An overview of cross-sectional studies of mortality and air pollution and related statistical issues

Environment International, Vol. 9, pp. 177-194, 1983

0160-4120/83 $3.00 + .00 Copyright © 1983 Pergamon Press Ltd.

Printed in the USA. All rights reserved.

AN OVERVIEW OF CROSS-SECTIONAL STUDIES OF MORTALITY AND AIR POLLUTION AND RELATED STATISTICAL ISSUES Paolo F. Ricci and Ronald E. Wyzga Electric Power Research Institute, Palo Alto, California 94303, USA

(Received 25 May 1980; Accepted 31 January 1983) A heuristic framework is developed for assessing statistical studies of air pollution and mortality which use the General Linear Model as a principal tool for analysis. Arrays that systematically compare the information of each study are introduced. In addition, a number of studies are contrasted against several statistical criteria to assess the reliability of the study results. Reliability is considered in terms of statistical estimation theory and practice. Finally, there is a discussion of several points of statistical contention which can affect the results of these studies. The conclusion is that statistical studies that use the General Linear Model have made limited use of available methods and tests. Several issues are raised which will require additional research.

Introduction The objective of this paper is to develop a framework through which the statistical and biological bases of health studies can be assessed, to enhance the comparisons and understanding of results of various studies, and to indicate the shortcomings of existing studies that frustrate such an assessment. No attempt is made to review every study of the health effects of air pollution. Criteria documents and other reviews serve this purpose. The discussion is limited to 1) selection of those studies which use the General Linear Model (GLM) to estimate the health effects of air pollution, and 2) comments on some of the methodological issues that need be addressed by those studies. Of three types of mortality studies (episode studies, cross-sectional studies, and time series studies), this paper considers only cross-sectional ones. These are nonexperimental studies which statistically examine geographic changes in mortality rates and relate these changes to differences in air quality, while controlling for socioeconomic, environmental, demographic, and other factors. The cross-sectional studies reviewed are shown in Table 1, which is a framework used to compare the qualitative content of the studies, as reported by the authors. Study omissions are, in part, voluntary to keep the review manageable and, in part, inadvertent.

The table describes the following factors for each study: the unit of analysis, the period of study, the sample size, the spatial extent of the study, the form of the model used in the study, the dependent and independent variables, the estimating techniques, and the range of reported summary statistics. Perhaps the most striking fact observed from Table 1 is the lack of uniformity in the specification of the independent variables. Although the choice of the air pollution variables is probably determined by data availability, the variability across studies of air pollutants considered is noteworthy. For example, although either suspended particulate or sulfur compounds are included by all researchers, only Lave and Seskin (1977), Mendelsohn and Orcutt (1979), EPA (1979), Hickey et al. (1977), and Schwing and McDonald (1976) include NOx in their studies. A proxy for smoking is included by Chappie and Lave (1981), Gerking and Schulze (1981), EPA (1979), Lipfert (1978), McDonald and Schwing (1973), and Riggan et al. (1972); trace metals are included by Hickey et al. (1977), EPA (1979), and Lipfert (1978). Indoor air pollution is excluded entirely. Lipfert (1978) includes benzo(a)pyrene; Hickey et al. (1977) include water hardness; Riggan et al. (1972) and Gregor (1976) include dustfall. The number of socioeconomic, climatic, demographic, and other variables considered varies from 177

178

Paolo F. Ricci and Ronald E. Wyzga

5

d

6,

N

~a

~

~

.~ N > >

.

.°

.

~

- ~° o

~.z~

o

.~

~

~o~ ~

•~ o

-~~§~o

Z

~.~ ~

~

~£

=

m~

~" ~ ~.o~

o.~

d

d

z

8

z

Z

~-

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

~

Z

Z

Z

S.

z

~-

Z

;~

Z

~"

Z

Z

~'.

Z

Z

~

Z

;~

d

Z

Z

Z

Z

~

Z

Z

0 Z

Z

Z

Z

Z

Z

Z

Z

Z

0

Z

Z

Z

Z

;~

Z

Z

Z

0 .=

d

0

._~ -,,i

,.o

Z

f--

o'~ ~

~ ~

:~

~ = ' ~ .~.r~ ~ ~ _

o

~o_

~.-

'~z~.

._~ .-1 ~

,..-1

.~

.~

,..-1

~-~ ~. ~ -.. "N

~o

~, . ~ , ~

,~,

.=

~

~

g < f-

Overview of cross-sectional studies

179

~,~ _

Z

~~

~

~. :g

~

~

~. e,

~,

Z~

Z

Z

~

. ..~ ~, ~ "~ {

-~? :_~ ~..~

^'

d

o o ~

~-~-- ~ o ~ ._.~.~, o ~

.....

~

Z

~

~

^,

~

Z

Z

H 8

o ~ o o . ~ ~

.'.o

_ "~

Z'r

~

Z~u

~

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

;~

Z

;~

Z

;:"

Z

~

Z

Z

;~

Z

~-

z

Z

~

Z

Z

;~

z

z

z

Z

Z

Z

Z

z

z

~"

z

Z

;~

Z

Z

z

z

z

z

-~ ~, ~

'~ ~

.~.~

u

~~~

~.

m'"~

.~

N,.

.-._

oNg

g ,...a

,..l

,...1

,.4

~.o

o

~"

~,

~,

"~ ~ = ' ~

~

u.

0°

.~-~

.

~o ~.~--r

~ ~:~~

~ ~ .~ ~.-

180

Paolo F. Ricci and Ronald E. Wyzga

~'~=~

-~ .-~ 0

o = .~ .o

"" ~ ' ~

~.:

0

~

E

._~ ~ ~" ~. -u . ~ ' ~ .

o

.

~

o

~o.~E8 ° ~ ° ° ~ °

o Z

~o

Z Z e,,

-~, 0 Z

,A Z 0 Z

Z

0 0

Z

O

o o~ .o

g

.~ . e ~ II

.=

II

.o

F~

~

o o=

Z E

"~'=

Z

-~ -~ ~ "~ ~'~ ~.~ u.~

<

~.~ o

Overviewof cross-sectionalstudies none to 10 or more. For example, Lipfert (1978) and Mendelsohn and Orcutt (1979) employ regional variables. However, Hickey et al. (1977), Riggan et al. (1972), Schwing and McDonald (1976), Koshal and Koshal (1974), Thomas (1975), and Liu and Yu (1976) do not include socioeconomic variables. Climatic and weather variables are not included by Smith (1976), Lipfert (1978), Bozzo et al. (1977), Hickey et al. (1977), or Riggan et al. (1972). Nutrition is considered by Chappie and Lave (1981), Gerking and Schulze (1981), and EPA (1979); these studies also control for smoking. The dependent variables vary from the total unadjusted mortality rate to the age-sex-race specific mortality rates. In some cases, cause-specific mortality has been considered as well. In general, the functional specification of the dose-response function that relates mortality to the independent variables is either linear or log-linear. In all cases the method for obtaining estimates of the parameters of the dose-response model is the general linear statistical model. The models themselves are either single equations or two simultaneous equations with an identity.

Overview of the Data The majority of socioeconomic and demographic data used in the work reviewed were available through the decennial census. Unfortunately, census data for diet, exercise, drinking, and smoking are nonexistent. To overcome this problem McDonald and Schwing (1973), Lipfert (1978), Mendelsohn and Orcutt (1979), and Riggan et al. (1972) estimated cigarette consumption from cigarette sales, independent surveys, and extrapolation of national data. Chappie and Lave (1981) use SMSA (Standard Metropolitan Statistical Area) expenditures per capita for cigarette and alcohol use; EPA (1979) uses state data for cigarette consumption. The level of ambient air pollution exposure of an entire area, such as an SMSA, is often derived from aerometric data from a single monitoring station, typically located in the center of an urban area. Yet, the spatial pattern of pollution may be such that, if the central city is more polluted than the suburbs, the measure of air pollution will be biased upward (i.e., the aggregate mean will be higher than the true mean). This will not induce a downward bias in the estimated response coefficient if the relative difference in air pollution values between the measured areas and the weighted population exposure is similar for all observations. In addition, the air pollution measurements are affected by (a) lack of sampling design for locating recording instruments; (b) changes in the analytical (chemical) methods; and (c) contamination occurring among the chemical species sampled and filters used in sampling. These concerns implicate the comparability of air pollution measures for different areas. There

181 may be physicochemical reasons for characterizing pollutants as a complex and interrelated unit, in addition to considering them singly. For example, NO2, 03, and peroxyacetylnitrate (PAN), as well as SOx and TSP, interact chemically, thereby allowing synergistic or antagonistic effects. Different averaging times are used to report the concentrations of pollutants. Although there is some relationship among concentrations of various averaging times, the coefficients estimated in the studies cannot be easily compared from one study to another when different averaging times are used. The location and number of instruments to monitor air pollutants have changed both over time and place, making it difficult to interpret some air pollution measures. Aside from considering physicochemical properties of air pollutants, it may be that there are regional associations between demographic characteristics and concentrations of the pollutants. For example, there is an increase in ambient sulfate levels from the Southwest to the Northeast. The increase coincides with increases in the urban population density, city age, and measures of income per capita. There also are microscale associations to be expected. If these systematic variations occur, the situation becomes more complicated, because the variation detected by the cross-sectional models might be due to simultaneous changes with factors which have been omitted from the model and which may be positively associated with the included independent variables. The units of analysis chosen by the researchers vary from city to census tracts to states. The choice of one unit of analysis over another (aggregated data) is expected to affect the overall variation. The model may show that it is less subject to uncertainty.

Framework for Statistical Review Introduction

A systematic approach for review ensures a common basis for comparing the study results. The flow-charts in Figs. 1 and 2 present a systematic approach that links several decisions associated with statistical estimation and hypothesis testing. The framework is general and applies to the estimation of the parameters of the dose-response models through use of the General Linear Model (GLM). Figure 1 includes time-series considerations because some cross-sectional studies utilize this kind of data. The discussion, however, is aimed primarily at cross-sectional models without lagged variables. The flow-charts establish initial paths to determine their "statistical adequacy"; i.e., the level to which violations of the assumptions underlying the more common statistical estimation techniques have been addressed within each study. This is admittedly a difficult task

182

Paolo F. Ricci and Ronald E. Wyzga

® Y~ d

~2

.o

~

4.a.~

ql

go~

o ip

o-

oi

E

E ~ I1; I1/

E,-I ~-oo

[

~.~

.o

t ~m N F

o~ z

0

E

N ~ m

~

T

~q

0./

°E

0

"~ ~.~

g

I---

f •

e-,

ohr~.

,-~oh

T

u~

,d u

~ o

I--

o I:~i i . u

?

g

N,--I

~_~

~

.~

v w

o P.,. e~ ,-.-i

o"~

oo~

~ . "~

o

o ~.E

._~

E 0

~

~.~

g

~ o~ ,~

o ~

~

~

0

o

o ~

~

o °~

~- ~ "~-~

g- J .o

~ ' ~ ~.. m ~

o

Overviewof cross-sectionalstudies

183

since the basis for study review must be taken from published results only and there is considerable subjectivity involved. Hence, the reader is urged to consider this effort to be preliminary. The framework consists of the following: 1. An array that permits consistent scrutiny of the key characteristics of each study (Table 1). 2. Flow-charts that link modeling to estimation (Figs.

context of nonexperimental work, the effect of air pollution on human health can be studied by considering a function relating the mortality rate to pollution and other independent variables: MR = f (demographic, socioeconomic, air pollution, climatic, regional, behavioral, diet, personal habit, hygiene variables) + U,

(2)

1-3). 3. Tabular description of the level of treatment afforded common statistical assumptions (Table 2). 4. An initial suggestion for aggregating statistically the estimated parameters of several equations across several studies. Most elements of Figs. 1-3 contain a key reference or references, which can provide the reader with an overview of the subject of that element. Figures l and 3 provide the kernel of the methodology for reviewing the studies, since they link estimation and hypothesis testing to model and variable specification. Since some recent studies include the use of simultaneous equations, Fig. 3 is included for completeness, but without much detail. The estimation of the model parameters and hypothesis testing require assumptions which, if violated, can lead to erroneous results. Figure 2 provides a mechanism to determine the accuracy of estimation, on the basis of some of the diagnostic and constructive tests.

Statistical modeling issues In matrix form, the simplest general linear model (of which regression analysis is an example) is Y = XB + U,

(1)

Where Y is a vector of observations on the independent variable, X is a matrix of observations on the explanatory variables, B is the vector of parameters to be estimated, and U is a vector of random errors. It should be noted that Eq. (1) is linear in B, although the variables may be nonlinear. The parameters of Eq. (1) are commonly estimated through the (ordinary) least squares (LS) method, given a sample of observations, Y and X, under several assumptions that are to be tested after estimation. For the studies we have reviewed, the LS is the technique preferred by the authors. For the included studies, the dependent and independent variables are developed from human physiology, economics, the environment, and other considerations. However, the interactions among these variables are not well understood. Researchers can, at best, make use of existing knowledge and experience to relate health effects and pollution. This knowledge suggests that, in the

where MR is the vector of mortality rates, (e.g., total male, total female, age-sex-race adjusted mortality rate, etc.), and U is the (additive) error structure. The models reviewed here use prior knowledge to specify models or variables, but only to a limited extent. Linear functions are most often applied because they are simpler and can be thought of as a first approximation to more complex functions. Of course, most biological phenomena are not linear, so it is probably incorrect to consider the linear function except perhaps over a limited range of observations. The estimation from alternative models may yield different results, especially at low doses. The true specification is unknown; it is therefore probably wise to consider several alternative specifications and investigate whether the results remain robust across specifications. Table 3, which subjectively summarizes the performance of the various studies, shows that 3 out of the 17 studies reviewed do not consider alternative specifications. In any case, it is quite apparent that the adequacy of specification for the studies reviewed is, to say the least, quite variable. The objective is one of accurate specification. One symptom of the specification error is the lack of independence between the independent variables and the residuals, as discussed in Table 2 (line C). This might reflect the fact that relevant independent variables are omitted from the estimated model. In this case, the estimates of the regression coefficients, B, are biased and the inference is inaccurate since the estimates of the residual variance will be biased upwards (Johnston 1972). Realistically, no study can include all relevant variables, and the error term is assumed to include random effects imparted by less than full knowledge. Figure 1 includes recent or pivotal statistical references on material useful in practice in constructing and testing models. The level of detail in Fig. 1 is relatively low; many of the boxes represent a subdiscipline, within statistical estimation theory, or consist of well-known estimation procedures (e.g., maximum likelihood or least squares). The objective is to indicate how a researcher might attempt to improve statistically obtained results, especially when (a) the specification of the model (i.e., the function and the independent variables) is unknown, (b) there is no control over the data, and (c) the sample size is "small." Since decision-

184

Paolo F. Ricciand Ronald E. Wyzga

makers have been influenced by estimates of lives lost or o f morbidity cases, the accuracy of the estimates is of great importance. Table 1 suggests that underspecification is among the key issues o f concern. The choice of the statistical estimators in the analysis depends on whether the researcher's concern is with bias or overall precision. If the specification of the model is correct, under squared error loss, the least squares (LS) estimator is unbiased and minimax, that is the estimator minimizes the maximum expected loss. The risk function is

o(fl,/~)= I I(~, ~)f(Y lfl) dy, Y

where /~ is the unknown parameter vector, ~ is the estimator, and y is a random observable outcome. However, the review of the literature suggests that the LS estimator may not be unbiased (see Table 4). Thus, it makes sense to consider other classes of estimators. Specification error is closely related to the choice of restrictions placed on the null hypothesis. If the prior information is correct, the (restricted) LS estimator is superior (more precise) to the unrestricted estimator. Pretest estimation (exclusion or inclusion of independent variables through the restricted null hypotheses) affects the true level of significance under which hypotheses are tested (the nominal level of significance being that level which the investigator uses). Judge et al. (1980) note that the inaccurate specification of the restrictions on the null hypothesis affects the level of significance, the critical level, and the performance of the estimator. The restricted estimator is unbiased only if the restrictions are correct. For the risk function discussed here, the estimators generally are not dominant over the entire parameter space. James and Stein estimators may be an improvement on the approaches taken by those reviewed here. Using the squared error loss criterion, when the number of parameters is strictly greater than 2, this class of estimators has been shown to be an improvement over least squares (Judge et al., 1980). There is no absolute prescription for assessing the performance of one estimator relative to another. Performance depends on the risk function and on the judgment of what constitutes an acceptable risk. In general, this judgment depends on the objective of the analysis, since one estimator does not normally dominate another entirely. In some cases the LS estimator is dominant over some portion of the parameter space, but most often this is not so. In the dose-response work we have reviewed, reference is often made to animal tests (in vivo tests). These tests may provide information on the likely direction of a health effect, though the magnitude is subject to extrapolation from animal to human. If this is the case,

the restrictions posed on the null hypothesis should include this information. This paper cannot attempt to show the effect that exactly restricted or inequality restricted estimators, Steinrule estimators, or other estimators (such as Bayesian) may have on the various specifications shown in Table 1. Our review shows that at least 11 of the studies consider alternative sets of independent variables in their analyses. Nonexperimental data preclude the estimation of a fully specified model, even if causation were known, because the full set of relevant independent variables is not known. The unresolved question is the extent to which the bias induced by omitting relevant variables may affect results. A practical method to explore the nature of association through alternative model and variable specifications may be found in pretesting, but as Schwing and McDonald (1976) suggest, there is concern that bias may be introduced through such a method by choosing a preliminary test estimator based on only part of the data. In doing so, the researcher has selected an estimator which, under quadratic loss function, performs well if the null hypothesis is correct; however, if the hypothesis is not correct, the results will be poor over a considerable interval of the parameter space. For this reason T o y o d a and Wallace (1979) recommend the use of the complete data for pretesting. Later replication should not, however, be discouraged. Epidemiological work normally does not exist in vacuo; experimental evidence from in vitro, in vivo, and human clinical studies may be available. This information may be useful to provide upper or lower bounds on the magnitude and direction of the effect of a pollutant. Experimental data can be included through restricting the null hypothesis. If the restriction (the hypothesis) is correct, the restricted LS estimators are unbiased and have smaller variances and mean squared errors than do the unrestricted LS estimators (Judge and Bock, 1978). However, if the restriction is not correct, the restricted estimator is adversely affected. Upper or lower bounds on the coefficients of an epidemiologic model may be derived from previous epidemiological studies. These restrictions would provide an indication of the stability of the estimation, by incorporating past evidence with new evidence. Bayesian estimation techniques appear promising, in that they take advantage of prior information and experimental information (see Fig. 1). In using previous studies, one must be aware o f their limitations and ensure that erroneous restrictions are not imposed. The appeal of the Bayesian approach is that it allows prior judgment to be introduced probabilistically rather than absolutely; in addition, experimental or new evidence can be used with prior information. Once the parameters are estimated, tests of significance for the parameters, under the null hypothesis, are

Overview of cross-sectional studies

185

applied at a prespecified level of significance, a. Hence the probability of rejecting the hypothesis, when true, is some a-level. In the case of k regressors, the probability p(k) of rejecting at least one hypothesis out of k is

p(k) = 1 - (1 - a ) k,

(3)

under the assumption that the k statistics are independent. The result is that for k tests, the level of significance is changed from a to (1 - (1 - a)k). If the k statistics are not independently distributed, it can be shown that an upper bound on the overall level of significance for the simultaneous test is the sum of each individual level of significance. This problem is part of the multiple comparisons problem which alters the level o f significance, when the same data are used in estimating alternative specifications of the same model (see Table 2, line J). This statistical issue is important in those studies that consider alternative causes of death, age groups, or geographic regions in separate regressions, on the same data. Among the studies reviewed here, all but one ignored the multiple comparisons problem, despite its relevance. The multiple comparisons problem is o f less concern when alternative specifications are considered because the test statistics are not independent, if the same data are used.

Diagnoses and remedies Once the model has been selected, various implications of estimation can be discussed. The question o f specification of the dose-response model has an empirical side which can be developed after estimation through the least squares. These tests include the analysis of residuals (i.e., the difference between the estimated values and the actual values of the dependent variable). Such analysis can be undertaken by visual inspection or by statistical tests, some of which are nonparametric. Figure 2 shows some of the most commonly used diagnostic tests and remedies in statistical work. The logic of Fig. 2 follows the assumptions underlying the ordinary least squares (OLS) estimators, simply because most researchers have utilized these estimators. Figure 2 is linked to Table 2; in the first row of Table 2, each assumption is identified by a capital letter, which will be used as reference in the text below. Table 2 lists several statistical considerations and provides a subjective indication of the importance of these on the obtained results. We have ranked the issues on a scale of very important (V), important (I), or contributing (C) to the interpretation of study results. This ranking reflects the subjective opinion of the authors; the significance of the ranking depends upon the application of the study, be it scientific investigation or policymaking. Depending on the use, statistical ineffi-

Table 2. S u m m a r y o f statistical points of contentions and effects on the ordinary least squares and their assumptions, a Code to Text of Paper (A)

Statistical Points o f Contention Linear combination of independent variables (multicollinearity)

Rank V

Likely Effects on Estimation and Testing

Areas of Applicability

The variance of B is increased

Some (not all) coefficients are affected

Estimates of B are sensitive to data, additions or deletions

Point estimates are less affected t h a n interval estimates

Difficult to separate the effect o f collinear variables (B)

Relevant omitted variables

V

Estimates of B are biased if stochastically dependent

See text for discussion of specification error

The estimated residual variance is biased upwards (C)

The independent variables and error are correlated

I

Inconsistent estimators of B are obtained

Refer to (E) below

Some coefficients may be biased (D)

Errors are correlated

C

Inefficient estimates of B R 2 is increased unwittingly

Coefficients of the dose-response are not biased, if the regressors are nonstochastic (Smith 1974)

Estimated variances are biased (E)

Substitution of a variable for another (instrument variables)

I-V

Inconsistent estimates of B result

Consistent estimators of B will be obtained if the instrument variables are uncorrelated in the limit with the error term and measurement error, (Johnston, 1972). See (D) above Smaller asymptotic bias for least squares

186

Paolo F. Ricci and Ronald E. Wyzga Table 2. (Continued)

Code to Text of Paper (F)

(G)

Statistical Points of Contention

Rank

The variance of the errors is not constant (heteroscedasticity)

C

Normality assumption for the error does not hold

I

Likely Effects on Estimation and Testing The estimators are not efficient The estimated variances are biased

Areas of Applicability The estimated coefficients are unbiased and consistent

Hypothesis tests under F and t statistics may not be valid in finite samples

ML yields asymptotically efficient estimators if the distribution of the error is known

The ordinary least squares estimators may not be efficient

See text

Sampling properties of estimators discussed may not hold The variance of economic variables may not be finite (H)

Errors in the dependent and independent variables

I-V

Variance is larger

Asymptotic bias may not be a problem

The estimates of regression coefficients are biased and inconsistent

Assumptions or knowledge about error variances may help estimation Restrictions on the null hypothesis may identify the model

F statistics are affected (1)

Missing observations

I

If observations are not missing at random, the estimates of B can be biased

Randomly missing observations do not change bias See instrument variables

Efficiency is reduced

(J)

Adjustment for several hypothesis tests

I

Incorrect interpretation of hypothesis test results, since level is changed

See text

(K)

Model specification (functional form of model)

V

The estimated coefficients are biased

See text

The variance of the residuals is biased aThe references from which this table is constructed include Judge et al. 0980); Johnston (1972); Theil (1971); Pindyck and Rubinfeld (1976).

ciency m a y be a c c e p t a b l e ; however, statistical inconsistency is u n a c c e p t a b l e . F i n a l l y , bias w o u l d also be generally u n a c c e p t a b l e . T r a d e - o f f s a m o n g bias, efficiency, a n d consistency exist a n d m u s t be considered by the researcher, especially when prescriptive a c t i o n s d e p e n d o n e s t i m a t i o n . O f the a s s u m p t i o n s , the r a n k c o n d i t i o n (A) is t h a t which m a y affect the O L S e s t i m a t i o n p r o c e d u r e fund a m e n t a l l y . It m a y be d e t e c t e d when the value o f the d e t e r m i n a n t o f the X TX m a t r i x is at o r n e a r zero. T h e p r o b l e m arises when one o r m o r e i n d e p e n d e n t variables are an exact o r n e a r l y exact linear c o m b i n a t i o n o f a set o f i n d e p e n d e n t variables. A d i a g n o s t i c test to detect m u l t i c o l l i n e a r i t y is suggested b y F a r r a r a n d G l a u b e r (1967). O t h e r m o r e recent tests exist. G e n e r a l l y , m u l t i c o l l i n e a r i t y affects the p r e c i s i o n o f the e s t i m a t o r , the v a r i a n c e o f which is increased. Inference a n d h y p o t h e s i s testing are j e o p a r d i z e d since the

s a m p l i n g p r o p e r t i e s o f several e s t i m a t o r s a r e not k n o w n . P o i n t e s t i m a t i o n is c e r t a i n l y less affected. T w o o f the 15 studies reviewed ( T h o m a s , 1975; Schwing a n d M c D o n a l d , 1976) have a p p r o a c h e d the m u l t i c o l l i n e a r i t y p r o b l e m t h r o u g h use o f the ridgeregression p r o c e d u r e . T h e basis for the r i d g e - r e g r e s s i o n (RR) a p p r o a c h is to m i n i m i z e the v a r i a n c e o f the e s t i m a t e d coefficients at the expense o f o b t a i n i n g b i a s e d coefficients. T h e essence o f the R R m e t h o d is to o b t a i n an a l t e r n a t i v e to the O L S e s t i m a t o r o f B, in the m o d e l Eq. (2), so that, for s o m e k _> 0:

B(RR) = ((X~X) + k l ) - ' X T Y , where the caret indicates the vector o f e s t i m a t e d p a r a m e t e r s a n d the X in d e v i a t i o n f o r m . W h e n k = 0, B(RR) is the O L S estimate. A s k increases, 13(RR) tends

Overview of cross-sectional studies

187

K 0

3(U 4

C,.

0 __

~

© 0~

o~

~

c~ r--

~

•.

E~

N

~

~

~

0

m

•

~

~

o

~

0 4-3

~ 0

~

0

~

4J

m

<

r"

t~

-,-.I

b~

01

0

0

OI 0

'U

~

• "Fk ~4 r ,

Xl

• b~

~o ~0

0 ~

• u~

-m . ~

'-'

0

q3.-~, ~

/

~q,-I

I

~

~

~

0

0-,~

~

~

o

,--I CO OOh

~ O

~CO ~Oh

,. >+

• ,-~

"'q •

c' • ~1

<

0

0

0

~

0

0

0

~

0

"~ -~

,.c:

o

o

~

o

~ 0 ,~.C ~4J

.o

0

0

0

121

-~

saA

~., U~ 0

"~

X

11', M

~

\

-

__ ~_

\

"13

.9 0

t o co oh

0

0

~

0

0 " mm

.,4

~: •

~qC3

..=

0

O~

C~

o

v

v

r~ • •. C~

.~

~

~

cq

~

0

-~

0

•

u~

.q

~¢j

-

•

•

•

0

I-+

8

qJ

n3 ,.+

~I~4

tn to

D.+

o

©,--I

O~

¢J ~

.,4

0

0

0

0

×1| o

° ~ .~

C~ O

<

0

r..fl

0

0

0 O~ o

.o Lq O~ v

0 aJ

m

~.o °° E"+ U'J

Cn~ C

£--;

0

Q3

"~ ~

0 <

0

I

~

188

to zero. The criterion for estimating ridge-regression parameters is the minimization of the sum of mean square errors. Vinod (1977) and others provide further exposition on the meaning of k and a critique of the ridge-regression procedures. The generalized ridge regression estimators can be linked to the Stein-rule estimators through the "generalized" ridge-regression estimator: fi* = (X~X - diag(k,))-' X~X = GAG T = ZIIoLS, where IlOLs is the OLS estimator which includes shrinkage fractions. Those "fractions" depend on the eigenvectors of a transformation of X which allows OLS estimation of the transformed canonical model, provided that the matrix of observations, X, is the nonsingular correlation matrix. The ordinary ridge estimate can be shown to be a function of the OLS estimator and the appropriate eigenvalues. Given the canonical form, prior information can be included in the estimation (under the normality assumption) as outlined by Weisberg (1980). The relationship of the ridge regression estimator to Bayesian methods is summarized by Weisberg (1980) and Judge et al. (1980). A second issue commonly affecting the estimation method used is whether the assumption of normally distributed error holds. Of the 17 studies reviewed, only five discuss the assumption of normally distributed error. The assumption of normally distributed error term ensures the following (Judge et al., 1980): 1. The least squares estimator b = (XrX)-IXTy is unbiased minimum variance from within the class of all unbiased estimators, asymptotically efficient and consistent. 2. The variance estimator #2 = ( y _ X b ) r ( y _ X b ) / ( T - K ) is best quadratic unbiased; that is, it has minimum variance of all estimators of o2 that are unbiased and quadratic in y; also it is asymptotically efficient and consistent. 3. The respective distributions for b and ( T - K ) # V o ~ are normal and x 2 ( T - K ) and, furthermore, they are independent. 4. The F test on a set of linear restrictions Iq~ = r, and t tests on the individual coefficients, are justified in finite samples. Here, the number of independent variables is K, and the sample size is T. If non-normality is present it may be due to the existence of outliers, distributions with infinite variance or simply distributions other than normal. Robust estimators (Hogg, 1979) may be an answer. In general the normality question is affected by outliers. The decision to remove outliers cannot be made solely on statistical grounds; theory, knowledge of physicochemical processes, analytical instrumentation, or other considerations of validity of data may provide good

Paolo F. Ricci and Ronald E. Wyzga

reasons for omitting outliers. There are methods for attacking this issue statistically (Andrews, 1974). In some cases, the data may be subjected to the Box-Cox transformation so that normality is introduced (Zerembka, 1974). Robust techniques have been developed to analyze data which are not well behaved (Table 2, line G) (Hogg, 1979). A typical example of a robust technique is the "jackknife" method, in which data are sequentially eliminated to obtain successive estimates of the parameter being sought. The deletion and estimation process determines the distribution of the parameters for further analysis. Other robust approaches include pretesting the model on a portion of the data, then fitting the "clean" model to the remaining data. Of the studies reviewed, apparently only four applied robust estimation techniques. A third statistical issue arises when the dependent and independent variables are heterogeneous. For example, air pollution measurements, census data, meteorological measurements, medical diagnoses and other data have errors; thus, an error structure is associated with both the dependent and the independent variables (Table 2, line H). The magnitude of the errors affecting the variables depends on the data used in the estimation. The errors in the independent variables produce biased parameters, as discussed by Johnston (1972) and Griliches (1974). Smith (1976) extensively considers the impact of the errors-in-variable in the context of air pollution and mortality. The consideration of errors in the independent variables leads to another issue, that of the unavailability of data for specific dependent or independent variables. In this case, the researcher uses proxies in lieu of the variables of choice. This approach, known as the "instrument variables" approach, consists of adopting variables which have high correlation with the desired variable (the correlation ideally being one; see Table 2, line E), but which are uncorrelated with the error, U. In epidemiologic work, there is often a need to pool cross-sectional and time-series data. This is due either to the paucity of information in any one period of time or to the health impacts of past pollution over an extended period of time. The effect of pooling is that the stochastic disturbance may be correlated with the independent variables, thus violating the assumption that the independent variables are uncorrelated with the disturbance. Serial correlation can sometimes be treated by assuming a first order autoregressive error structure: U, =

QU,-I d- ~'t,

where t is time (see also Newbold and Davis, 1978). The issue of autocorrelation appears when a researcher uses lagged dependent variables. The issue is addressed in Figs. 1 and 2. Whether it is reasonable to assume that a

Overview of cross-sectional studies

189

first (or any other) order error structure is acceptable depends on (a) whether the data of interest are of the appropriate time interval, and (b) whether the error term reflects lack of knowledge about the model's specification or about the influence of independent variables whose effect changes over time (see Maddala, 1977). Autocorrelation will not lead to biased LS estimators, but in general the estimators will not be efficient (Judge et aL, 1980). Another data-related problem which affects the precision of the estimates obtained through the OLS method occurs when the observations belong to distributions with different variances. When this occurs, the homoscedasticity assumption (line F in Table 2) of the OLS does not hold. A remedy to the problem of heteroscedasticity, suggested by Johnston (1972), is weighted least squares. Earlier, Glejser (1969) had suggested an alternative to weighting the observations. This alternative consists of estimating the covariance structure of the disturbances by forming a polynomial in X , : U = Vp(X,),

when V is the multiplicative error. The error covariance matrix is obtained by taking the absolute value of each residual, a linear function of the appropriate indepen-

dent variable, and testing the significance of the coefficients of the linear function. The heuristic (and tentative) linkages Table 3 links the framework to the studies reviewed by indicating the extent to which the studies observed the various statistical considerations given in Table 2. Table 3 uses three subjectively defined descriptors: 1. iI, which indicates that the study implicitly accounts for the points of contention; 2. el, which indicates that the study explicitly accounts for the points of contention; and 3. E, which indicates that, on the basis of the material reviewed, there is a subjective judgment that the study does not include a point or points of contention.

Of course, only the authors of the various studies can legitimately classify their studies; nevertheless, a threescore method is used here. When the authors clearly addressed a point of contention it was rated el. If there was mention of the point, without actual application, it was rated iI. An omission was rated E. Clearly, this scoring method is imperfect and is based upon reported findings. Table 3 limits consideration to 8 of the 11 statistical points of contention given in Table 2.

Table 3. Points of contention addressed by the researchers in the statistical estimation of dose-response models,a Ranking (in parentheses).

Model (1)

Variable (2)

Errors in the Measurement of the Independent Variables (4)

el

iI

el

el

el

el

E

none needed

el

E

E

E

E

el

E

no

el E

E el

E E

E E

E E

el E

E E

no no

il

eI

E

E

il

el

E

no

il

E

E

E

E

E

E

no

E

el

E

E

E

el

E

no

el el iI il iI

el el el E il

E E E E E

el E il el E

el E E E E

E eI il el E

E E E E E

no no no no no

iI E

il el

E E

E E

E el

E il

E E

no no

el

el

E

E

E

E

E

no

Specification

References Smith (1976) Schwing and McDonald (1976) McDonald and Schwing (1973) Bozzo et al. (1977) Hickey et aL (1977) Riggan et aL (1972) Koshal and Koshal (1973) Mendelsohn and Orcutt (1979) EPA (1979) Liu and Yu (1976) Gregor (1976) Thomas (1973) Kitagawa and Hauser (1973) Lipfert (1978) Lave and Seskin (1977)

Constant Variance for the Error Term (8)

Normality of Error Term (6)

Correlation Among the Independent Variables (2)

Correlations Within the Error Term (Spatial) (7)

Adjustment for Several Hypothesis Tests (4)

al included (il is implicitly included; eI is explicitly included); E is excluded. See text for further discussion.

190

Paolo F. Ricciand RonaidE. Wyzga

The information contained in Table 3 indicates that considerably more statistical information could have been reported in the work reviewed here.

Multiequation estimation As multiple equations have only recently been introduced to the mortality air pollution problem (Chappie and Lave, 1981; Gerking and Schulze, 1981; EPA, 1979), this section is short and merely introduces the subject. References are widespread (Johnston, 1972; Zellner and Theil, 1962; Klein, 1973). The structure of a multiequation model is as follows: yI" + xB = e,

(4)

where I" is a square and invertible matrix of the coefficients for the endogenous variables, B is a matrix of coefficients for the exogenous variables, x is the vector of exogenous variables, y is the vector of endogenous variables, and e represents the disturbances. This model is readily changed to the so-called "reduced" form (assuming the correct specification for the equations in the model) for statistical estimation: y =

-xBI"-'

+ el"-L

(5)

The multiequation figure, Fig. 3, links the mathematical aspects of identifying the equations forming the system to some of the methods for obtaining estimates of parameters. Identification consists of ensuring that the simultaneity that characterizes the equations does not affect the estimation process (in the sense of obtaining unique estimates for the parameters of the model). Broadly, completeness and the necessary and sufficient conditions for identification involve the rank conditions of I" and B and the variance-covariance matrix of the system of equations. The estimation of parameters of simultaneous equations through the OLS yields inconsistent and biased estimates, since the condition that the cov(xu) = 0 does not hold, and the regressors may not be predetermined. In other words, the model contains variables that are at once predetermined and endogenous. Hence, other methods are used. We recognize that multiequation construction and estimation is a difficult area, essentially new in the study of mortality by air pollution. It is also an area which requires continued analytical and empirical research. With these concerns in mind, we limit our discussion to points which may be of interest to future work. In some instances, the parameters of a system, quite narrowly defined, can be estimated through the proxy-variable approach in a modification of the OLS. To the extent that the 2SLS, an interated application of the OLS, can be assumed to be consistent, the 2SLS may be an appropriate estimator. However, since the 2SLS is an equation-by-equation estimator, its efficiency may be

questionable. In this case, likely candidates are the 3SLS or the Full Information Maximum Likelihood method. The 3SLS is based on the generalized least squares approach, described in Zellner and Theil (1962). The 3SLS method, if the system is overidentified and the sample is large, uses all of the information relevant to the identification as utilized in the estimation, yielding a more efficient estimator. If the model is just identified, the 3SLS will not yield improved estimators over the 2SLS. If the errors of each equation forming the system are correlated, an efficient estimator may be obtained through the estimation of all equations considered as a larger equation. Here, some information about the variance-covariance matrix for the population must be known. Generally, however, the variance-covariance structure is calculated from sample information. When this structure has nonzero covariances, the advantage of forming a larger system is a gain in efficiency. However, unless the system is fully recursive, the OLS estimators will be inconsistent. In a fully recursive system, the dependent variables are determined sequentially, given that the errors are independent, among the equations of the system. If the structure which relates mortality rates can be assumed to be recursive, rather than simultaneous, then the OLS is an appropriate procedure. In both single- and multiple-equation systems, the definition of a small sample, and the effects of using a small sample, are important considerations. The performance of the estimators for simultaneous equations, given a small sample, is not generally known. Moreover, as Zellner (1979) has noted, even if a structural econometric model is correctly specified, various asymptotically justifiable estimators of the same parameter can assume notably different values. Little information is available on this issue; most of the attention has been directed to finite sample properties of the estimators. Zellner (1971) describes the Bayesian estimators, in the context of single-equation analysis, 2SLS, and 3SLS, as well as other estimators and hypothesis testing, under finite sample and large samples.

Statistical "meta-analysis" A possible avenue to pool the results from several studies to see whether a "higher" level of information can be found from literally hundreds of equations is meta-analysis. It consists of aggregating information provided by the coefficients of the estimated equations across studies. This is discussed briefly in the paper by Ricci and Wyzga, prepared for the 1980 DOE statistical symposium, and is summarized as an outline in Fig. 4.

Conclusions Statistical tools, largely multiple regression analysis, were used in the studies reviewed to obtain estimates of

Overview of cross-sectional studies

191

!.4 0

~.5 I

=

m

t

c: 0

4~

t~

O~ ~0[~

0

0

0

4a I~o ~ ~ ~t~

© 0 ~

o

•

•

~o

~

0

-,~ ~

~'~

0 0 0

•

t~ ~ ~

•~

0

m ,. O~11

O

m ~m O O

0 0 O

0

I1)

I

O 0 .~

• .r-I

'lJ O

0

0.~

0 ,.~

O

tO

O~ r~ -~

~ ~

.~ ..~ 0

O

•~

H

~

0

~t

c: 0

~4 C1

/

....

0 -~

044

©

~.~

~0-~0 ~ - ~

~0~

0

F..fl

.g

O ig O 0 . ~ g-i

0

%

r~ 0 01

c~ 0

~

~

oo

.~ OJ ¢J-,~ 0"~

mm

~0~

-I~

,1J

m

m 0

"~

4.1 q:Jom 0 0 0 Z O ~

~'~

~mo

0 0

0" :~ 0 F.fl ~ C)

°1 O .,...I -~

- ~ ~ cO .,-.I

O

1

®,.----

®

g r,.

O

"~ O O

.~ N N ,--I -~ O .~ •

m 0-,-~ ~o-iJ

.,-4

~1

~0~

m ~

L

0 0

~ . ~

®

192

Paolo F. Ricci and Ronald E. Wyzga Table 4. Model alternatives and statistical implications, a States of World Investigators Actions Assume statistical model: y = X,~, + X~;3~ + e Assume statistical model: y = X~/3, + X,B~ + e; ~3, = 0

True Model

True Model

y = X , ~ , + X~B, + e

y = x,/3, + x~3~ + e; ~3~ = 0

b is minimum variance unbiased b* is biased, but its sampling variability is less than that of the unbiased estimator b*o' is biased upward

b is unbiased but not minimum variance b* is minimum variance unbiased

aFrom Judge et al. (1980); b is the least squares (maximum likelihood) estimator; b* is the (hypothesis) restricted least squares estimator.

the mortality-air pollution association. It is fair to state that the research reviewed shows that the multiple regression method is not always carefully applied. In general, the studies we have reviewed use different data, covariates, levels of aggregation, and units of analysis. None of the studies formally introduced the information provided by toxicological and experimental studies. The discussions outlined in the sections of this paper are not exhaustive. They are meant to guide the reader to some of the areas of statistical concern, and to begin the dialogue on structuring analysis in the type of studies we review. Once the issues are clarified, it may be possible to initiate steps to integrate the results from studies as we suggest in Fig. 4. Note that in Fig. 4 the summation indicates that information is aggregated. Finally, we feel that the evident lack of consistency raises questions about the usefulness of the individual results reported, s e n s u policy analysis. Certainly, the in-

formational content of the studies is more uncertain than the results indicate. Areas for further research include: 1. The review of small sample properties of the GLM estimators within the context established in this paper. 2. An application of the methodology for statistical meta-analysis to relevant regression equations found among the studies. 3. An attempt at including experimental studies within the context of the GLM. 4. A fundamental investigation on the validity of air pollution mortality studies (again, in the context of the GLM) given such omitted variables as indoor air pollution, diet, exercise activities, and more accurate measures on smoking.

study (i) the nasxs o

~

I.Ithe

individualh

tallty rate, and physlo- ~stuaies.

T

/I;.~-.~aat~

I A

[

by

[ i , t i = , (e.,J., J t's, etc.)

I[ II ~ = a t ,

I I Ir~,,w~

de-

I

i m ~ of rex- I I tyre lr~ces I H e v a n t wut-tabZm~ [ [e.g, e ~ l J

, remits m ~ s ~kth) • • •

I~ttlnated ~luatietas Smmmry S t a t i s t i c s Test Results

• Summaries of Data • Etc.

~ n n i n e the size o f po~iti~ n~jati~ effects (e.g.

of statistical assumptions

of association) (k + I)

Heuristic filter (i.e. consistency Analytical

at the or-

able t h e o r y )

metho~ologica/filter~late_(e'g"

nominal

nmss) Rs-analysls on the basis of

Data Bases

T

I~

OK? of air pollution variables normallzet by averaqing times (e.g. , through the method of Larsen).

Fig. 4. Toward an aggregation of results.

tl t h e la

clullnlllmal~s~s~c of ~wliv~lual st~li~

Overview of cross-sectional studies

There is a "Catch-22" paradox in our work, since we do not know the marginal improvements from using the methods we suggest: We add sophistication to a problem which may in reality be moot. Adding precision to an inaccurate device is certainly counterproductive. Yet if decisions are taken on the basis o f a device, the device itself should be well developed. On the basis of this review we are not entirely sure that reliability has been assured nor that the device is accurate.

Acknowledgement-This paper is modified from "A Statistical Review of Cross-Sectional Studies of Ambient Air Pollution and Mortality," 1980 Department of Energy Statistical Symposium, Berkeley, CA, October 1980. This paper is the sole responsibility of the authors and should not be attributed to EPRI.

References Aigner, D. J. and Goldberger, A. S. (1979) Latent Variables in SocioEconomic Models. North-HoUand, Amsterdam. Andrew, D. F. (1974) A robust method for multiple linear regression, Technomet. 16, 523-531. Belseley, D. A., Kuh, E., and Welsh, R. E. (1980) Regression Diagnostics. John Wiley, New York. Box, G. E. P. and Watson, G. S. (1972) Robustness to non-normality of regression tests, Biomet. 6S, 318-335. Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis. Holden-Day, San Francisco, CA. Bozzo, S. R., Novak, K. M., Galdos, F., Hakoopian, R., and Hamilton, L. D. (1977) Mortality, Migration, Income, and Air Pollution: A Comparative Study. Preprint of paper presented at the American Association o f Public Health Annual Meeting, Washington, DC. Carroll, R. J. (1979) On estimating variances of robust estimators when the errors are asymmetric, J. Am. Statist. Assoc. 74, 674-679. Chappie, M. and Lave, L. (1981) The health effects of air pollution: A reanalysis, Journal o f Urban Economics (to be published). Dhrymes, P. J. (1971) Distributed Lags: Problems o f Estimation and Formulation, Holden-Day, San Francisco, CA. Draper, N. R. and Smith, H. (1968) Applied Regression Analysis. John Wiley, New York. Draper, N. R. and Van Nostrand, C. (1979) Ridge regression and James-Stein estimation: Review and comments, Technomet. 21, 451-466. Environmental Protection Agency, Office of Health and Ecologic Effects (1979) Methods Development for Assessing Air Pollution Control Benefits. Vol. I. Experiments in the Economics o f Air Pollution Epidemiology. EPA-600/5-79-001a, Washington, DC. Farebrother, R. W. (1979) Estimation with aggregated data, J. Economet. 10, 43-55. Farrar, D. E. and Glauber, R. R. (1967) MuiticoUinearity in regression analysis: The problem revisited, Rev. Econ. Statist. 49, 92-107. Frost, P. A. (1978) Proxy variables and specification bias, Rev. Econ. Statist. 85, 323-325. Gerking, S. and Schulze, W. (1981) What do we know about benefits of reduced mortality from air pollution control? American Economic Association Papers and Proceedings, 1981, pp. 228-234. Glejser, H. (1969) A new test for homoscedasticity, J. Am. Statist. Assoc. 64, 316-323. Gregor, J. (1976) Mortality and air quality, in The 1965-1972 Allegheny County Experience. Center for the Study of Environmental Policy, Pennsylvania State University, University Park, PA (unpublished). Griliches, Z. (1967) Distributed lags: A survey, Econometr. 29, 65-73. Griliches, Z. (1974) Errors in variables and other unobservables, Economet. 42, 971-998. Judge, G. G. and Bock, M. E. (1978) The Statistical Implications o f

193 Pre-Test and Stein-Rule Estimates in Econometrics. NorthHolland, Amsterdam. Judge, G. G., Griffiths, W. E., Carter Hill, R., and Lee, T.-C. (1980) The Theory and Practice o f Econometrics. John Wiley, New York. Hickey, R. J., Boyce, D. E., Clelland, R. C., Bowers, E. J., and Slater, P. B. (1977) Demographic and Chemical Variables Related to Chronic Disease Mortality in Man. Technical Report No. 15, Department of Statistics, University of Pennsylvania, Philadelphia, PA. Hinich, M. J. (1979) Estimating the lag structure of a nonlinear time series model, J. Am. Statist. Assoc. 74, 449-452. Hogg, R. V. (1979) Statistical robustness: One view of its use in applications today, Am. Statist. 33, 108-115. Huber, P. J. (1972) Robust statistics; A review, Ann. Math. Statist. 43, 1041-1067. Jeong, K. J. (1978) Estimating and testing a linear model when extraneous information exists, Int. Econ. Rev. 19, 541-543. Johnston, J. (1972)Econometric Methods, McGraw-Hill, New York, NY. Jonneda, R. K. and Fegley, K. A. (1974) Path analysis in systems in science, IEEE Trans. Syst. Man Cybern. SMC-4, 418-424. Kitagawa, E. M. and Hauser, P. M. (1973) Differential Mortality in the United States: A Study in Socio-economic Methodology. Harvard University Press, Cambridge, MA. Klein, L. R. (1973) A Textbook o f Econometrics. Row-Peterson, Evanston, IL. Kmenta, Jan (1971) Elements o f Econometrics. MacMillan, New York, NY. Koshal, R. K. and Koshal, M. (1973) Environment and urban mortality--An economic approach, Environ. Pollut. 4, 247-259. Koshal, R. K. and Koshal, M. (1974) Air pollution and the respiratory disease mortality in the United S t a t e s - A quantitative study, Soc. Indicators Res. 1, 263-278. Lave, L. B. and Seskin, E. P. (1977) Air Pollution and Human Health. The Johns Hopkins University Press, Baltimore, MD. Lawless, J. F. (1981) Mean squared error properties of generalized ridge estimators, J. Am. Statist. Assoc. 76, 462-466. Learner, E. E. (1978) Specification Searches. John Wiley, New York, NY. Lipfert, F. W. (1978) The association of human mortality with air pollution: Statistical analyses by region, by age, and by cause of death. Unpublished Ph.D. Diss., Union Graduate School, New York, NY. Liu, B. C. and Yu, E. S. (1976) Physical and economic damage functions for air pollutants by receptor. U.S. EPA Report EPA-600/5-76-011, U.S. Environmental Protection Agency, Corvallis Environment Research Lab, Corvallis, OR. Loeb, P. D. (1976) Specification error tools and investment functions, Econometrica 44, 185-193. Maddala, G. S. (1971) The use of variance components models in pooling cross-section and time-series data, Economet. 39, 341-358. Maddala, G. S. (1977) Econometrics. McGraw-Hill, New York, NY. McDonald, G. C. and Schwing, R. C. (1973) Instabilities of regression estimates relating air pollution to mortality, Technomet. 15, 463 -481. Mendelsohn, R. and Orcutt, G. (1979) An empirical analysis of air pollution dose-response curves, Environ. Econ. Manag. 6, 85-106. Morrison, D. F. (1967)Multivariate Statistical Methods. McGrawHill, New York. Newbold, P. and Davies, N. (1978) Error mis-specification and spurious regression, Int. Econ. Rev. 19, 513-519. Pindyck, R. S. and Rubinfeld, D. L. (1976) Econometric Models and Economic Forecasts. McGraw-Hill, New York. Quade, D. (1979) Regression analysis based on the signs of residuals, J. Am. Statist. Assoc. 74, 411-417. Quandt, R. E. (1973) A comparison of methods for testing nontested hypotheses, Rev. Econ. Statist. 74, 92-99. Ramsey, J. B. (1969) Tested for specification errors in classical linear least squares regression analysis. R. Statist. Soc., Ser. B 31, 350-371. Ramsey, J. B. and Gilbert, R. (1972) A Monte Carlo study of some small sample properties of test for specification errors, J. Am. Statist. Assoc. 67, 180-186.

194 Riggan, W. F., Nelson, W. C., Sharp, C. R., Hammer, D. J., Buecheley, R. W., and Van Bruggen, J. (1972) Relationships Between Air Pollution and Mortality in Census Tracts Near Aerometric Monitoring Stations. Office of Research and Monitoring, U.S. Environmental Protection Agency, Washington, DC. Schwing, R. C. and McDonald, C. C. (1976) Measures of association of some air pollutants, natural ionizing radiation, and cigarette smoking with mortality rates, Sci. Tot. Environ. 5, 139-169. Shappiro, S. S. and Wilk, M. B. (1965) An analysis of variance test for normality, J. Am. Statist. Assoc. 63, 1343-1372. Sims, C. (1974) Distributed lags, in Frontiers o f Quantitative Economics, M.D. Intrilligator and A.D. Kendrick, eds. pp. 215-249. North-Holland, Amsterdam. Smith, Baud and Campbell, J. M. (1978) Aggregation bias and the demand for housing, Int. Econ. Rev. 19, 495-505. Smith, V. Kerry (1974) Multivariate analysis: Theory and practice, Socio-Econ. Plan. Sci. 8, 77-94. Smith, V. K. (1976) The Economic Consequence o f Air Pollution. Ballinger, Cambridge, MA. Taub, A. J. (1979) Prediction in the context of the variance-components model, J. Economet. 10, 103-107. Theil, H. and Goldberger, A. S. (1960) On pure and mixed statistical estimation in economics, Int. Econ. Rev. 2, 65-78.

Paolo F. Ricci and Ronald E. Wyzga Theil, H. (1971)Principles o f Econometrics. John Wiley, New York, NY. Thomas, T. H. (1975) An investigation into excess mortality and morbidity due to air pollution. Unpublished Ph.D. Diss., Purdue University, La Fayette, IN. Toyoda, T. and Wallace, T. D. (1979) Pre-testing on part of the data, J. Economet. 10, 265-273. Vinod, H. D. (1977) A survey of ridge regression and related techniques for improvements over ordinary least squares, Rev. Econ. Statist. 71, 835-841. Weisberg, S. (1980) Applied Linear Regression. John Wiley, New York, NY. Zellner, A. and Theil, H. (1962) Three-stage least squares: simultaneous estimation of simultaneous equations, Econometrica. 30, 54-78. Zellner, A. (1971) A n Introduction Bayesian Inference in Econometrics. John Wiley, New York. Zellner, A. (1979) Statistical analysis of econometric models, J. Am. Stat. Assoc. 74, 628-643. Zerembka, P. (1974) Transformations of variables in econometrics, in Frontiers in Econometrics, P. Zerembka, ed. Academic Press, New York.