
= (1 + exp( L1t
= L( =
Mt ) Xt
0
0
Xt
(1 + exp(
0
1
Mt ))
(20)
Mt ) Xt
Mt ))
1
(21)
Under the null hypothesis that the original speci…cation is correct and there is no omitted variables, = 0. We can easily con…rm that a particular case of this test occurs when we consider Mt to be powers of ^ Xt , which turns out to be the RESET-like test suggested by Pagan and Vella(1989). If the residuals ut are heteroscedastic with a particular pattern: G0t (15) and G1t (16) are transformed in, respectively: L0t
= L(
0
exp(wt0 )
= (1 + exp(
L1t
= L(
exp(wt0 )
exp(wt0 )
0
exp(wt0 )
= (1 + exp(
exp(wt0 )
Xt ))
Where wt is a vector of some explanatory variables and that there is homoscedasticity, = 0.
2
: exp(2wt0 ) 3
, the previous
1
(22)
Xt )
0
exp(wt0 )
=
Xt )
0
exp(wt0 )
2 t
exp(wt0 )
Xt ))
1
(23)
is a vector of coe¢ cients. Under the null
Although Murphy(1996) didn’t mention any particular form for wt , we will consider wt as powers of ^ Xt , where ^ is the estimate of under the null hypothesis. If the distribution of the disturbances ut follows a more general shape not necessarily symmetric, then the correct G0t and G1t are, respectively:
8
L0t L1t where
= L( = L(
0
Xt ) = (1 + exp(
0
Xt ) = (1 + exp(
Xt ))
0
0
(24)
Xt ))
> 0: Clearly, if we want to test the hypothesis of symmetry then
(25) = 1:
To obtain the test statistic necessary for the purpose mentioned, Murphy de…nes:
where j=0,1,2 and
0
= ( 0;
0
;
0
ujt
=
w ^jt
=
; 0 ; ).
(Yj Pjt ) p Pjt @Pjt 1 :p @ Pj
(26)
(27)
Finally, the LM test statistic is no more than the explained sum of squares resulting from the auxiliary regression of the ujt on the w ^jt , across all three alternatives and T individuals. As it should be evident, ( 0 ; 0 ) are replaced for their estimates, under the null that there is no mis-speci…cation on the initial Ordered Logit. By Murphy(1996) we can also know that this test-statistic has a chi-squared distribution, with as much degrees of freedom as the number of restrictions tested. As a personal preferred methodology, the di¤erent hypothesis are tested separately, because as Murphy(1996) notices "single or joint LM test statistics, incorrect functional form and asymmetry may be readily calculated."
2.3
Data
For this study, we use the National Health Survey, collected in Portugal during the years of 1995 and 1996 (INE, 1997). It was made to 49 718 individuals of …ve di¤erent Portuguese geographic regions (named North, Center, Lisboa and Vale do Tejo, Alentejo and Algarve). Its purpose was to obtain personal information in several domains, such as personal characteristics, consumption habits and health related issues.10 From the 195 answers given by each person inquired in this survey, we select 17 variables, sometimes constructed based on multiple and related questions. After the exclusion of records for which there are no information for at least one variable of interest, we remain with a sample of 16 383 observations. Concerning the econometric procedures used, we begin by treating explanatory variables as dummies variables, as they refer to qualitative and/or scaled variables. Afterwards, we transform some of the regressors as continuous, attempting not only to achieve an extensive interpretation, but also to obtain evidence of estimation stability. 1 0 Being more accurate, the survey concerns the following general topics: demographic characterization, general health information, temporary incapacity, long term incapacity, chronic diseases, medical care, expenditure and income, tobacco consumption, food and drink consumption, children health, physical activity.
9
The explanatory variables can be summarized as: Variable Medical Exams 11 Exam Drinks 12 Dr1 Dr2 (basis) Cigarettes 13 Cig1 Cig2 Cig3 (basis) Marital Status M s1 M s3 (basis) Schooling Sch0 Sch1 Sch2 Sch3 (basis) Exercise Exer1 Exer2 (basis) Working Hours W h2 W h3 W h4 (basis) Age Age2 Age3 Age4 (basis)
Brief Description at least one medical exam often drank per week drank once a week or once a month (rarely or never drunk ) 1 to 10 cigarettes a day 11 to 40 cigarettes a day 41 to 100 cigarettes a day (did not smoke) married separate or widow(er) (single)
1 5 10 (13
0 years of approval to 4 years of approval to 9 years of approval to 12 years of approval to 24 years of approval)
1 to 3 days a week to exercise 4 to 7 days a week to exercise (no physical exercise regularly) 30 to 40 hours a week 40 to 50 hours a week 50 to 80 hours a week (at most 30 hours a week ) 30 to 45 years old 46 to 65 years old more than 65 years old (at most 30 years old )
1 1 Here
we consider X-ray, ecography, clinical analysis or electrocardiogram. Period of analysis: previous 3 months. includes: wine, beer, husk, brandy, port wine, liqueur, whisky and gin. Period of analysis: previous 12 months. 1 3 It includes cigarettes, tobacco-pie and cigars. Period of analysis: previous 2 weeks. 1 2 It
10
Variable (cont.1) Body Mass Index 14 BM I1 BM I3 BM I4 (basis) Global Capacity Index 15 GCI1 GCI2 (basis) Sleeping Pills 16 Spills Reimbursement of Expenditures Rc Per-Capita Family Income 18 P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 (basis) Meals M e1 M e3 (basis) Gender Fe Health Evaluation 19 He3 He4 He5 (basis) Sub-Systems Ss (basis)
Brief Description low weight pre-obese obese (ideal weight ) GCI below 70 GCI between 70 and 90 (GCI above 90 ) took sleeping pills 17
received …nancial support at most 20.000 escudos per month 20.000 to 40.000 escudos per month 40.000 to 70.000 escudos per month 70.000 to 100.000 escudos per month 100.000 to 200.000 escudos per month (more than 200.000 escudos per month) at most 2 meals a day more than 3 meals a day (exactly 3 meals a day ) female reasonable ill very ill (good or very good ) sub-system bene…ciary (only NHS bene…ciary )
In a second approach, some groups of dummies variables are treated jointly as a unique regressor. 1 4 BMI is a measure of obesity that can be calculated from the ratio of the weight (in kilograms) to the square of the height (in meters). The World Health Organization established that a person shall be considered as having low weight if his BMI is at most 20, ideal weight if his BMI is between 20 and 25, pre-obese if his BMI lies between 25 and 30, and obese if his BMI is at least 30. 1 5 GCI aggregates information from 21 di¤erent questions made, all related with long duration incapacity. The construction of GCI uses one pre-scored multi-attribute health status classi…cation system, namely the Quality of Well-Being (QWB) (see Drummond et al., 1987). Using QWB scale, individuals are classi…ed according to four attributes: mobility (MOB), physical activity (PAC), social activity (SAC) and symptom-problem complex (CPX). For each category, we give a score between -1 and 0, according to the limitations felt. The well-being score for each individual is then calculated as: GCI = (1 + M OB + P AC + SAC + CP X)100: The result is a value above 0 (dead) and at most 100 (perfect health). 1 6 Period of analysis: previous 2 weeks. 1 7 Financial supports for expenditures related with consultation, internment, surgical operations, treatments or other health expenditures. 1 8 Period of analysis: previous month. 1 euro = 200.482 escudos 1 9 It referes to how the person inquired evaluates his own health state.
11
These variables are: Cigarettes, Schooling, Exercise, Working Hours, Age,20 BMI, GCI, Per-Capita Family Index and Meals. As far as the sample used in the present study is concerned, it is considered interesting to know how it may be characterized accordingly to the variables chosen as regressors. Therefore, from the 16 383 individuals inquired: Variable Medical Exams Exam Drinks Dr1 Dr2 (basis) Cigarettes Cig1 Cig2 Cig3 (basis) Marital Status M s1 M s3 (basis) Schooling Sch0 Sch1 Sch2 Sch3 (basis) Exercise Exer1 Exer2 (basis) Working Hours W h2 W h3 W h4 (basis)
In the Sample 31.1% did medical exams in the previous 3 months 40.6% drank often, per week 8.17% drank once a week or once a month 51.23% rarely or never drank liquors in the previous 12 months 4.54% smoked 1 to 10 cigarettes a day 11.37% smoked 11 to 40 cigarettes a day 0.4% smoked 41 to 100 cigarettes a day (83.69% didn’t smoke in the previous 2 weeks) 71.97% are married 11.56% are separated or widow(er) (16.47% are single) 3.333% have 0 years of approval 60.269% have 1 to 4 years of approval 21.071% have 5 to 9 years of approval 8.393% have 10 to 12 years of approval (6.934% have 13 to 24 years of approval) 5.39% practised exercise 1 to 3 days a week 2.57% practised exercise 4 to 7 days a week (92.04% didn’t practise exercise regularly) 19.14% worked 30 to 40 hours a week 18.49% worked 40 to 50 hours a week 6.6% worked 50 to 80 hours a week (55.77% worked at most 30 hours a week)
2 0 There
may exist some non-linear e¤ect from age, namelly a higher use of medical care in younger and senior citizens. To test such hypothesis, we include the square of age also as a regressor.
12
Variable(cont.1) Age Age2 Age3 Age4 (basis) Body Mass Index BM I1 BM I3 BM I4 (basis) Global Capacity Index GCI1 GCI2 (basis) Sleeping Pills Spills Reimbursement of Expenditures Rc Per-Capita Family Income P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 (basis) Meals M e1 M e3 (basis) Gender Fe (basis) Health Evaluation He3 He4 He5 (basis) Sub-Systems Ss (basis)
In the Sample 28.28% were from 30 to 45 years old 37.75% were from 45 to 65 years old 18.45% were more than 65 years old (15.52% were less than 30 years old) 6.34% had low weight 38.47% were pre-obese 12.98% were obese (42.21% had an ideal weight) 15.64% had a GCI below 70 47.37% had a GCI between 70 and 90 (36.99% had a GCI between 90 and 100) 14.65% took sleeping pills in the previous 2 weeks 0.57% received …nancial support in the previous 2 weeks 8.78% had at most 20.000 escudos, month 14.14% had between 20.000 and 40.000 escudos, month 28.8% had between 40.000 and 70.000 escudos, month 15.6% had between 70.000 and 100.000 escudos, month 21.19% had between 100.000 and 200.000 escudos, month (11.51% had more than 200.000 escudos, month) 9.96% ate at most 2 meals a day 10.56% ate more than 3 meals a day (79.48% ate 3 meals a day) 59.29% were women (40.71% were men) 44.25% considered to have a reasonable health state 18.95% considered to be ill 4.3% considered to be very ill (32.5% considered to have good or very good health) 17.87% were bene…ting from a sub-system (82.13% were bene…ting only from NHS)
Although 16 383 observations represents only 0.16% of total Portuguese population, we may have some statistical con…dence in conclusions presented, from a theoretical point of view. To test results’ representativeness, we could think on comparing sample characteristics with those of the Portuguese population. Nevertheless, this would end up to be quite di¢ cult, giving speci…city of some issues considered, for which the National Survey used is the main possible source of information. Having in mind this potential limitation of the data used when arguing results interpretation, we may proceed
13
with our empirical work.
2.4
Estimation and Results
2.4.1
21
Binary Logit
General Results Initially, we estimate a Binary Logit considering all regressors as dummies. The results obtained are in accordance with the ones for Binary Probit. Regarding speci…cation tests, when checking the signi…cance hypothesis, the results favour the insigni…cance of Marital Status, Exercise and BMI, among regressors. For the homoscedasticity test, the Godfrey methodology sheds some light on possible causes for non-homoscedasticity. In fact, when we run Godfrey test, Per-Capita Family Income and Age seem to be associated with residuals variance. Nevertheless, when we transform original binary models according to this new information, we obtain estimations that are very similar to the initial ones. Finally, a Binary Logit is estimated considering some regressors as continuous variables, as well as a Binary Probit. Again results are in line with each other. Since here we are mainly concerned with medical demand behavior among sub-systems, general results regarding all explanatory variables are presented in Annex, section 5.2.1.. Behavior in Sub-Systems Regarding the coe¢ cient of sub-systems dummy variable, the estimation result is the following: variable Ss
Binary Logit coe¤ . -.139288
Which means that the impact on each category of the dependent variable is given by: dummy Ss= 0 Ss= 1 change
p0 = P r(Yt = 0) 0.6830 0.7124 0.0294
Here we …nd that having the additional bene…t of a health sub-system is linked with a greater tendency to demand the medical services. In order to …nd reasons for the empirical result achieved, we must certainly have to consider that we may be dealing with a net demand e¤ect of di¤erent causes. On the one hand, accepting that the individual’s behavior towards medical appointments may be considered an indicator for his real health state, the argument that sub-systems favour a better health state through preventive medicine could lead us to forecast the opposite e¤ect than the one we achieve. That is to say, with better prophylatic medicine, we would expect that among sub-systems users the demand for medical services would be smaller, comparing with NHS. On the other hand, turning to other level of arguments that stand up for sub-systems being more accessible to their bene…ciaries, we can easily explain the estimation results of higher propensity to demand medical care. Therefore, without questioning the better preventive medicine argument, we may interpret the em2 1 The estimation presented makes use of the software TSP, version 4.4, programmed by Chris Ernest Hall, Clint Cummins and Daniel Perman, 1996-1998.
14
pirical results obtained as a net e¤ect, by which it dominates the fact that "higher accessibility induces a larger demand". 2.4.2
Ordered Logit
General Results Following the methodolgy proposed, we now consider a dependent indicator variable with three categories.22 The speci…cation tests made to the Ordered Logit model considering omitted variables, homoscedasticity and symmetry of the residuals, follow the approach proposed by Murphy(1996). As far as omitted variables are concerned, the test value favours the inclusion of Health Evaluation among regressors. However, the results achieved for symmetry and homoscedasticity tests are disappointing, as they strongly reject both assumptions for the residuals estimated (please refer to Annex, section 5.2.2.). Owing to the serious consequences of residuals heteroscedasticity, early stated, we took its resolution aim through several alternative methods (please refer to Appendix, section 5.2.2.). However, in spite of all essays to overstep the heteroscedasticity problem, the conclusion obtained from the speci…cation tests is always discouraging (when divergence of estimations caused by numerical errors hasn’t stopped us earlier). Nevertheless, in all possible estimations for Ordered Logit model, coe¢ cients signs and magnitudes are quite stable (and in accordance with Ordered Probit as well as LPM), which give us some encouragement to pursue our analysis. As stated for the binary approach, we present in Annex results for all variables, which allow us to concentrate here in our main issue: medical care demand among subsystems. In section 5.2.2., we can see signi…cance tests made for Ordered Logit, as well as estimation outcomes. Behavior in Sub-Systems In the case of three possible types of medical demand behavior, the estimation of the coe¢ cient for sub-systems bene…ciaries is given by: variable Ss
Ordered Logit coe¤ . -.313675
which means that the impact on the probability of each category of the dependent variable can be estimated as: dummy Ss=0 Ss=1 change
p0 = P r(Yt = 0) :0862 :1143 :0281
p1 = P r(Yt = 1) :4903 :5363 :0461
p2 = P r(Yt = 2) :4235 :3493 :0742
As we can conclude, the values given above are totally in accordance to what we may conclude from the binary analysis. In fact, we reinforce the evidence that having the additional bene…t of a health sub-system is linked with a greater propensity to demand medical care. It is also interesting to notice that this e¤ect is causing a higher increase for general practitioner demand than it is towards specialists. Again, we may interpret these results as an evidence of the higher accessibility of sub-systems services, when comparing to NHS. Results obtained from Ordered Probit and LPM allow us to achieve similar conclusions. 2 2 Please remember that the dependent variable takes the value Y = 0 when the last medical appointment was to a t specialist, Yt = 1 when the last medical appointment was to a GP, and Yt = 2 when there was no medical appointment in the previous three months.
15
A general analysis of estimations’outputs for all explanatory variables, comparing the three econometric approaches, is presented in Annex section 5.2.4..
3
Induced Behavior ?
With evidence that the probability of demanding medical care is related with the health system chosen, we now look for signs that this di¤erent demand behavior may be induced by the health system itself. To achieve this purpose, we re-estimate Ordered Logit model allowing di¤erent threshold coe¢ cients for sub-systems and for NHS protection. In fact, if there exist di¤erent thresholds, we may interpret it as an evidence that citizens request medical services, and even request di¤erent medical cares (GP versus specialized) from distinct health states felt, depending on the health system they use. According to this reasoning, we may change the model formalization to: 8 > 0, the last medical appointment was to a specialist, if Ht 1 > > > > 1, the last medical appointment was to a general practitioner (GP), < Yt = if 1 < Ht 2 > > > 2, in the last three months the person inquired didn’t go to any > > : medical appointment, if Ht > 2 where
1
and
2
for NHS, we
have possible distinct values, regarding the health system. In fact:
N HS 1
and
N HS 2
;
for sub-systems, we would like to introduce: – –
SS 1 SS 2
= =
N HS 1 N HS 2
+ Ds1 = Ds1 , where Ds1 is a parameter to estimate; + Ds2 , where Ds2 is another parameter to estimate.
Taking into consideration this new approach, the probability of each category for the indicator variable can be re-written as:23
p0
= P r(Yt = 0) = P r(Ht = P r(ut
Ds1 Ss
= F (Ds1 Ss p1
= P r(Yt = 1) = P r(Ds1 Ss = P r(ut
+ Ds2 Ss
= F ( + Ds2 Ss p2
0
0
Ht Xt )
0
Xt )
Ds1 Ss) = Xt )
1
= 1
P r(ut
(28)
+ Ds2 Ss) P r(ut
Ds1 Ss
F (Ds1 Ss
= P r(Yt = 2) = P r(Ht =
Xt )
0
Xt )
Xt )
0
(29)
+ Ds2 Ss)
+ Ds2 Ss
F ( + Ds2 Ss
0
0
0
Xt )
Xt )
(30)
HS HS Due to non-convergence of estimating process, we can not estimate N ; N , Ds1 and Ds2 1 2 simultaneously. In fact, estimation constraints force us to divide this empirical research into two steps: 2 3 Remember
that Ss is a dummy variable indicating that the inquired person is a sub-systems’bene…ciary.
16
1. normalizing 24 2:
N HS 1
estimation output N HS 2 SS 2
2. considering
N HS 2
estimation output N HS 1 SS 1
= Ds1 = 0 (which means that
Ordered Logit estimation 2:89919 1:86557
SS 1
= 0) and allowing to estimate di¤erent
Ordered Probit estimation 1:60977 1:060455
= Ds2 = 0 (which means that Ordered Logit estimation -2:89919 -1:86557
SS 2
= 0) and allowing to estimate di¤erent
1:
Ordered Probit estimation -1:60977 -1:06046
At …rst glance, we may be tempted to develop an interpretation, comparing health state felt by bene…ciaries of each system, when they take the decision of consulting a GP rather than a specialist, or consulting a GP instead of not demanding any medical service. However, if we take a careful look, we conclude that the normalization imposed, due to estimations constraints, only gives information about the relative importance of GP demand. That is to say, both steps argue in favor of a higher demand of GP appointments on NHS, when comparing with sub-systems.
4
Conclusion
Summarizing our main purpose, we are looking for evidences that support or refute the argument that, in Portugal, sub-systems are associated with a higher health state of their bene…ciaries, when compared with National Health System (NHS). Audacious though it may be considered, this purpose becomes more attainable when we think over individual medical appointment behavior as an indicator for the real health state. In fact, during our study, the intuitive association of these ideas is strengthened by the reasoning that, the worse someone evaluate his health state, the higher the probability to demand a medical appointment and, may even higher, the propensity to consult a specialist doctor di¤erent from GP. Therefore, considering medical behavior as a health state indicator, we seek to investigate whether being a sub-system bene…ciary or not is associated with di¤erent probability for medical care demanding. Particularly regarding the argument that sub-systems are associated with a higher health state, it would then be expected a higher probability of didn’t go to any medical appointment in the previous three months among their bene…ciaries, and in the case where those services are demanding, a greater propensity to consult a general practitioner than a specialist doctor. Nevertheless, econometrics estimations point out in the opposite direction, linking sub-system protection with a higher probability of went to some medical appointment. This outcome may be, however, not totally surprising. In fact, being oriented to serve more restricted segments of population and, therefore, doesn’t su¤er much from strangling services supply, sub-systems ‡uent accessibility may induce a larger demand for medical cares. If this increment in demand is to be considered a moral hazard problem, can only be con…rmed comparing the actual level of treatments required to an optimal value, which requires an utility function estimation. This is an interesting issue to be developed in future studies. Finding evidence of di¤erent medical demand behavior associated with the health system chosen, another attractive and connected question is to investigate whether the health system chosen can induce 2 4 General
results are presented in Annex, section 5.3.1. and 5.3.2..
17
the threshold health state underlying di¤erent medical demand behavior. If this supply e¤ect exists, then it may be expectable to …nd distinct medical utilization pattern, since dissimilar health states would take citizens to request medical services, and even to request di¤erent medical cares (GP versus specialized). The outcome of this second investigation unables us to distinguish between sub-systems and NHS, regarding the health state threshold of specialized and general practice consultations. However, it seems quite robust the fact that sub-systems bene…ciaries have a relative lower demand of GP appointments, comparing with NHS users.
18
5
Annex
5.1
Sub-Systems
According to Baptista(1999) some requirements must be ful…lled by an entity, in order to be considered as a health sub-system, namely: 1. To be recognized as a sub-system by the organization-leader, as well as to be part of its management strategy; 2. To have a bene…t’s charter, where rights and obligations of users are de…ned; 3. To render medical assistance, directly or indirectly, to its bene…ciaries; 4. To be responsible for medical services, for its continuity and its …nancing. In general, the sub-systems may render the following services (Alberto, 1999): their own medical services, where the bene…ciaries pay a reduced duty value; agreed medicine, where the sub-system pays part of the treatment, according to a convention previously stipulated; independent medicine, where there is no previous settlement between the entity that render the medical service and the sub-system, but where the bene…ciary can receive a partial reimbursement; partial support for medical drugs cost, under the values legally established. Baptista(1999) identi…es sixteen Portuguese entities that …ll up the necessary requirements to have the status of sub-systems, having bene…ts of around 22% of the total Portuguese population. For those who may have a special interest in obtaining more information regarding Portuguese health system, one possible suggestion is the country report at http:www.observatory.dk.
5.2 5.2.1
Sub-systems as regressor Binary Logit Model
For testing the hypothesis of a set of restrictions R = q on the coe¢ cients , the statistic of Wald (W ) test is given by: W = (R ^
q)0 fRV R0 g
1
(R ^
q)
(31)
where V is the estimated asymptotic covariance matrix of ^ . The statistic of this test follows a chisquared distribution with the number of degrees of freedom being equal to the number of restrictions we are testing. For the same null hypothesis, the Likelihood Ratio (LR) test suggests a di¤erent statistic: LR =
^r 2(ln L 19
^u) ln L
(32)
^ r refers to the likelihood function evaluated under the null hypothesis estimates and L ^ u is where L calculated in the likelihood function evaluated at unrestricted maximum likelihood estimates. As before, the statistic of this test follows a chi-squared distribution with the number of degrees of freedom being equal to the number of restrictions tested. Binary Logit with Dummies Regressors variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 M s1 M s3 Sch0 Sch1 Sch2 Sch3 Exerc1 Exerc2 W h2 W h3 W h4 Age2 Age3 Age4 BM I1 BM I3 BM I4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 M eals1 M eals3 Fe HE3 HE4 HE5 Ss
estimated coe¢ cient 1.16745 -3.72271 .153913 .024841 .137902 .213378 .029374 -.068046 -.079802 .020671 .172285 .074917 -.072834 -.105984 -.069834 .154647 .048634 .196717 .207516 .246639 -.078854 .035599 -.865891E-02 -.050974 -.635177 -.483421 -.801538 -.555153 .179253 .025890 -.581428E-03 -.066334 .789233E-03 .249837 -.351201 -.234641 -.503476 -1.27828 -1.32308 -.139288
t-statistic 9.39976 -44.7241 3.17215 .326312 1.40744 3.13603 .093589 -.964317 -.851276 .133498 1.70963 .778686 -.700279 -1.14476 -.517684 2.65460 .831539 2.27320 2.86503 3.12313 -.855762 .414558 -.187165 -.763308 -8.44072 -10.4165 -11.9024 -1.68307 1.69035 .270590 -.694810E-02 -.756297 .985648E-02 3.60250 -5.31062 -4.54173 -10.1561 -17.3235 -9.90781 -2.27389
20
pvalue [.000] [.000] [.002] [.744] [.159] [.002] [.925] [.335] [.395] [.894] [.087] [.436] [.484] [.252] [.605] [.008] [.406] [.023] [.004] [.002] [.392] [.678] [.852] [.445] [.000] [.000] [.000] [.092] [.091] [.787] [.994] [.449] [.992] [.000] [.000] [.000] [.000] [.000] [.000] [.023]
Testing the existence of homoscedasticity among the residuals for Binary Logit, according to methodology proposed by Godfrey(1988), we develop the test considering di¤erent alternatives for Zt vector and the conclusion derived from the test changes in response: when Zt = [P cInc1t , P cInc2t , P cInc3t , P cInc4t , P cInc5t ], the homoscedasticity test rejects this hypothesis, null hypothesis homoscedasticity
test statistic CHISQ(5)=11.20806
upper tail area .04741
when Zt = [P cInc1t , P cInc2t , P cInc3t , P cInc4t , P cInc5t , Age2t , Age3t , Age4t ], the homoscedasticity test only rejectes this hypothesis to at least 95% signi…cance test, null hypothesis homoscedasticity
test statistic CHISQ(8)=14.30105
upper tail area .07425
when Zt = [Age2t , Age3t , Age4t ], the homoscedasticity hypothesis is not rejected, null hypothesis homoscedasticity
test statistic CHISQ(3)=2.057993
upper tail area .56046
from what we deduce that one source of heteroscedasticity issue can be variable Per Capita Family Income and, consequently, we pursue transforming the original binary Logit with this new clue:
p0
= P r(Yt0 = 0) = P r(ut = =
p1
(
i Xit )
0
1
(
0
k
+
(
0
k
+
where 2
(
0
+
k Xkt )
(33) (
0
Xkt )
2
=
=1
Xkt )
= P r(Yt0 = 1) = 1 =
k Xkt )
0
: exp(2Zt0 ) 3
k Xkt )
(34)
(35)
and Zt = [P cInc1t , P cInc2t , P cInc3t , P cInc4t , P cInc5 ]. However, as we conclude from the following table, the estimation outcome when we apply this transformation is very similar to the result already achieved without the transformation:
21
variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 M s1 M s3 Sch0 Sch1 Sch2 Sch3 Exerc1 Exerc2 W h2 W h3 W h4 Age2 Age3 Age4 BM I1 BM I3 BM I4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 M e1 M e3 Fe HE3 HE4 HE5 Ss (P cInc1 ) 1 2 (P cInc2 ) 3 (P cInc3 ) 4 (P cInc4 ) 5 (P cInc5 )
estimated coe¢ cient 1.05675 -3.25070 .130925 .026426 .136006 .193472 .013070 -.054353 -.058318 .832051E-02 .136127 .057175 -.063753 -.099679 -.060957 .130969 .044234 .173191 .182290 .217398 -.060696 .036963 -.839859E-02 -.047677 -.545450 -.418529 -.702914 -.512331 .106437 -.019718 -.051788 -.097905 -.045103 .217734 -.299462 -.203445 -.437823 -1.09839 -1.12938 -.124506 -.364388 -.255767 -.390651 -.273913 -.311308
t-statistic 8.99439 -19.9713 3.08043 .400311 1.59284 3.22795 .047238 -.891245 -.722971 .061838 1.50462 .662583 -.685104 -1.22045 -.518015 2.56264 .875303 2.30240 2.89589 3.16158 -.761364 .497745 -.210007 -.827157 -7.78853 -9.33028 -10.6411 -1.72572 1.09713 -.222764 -.664461 -1.20046 -.596288 3.56783 -5.04644 -4.44017 -9.21621 -13.1383 -8.82796 -2.31382 -2.26593 -1.82331 -3.21684 -2.02457 -2.51960
22
pvalue [.000] [.000] [.002] [.689] [.111] [.001] [.962] [.373] [.470] [.951] [.132] [.508] [.493] [.222] [.604] [.010] [.381] [.021] [.004] [.002] [.446] [.619] [.834] [.408] [.000] [.000] [.000] [.084] [.273] [.824] [.506] [.230] [.551] [.000] [.000] [.000] [.000] [.000] [.000] [.021] [.023] [.068] [.001] [.043] [.012]
Binary Logit with some Continuous Regressors variable Constant Exam Dr1 Dr2 Cigarettes M s1 M s3 Schooling Exercise W Hours Age Age2 BM I GCI SP ills Rc P cIncome M eals Fe HE3 HE4 HE5 Ss
estimated coe¢ cient .060531 -3.72101 .152151 .027042 .896276E-02 -.100297 -.076157 -.018449 -.011282 .225446E-02 .044843 -.469215E-03 -.776807E-02 .014966 -.808601 -.529289 -.977307E-07 -.264247 -.239310 -.511635 -1.27720 -1.31877 -.136045
t-statistic .298804 -44.7846 3.14535 .356514 3.50737 -1.43038 -.816151 -2.44592 -.580333 2.26234 5.38750 -5.78031 -1.44544 9.81556 -12.0220 -1.61120 -.467639 -6.64175 -4.73405 -10.3084 -17.3965 -9.98696 -2.26796
pvalue [.839] [.000] [.002] [.721] [.000] [.153] [.414] [.014] [.562] [.024] [.000] [.000] [.148] [.000] [.000] [.107] [.640] [.000] [.000] [.000] [.000] [.000] [.023]
Following a methodology similar to that already used, we submitte this Binary Logit model to heteroscedasticity test proposed by Godfrey(1988) and we …nd evidence that Per Capita Family Income and Age can be two main causes for non-existence of homoscedasticity, as the Godfrey test result when Zt = [P cIncome, Age] is: null hypothesis homoscedasticity
test statistic CHISQ(2)=15.19109
upper tail area .0005
Transforming the original binary Logit model following this new suggestion, we estimate the model in expressions (33), (34) and (35), but where Zt = [P cIncomet , Aget ]. Again, the estimation outcome is very similar to the result already achieved without the transformation:
23
variable Constant Exam Dr1 Dr2 Cigarettes M s1 M s3 Schooling Exercise W Hours Age Age2 BM I GCI SP ills Rc P cIncome M eals Fe HE3 HE4 HE5 Ss (P cIncome) 1 2 (Age) 5.2.2
estimated coe¢ cient .046323 -3.93206 .154424 .035731 .999484E-02 -.102996 -.077757 -.019157 -.012728 .238625E-02 .047312 -.491966E-03 -.855810E-02 .015572 -.853410 -.601051 .197187E-07 -.273467 -.245671 -.541224 -1.33164 -1.36834 -.149336 .767160E-06 .194206E-03
t-statistic .145367 -15.5960 3.02599 .448295 3.61011 -1.40388 -.797710 -2.36676 -.617870 2.29018 4.62322 -4.90951 -1.52081 8.42848 -9.27527 -1.66199 .081252 -6.18407 -4.37106 -9.18899 -11.4327 -8.24248 -2.31540 2.73030 .079671
pvalue [.884] [.000] [.002] [.654] [.000] [.160] [.425] [.018] [.537] [.022] [.000] [.000] [.128] [.000] [.000] [.097] [.935] [.000] [.000] [.000] [.000] [.000] [.021] [.006] [.936]
Ordered Logit Model
Speci…cation Tests The results for speci…cation tests proposed by Murphy(1996) are: null hypothesis omitted variables(HEi ) symmetry homoscedasticity
test statistic 289.5315 1086.778 1110.101
critical value CHISQ(3), 5%=7.82 CHISQ(1), 5%=3.84 CHISQ(2), 5%=5.99
upper tail area .00000 .00000 .00000
Temptatives to solve heteroscedasticity problem: 1. In the …rst place, we consider that variance of the residuals follow a particular pattern: 2t = 2 : exp(2wt0 : ) , where di¤erent functions of wt are considered as alternatives. At …rst, it is used ^ Xt 3 and powers of it. A second e¤ort considers wt as Per Capita Income Family and powers of it, as well as Age and powers of it. Although the conclusion of the test always rejects homoscedasticity, the value of the statistic test for heteroscedasticity fall for 684:5369 when wt = Per capita income family, which may be an indicator that this variable can be one source for not having homoscedasticity. Unfortunately, subsequent investigations on that direction don’t improve those results. 2. Afterwards, we estimate a regression of the sum of square residuals of LPM (with dummies re24
gressors, including HE and White estimator for variance) into whole set of explanatory variables of our original model and, iteratively, we take out those that don’t seem signi…cant. The result is the estimation of the following relation:
^2
=
145:96Exam + 459:6Dr1 + 588:6Dr2 + 516Cig1 + 567Cig2 + 796:9Cig3 + +807:6M s1 + 563M s3 + 1375:2Sch0 + 1441:8Sch1 + 1738:4Sch2 + +1943:9Sch3 + 922:25Exerc1 + 643Exerc2 + 434:2W h2 + +616:6W h3 + 492:3W h4 + 679:86Age2 + 906Age3 + +1111:2Age4 + 623:9BM I1 + 190:9BM I3 + 165:4BM I4 + 67:8GCI1 + +182GCI2 + 98:4HE3 + 39:3HE4 + 35:1HE5 + 790:4SSystem + +1696:6P cInc1 + 1678P cInc2 + 1512:4P cInc3 + 1320P cInc4 + +1171:9P cI5 + 184:6M eals1 + 278:5M eals3 + 684:2F e
However, transforming the Ordered Logit model (with dummies regressors) according to this estimated relation (dividing the parameters to estimate by the square root of exponential value of the previous regression), results in non-estimated Logit owing to numerical errors of the process. The same disappointing result is obtained when we make the same transformation, but treating the coe¢ cients of the previous relations as parameters to be estimated and giving the shown values as starting-up values. Di¤erent relations are estimated, sometimes introducing new variables, sometimes leaving out others, but unfortunately we always end up with numerical errors in estimation, even if we give di¤erent starting values for estimation process. 3. A similar methodology is proceeded for Ordered Logit with some continuous regressors, but unfortunately the same non-convergence of estimation procedures are an unsurpassable obstacle. 4. Estimating the Ordered Logit with dummies regressors and, among them, powers of ^ X, like RESET-Ramsey test suggest, result in values too much peculiar to be taken seriously. In fact, almost all explanatory variables now turn out to have very high p-values, except ( ^ X)2 . The odd result obtained is not alleviated with the conclusion achieved from the Murphy homoscedasticity test, as we conclude for the presence of heteroscedasticity in the residuals: null hypothesis homoscedasticity
test statistic 896.6859
critical value CHISQ(2),5%=5.99
upper tail area .000
5. Similar methodology is applied to Ordered Logit model with some regressors treated as continuous and the result for RESET-Ramsey test is that like before. In the estimatives obtained, almost every parameter seems irrelevant, besides ( ^ X)2 : We also reject homoscedasticity for residuals (as can be seen from the table below). Therefore, we reject the inclusion of ^ X among regressors. null hypothesis homoscedasticity
test statistic 906.9311
critical value CHISQ(2),5%=5.99
upper tail area .000
6. Also considering some explanatory variables as continuous, we introduce powers of some of those regressors and cross-products of them and in each step we not only evaluate the coe¢ cient significance but also test the presence of homoscedasticity in the error term. This last test, as before, 25
follows Murphy(1996) approach, but alternative vector of variables wt (please refer to expressions 22 and 23) in each estimation process. The "prettiest"’outcome25 is achieved when regressors set don’t include any explanatory squared variable neither cross-product of variables and wt = Per capita family Income (treated as continuous regressor). However, we still conclude in favour of heteroscedasticity for residuals variance, as can be inferred from the table below: null hypothesis homoscedasticity
test statistic 684.5369
critical value CHISQ(2),5%=5.99
upper tail area .0000
Signi…cance tests null hypothesis slopes joint insigni…cance
test statistic Wald=4666.32522 LR=5899
critical value CHISQ(39), 10%=50.65977
The results leave us without doubts, as they strongly reject the global insigni…cance of all slopes coe¢ cients. Testing the partial insigni…cance of some coe¢ cients that appeared irrelevant, we don’t reject the insigni…cance of explanatory variables as Marital Status, Exercise and BMI, which can be con…rmed from the following values of Wald test (conclusions are very similar to those obtained in Ordered Probit and LPM): variable Dr1 , Dr2 Cig1 , Cig2 , Cig3 M s1 , M s3 Sch0 , Sch1 , Sch2 , Sch3 Exer1 , Exer2 Age2 , Age3 , Age4 BM I1 , BM I3 , BM I4 P cInc1 , P cInc2 , P cInc3 , P cInc4 , P cInc5
test statistic 13:44985 9:27854 2:36915 46:73851 4:42936 12:95293 :81109
upper tail :00120 :02581 :30588 :000 :10919 :00474 :84681
critical value 5.99 7.82 5.99 9.49 5.99 7.82 7.82
31:96046
:00001
11.07
The hypothesis that Marital Status, Exercise and BMI are not signi…cant for our model, is also corroborated through Likelihood Ratio test: null hypothesis Marital Status, Exercise and BMI joint insigni…cance
test statistic
critical value
LR=7.4
CHISQ(7),10%=12.02
2 5 "prettiest"
is an excessive language term, meaning that this outcome was associated with the lowest value for heteroscedasticity statistic test.
26
First Ordered Logit variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 M s1 M s3 Sch0 Sch1 Sch2 Sch3 Exer1 Exer2 W h2 W h3 W h4 Ag2 Ag3 Ag4 BM I1 BM I3 BM I4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 M e1 M e3 Fe Ss
estimated coe¢ cient 3.16095 -2.16571 .179251 .083205 .096403 .158107 .015941 -.066466 .017441 .352463 .299758 .178404 .012313 -.061932 -.147195 .215567 .162698 .219452 .025966 .181583E-02 -.154659 .033947 .022819 -.014405 -.714971 -.492488 -.504648 -.650689 .224358 .172823 .088734 .019575 .036129 .111684 -.287879 -.255029 -.274961
t-statistic 30.0331 -56.3042 4.68201 1.37386 1.21828 2.79176 .059022 -1.15945 .237546 2.99640 3.68959 2.28279 .144050 -.830656 -1.44478 4.63806 3.41331 3.19375 .431540 .028352 -2.12396 .493784 .627021 -.282947 -13.3914 -13.1362 -11.1447 -3.16644 2.71807 2.30818 1.34978 .283814 .572591 2.07223 -5.54454 -6.23248 -5.74338
pvalue [.000] [.000] [.000] [.169] [.223] [.005] [.953] [.246] [.812] [.003] [.000] [.022] [.885] [.406] [.149] [.000] [.001] [.001] [.666] [.977] [.034] [.621] [.531] [.777] [.000] [.000] [.000] [.002] [.007] [.021] [.177] [.777] [.567] [.038] [.000] [.000] [.000]
For the boundary coe¢ cient, the estimation is: variable
estimated coe¢ cient 2.63991
t-statistic 85.1092
pvalue [.000]
These results are obtained using the Newton’s method for the maximization of the maximum likelihood function and for analytic second derivatives.
27
Ordered Logit with Health Evaluation among Dummies Regressors variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 M s1 M s3 Sch0 Sch1 Sch2 Sch3 Exer1 Exer2 W h2 W h3 W h4 Ag2 Ag3 Ag4 BM I1 BM I3 BM I4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 M e1 M e3 Fe HE3 HE4 HE5 SS
estimated coe¢ cient 3.22252 -2.10020 .141427 .061165 .104791 .164297 .114483 -.038154 .038471 .509078 .432101 .241798 .031261 -.106354 -.169592 .157975 .096364 .157262 .091731 .155751 .872801E-02 .041718 .026405 .314809E-02 -.416181 -.351543 -.399845 -.576922 .342001 .272194 .177310 .091330 .082324 .131582 -.300557 -.250598 -.498716 -.943939 -.933933 -.311079 2.67005
t-statistic 30.4320 -54.3071 3.66740 1.00500 1.31811 2.88446 .419567 -.661399 .520928 4.29345 5.26460 3.07408 .364292 -1.42036 -1.65705 3.36724 2.00106 2.26749 1.51037 2.38887 .118031 .603823 .721375 .061414 -7.33632 -9.07345 -8.70591 -2.80218 4.10489 3.60376 2.67153 1.31406 1.29566 2.42890 -5.75065 -6.08813 -11.7873 -16.5549 -10.6824 -6.45682 85.2086
pvalue [.000] [.000] [.000] [.315] [.187] [.004] [.675] [.508] [.602] [.000] [.000] [.002] [.716] [.156] [.098] [.001] [.045] [.023] [.131] [.017] [.906] [.546] [.471] [.951] [.000] [.000] [.000] [.005] [.000] [.000] [.008] [.189] [.195] [.015] [.000] [.000] [.000] [.000] [.000] [.000] [.000]
28
Ordered Logit with some Continuous Regressors variable Constant Exam Dr1 Dr2 Cigarettes M s1 M s3 Schooling Exercise W Hours Age Age2 BM I GCI SP ills Rc P cIncome M eals Fe HE3 HE4 HE5 Ss
estimated coe¢ cient 3.21741 -2.10569 .134463 .057685 .674699E-02 -.084075 .012031 -.045783 -.022586 .231583E-02 .022251 -.220649E-03 -.148483E-02 .997229E-02 -.401987 -.569596 -.438471E-06 -.191637 -.246821 -.513385 -.942028 -.917786 -.311610 2.66541
t-statistic 13.3137 -54.5272 3.49012 .950779 3.17630 -1.47242 .164074 -7.55963 -1.52853 2.90597 3.33010 -3.42241 -.355962 8.02015 -8.75731 -2.76551 -2.64704 -6.22247 -6.10766 -12.1408 -16.5372 -10.5606 -6.59540 85.1944
29
pvalue [.000] [.000] [.000] [.342] [.001] [.141] [.870] [.000] [.126] [.004] [.001] [.001] [.722] [.000] [.000] [.006] [.008] [.000] [.000] [.000] [.000] [.000] [.000] [.000]
Ordered Logit with Signi…cant Dummies Regressors variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 Sch0 Sch1 Sch2 Sch3 W h2 W h3 W h4 Age2 Age3 Age4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 M e1 M e3 Fe HE3 HE4 HE5 Ss
estimated coe¢ cient 3.16997 -2.10207 .141052 .061268 .106068 .173217 .137664 .521764 .441650 .247296 .031330 .158543 .100583 .155546 .087883 .157792 .025993 -.414771 -.352416 -.397482 -.577583 .362850 .278478 .186779 .096792 .089109 .136595 -.302757 -.233666 -.496047 -.941976 -.930205 -.313675 2.66921
t-statistic 32.0384 -54.3679 3.66587 1.00736 1.33678 3.06227 .505083 4.45487 5.52084 3.19181 .365752 3.38158 2.09133 2.24480 1.54938 2.57395 .376240 -7.35261 -9.11936 -8.68182 -2.80914 4.44844 3.73033 2.83651 1.39813 1.40547 2.52692 -5.79593 -5.82993 -11.7519 -16.5434 -10.6531 -6.51446 85.2072
pvalue [.000] [.000] [.000] [.314] [.181] [.002] [.614] [.000] [.000] [.001] [.715] [.001] [.036] [.025] [.121] [.010] [.707] [.000] [.000] [.000] [.005] [.000] [.000] [.005] [.162] [.160] [.012] [.000] [.000] [.000] [.000] [.000] [.000] [.000]
30
Ordered Logit with some Continuous Regressors and Signi…cant Explanatory Variables
variable Constant Exam Dr1 Dr2 Cigarettes Schooling W Hours Age Age2 GCI SP ills Rc P cIncome M eals Fe HE3 HE4 HE5 Ss
5.2.3
estimated coe¢ cient 3.20024 -2.10755 .132664 .059511 .718954E-02 -.045021 .236130E-02 .017845 -.174902E-03 .010016 -.397668 -.569519 -.476251E-06 -.195164 -.230676 -.513551 -.940972 -.914565 -.313918 2.66463
t-statistic 14.4587 -54.5876 3.44840 .981015 3.40468 -7.59863 2.96814 2.92232 -2.94833 8.09725 -8.68853 -2.76827 -2.90384 -6.35368 -5.85024 -12.1545 -16.5280 -10.5303 -6.64751 85.1915
pvalue [.000] [.000] [.001] [.327] [.001] [.000] [.003] [.003] [.003] [.000] [.000] [.006] [.004] [.000] [.000] [.000] [.000] [.000] [.000] [.000]
Linear Probability Model (LPM)
In LPM, the discrete dependent variable Yt (medical Behavior) for each individual t is a linear function of K explanatory variables Xkt : Yt =
0
+
1 X1t
+ ::: +
K XKt
+ vt
t = 1; :::; T
(36)
where it is assumed that E(vt ) = 0 and, as a result: E(Yt j Xt ) =
0
+
1 X1t
+ ::: +
K XKt
(37)
Letting
p0
= P r(Yt = 0);
p1
= P r(Yt = 1);
p2
=
1
p0
p1 = P r(Yt = 2)
then (37) is equivalent to: E(Yt j Xt ) = 2
31
2p0
p1
(38)
Since any probability must lie between 0 and 1 and p0 + p1 + p2 = 1 , we then know that E(Yt j Xt ) must lie between 0 and 2. The attractiveness of using LPM is directly connected with the simplicity of estimating it by OLS. However, as Gujarati(1995) said: "as a mechanical routine, we can do this. But now we must face some special problems." These problems arrive because: 1. To estimate the model through OLS, we assume that the residuals vt are normally distributed and for statistical inference we need that assumption. Nevertheless, Yt is a discrete variable with three categories and so the disturbances are:
vt =
8 <
1 X1t
0
1 : 2
0 0
1 X1t 1 X1t
:::
K XKt
, when Yt = 0 X , when Yt = 1 Kt K X , K Kt when Yt = 2
::: :::
(39)
from what it becomes clear that the normality assumption is not ful…lled. Although this appears to be a strong limitation for the use of LPM, it isn’t so because the OLS estimates still remain unbiased and, if we turn to the central limit theorem, it can be proved that as sample size increases inde…nitely, the OLS estimators tend to be normally distributed (see Malinvaud, 1966). In the present study, the sample size (16 383 observations) gives us some comfort to apply this limit result. 2. It can no longer be assumed that the residuals vt are homoscedastic, because their variance depends on the conditional expectation of Yt , which in turn depends on the value of Xkt . This can easily be seen, because from (39) and from the assumption that E(vt ) = 0, we obtain: var(vt )
= E(vt2 ) = P (Yt ):(Yt = p0 :( +(1
0
p0
0
Xt )2 + p1 :(1 p1 ):(2
0
Xt )2 = 0
Xt )2 +
Xt )2
(40)
In the presence of heteroscedasticity, OLS estimators still remain unbiased, but are not e¢ cient. Nevertheless, "the problem is not unsurmountable" argued Gujarati (1995), p pointing one way of resolving it by transforming the data, dividing both sides of the model (36) by var (vt ) : Y p t =p 0 + var (vt ) var (vt )
Xkt
vt +p var (vt ) var (vt )
kp
(41)
As the real value of var (vt ) is unknown, it should be replaced by an estimation of it, that can be obtained from the following procedures: 1st. calculation of an approximate value for p0 and p1 , using the proportion of Yt = 0 and Yt = 1, respectively, in the sample used; 2nd. estimation of var(vt ) substituting the estimations obtained in the previous step, into expression (40); 3rd. running the OLS regression on the model (41).
32
If it is not possible to improve estimations with the transformation proposed by Gujarati(1995), we can carry on using White’s result, who states that it is still possible to obtain a consistent estimator for the variance of the coe¢ cients, without actually specifying the type of heteroscedasticity, using as an estimate of the true variance: V\ ar( ^ ) = T:(X 0 X) where S0 = T1 : (k + 1) columns.
PT
t=1
1
:S0 :(X 0 X)
1
(42)
vt2 :xt :x0t , T is the number of individuals (sample size) and xt is a row-vector of
3. The main objection for the use of the Linear Probability Model arrives when we verify that the conditional expected value of the dependent variable must lie between 0 and 2, as is deducted from expression (38), but nothing guarantee us that estimations obtained through the above procedures, satisfy this restriction. However, there are two alternative ways to overcome this obstacle. One is to truncate values below 0 and above 2, converting to the value 0, if the expectation is less than 0, and to 2 if the value is greater than 2. Another is to develop an estimating technique that guarantees that the estimated conditional expectation is between the desired values. These alternative techniques turn out to be the other two models analyzed afterwards: Ordered Logit and Ordered Probit. 5.2.4
Interpretation of Coe¢ cients
In order to have an easier comparison between the three econometric approaches, we present the results obtained: in LPM, using White estimator for variance and only considering the regressors which insigni…cance is rejected; in Ordered Probit with dummies variables whose insigni…cance are rejected by Wald tests; in Ordered Logit, only taking into account the regressors that are classi…ed as signi…cant through Wald tests developed; also regarding Ordered Logit results, it is considered of interest to present here the impact of each dummy variable on the probability of every medical behavior category, following Greene(1993)26 . 2 6 In this approach, we compare the probabilities of the dependent variable when the regressor (dummy) takes its two di¤erent values, holding the other explanatory variables at their sample means, taking advantage of expressions (10), (11) and (12).
33
Medical Exams variable Exam dummy Exam=0 Exam=1 change
LPM coe¤ . -.673732
Probit coe¤ . -1.18310
p0 = P r(Yt = 0) :0468 :2865 :2397
Logit coe¤ . -2.10207
p1 = P r(Yt = 1) :3677 :5663 :1986
p2 = P r(Yt = 2) :5855 :1472 :4383
Having done medical exams in the last three months is associated with a less healthy index, as the coe¢ cient of this variable shows an increase on the probability to having been to a specialist consultation(Yt =0) and having been to a GP appointment (Yt =1), and a negative impact on the chance to having not consulted any doctor in the last three months (Yt =2). However, the use of Exam dummy variable may need additional comments. In fact, medical exams are generally preceded by a medical appointment, which may be an indicator of some endogeneity e¤ect. Therefore, we should be careful when including and interpreting the coe¢ cient of Exam regressor, despite its sign being the one we logical expect to …nd. Drinks variable(cont.1) Dr1 Dr2 dummy Dr1 =0 Dr1 =1 change Dr2 =0 Dr2 =1 change
LPM coe¤ . .044268 .022461
p0 = P r(Yt = 0) :0956 :0840 :0115 :0911 :0862 :0049
Probit coe¤ . .087152 .041214
Logit coe¤ . .141052 .061268
p1 = P r(Yt = 1) :5083 :4857 :0227 :5002 :4902 :0099
p2 = P r(Yt = 2) :3961 :4303 :0342 :4087 :4236 :0149
Drinking alcohol more than rarely (Dr1 and Dr2 ) seems to be associated with a higher propensity to having not consult any doctor in the last three months (Yt = 2) and with a smaller tendency to having been to a specialist consultation (Yt = 0), comparing to rarely or never drunk in the previous twelve months (the basis). Although this result may seem a little strange, two explanations can be proposed: only people who do not feel very ill or that just don’t care much about their health state are able to frequently drink liquors. But if they don’t have much concern with their health-being, they won’t probably demand many medical services, thus linking their drink consumption with their medical behavior. the negative e¤ect of consumption of alcoholic drinks on health may be such a well known fact, that those who drink more are able to establish a natural link between that consumption and a worse health possibly felt. Therefore, knowing in advance the most possible solution of their illness (decrease liquors consumption), they don’t feel the need to demand additional medical care.
34
This second potential explanation may also explain why the impact on medical demand is larger on Dr1 than on Dr2 (more frequent consumption, less contemporaneous demand of medical services). Nevertheless, it would be interesting to analyze the evolution of medical care demand by the di¤erent groups of drinkers, since it would be expectable to …nd a negative e¤ect of drinks-consumption into health state, in long term. Unfortunately, for the moment we just have the data obtained in one moment of time. Cigarettes variable(cont.2) Cig1 Cig2 Cig3 dummy Cig1 =0 Cig1 =1 change Cig2 =0 Cig2 =1 change Cig3 =0 Cig3 =1 change
LPM coe¤ . .039126 .044279 .020548
p0 = P r(Yt = 0) :0911 :0827 :0084 :0924 :0788 :0135 :0908 :0800 :0107
Probit coe¤ . .078868 .102002 .051634
Logit coe¤ . .106068 .173217 .137664
p1 = P r(Yt = 1) :5002 :4827 :0174 :5025 :4737 :0288 :4995 :4766 :0229
p2 = P r(Yt = 2) :4087 :4346 :0259 :4051 :4475 :0423 :4098 :4434 :0337
Having smoked in the previous two weeks (Cig1 , Cig2 and Cig3 ) suggests the presence of a higher tendency to having not consulted any doctor in the last three months (Yt = 2), specially for those who smoked between 11 and 40 cigarettes a day (Cig2 ), when compared to had not smoked (the basis). When considering the number of cigarettes smoked per day as a continuous variable, this positive relation is reinforced. Like before, this apparently strange e¤ect may be caused, on the one hand, by the fact that in general someone is able to smoke (or admit to had smoked) if he doesn’t feel sick or, alternatively, if he just doesn’t care. On the other hand, even if someone does not feel very healthy, he may just relate it to the known fact of being a smoker and therefore doesn’t need to demand a medical care, which would inform him of something he already knew and probably didn’t want to give it up. Also here, it would be interesting to study the evolution in time of medical care demand, for di¤erent smoking habits. What should then be expected is to …nd an increasing demand of medical services as smoking habits persist in time. However, at the present time, such information is not available. It is interesting to notice that for all Cigarettes dummies, the negative impact on demanding a GP consultation is larger than on demanding a specialist advice. That is to say, smoking causes a larger decrease in GP demand than in specialist demand. In fact, from the base assumption that demanding a specialist consultation is associated with a worse health state than the one that leads to a GP demand, we can extrapolate that a current smoker doesn’t ask a medical care because he already knows one possible cause for it, as long as his health level is not so low. Therefore, the smoker decision of not going to a medical appointment is stronger for low levels of illness.
35
Schooling variable(cont.3) Sch0 Sch1 Sch2 Sch3 dummy Sch0 =0 Sch0 =1 change Sch1 =0 Sch1 =1 change Sch2 =0 Sch2 =1 change Sch3 =0 Sch3 =1 change
LPM coe¤ . .174469 .140012 .077018 .012057
p0 = P r(Yt = 0) :0922 :0568 :0353 :1152 :0773 :0380 :0951 :0759 :0193 :0909 :0884 :0026
Probit coe¤ . .344520 .282255 .154718 .022113
Logit coe¤ . .521764 .441650 .247296 .031330
p1 = P r(Yt = 1) :5021 :4082 :0939 :5374 :4698 :0676 :5075 :4663 :0412 :4998 :4948 :0050
p2 = P r(Yt = 2) :4057 :5349 :1292 :3474 :4529 :1055 :3973 :4578 :0604 :4093 :4168 :0076
Having completed less than 13 years at school (Sch0 , Sch1 , Sch2 and Sch3 ) seems to be associated with a better health indicator, comparing to those that had approval in a larger number of schooling years. In fact, the less literate a person is, the bigger seems to be the probability of not having been to a medical appointment. This e¤ect may be due to the fact that more educated people are, in general, more sensitive to the importance of medical watchfulness, thus requiring medical advice more often. The same conclusion may be obtained when the number of school years with approval is taken as a continuous variable, as it has a negative in‡uence in our health indicator. Again, it is worth noting that this negative impact of lower schooling is higher on GP demand than on specialist demand, which may reinforce the intuition of a specialist consultation being associated with a worse health state and therefore less sensitive to education e¤ects. Working Hours variable(cont.4) W h2 W h3 W h4
LPM coe¤ . .047066 .028957 .045321
Probit coe¤ . .092339 .061236 .094786
Logit coe¤ . .158543 .100583 .155546
36
dummy W h2 =0 W h2 =1 change W h3 =0 W h3 =1 change W h4 =0 W h4 =1 change
p0 = P r(Yt = 0) :0933 :0807 :0126 :0923 :0842 :0081 :0916 :0794 :0121
p1 = P r(Yt = 1) :5042 :4781 :0261 :5023 :4860 :0164 :5010 :4751 :0259
p2 = P r(Yt = 2) :4026 :4412 :0386 :4054 :4299 :0245 :4074 :4454 :0380
Working more than 30 hours weekly, on average (W h2 , W h3 and W h4 ) reveals to be linked to a healthier index, comparing with having worked at most 30 hours a week (the basis), as the former behavior reveals a higher propensity to having not consulted any doctor in the last three months (Yt =2). Considering the number of hours worked per week as a continuous variable (WHours), we reinforce the conclusion that working more is associated with a lower propensity to have a medical appointment. Reasonable judgements for this outcome may be: on the one hand, those who work more have a higher opportunity cost of their time, thus turning to medical commitments less frequently; on the other hand, someone works more if he feels healthier or even if he does not feel very well he knows that it would probably be because of working too much and therefore doesn’t need to go for a medical appointment to …nd the source of his problem. As before, the reduction in the specialist demand probability caused by working more hours is less than the e¤ect produced in GP consultation. Once again, this can be an evidence of a worse health state associated with specialist demand, thus turning this type of medical service less sensitive to the amount of hours spent at work. Age variable(cont.5) Age2 Age3 Age4 dummy Age2 =0 Age2 =1 change Age3 =0 Age3 =1 change Age4 =0 Age4 =1 change
LPM coe¤ . .020849 .046698 .013118
p0 = P r(Yt = 0) :0928 :0857 :0071 :0958 :0829 :0128 :0911 :0890 :0021
Probit coe¤ . .050587 .098466 .028113
Logit coe¤ . .087883 .157792 .025993
p1 = P r(Yt = 1) :5033 :4891 :0142 :5087 :4832 :0255 :5001 :4960 :0042
p2 = P r(Yt = 2) :4039 :4252 :0213 :3956 :4338 :0383 :4087 :4150 :0063
Being more than 30 years old (Age2 , Age3 and Age4 ) and specially being between 45 and 65 years old (Age3 ) seems to be associated with a healthier indicator than being at most 30 years old (the basis). This e¤ect not only goes in favour of our initial intuition, that lead us to expect a non-constant e¤ect from age on demanding medical services, but is also reinforced when age is taken as a continuous 37
variable. In fact, from the continuous usage of this same regressor and its square, we …nd out a negative and signi…cant e¤ect from the square of age, which means that on "tails-age" (youth and old-age) there is a greater propensity to consult a general practitioner and, even greater, to consult a specialist. Global Capacity Index variable(cont.6) GCI1 GCI2 dummy GCI1 =0 GCI1 =1 change GCI2 =0 GCI2 =1 change
LPM coe¤ . -.116525 -.096175
p0 = P r(Yt = 0) :0855 :1240 :0385 :0779 :1072 :0294
Probit coe¤ . -.242679 -.205600
Logit coe¤ . -.414771 -.352416
p1 = P r(Yt = 1) :4888 :5473 :0585 :4714 :5269 :0555
p2 = P r(Yt = 2) :4257 :3286 :0970 :4508 :3659 :0849
Taking the Global Capacity Index as a measure of well-being, the above coe¢ cients present the values we expect. In fact, we anticipate that those people who have a lower GCI would have a lower propensity to not going to a medical appointment (Yt =2), comparing with those who have a GCI between 90 and 100 (the basis). When we treat GCI as a continuous variable, we reinforce this intuitive result that the lower GCI, the higher the propensity to consult a doctor. Nevertheless, the impact of GCI on the variation of probability is still higher for GP consultations than for a specialist. Sleeping Pills variable(cont.7) SPills dummy SP ills=0 SP ills=1 change
LPM coe¤ . -.128941
p0 = P r(Yt = 0) :0860 :1229 :0368
Probit coe¤ . -.231619
Logit coe¤ . -.397482
p1 = P r(Yt = 1) :4899 :5461 :0562
p2 = P r(Yt = 2) :4240 :3310 :0930
Thinking of taking sleeping pills as a stress/nervousness indicator, we easily agree about the negative impact that this behavior has in health state, as it is evident from the higher tendency to demand a medical appointment. Attending to probability changes of each type of medical services, we again …nd that this e¤ect is greater for GP than for specialists. Reimbursement of Expenditures variable(cont.8) Reimbursement
LPM coe¤ . -.156848
Probit coe¤ . -.297990
Logit coe¤ . -.577583
38
dummy Rc=0 Rc=1 change
p0 = P r(Yt = 0) :0905 :1505 :0601
p1 = P r(Yt = 1) :4989 :5683 :0695
p2 = P r(Yt = 2) :4107 :2812 :1295
As expected, the fact that someone receives …nancial support for medical treatments increases the probability of demanding medical services, as they are now more accessible. Nevertheless, it is not totally unreasonable to have some caution when considering this variable among regressors. In fact, it may happen that this variable captures two misleading e¤ects: on the one hand, having received …nancial support for medical treatments is a natural indicator of the need of those cares, which may point the existence of some endogeneity among this question; on the other hand, since NHS consultations are almost free of charge, having received …nancial support for medical treatments may be highly correlated with the decision of going to a private appointment and therefore may capture some income e¤ect, also re‡ected in the next variable. Per Capita Family Income variable(cont.9) P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 dummy P cInc1 =0 P cInc1 =1 change P cInc2 =0 P cInc2 =1 change P cInc3 =0 P cInc3 =1 change P cInc4 =0 P cInc4 =1 change P cInc5 =0 P cInc5 =1 change
LPM coe¤ . .118377 .091994 .061465 .035436 .026261
p0 = P r(Yt = 0) :0934 :0669 :0265 :0940 :0728 :0212 :0953 :0803 :0149 :0920 :0842 :0078 :0923 :0851 :0072
Probit coe¤ . .234243 .179366 .123148 .071108 .056259
Logit coe¤ . .362850 .278478 .186779 .096792 .089109
p1 = P r(Yt = 1) :5044 :4415 :0629 :5056 :4584 :0471 :5078 :4773 :0305 :5018 :4860 :0158 :5024 :4879 :0144
p2 = P r(Yt = 2) :4022 :4916 :0894 :4004 :4687 :0683 :3969 :4424 :0455 :4062 :4298 :0235 :4053 :4270 :0216
This group of dummies indicates that the lower the per capita family income, the higher the tendency of not going to a medical appointment (Yt =2), which is the same as to say that, a smaller per capita family income decreases the probability of demanding a GP or a specialist consultation. It is interesting to notice that this negative e¤ect of lower income is more evident for GP services than for specialist cares. Results are consistent with the ones obtained when per capita family income is considered a continuos variable.
39
Possible explanations for this e¤ect may be that people with higher income are more sensitive to the importance of medical watchfulness or merely that it is detected an income e¤ect by which the higher the income, the larger the demand of medical care. Meals variable(cont.10) M e1 M e3 dummy M e1 =0 M e1 =1 change M e3 =0 M e3 =1 change
LPM coe¤ . .043325 -.097518
p0 = P r(Yt = 0) :0919 :0811 :0108 :0881 :1157 :0276
Probit coe¤ . .084956 -.187625
Logit coe¤ . .136595 -.302757
p1 = P r(Yt = 1) :5015 :4790 :0226 :4942 :5380 :0438
p2 = P r(Yt = 2) :4066 :4399 :0333 :4176 :3463 :0713
Comparing to eat exactly 3 meals a day (the basis), the inquiries that have less than 3 meals a day (M e1 ) reveal to have a healthier index as they have a greater propensity to not having consulted a doctor (Yt =2), reversing this e¤ect when we consider the people that eat more than 3 meals a day (M e3 ). This result of negative relation between the number of meals taking per day and demand indicator is con…rmed when we treat this group of dummies as only one continuous regressor. One natural proposition that may be taken from these values is that people who eat more are those that have a more concern with their health state and, therefore, also a higher propensity to demand health services. Once more, having been to a specialist consultation or to a GP have impacts that point in the same direction, but again the probability to go to a specialist changes less with meals consumption. Female variable(cont.11) F emale dummy F e=0 F e=1 change
LPM coe¤ . -.068784
p0 = P r(Yt = 0) :0799 :0989 :0190
Probit coe¤ . -.138930
Logit coe¤ . -.233666
p1 = P r(Yt = 1) :4763 :5140 :0377
p2 = P r(Yt = 2) :4438 :3871 :0567
Being a woman increases the probability of demanding medical services and this raise is higher in the probability of GP care than on specialists. Health Evaluation variable(cont.12) HE3 HE4 HE5
LPM coe¤ . -.133097 -.276773 -.279069
Probit coe¤ . -.286328 -.550742 -.550371
Logit coe¤ . -.496047 -.941976 -.930205 40
dummy HE3 =0 HE3 =1 change HE4 =0 HE4 =1 change HE5 =0 HE5 =1 change
p0 = P r(Yt = 0) :0742 :1163 :0421 :0770 :1763 :0993 :0875 :1955 :1080
p1 = P r(Yt = 1) :4620 :5387 :0767 :4693 :5791 :1098 :4929 :5826 :0897
p2 = P r(Yt = 2) :4638 :3450 :1188 :4537 :2446 :2091 :4196 :2219 :1977
As known in advance, the fact that someone doesn’t evaluate his own health state as good or very good (the basis), increases the probability of demanding medical consultation. LPM coe¢ cients state that the worse self-health evaluation is, the greater the propensity to visit a general practitioner and a specialist doctor, which is totally in harmony with it should be expected.
Sub-systems variable(cont.13) Ss
LPM coe¤ . -.099890
Probit coe¤ . -.192447
Logit coe¤ . -.313675
All estimations regarding sub-systems coe¢ cient point out that having the additional bene…t of a sub-system increase the probability of having demand a medical appointment in the previous three months. Further comments are developed in section 2.4.2..
41
5.3
Sub-systems with di¤erent threshold coe¢ cients
5.3.1 SS 1
Ordered Logit Model
=
N HS 1
SS 2
=
+ Ds1 Ss = Ds1 Ss (since we normalized
N HS 2
N HS 1
+ Ds2 Ss
Ordered Logit with Signi…cant Dummies Regressors variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 Sch0 Sch1 Sch2 Sch3 W h2 W h3 W h4 Age2 Age3 Age4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5 M e1 M e3 Fe HE3 HE4 HE5 Ss N HS 1
Ds1 N HS 2
Ds2
estimated coe¢ cient 3.34137 -2.12182 .140543 .064912 .105208 .167590 .139267 .515662 .439727 .244318 .030720 .153936 .115527 .163576 .081050 .157380 .021549 -.420398 -.350738 -.403848 -.499571 .356638 .270690 .173350 .088440 .083599 .139401 -.312794 -.225716 -.482666 -.950439 -.960683 -.951235 0 0 2.89919 -1.03362
t-statistic 33.5049 -54.3481 3.63528 1.06372 1.32413 2.96042 .510594 4.38716 5.51985 3.16900 .360891 3.28147 2.38980 2.34571 1.42663 2.56124 .310767 -7.40499 -9.05877 -8.75115 -2.44585 4.35205 3.61506 2.62948 1.27723 1.31971 2.56644 -5.96722 -5.61023 -11.4419 -16.6090 -10.8900 -15.7105 80.3797 -17.0599
pvalue [.000] [.000] [.000] [.287] [.185] [.003] [.610] [.000] [.000] [.002] [.718] [.001] [.017] [.019] [.154] [.010] [.756] [.000] [.000] [.000] [.014] [.000] [.000] [.009] [.202] [.187] [.010] [.000] [.000] [.000] [.000] [.000] [.000] [-] [-] [.000] [.000]
then, we have that: 42
= 0)
SS 1
=
N HS 2 SS 2
5.3.2 SS 1
N HS 1
=0
= 2:89919
= 2:89919
1:03362 = 1:86557
Ordered Probit Model
=
N HS 1
SS 2
=
+ Ds1 Ss = Ds1 Ss (since we normalized
N HS 2
N HS 1
+ Ds2 Ss
Ordered Probit with Signi…cant Dummies Regressors variable Constant Exam Dr1 Dr2 Cig1 Cig2 Cig3 Sch0 Sch1 Sch2 Sch3 W h2 W h3 W h4 Age2 Age3 Age4 GCI1 GCI2 SP ills Rc P cInc1 P cInc2 P cInc3 P cInc4 P cInc5
estimated coe¢ cient 1.80446 -1.19132 .087433 .044156 .082161 .099933 .051864 .339871 .278467 .147973 .018967 .089522 .067957 .098398 .047116 .098843 .026960 -.244359 -.205421 -.234880 -.268364 .232243 .176325 .117217 .067215 .053649
t-statistic 31.8981 -55.8363 3.85506 1.23510 1.76397 3.04065 .332441 4.89012 6.03735 3.31890 .386023 3.26330 2.40753 2.42124 1.42916 2.76323 .664776 -7.32584 -9.11221 -8.65493 -2.19481 4.83126 4.02847 3.04926 1.66472 1.45719
pvalue [.000] [.000] [.000] [.217] [.078] [.002] [.740] [.000] [.000] [.001] [.699] [.001] [.016] [.015] [.153] [.006] [.506] [.000] [.000] [.000] [.028] [.000] [.000] [.002] [.096] [.145]
43
= 0)
variable(cont.) M e1 M e3 Fe HE3 HE4 HE5 Ss N HS 1
Ds1 N HS 2
Ds2
estimated coe¢ cient .085690 -.191603 -.134391 -.282148 -.554585 -.563375 -.500317 0 0 1.60977 -.549315
t-statistic 2.68369 -6.29969 -5.69976 -11.5203 -16.6129 -10.8501 -14.7868 88.7988 -16.2378
then, we have that: N HS 1
=
N HS 2
= 1:60977
and
SS 2
SS 1
=0
= 1:60977
:549315 = 1:060455
44
pvalue [.007] [.000] [.000] [.000] [.000] [.000] [.000] [-] [-] [.000] [.000]
6
References (Baptista, Carlos, 1999), "Uma Abordagem Quantitativa: Os Subsistemas de Saúde, Compromisso na E…cácia e na Qualidade", Papel dos Sistemas Privados de Saúde num Sistema em Mudança, Associação Nacional de Sistemas de Saúde, 1999, pp. 26-64 (Becker and Kennedy, 1992), "A Graphical Exposition of the Ordered Probit", Econometric Theory 8, 1992, pp.127-131 (Cox, D., 1970), Analysis of Binary Data, London: Methuen, 1970 (Crawford et al., 1998), "Simple Inference in Multinomial and Ordered Logit", Econometric Reviews 17(3), 1998, pp. 289-299 (Davidson, J., and Mackinnon, J., 1993), Estimation and Inference in Econometrics, New York: Oxford University Press, 1993, ch. 15 (Drummond et al., 1987), pp. 157-165 (Finney, D., 1971), Probit Analysis, Cambridge: Cambridge University Press, 1971 (Godfrey, L., 1988), "Tests for Qualitative and Limited Dependent Variable Models", Misspeci…cation Tests in Econometrics, Cambridge: Cambridge University Press, 1988, ch.6 (Greene, William H., 1993), Econometric Analysis, Mamillan, New York, 1993 (Gujarati, Damodar N., 1995), Basic Econometrics, McGraw-Hill International Editions, 3rd Edition, 1995 (Instituto Nacional de Estatística, 1997), Inquérito Nacional de Saúde 1995-1996, Portugal, 1997 (Leal, António de Sá, 1999), "Os Sistemas Privados de Saúde em Portugal", Papel dos Sistemas Privados de Saúde num Sistema em Mudança, Associação Nacional de Sistemas de Saúde, 1999, pp. 11-16 (Malinvaud, E., 1966), Statistical Methods of Econometrics, Rand McNally and Company, Chicago, 1966, pp.195-197 (Murphy, Anthony, 1996), "Simple LM Tests of Mis-Speci…cation for Ordered Logit Models", Economic Letters 52, 1996, pp. 137-141 (Pagan, A., and Vella, F., 1989), "Diagnostic Tests for Models based on Individual Data: a Survey", Journal of Applied Econometrics 4, 1989, S29-S59 (Reis, Vasco Pinto, 1999), "O Sistema de Saúde Português: Donde Vimos, Para Onde Vamos", Livro de Homenagem a Augusto Mantas, Associação Portuguesa de Economia da Saúde, Lisboa, 1999, pp. 261-297 (Yatchew, A., and Griliches, Z., 1984), "Speci…cation Error in Probit Models", Review of Economics and Statistics 66, 1984, pp. 134-139 (Zavoina, R., and McElvey, W., 1975), "A Statistical Model for the Analysis of Ordinal Level Dependent Variables", Journal of Mathematical Sociology, Summer 1975, pp.103-120
45
