Correcting for ratio variable correlation: Examples using models of mortality

SOCIAL

SCIENCE

RESEARCH

13, 268-286 (1984)

Correcting for Ratio Variable Correlation: Examples Using Models of Mortality BRIAN F. PENDLETON University

of Akron

The major purpose of this paper is to examine one part of the ratio variable correlation problem, correlated denominators, using path models of young, middleage, and older-age mortality, each with 11 sociodemographic and cause-of-death independent variables. The models are calculated using a traditional ratio approach and a residual variable approach for a sample of counties. The methodological discussion focuses on differences in coefficients for the two approaches in terms. of significance, magnitude, and hypothesis support. It is concluded that the residual approach is worth pursuing as a viable statistical solution to the correlated denominators component of the ratio variable correlation problem.

Much of the research used to study health status, social behavior, and mortality relies on the use of aggregate data. A major controversy exists in this area. Ratio variables often are constructed for the purpose of controlling an extraneous variable (e.g., population size or per capita characteristics; cf. Uslaner, 1976, and Lyons, 1977). Evidence indicates that the correlation of two or more ratios that have common, or highly correlated, components can sometimes lead to statistical spuriousness and conceptual ambiguity (Fuguitt and Lieberson, 1974; Schuessler, 1973, 1974; Bollen and Ward, 1979; Anderson and Lydic, 1977, 1978; Atchley, Gaskins, and Anderson, 1976; Pendleton, Warren, and Chang, 1979; Pendleton, Newman, and Marshall, 1983). Little empirical evidence exists, however, for examining the degree and consequences of this spuriousness when actual data are used. The major purpose of this paper is to examine the problem of ratio variable correlation for multivariate path models of young, middle-age, and older-age mortality and to test a residual procedure designed to “correct” for the spuriousness resulting from ratio variable correlations. Briefly reviewed first is the problem of ratio variable correlation, followed The author thanks Richard D. Warren. H. C. Chang. David L. Rogers. and an anonymous reviewer for extremely helpful comments on earlier versions of this paper. Requests for reprints should be sent to Brian F. Pendleton, Department of Sociology, University of Akron, Akron, OH 44325. 268 0049-089X/84 $3 .OO Copyright All rights

0 1984 by Academic Press. Inc. of reproduction in any form reserved

RATIO

VARIABLE

269

CORRELATION

by a description of the multivariate model of mortality. Hypotheses, methods and variables, and results follow, and a discussion of the results and implications ends the paper. RATIO VARIABLE

CORRELATION

Recent social science research shows that at a purely empirical level when two ratios are correlated, and two or more of the ratio components are highly correlated, the resulting correlation will be spuriously inflated. For example, if two ratio variables are computed, a = b/c and d = el f, and the denominators are highly correlated (i.e., rcf # 1.00 but is high) the correlation between the ratios is (Pearson, 1897; Fuguitt and Lieberson, 1974, Pendleton er al., 1979) rbeVbVe rad

=

(vi

+

-

vz

-

recVeVc

-

2rbcvbvc)"2

rbfvbvf (v:

+

vj

rcfvcvf -

2refvevf)"*

(1)

where rod = correlation between blc and elf, rb,, rd, rbe, r,, , rbf, rcf = product-moment correlation coefficients, vb, V,, V,, Vf = coefficients of variation (mean divided by the standard deviation) for variables 6, e, c, and f, respectively. Even though the researcher’s original intention may have been to examine the association between b and e, the introduction of highly correlated control (or deflating) variables c and f, introduces a statistical dependency (Pearson, 1910; Yule, 1910; Vanderbok, 1977; Bollen and Ward, 1979; Fuguitt and Lieberson, 1974; Schuessler, 1974; Pendleton et al., 1983). Even if it is assumed in Eq. (1) that rbe = r,, = rbf = rbc = 0 (i.e., all intercorrelations are equal to 0), the correlation ref between ratios a and d is greater than zero because of the mutual dependencies on c andf, the highly correlated denominators. In this case, the spurious correlation would be equal to (Schuessler, 1973; Pearson, 1897; Fuguitt and Lieberson, 1974; Pendleton et al., 1979) rcfVJf rad

=

(VZ

+

vy

(VZ

+

byn

(2)

It also has been shown that when the denominators are the same the statistical dependency still exists and that this problem, both for highly correlated denominators and for equal denominators, extends from bivariate correlation to multivariate statistical techniques based on correlation matrices (cf. Fuguitt and Lieberson, 1974; Pendleton et al., 1979, 1983; Rangarajan and Chatterjee, 1969; Pearson, 1897). It appears, however, that “spuriousness” is a major problem only when model testing, not model building (Pendleton et al., 1983; see also Kasarda and Nolan, 1979). Although a number of solutions have been investigated, including logarithms (O’Conner, 1977; Schuessler, 1973, 1974), partial correlation

270

BRIAN

F. PENDLETON

(Kuh and Meyer, 1955; Przeworski and Cortes, 1977; Fleiss and Tanur, 1971), part correlation (Logan, 1971, 1972), and principle components analyses (Atchley et al., 1976), debate about the efficiency and bias of such solutions continues (Kasarda and Nolan, 1979; Anderson and Lydic, 1978; Atchley and Anderson, 1978; Long, 1979a; Wright, 1979; Pendleton et al., 1983; Bollen and Ward, 1979). Recently, it has been suggested that each ratio variable may be expressed as a residual in which the numerator has been regressed on the denominator (Freeman and Kronenfeld, 1973; Pendleton et al., 1979, 1983; Bollen and Ward, 1979; Vanderbok, 1977; Fuguitt and Lieberson, 1974). This study is an attempt to assess the empirical and hypothesis testing differences between the traditional use of ratios and the residual form of control for ratio variables. For the sake of parsimony, ratios with correlated denominators are used. Mortality models for young, middleage, and older-age mortality will be expressed both in the traditional ratio form and as residuals and used as the substantive example. THE MORTALITY

MODEL

Briefly discussed below is a simple 11-variable mortality model. It will provide an empirical test of the ratio variable correlation problem where correlated denominators have purposefully been built in. Variables were chosen that were prevalent in the literature and could easily be calculated as rates and residuals to allow for comparisons. Because previous research rarely defined situations for only a young, middle-age, or older-age population, hypothesized relationships are applied to all three age groupings with support or refutation to be determined inductively in the findings. Substantive characteristics of the model construction are discussed in greater detail by Pendleton (1977, pp. 16-49; see also Kitagawa and Hauser, 1973; McGirr, 1976a, 1976b). Social Class Although social class measures often are crude and the dividing lines between class stratums are arbitrarily made (Roberts, McBee, and MacDonald, 1970; Hodge and Siegal, 1%8), an inverse relationship between social class and mortality has been documented by a number of studies (Antonovsky, 1967; Sheps and Watkins, 1947; Griffiths, 1971; Roberts et al., 1970). There are three middle-range concepts employed most often to represent social class: education, occupation, and income. The weight of the evidence points to an inverse relationship between these measures of social class and mortality at all ages. Kitagawa and Hauser (1968, 1973), Stockwell (1963), and Upchurch (1962) all report a general inverse relationship between education and mortality. Schwirian and Lagreca (1971) report the influence of education on mortality operating through soundness of housing (see also McGirr,

RATIO

VARIABLE

CORRELATION

271

1976b). Persons engaged in higher status occupations generally experience lower overall mortality (Sly and Chi, 1972; King, 1971; Brenner, 1971; Metropolitan Life Insurance Co., 1959) but this direction is not a universal finding (e.g., Sauer and Parke, 1974; Stocks, 1938; Martin, 1951) and results appear to differ somewhat depending upon causes of death investigated (e.g., Stern, 1951; Tuckman, Youngman, and Kreizman, 1965; Logan, 1954). Income (or economic status) almost invariably is negatively associated with mortality (Coulter and Guralnick, 1959; Patno, 1960; Stockwell, 1963; Yeracaris, 1955; Altenderfer, 1947; Sly and Chi, 1972; Ellis, 1957; Coombs, 1941; McGirr, 1976b; Stem, 1951), and Schwirian and Lagreca (1971) conclude that income influences mortality through housing quality. Urban-Rural

Residency

Rural areas generally enjoy lower overall mortality, but this differential appears to be decreasing over time (Price, 1954; Nam, 1968; Syme, Borhani, and Buechley, 1966; Pendleton and Chang, 1979; Kitagawa and Hauser, 1973). Most authors, however, find urban areas to be characterized by higher death rates from most causes of death (Arriga, 1967; Wiehl, 1948; Hamilton, 1955; McGirr, 1976a, 1976b; Stern, 1951; Stocks, 1938; Dot-n, 1959; Hitt and Bertrand, 1951; McMahan, 1951). Marital

Status

Research shows married populations to have lower mortality than single, divorced, or widowed populations (Nam, 1968; Sheps, l%l; Geerken and Gove, 1974; Gove, 1973; Young, Benjamin, and Wallis 1970). This inverse relationship generally holds for all age categories and both sexes (Shurtleff, 1955). Sex Among all the differentials of mortality, the most universally supported is sex (Dom, 1959; Nam, 1968; Wingard, 1982). Research invariably shows lower mortality for females (Price, 1954; McMahan, 1951; Spiegelman, 1967; Madigan, 1957; Enterline, 1961; National Center for Health Statistics, 1973). Some authors posit social class and marital status as variables intervening between the sex differential and mortality (Patno, 1960; Yeracaris, 1955; Kitagawa and Hauser, 1968; Stocks, 1938; Logan, 1954; Martin, 1951; Gove, 1973; Geerken and Gove, 1974). Health

and Medical

Care and Facilities

Where medical or health facilities and care are relatively unavailable to the population, mortality is higher. Sound health and the availability and utilization of medical facilities for emergency and durational care are inversely associated with mortality (Dorn, 1959; Stockwell, 1961; Sly

272

BRIAN

F. PENDLETON

and Chi, 1972; Schwirian and Lagreca, 1971). There is some evidence that social class is antecedent to health and medical care and facilities (Upchurch, 1962). Housing Housing density (overcrowding) and poor housing conditions usually are linked positively with mortality (Ellis, 1957; Stockwell, 1%3; Coombs, 1941). In the past, this relationship was especially strong when mortality was measured as the incidence of acute (infectious) diseases (Benjamin, 1965; Mabry, 1958; Ellis, 1957). There also is evidence to indicate that housing serves as an intervening variable between education and income levels and mortality (Schwirian and Lagreca, 1971). Causes of Death Three causes of death are delimited to represent the composition of overall adult mortality: acute, chronic, and social (World Health Organization, 1957; Weatherby, Nam, and Isaacs, 1983; Hillery, Ludtke, and Weisbuch, 1968; Roberts et al., 1970). Maternal deaths, infant deaths and other symptoms, senility, and ill-defined conditions are not directly relevant to this study and are not considered further. Greater control of acute diseases has lowered their relative contribution to overall mortality (Dauer, Korns, and Schuman, 1968; Spiegelman, 1967; Ellis, 1957). Chronic diseases are characterized by a degeneration of normal physiological processes. Mortality from degenerative diseases has increased in relative importance during the past few decades. The third major category of causes of death is social. Included here are mortality vehicle accidents, all other accidents, suicide (and self-inflicted injury), and homicide (and war). Social mortality other than motor vehicle accidents is somewhat culture bound largely because available sociocultural materials and economic factors define the possibilities of death (Hillery et al., 1968). Figure 1 graphically displays the analytical relationships between the dependent variables, causes of death, and young, middle-age, and olderage mortality. The literature review suggests the following multivariate hypotheses. TH.l. Social class as measured by income, educational and occupational characteristics will exhibit significant inverse relationships in multivariate models with young, middle-age, and older-age mortality due to acute, chronic, and social causes. TH.2. Residency, marital, health and medical, sex, and housing status will exhibit significant relationships in multivariate models with young, middle-age, and older-age mortality due to acute, chronic, and social causes.

RATIO VARIABLE Sex

Social Class (Education) Social Class (Occupation) Social Class (Income)

Housing

Acute Causes of Death Chronic Causes of Death

Marital Status

Residency

273

CORRELATION

Health and Medical Care and Facilities

Mortality A. Young B. Middle-Age C. Older-age

Social Causes of Death

FIG. 1. Analytical variables and orders of priority for models of young, middle-age, and older-age mortality. For the sake of clarity directional arrows are left out. However, the models may be read as path models with temporal movement from left to right and downward. Exceptions to this causal inference are the relationships between sex and marital status, and among acute disease, chronic disease, and social causes of death: these are associations with no causal implications.

MEASUREMENT AND METHODS The three dependent variables and 11 independent variables just reviewed are measured with aggregate data from all 99 Iowa counties for 1970.’ Discussed below are the procedures for dependent and independent variable measurement. Dependent Variables: Young, Middle-Age,

and Older-Age Mortality

Correlates and causes of death are known to differ with various age groups, but rarely has an attempt been made to divide adult mortality into sociologically meaningful groups and identify socioeconomic antecedents to causes of death. The division of adult mortality into young, middle-age, and older-age mortality can aid in the interpretation of mortality differentials at meaningful life cycle levels. “Young” mortality refers to the standardized death rate for persons 20 to 39 years of age. Various methods for age standardization are available (Daniel, 1974; Kitagawa, 1955, 1964; Wunsch and Termote, 1978). The direct method is used in this study (U.S. Bureau of the Census, 1975, p. 419) in which

where m, = age-adjusted standardization rate, m, = the age-specific death rate for a P, particular area at a given time, P, = the standard population at each age, and P = EP, = the total standard population. I Data are from Project 1972 at Iowa State University, Dr. David L. Rogers, principal investigator, and Pendleton (19771, drawing from the Iowa State Department of Health (19771, Taylor (19771, and various Bureau of the Census publications.

274

BRIAN

F. PENDLETON

Both the minimum and maximum age values of this age group are by no means universally accepted. Past research (e.g., Weiss, 1976; U.S. Department of Health, Education, and Welfare, 1974; Metropolitan Life Insurance Co., 1977; Neugarten, Moore, and Lowe, 1968), however, indicates that a satisfactory median for establishing the lower age limit for “young” is 20 years of age. “Middle-aged” mortality refers to the standardized death rate for persons 40 to 59 years of age. Research by Dorn (1959), Spiegelman (1967), Weiss (1976), the Metropolitan Life Insurance Company (1977), Neugarten et al. (1968), and Neugarten (1974) support a division between young and middle age at 40 (see also Bogue, 1969, and U.S. Bureau of the Census, 1975, p. 473). “Older-aged” mortality refers to the standardized death rate for persons 60 to 75 years of age. Criteria for identifying an older segment of a population are perhaps least agreed upon for any substantive age grouping. The Metropolitan Life Insurance Company (1960a, 1960b) has used 60 to 80 and 65 to 84 years of age, respectively, to define the older segment of the United States population, and Weatherby et al. (1983) use 85 + in a cross-national context. Demographic research by Spiegelman (1967), Bogue (1969), and, for country mortality, by McGirr (1976b) and Van Es and Bowling (1976) has defined the percent of the population over age 65 as an elderly population. Neugarten (1974) and Neugarten et al. (1968), however, found people’s perception of “old-age” actually was lower. An upper limit of 75 is chosen because the impact socioeconomic factors have in determining mortality lessens considerably after the average expectation of life age is passed and because of people’s perceptions of a difference at age 75 (Neugarten, 1974). Independent

Variables

When calculating each independent variable as a rate, the denominator, or deflating factor, is a population variable (e.g., total county population, population 14 and older, population 20 and older). Residuals are calculated by regressing the numerator on the denominator. The resulting regression coefficient is then used to calculate a predicted numerator; the residual is the difference between the observed numerator and the predicted numerator (the residual approach is discussed by Bollen and Ward, 1979; Pendleton et al., 1979, 1983; Vanderbok, 1977). For example, if X = Y/Z, a residual variable X is calculated as Y = 6, + b,Z + e

(4)

^y = b, + b,,=Y

(5)

X=Y-

^u

(6)

RATIO

VARIABLE

CORRELATION

275

where Y = numerator (criterion variable), Z = denominator (deflating variable), b0 = intercept, bl = regression coefficient, e = error,Y = predicted Y, & = b, = regression coefficient from Eq. (5) where Z, the denominator or deflating variable, has been entered for control purposes; and F = residual variable. Each of the 11 independent variables was calculated as a rate, and then as a residual, for 1970. The discussion below explains the operationalization of these variables; findings in which the three mortality models (young, middle-age, older-age) for 1970 expressed as ratios are compared to those expressed as residuals follow. Notice that the denominators used for all variables are highly correlated (e.g., county population, population 14 and older, population 20 and older, total number of families), which meets the example described earlier in Eqs. (1) and (2).

Sex is calculated as the proportion of the county’s total population that is over age 14 and female. The residual is calculated from the number of females regressed on the total county population. The proportion of the county population age 14 and over and married is chosen to represent marital status. The contribution of persons below age 14 to marital status is negligible. The residual calculation is the regression of number of married on the population age 14 and older. High school graduates among the population 20 years of age and older are chosen to represent the general educational status of a county. Most high school educations are completed by age 20 while many college educations (the most viable alternative) continue into the late 20s well beyond the lower age boundary for the young mortality model. The ratio calculation is the number of high school graduates divided by the population 20 and older. Residual calculation is done by regressing the number of high school graduates on the population 20 and older. Occupations designated as white-collar (Bergel, 1962) for 1970 were chosen for an occupational measure and include professional, technical, and kindred workers; managers, officials, and proprietors (except farm); clerical and kindred workers; and sales workers. The ratio calculation for the proportion of the labor force engaged in white-collar occupations is the sum of males and females employed in white-collar jobs divided by the total employed labor force. Residuals are obtained by regressing those employed in white-collar occupations on the total employed labor force. Income is measured by the proportion of all families above the state median family income level for 1970. The ratio calculation is the number of families in the county above the state’s median family income level for 1970 (median = $6664) divided by the total number of families in the county. Residuals are calculated by regressing the number of families above the median family income level on the total number of families

276

BRIAN F. PENDLETON

in the county. Residency is represented by the proportion of a county’s population that is urban and uses the number of people residing in urban areas and total population. Housing density is the proportion of all occupied housing units with more than 1.0 persons per room. Housing units with more than 1.0 persons per room are considered to be “overcrowded.” Ratios are calculated by dividing the number of occupied housing units with more than 1.0 persons per room by the total number of occupied housing units. Residuals are the regression of the former on the latter. The number of medical doctors available in the county accurately reflects both health care availability and the proximity of medical care facilities (e.g., clinics, hospitals, or private office practices). The ratio calculation for 1970 is the number of medical doctors divided by the total population; residuals are again obtained by regressing the former on the latter. Acute and chronic diseases and social causes of death were identified from work by Hillery et al. (1968), Roberts et al. (1970), and the World Health Organization (1957).’ The ratio calculation is the sum of all deaths due to acute diseases divided by the total number of deaths. Residuals are obtained by regressing the number of acute disease deaths on the total number of deaths. Data used for acute diseases and the total number of deaths are 3-year averages. The logic of these calculations is the same for chronic diseases and social causes of death. FINDINGS Correlations

Table 1 displays the zero-order correlations for all variables for 1970. Coefficients above the diagonal are based on variables calculated as rates. Coefficients below the diagonal are based on variables calculated as residuals. Note that the correlations for the three mortality ages are the same both for ratios and residuals because the mortality rates have been standardized and, thus, are the same. Following are tables displaying coefficients for OLS equations based on variables calculated as rates and residuals. The variables should be read vertically as one continuous equation. ’ Acute diseases include influenza and pneumonia; all forms of tuberculosis; syphilis and its sequelae; alI forms of dysentery; bronchitis, emphysema, and asthma; meningococcal infections, poliomyelitis; meningitis; and all other infectious annd parasitic diseases. Chronic diseases include diseases of the heart; hypertension, cerebrovascular disease, arteriosclerosis, all other diseases of arteries, arterioles, and capillaries; all other major cardiovascular diseases; malignant neoplasms, including neoplasms of lymphatic and hematopoietic tissues; diabetes mellitus; peptic ulcer and ulcer of the stomach and duodenum; cirrhosis of the liver; nephritis and nephrosis; and vascular lesions affecting the central nervous system. Social causes of death include motor vehicle accidents, all other accidents. suicides, homicides, and all other external causes.

age

Y0Ulg

.014 .344***

- .025 .371*** ,141

- .408***

.743*** .436*** ,124 -.189 .240* ,063

,119 -

,151

.138

Sex StatUS

-

.034 ,120

.ool

- .614***

.550*** - .054 - ,028 ,099 ,059

- ,029 - ,018 - .ool

- .368***

,186 .322*** - ,084 .154 - .346***

,103 .I80

SK?

,089

Education

- ,103 .236** - ,039

,142

- .513*** .425*** - .400*** .555*** - .5%***

.2w* - ,081

,177

-.169

occupation

,013 -.130 - ,013

0.36

- .217* .462*** .679*‘* - .372*‘* ,078

.306** - ,057

.03 1

- .019

Income

** pc.01.

*** p e .OOl.

,115 ,042

-.004

-.I46

- .389*** - .300** ,181 .305** .381*** -

,232’ - ,075

.231*

- ,034

HOUSing

- ,078 - .649***

-

.076

-

,024 - ,220’ - .075

- .095 - ,030 ,036 ,064 .114 ,158

.141 ,030

.207*

.086 -.124

.166

- .039

Social causes of death

with

-.l% -sum

-.161

-.119 ,009 - ,091

,150 - .220 -

.177

.224* ,100 .219*

399 - ,089 - ,068 .236* -440 ,151

,053 ,182

- .067

-.187

Chronic causes of death

Calculated

Acute causes of death

- ,026

Mortality

- .474*** ,134 .491*** .243* .370*** .121

,024 -.170

,038

-.121

Health and medical

and Older-age

based on variables calculated as residuals are below

-.I70 .235**

- ,023

- .464*** ,181 .79v** .620*** - .250**

.435*** ,060

,216’

-.117

Residency

TABLE 1 Causes of Death, and Young, Middle-age, Ratios and Residuals for 1970”

- ,023 .511***

- ,002

.104

Characteristics,

based on variables calculated as rates are above the diagonal. Correlations

- .036 -.I11

-.190 ,047 .393***

-.167

-.090

.117

.064 -.loo

.226* ,016 - .231 .122 -.006 .126

.131

-

.357***

.133

.158 ,024 -.143 -.177 - xl95 .176

.357*** ,158

-.I02

Olderage Mortality

,144 .I55 .082 .077 .I24 .014

.I33 .194

- .I02

-

Middleage Mortality

Matrix for socioeconomic

Mortality

’ Correlations the diagonal. * p =G .05.

mortality Sex status Marital status Education Occupation Income Residency Housing Health and medical Causes of death Acute Chronic Social

Older

mortality Middle-age mortality

Young

Correlation

278

BRIAN F. PENDLETON

Ratio and Residual Standardized Partial Coejjicients

Table 2 displays standardized partial regression coefficients for young, middle-age, and older-age mortality models calculated with ratios for 1970. The models of mortality display some interesting direct effects. Between 16 and 26% of the variation in young, middle-age, and older-age mortality is accounted for by 1970 socioeconomic characteristics and causes of death. The strongest predictor variable is white-collar occupations with young mortality (it is in the hypothesized direction); white-collar occupations and income with middle-age mortality (the latter is in the hypothesized direction); and urban residency with older-age mortality (it is in the hypothesized direction). TABLE 2 Regression Coefficients for Socioeconomic Characteristics, Causes of Death, and Young, Middle-age, and Older-age Mortality Calculated with Ratios for 1970 Dependent variables 1970 Independent variables 1970 Sex status Marital status

Young mortality”

Middle-age mortality”

Older-age mortality”

,227’ (.W - ,289’ .24@ (.019) - .485* (.020) .137b (.012) .04Sb (.ow - .009 f.044

.020 (.095) ,282’ C.044) .lOSb ( .029) ,432’ (.030) - ,458’ (.018) .134b (.007) .341b (.067)

- .071 (.258) .281b (.119) .017 (.078) ,016 (.081~ - .006 (.048) .519* (.018) .134 (.183)

- ,080 (.076)

- ,068 (.116)

- .075 (.315)

- .07Sb 1.003) - ,233’ (.oOl) - .0806 (.003) .16

.161 (.004) - .056’ (.OOl) .0936 C.004) .23

103b (.Oll) .061b (.004) .027b (.012) .26

(.02) Education Occupation Income Residency Housing Health and medical care facilities Causes of death Acute Chronic Social R2

a The first number is the standardized partial regression coefficient (direct effect). The numbers in parentheses are standard error. ’ Coefficient is at least twice its standard error and is significant.

RATIO VARIABLE

279

CORRELATION

Table 3 presents the standardized partial regression coefficients for socioeconomic characteristics, causes of death, and young, middle-age, and older-age mortality calculated with residuals for 1970. Table 3 is organized the same as Table 1. Two models of mortality display R”s above .20; middle-age mortality is .24 and older-age mortality is .27. While only 14% of the variation in young mortality is accounted for by the 1970 socioeconomic characteristics and causes of death, all but one of the partial /Is are significant. Relationships correctly hypothesized are those with marital status, urban residency, housing density, and health and medical care and facilities. With the correlated effect of population size and number of deaths removed through TABLE 3 Regression Coefficients for Socioeconomic Characteristics, Causes of Death, and Young, Middle-age, and Older-age Mortality Calculated with Residuals for 1970 Dependent variables 1970 Independent variables 1970

Young mortality”

Middle-age mortality”

Older-age mortality”

Sex

.370b (.18 E-3) - .208’ (.lO E-3) .020b (.Ol E-5) .199b (.07 E-3) .22gb C.01) .126* (.Ol E-5) .130b (.40 E-3)

- ,082 (.27 E-3) .l95b (.15 E-3) .080* (.16 E-3) - .121b (.lO E-3) - .268’

- ,203’ (.74 E-3) .309b (.41 E-3) .040b (44 E-3) - .276b (.28 E-3) .0666 (.M) .153b (.Ol E-5) .Oli” (.16 E-2)

status

Marital status Education Occupation Income Residency Housing Health and medical care/facilities Causes

of death

Acute Chronic Social R’

(.W - ,108’ (.Ol E-5) .132b (.60 E-3)

- .172’ C.001)

-.117b

- .002 (.003) - .269’ C.004) -.151b (.09 E-3) .14

.157b (.ow .o86b (.007) .432b (.13 E-3) .24

(.ow

.065b (.58 E-2) .103” (.Ol) .172b

(.@a .443b (.36 E-3) .27

a The first number is the standardized partial regression coefficient (direct effect). The numbers in parentheses are the standard error. b Coefficient is at least twice its standard error and is significant.

280

BRIAN

F. PENDLETON

residualization, we find all three causes of death to be negatively related to young mortality and significant (although acute causes of death is not significant). Within the middle-age mortality model all but one of the partial p’s again are significant. Socioeconomic characteristics operating in hypothesized directions are sex status (p = - .08; but not significant) whitecollar occupations (p = - .12), county wealth or income (/3 = - .27), and housing density (p = .13). While all three causes of death are significant and positive, a special note is made about the large p for social causes of death among the middle-age (/3 = .43). Most important among the socioeconomic characteristics accounting for the R* of .27 in the older-age mortality model are sex status (p = - .20), marital status (/3 = .31), and social causes of death (p = .44). Sex status, white-collar occupations, urban residency, housing density, and health and medical care/facilities all are socioeconomic characteristics that are significant and in the hypothesized directions. All three causes of death are significant in the older-age mortality model. Most prevalant, however, are social causes of death (p = .44). Comparison of Rate and Residual Equations in Standardized Form

The young, middle-age, and older-age mortality models for 1970 rate and residual calculations offer insight to the apparent advantage of residual analysis. The R2’s for each mortality model are very similar but a number of differences are apparent when the direct effects are compared. When calculated with ratios, 14 of the 24 relationships between socioeconomic characteristics and young, middle-age, and older-age mortality are significant; 7 of these 14 are in the hypothesized directions. In contrast, when residuals are used, 23 of the 24 relationships are significant and 12 of these are in the hypothesized directions. The most consistently significant socioeconomic variables in the rates models are marital status and urban residency. In the residual analyses marital status, education, white-collar occupations, income, urban residency, housing, and health and medical care/facilities are significant in all three mortality models for 1970. For the most part, levels of significance and directions for the three causes of death and models of mortality equations using rates and residuals are the same. Two notable exceptions are the extremely high, positive p’s between social causes of death and middle-age and older-age mortality (~3 = .43 and .44, respectively), in the residual analysis. In summary, the use of residuals instead of ratios in 1970 resulted in a much greater number of relationships for socioeconomic characteristics and causes of death with mortality models to be significant. Also, more of these significant relationships using residuals tend to be in the hy-

RATIO VARIABLE

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pothesized direction.3 As noted by Pendleton et al. (1983) when model testing, the ratio variable correlation problem may be of major importance, and correcting for the induced spuriousness becomes a major priority for the researcher (see also Yule, 1910). Kasarda and Nolan (1979), a few years earlier, stressed the importance of the theoretical framework from which models are derived (see also Schuessler, 1973; Long, 1979a, 1979b; Macmillan and Daft, 1979). Interestingly, the magnitude between relationships displayed in ratio and residual analyses differ, but not with any patterned regularity. About half increase between ratio and residual comparisons and half decrease. Yet the number of significant relationships, which increase dramatically when one moves into the residual models, means correlated denominators can change dramatically the standard error of regression coefficients4 DISCUSSION More of the significant relationships, both hypothesized and not hypothesized, were found with equations using variables calculated as residuals. The multivariate mortality models were developed from extremely strong previous research. It may be assumed that the residual analysis more accurately described the hypothesized relationships and, thus, provided a more accurate empirical “picture” of the theory. Multiple benefits to the development of theories of socioeconomic epidemiology may be gathered from studies based upon residual, as opposed to more traditional, rate or ratio calculations. Socioeconomic epidemiology is the study of the extent to which differences in socioeconomic status account for differences in mortality, indicating gains that could be achieved in mortality reduction if socioeconomic conditions are improved. A number of multivariate relationships tested and generated in this study provide insight into, and raise questions about, relationships between county socioeconomic characteristics and young, middle-age, and older-age mortality. For example, the cross-sectional analysis for 1970 ratio and residual models show higher occupational status and social causes of death to be positively correlated with younger age mortality (with other factors controlled), rather surprising in view of past bivariate 3 It should be remembered that residuals for the three dependent variables, young, middle-age, and older-age mortality cannot be calculated. They are calculated and used as standardized rates both for rates and residual analyses. 4 It should be noted that the lack of changes in magnitude between ratio and residual models may be due to the substantive nature and ordering of variables in the mortality models rather than the lack of a correlated denominator effect manifested in the p coefficients. This interpretation is partially supported by a second look at the zero-order correlation coefficients (Table 1). More than half the comparisons show lower bivariate correlations for the residual correlations, empirically providing some support for the notion of spuriousness when there are correlated denominators.

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