American Association for Public Opinion Research by Juan Pardo

American Association for Public Opinion Research

Procedures for Evaluating Trends in Public Opinion Author(s): D. Garth Taylor Source: The Public Opinion Quarterly, Vol. 44, No. 1 (Spring, 1980), pp. 86-100 Published by: Oxford University Press on behalf of the American Association for Public Opinion Research Stable URL: http://www.jstor.org/stable/2748591 Accessed: 23/01/2009 14:07 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=aapor. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org.

Oxford University Press and American Association for Public Opinion Research are collaborating with JSTOR to digitize, preserve and extend access to The Public Opinion Quarterly.

http://www.jstor.org

Procedures for Evaluating Trends in Public Opinion D GARTH TAYLOR

HIS PAPER shows how to use the percentage difference models described by Davis (1976) and by Grizzle et al. (1969) to analyze trends in public opinion data. The first part of the paper is an attempt to persuade the reader that most of the interestingtheories of public opinion change can be thought of as statistical models which make formal predictions about percentages or percentage differences. The next part shows how to choose the appropriatemodel (or theory) to describe the change by examining a series of goodness-of-fit tests. Finally, a few detailed examples show the applicationof the method to patterns of change in public opinion data which are difficult to describe, both verbally and statistically.

Changeand ChangeModelsin Public OpinionResearch Trend analysis in public opinion research begins with a theory of change (or lack of change) to be tested or a data set (usually consisting of a series of percentages or percentage differences) to be described as showing a "significant" or "nonsignificant" pattern of

Abstract This article expresses a variety of theories of public opinion change in terms of a formal model for analyzing public opinion data. Statistical criteria are proposed whereby models of varying complexity can be accepted or rejected. The observation is made that certain substantive theories of change are more parsimonious than others in terms of their statistical predictions. The model for analyzing change presented here is based on the premise that the most parsimonious theories should be accepted or rejected first. D Garth Taylor is an Assistant Professor in the Department of Political Science at the University of Chicago and Senior Study Director at the National Opinion Research Center. Public Opinion Quarterly ? 1980 by The Trustees of Columbia University Published by Elsevier North Holland, Inc. 0033-362X/80/0044-086/$1.75

EVALUATING TRENDS IN PUBLIC OPINION

change. Usually the researcher has both. In our experience, an array of possible theories can be used to describe a set of data, and the ultimate theoretical conclusion depends on the most parsimonious statistical model which can be used to describe the data. It is important to recognize the predictions about the statistical patterns in public opinion data which are made in various theories of change. Following are some examples of theories of change that have appeared recently in the literature, and their associated statistical predictions. Theory1: Plus (a change, plus c'est la meme chose. This is a theory that things don't really change. Statistically,the predictionis that a set of proportions only departs randomlyfrom a fixed value or that a series of percentage differences are not statisticallydifferentfrom zero. For instance, Sullivan et al. (1979) claim that within a proper framew9rkfor conceptualizationand measurement,survey research shows that the level of tolerance in America has not changed in recent decades. Theory2: Social statics. The key propositionhere is that differencesendure. Some recent examples in the literatureare studies of enduringregional and culturaldifferences (Reed, 1972; Glenn and Simmons, 1967), enduringclass differences in political behavior (Alford, 1963), and the enduringeffects of education (Hyman et al., 1975). Each study argues that, controllingfor other changes in the society, the correlations between certain sets of variables remain constant. Theory3: Regular processes described by transitive verbs in gerundform. Studies on assimilating blacks into the electoral process (Converse, 1972), massifying or differentiatingsocial differences (Glenn, 1967) and diminishing social class differentials(Mayer, 1959) are in this category. Each study suggests steady, gradual change in intergroupdifferences-in some theories the differences are growing, while in others the differences are diminishing. The point is the regularityof the process of change and the suggestionthat the data mightbe describedby a linearpatternof increasingor decreasing differences in the population. Theory4: Social catastrophe. Analyses of fertilityexpectations (Blake, 1974), confidence in national leaders (Smith et al., 1979), and attitudes toward foreign involvements (Gallup, 1972) all argue that trends in these measures change erratically as the result of sudden developments in the world economic or political situation.The public opinion measures show a great deal of statistically significant"bounce," and the hypothesis is that the change cannot be described by any of the preceding, simpler models. In analyzing a set of public opinion data, we typically want to know which theory describes the change observed in the data. More formally, this question becomes, which of the statistical models associated with the four theories most parsimoniously describes the data?

D GARTH TAYLOR

The next section illustrates the procedures used to decide this question. Statistical Procedures

The steps in the statisticalanalysis are to derive the trend in public opinion which is predicted under each theory of change and then, using a chi-squaregoodness-of-fittest, decide which theory provides the best and most parsimoniousfit to the data. In this exposition the analyses will all be focused on describing trends in intergroup differences in public opinion (i.e., the object of attentionis the patternof change in a series of percentage differences). The procedures described here can be simplified to study trends in percentages. Table 1 shows the trend from 1965 to 1973 in supportfor abortion amongProtestants(excludingBaptists) and Catholics. We note that in 1965 both groups were predominantlyopposed to abortion for the Table 1. The Trend in Religious Differences in Response to the Question: "Tell me whether or not you think it should be possible for a pregnant woman to obtain a legal abortion if the family is poor and cannot afford another child." Data

AIP0721 12/65

Surveya Date

SRS870 11/65

GSS72 3/72

AIP0788 9/69

GSS73 3/73

Protestantsb

Percent No

80.6

79.4

75.3

51.5

42.8

(N)

(803)

(807)

(815)

(756)

(755)

Catholics Percent No (N)

83.8

83.2

82.6

62.1

60.3

(853)

(327)

(356)

(364)

(358)

Statistical Analysis Category difference (Base =Protestant)

Catholic

a) no difference

d =0

94.4

<.05

reject

b) constant difference c) linear change

d = dp d = a + bx

29.6 .3

4 3

<.05 >.05

reject accept

29.3

<.05

Hypothesis

Model

reductionfrom linear term

Decision

significant

Final Model Catholic: d = -1.34 + .021 (year-1900) a SRS870 is a survey conducted by the Survey Research Service of the National Opinion Research Center. GSS refers to General Social Surveys, conducted by the National Opinion Research Center, funded by the National Science Foundation. AIP0721andAIP0788are surveysconductedby the AmericanInsititueof PublicOpinion (Gallup). b Protestants,not includingBaptists.

EVALUATING TRENDS IN PUBLIC OPINION

stated reason (if the family is poor and cannot afford another child), and the religious difference is small (about 3 percent). By 1973 both groups were more in favor of abortion and the religious differences are greater (about 17 percent.) Which theory adequately and parsimoniously describes the pattern of religious differences shown in Table 1? To answer this question we need to determine:(a) the patternin the data predicted under each theory; (b) the appropriatestatistical procedure for deriving the predicted percentage differences under each model; and (c) the closeness (goodness-of-fit)between the predicted and observed percentage differences for each model. The Plus qa change theory predicts that the percentage difference is actually zero in each survey year and that the differences observed in Table 1 are nothing more than random sampling fluctuations. In this case, the predicted percentage difference is supplied by the theory-each difference should be zero-and so there is no further statistical estimation required.We will learn that this theory does not fit the data in Table 1 very well. The exact goodness-of-fit test is described below. The social statics theory predicts that the religious difference is constant-that there is some true, nonzero value for the religious difference which characterizes each survey and that the pattern of differences is really random fluctuation around this true difference. The intuitive solution is to take the average of the percentage differences as the pooled estimate of the "true" difference. For full statistical power, however, it is more efficient to take a weighted average of the survey results, where the results for any survey are weighted proportionallyto the amount of confidence we place in the accuracy of that survey. There are several reasons for weightingthe results for each survey proportionateto the amount of confidence we have in that survey. It can be the case, althoughit is not in Table 1, that surveys at different times differ greatly in the number of cases in the sample. It is an accepted standardof survey procedurethat proportionsfrom a larger sample should be given greater weight in reckoning the significant pattern in the data. We also note that there are two Gallup surveys and three NORC surveys represented in the time series in Table 1. These surveys may differ in the efficiency of the samplingprocedures, in the size of the sample clusters, or in other procedures such as interviewertrainingor interviewerinstructionswhich will have implications for the effective sample size (and hence error variance) of the survey results (Hansen, 1951; Glenn, 1975; Newman, 1976). The most efficient estimate of the constant true score predicted by the

D GARTH TAYLOR

social statics theory for describing the data in Table 1 is a weighted estimate of the pooled percentage difference for the five surveys (Goodman 1963). The procedure for estimating the weighted pooled percentage difference is as follows: 1. Compute the percentage difference and the variance of the percentage difference for each year. According to textbook formulas the percentage differenceis Pi - P2 and the varianceof the percentagedifferenceis the sum of the variances for each percentage. The variance of a percentage is (p* (1-p))IN where N is the numberof cases the percentageis based on. If the data are not from a probabilitysample or if there are other reasons to suspect the quality or effective sample size of the data, the place to quantify this reservationis in the estimate of the varianceof the percentagesfor that survey. The formulafor pooling the results of the surveys assigns less weight to the less reliable surveys. 2. The weights for pooling the percentage differences are inversely proportionalto the varianceof the percentagedifference.The weights are arrived at by takingthe reciprocalof the variancefor each percentagedifferenceand dividingthis by the sum of the reciprocalsof the variancefor each percentage difference, i.e.:

w -=

(1)

1 k

where W indexes the weights, V indexes the variances and k indexes the numberof surveys pooled for the weighted average. 3. The pooled weighted percentagedifference is the sum of the weight for each conditionalD times the difference for conditionalD. Dp=

(2)

Wk *Dk

The variance of Dp can be used to assess the statistical significance of the pooled weighted average. This statistic is: VP = X (Wk)2 k

(3)

In sum, the social statics theory predicts that each percentage difference is only randomly different from the weighted pooled percentage difference. The statistical test for this conclusion will be given below (the model does not fit the data in Table 1). The third theory, the transitive gerund theory, predicts that the

EVALUATING TRENDS IN PUBLIC OPINION

percentage differences are regularlybecoming larger or smaller. The rhetoricof this theory is easily stated in a linear regressionformat:the predictedpercentagedifferenceat any time is equal to a constant plus some factor adjusted for the amount of time elapsed between observations: +b * (Time)

D =a

(4)

where D indexes the predicted percentage difference.

In the linear regression analysis of the predicted percentage differences, the same considerations apply as earlier regardingthe importance of weighting the results of each survey inversely proportional to the variance of the percentage difference for that survey. Some explanations of how to do this appear in statistics texts under discussions of weighted least squares or heteroscedasticity (Wonnacot and Wonnacot, 1970). In the language of regression analysis, the goal is to minimize the weighted squared deviations from the regression line, where the weights are defined as before in Eq. (1). The formulasfor a weighted least squares analysis of the model in Eq. (4) are: b

=_k

*(Dk X k

a =D

where

Wk = Dk

Tk =

-D) *

* (Tk - T)

(T k -

(5)

7T)2

-b*T

(6)

the weight for each percentage difference as before the percentage difference in the kth survey the simple mean of the Dk the time (year and month) of the kth survey the mean of the Tk

The two important substantive comments to make about this proce-

dure are: (1) the difference between these estimates and the regular regression results is the presence of the weights in the numerator and denominator of Eq. (5); and (2) the same weighted least squares principles apply to more complex models for describing nonlinear patterns of change.

The transitive gerund theory predicts that the percentage differences in Table 1 are predicted (within samplingerror) by a regression model such as the one shown in Eq. (4). It turns out that this is the theory which is appropriate for describing the data in Table 1-the religious differences are growing gradually larger.

D GARTH TAYLOR

Before describing the goodness-of-fit tests that are used to choose between theories of change, it is useful to consider what the catastrophe theory would have predicted for the pattern of differences in Table 1. The catastrophe theory predicts that there are significant fluctuations in the data which are not captured by a constant difference model or a linear trend model for the percentage differences. Since a linear change model subsumes the constant difference model (the constant difference model is a linear change model with b = 0), the prediction of the catastrophe theory is essentially that the linear change model does not fit. We will see that the linear change model does fit the data in Table 1 and so the catastrophe theory is not appropriate there. The chi-square test for the goodness of fit for any model compares the observed percentage differences with the percentage differences predicted under the model. The number of degrees of freedom for the chi-square is equal to the number of surveys being compared minus the number of parameters estimated in the model for the predicted percentage differences. Under the plus qa change theory there are k tables (in our notation) and the theory provides the only parameter required for the analysis-the hypothesis is that all differences are equal to zero. Therefore there are k degrees of freedom for the plus ga change hypothesis. The social statics theory uses the data to estimate 1 parameter, the pooled weighted percentage difference and so there are k-i degrees of freedom for this test. The transitive gerund theory estimates two parameters-the slope and intercept of the regression equation-and so there are k-2 degrees of freedom here. The social catastrophe predicts that the linear regression model does not fit and so there is no further estimation for this theory: if the linear model fits we accept the transitive gerund theory, if it does not fit we accept the catastrophe theory. Once the predicted values are calculated, the goodness of fit of the theory is assessed using the following chi-square formula: 2

E k

where

(Dk -Ek)2 Vk

df = (k -p)

(7)

k = the number of tables being compared = the observed percentage difference in the kth table Ek = the expected percentage difference in the kth table under some model p = the numberof parametersestimated for the model Vk = the variance of the observed percentage difference in the kth table

EVALUATING TRENDS IN PUBLIC OPINION

The expected percentage difference is subtractedfrom the observed value, the difference is squared and divided by the variance of the observed percentage difference, and the results are added over the number of surveys available. Table 2 summarizesthe models that are being tested, the predicted values undereach theory of change, and the chi-squarecalculationfor each goodness-of-fit test. The sequence of parameters estimated and models tested is an extremely importantaspect of the general procedure presented here for evaluating trends in public opinion. The four theories of opinion change were not initially presented as a sequence or hierarchy of hypotheses. Indeed, they are substantivelydifferentinterpretationsof the process of change in public opinion. However, it is importantto recognize that from a statistical point of view, the theories are ordered in their complexity in at least two ways: as we move down the list the later theories require a greater number of parametersto be estimated in order to "explain" the patterns in the data; and, the theories constitute a nested hierarchyof hypotheses. We have already noted that the later theories "subsume" the earlier theories. For instance, if the plus qa change theory fits the data, then the social statics theory will also fit because the "no difference" theory is the same as a "constant difference" theory with the difference equal to zero. Thus, the principleof parsimonybecomes importantas the basis for our choice of theories to explain the data. The goal of the sequence of statistical tests is to choose the theoretical model which most parsimoniouslydescribes the data. If the simplest theory does not "fit" (i.e., if the chi-square is too large given the number of degrees of freedom), then we move on to test the next most complex model, and so on. The series of models is organized in such a way Table 2. A Summary of the Predicted Percentage Differences and the Chi-Square Test Under Each Hypothesis for Describing Change in Public Opinion Data

Hypothesis 1. Zero difference 2. Constant 3. Linear change

Predicted Values for Each Table

Zero

I (Dk - zero)2

(k)

Weighted average Weighted least squares estimate

E (Dk - pool)2 k

I (Dk k

(k-i)

Vk b)k2

(k-2)

D GARTH TAYLOR

that each successive model requires one more parameterto be estimated and hence uses up one more degree of freedom in explaining the data. The difference in chi-squaresbetween successive models is the test of significance (evaluated on one degree of freedom) for the additional parameter that was required for the more complicated model. Thus, a sequence of tests that begins with the "no difference" model and tests the goodness of fit of each successive model and the significance of the parameter which is added for each successive model produces an unambiguousprocedure for arrivingat the most parsimoniousmodel for describing a series of percentage differences in public opinion data. This sequence of tests is shown in Figure 1. Final Model (check for outliers)

Hypothesis No difference (df = kA) I

accept_

N. S.

Model Type 1

reject

Constant value accept (df = k-i)

X2 N.S.

Test reduction x2 Significant-.D= constant from No difference < D = constant model on one df N.S. borderline signif icance

reject

Linear trend(df = k-2)

2 accept X N.S. t

Tsfrom Constant x difference model

reject

on one df

Test reduction x2 from Constant difference model on one df

Significant-mD = a + b * (time) N.S. -Borderline linear

3 3a

trend Substantial linear component trend -Nonlinear

Significant-

N.S.

Explore further nonlinear or multiple linear modeIs Figure 1. Decision Rules for Evaluating Trends in Public Opinion

EVALUATING TRENDS IN PUBLIC OPINION

There are seven possible outcomes for the sequence of tests proposed here. If one accepts the hypothesis that all differences could be zero (theplus qa change theory) then one arrives at Model Type 1. If one finds that the "no difference" theory does not fit the data but the "constant difference" does, then one accepts the social statics explanation. Because of the character of the chi-square distribution,it is possible that the "constant difference" model is required to fit the data, but the pooled weighted percentagedifference is only of marginal statistical significance. This is determinedby subtractingthe chisquare for the "no difference" model from the chi-square for the "constant difference" model and assessing the difference in chisquareon one degree of freedom. This is the test of significanceof the pooled percentage difference. If the Dp is significant, we arrive at model type 2, if it is not we choose model type 2a. A similar set of proceduresapplies for testing the "linear change" model. The model may fit the data or it may not (in which case we accept the catastrophe theory). If the model does fit, then we assess the significance of the linear term (i.e., the slope) by subtracting chi-square and testing for significance on one degree of freedom. If the linear model does not fit, there is still the possibilitythat there is a significantlinear component to the change in the percentage differences. This is examinedusing the same differencein chi-squares.If the linear model does not fit the data, but the chi-square reduction for the linear component is significant then we choose model 3b. In the next part of the paper we will apply the decision rules to some examples of trends to be analyzed in public opinion. At the end of the section are a few furtherdetails and caveats of a methodological nature. Some Examples For the data in Table 1, the decision rules lead to the adoption of the linear change model (type 3). The sequence of chi-squaretests is shown in the section of the table labeled "statistical analysis." The "no difference" and "constant difference" models do not fit, the "linear change" model does fit, and the reduction from the linear term is highly significant. The final model states that the religious difference changed by about 2.1 percentage points a year between 1965 and 1973 (b = .021).

Table 3 shows the percentage of men and women approving of abortion if the woman became pregnantas a result of rape. In each survey the sex difference is not very large and the sign of the difference varies dependingon the year. Both these factors conspire to

D GARTH TAYLOR

Table 3. The Trend in Sex Differences in Response to the Question: "Tell me whether or not you think it should be possible for a pregnant woman to obtain a legal abortion if she became pregnant as a result of rape." Data . Surveya Date Male Percent No (N) Female Percent No (N)

SRS870 11/65

GSS72 3/72

GSS73 3/73

GSS74 3/74

40.4 (675)

20.8 (751)

15.6 (673)

14.5 (668)

40.2 (715)

21.0 (761)

17.2 (778)

12.6 (752)

Statistical Analysis Category Difference (Base =Male) Female

Hypothesis

Model

Decision

a) no difference

d=0

1.8

>.05

Final Model Female: d = 0 a SRS870 is a survey conducted by the Survey Research Service of the National Opinion Research Center. GSS refers to General Social Surveys, conducted by the National Opinion research Center, funded by the National Science Foundation.

make the "no difference" model the appropriateone for describing the data. Table 4 shows the trend in the percent favoring abortionin case of rape for three educational groups. For polytomous variables, we follow Davis's strategy (1976) and choose one category as the base category, analyzing the trend in the difference between the base category and each of the nonbase categories in the percent favoring abortion. In our example "less than high school education" is the base category. The statistical analysis shows that we accept a different model for each of the nonbase categories. For the difference between high school and less than high school education, the chi-squarefor the "constant difference" model is 10.7 with 3 degrees of freedom. This chi-squarewould be insignificantif we had applied correction factors to the variances of the percentages to account for the clusteringeffects in the constructionof the survey samples. The simple route to the chi-squarecorrectionis to divide the chi-square by some factor between 1.2 and 2 before assessing the significance of any comparison (Kish, 1965:162; Kish, 1957:159; Taylor, forthcoming). The exact correction factor depends on the particularsof the survey but an extremely conservative procedure is always to divide the chi-squares by 2 before the significance test. Looking at the trend in the college vs. less than high school dif-

EVALUATING TRENDS IN PUBLIC OPINION

Table 4. The Trend in Educational Differences in Response to the Question: "Tell me whether or not you think it should be possible for a pregnant woman to obtain a legal abortion if she became pregnant as a result of rape." Data Survey Date Education category Less than high school Percent No (N) High school graduate Percent No (N) College Percent No (N)

SRS870 11/65

GSS72 3/72

GSS73 3/73

GSS74 3/74

50.9 (709)

33.3 (589)

26.8 (519)

18.6 (473)

35.7 (342)

16.4 (477)

13.2 (470)

12.2 (426)

23.0 (330)

9.0 (442)

7.9 (458)

9.8 (469)

Statistical Analysis Category Difference (Base = <HS) High school graduate College

Decision

0 dp 0 dp

102.8 10.7 273.4 34.1

4 3 4 3

<.05

reject

<.05 <.05

reject reject

d = a + bx

14.4

<.05

reject

19.7

<.05

significant

Model

Hypothesis a) b) a) b) c)

no difference constant difference no difference constant difference linear change in difference reduction from linear term

d d d d

= = = =

Final Model High school

College

graduate

d = -.125

= .0130

d = -.189 d = -1.54

c= .0122 + .0187

(year-1900)

ference, we see the final model is type 3b. The linear model of change does not fit, but there is a substantiallinear component to the pattern of percentages. Therefore neither of the final models reported is adequate for describing the pattern in the data.- The constant difference is an inappropriatemodel because there is change over time in the percentage difference. The linear change model is inappropriate because there are significant deviations from the linear trend. The appropriate strategy for the analyst in this situation would be to explore further models or to construct a verbal explanation of the pattern of the percentage differences which takes account of the results shown in Table 4. OUTLIERS

After applying the decision rules in Figure 1, it is a good idea to examine the data for outliers before reporting a final model. When

D GARTH TAYLOR

comparingsurveys over time, outliers are a distinct possibility. "Wild observations" can come from slight differences in question wording or interviewer instructions, differences in samplingprocedures from study to study or among survey organizations, or from mechanical errors in processing older studies with incomplete documentation. There is no single, unambiguousprocedure for spotting outliers, but the following rule is consistent with the statistical literature (Anscombe and Tukey, 1963;Tukey, 1962):An individualobservation may be consideredan outlier if the chi-squarefor the deviationof that observation from the modeled estimate is too large. Therefore, the chi-square test for an outlier is: X2 (1 df) =

(Dk-Ek)2

(8)

The sampling correction should be applied to this formula, and so we might summarize the decision rule as follows: an observation should be studied more carefully and possibly rejected from the analysis if the corrected chi-square for the deviation of that observa-

tion from the modeled estimate is significantat the .01 level (6.6 or greater). ANOTHER METHODOLOGICAL PROBLEM

When applyingthe decision rules it is possible to arrive at a negative chi-square for the contribution of an added parameter. In this case the chi-square should be taken as zero for purposes of the analysis. This is not due to roundingerroror a fault in the logic of the system; rather, the reason lies in the distribution theory for the statistics and the fact that the methods we propose here are "approximate" methods. Conclusion The purpose of this paper is not to invent new statistical procedures. Rather,the goal is to take the weighted least squares model for percentage differences (Davis, 1976;Grizzle et al., 1969)and link the steps in this model to a series of theories which might be used to describe change in public opinion data. Like most calls for applied statistics in substantive research, we emphasize the ordering of hypotheses and the use of significance tests to unambiguouslychoose between alternative explanations.

EVALUATING TRENDS IN PUBLIC OPINION

The discussion addresses the question of how to find the most parsimonious description of a series of percentage differences. But the principles of the analysis apply to any statistic that is relevant to public opinion research (the only limitationbeing that the statistic to be analyzed must have a variance that can be calculated.) Thus, the procedures can be used to find the most parsimonious model for a series of percentages(althoughthe hypothesis that all the percentages are equal to zero is not of much interest.) Likewise, by analyzing trends in the logarithms of odds ratios, we are simply committing ourselves to a particular series of hypotheses in testing log linear models (Davis, 1975). Finally, other studies of trends in gamma statistics (e.g., Verba et al., 1976)might well have used some variant of the decision rules suggested here to analyze the data. References Alford, R. L. 1963 "The role of social class in American voting behavior," Western Political Quarterly16:180-94. Anscombe, F. J., and J. Tukey 1963 "The examination and analysis of residuals," Technometrics 5:141-60. Blake, Judith 1966 "Ideal family size amongwhite Americans," Demography3:154-73. 1974 "Can we believe recent data on birth expectations in the United States?" Demography 11:25-44. Converse, P. E. 1972 "Changein the AmericanElectorate," in Angus Campbelland Philip Converse (eds.), The Human Meaning of Social Change. New York: Russell Sage Foundation. Davis, James A. 1975 "The log linear analysis of survey replications," in Kenneth Land and Seymour Spilerman (eds.), Social Indicator Models. New York: Russell Sage Foundation. 1976 "Analyzing contingency tables with linear flow graphs," Sociological Methodology. San Francisco: Jossey Bass. Gallup, George 1972 The Gallup Poll: Public Opinion 1935-1971. New York: Random House. Glenn, Norval 1967 "Massification and differentiation:some trend data from national surveys," Social Forces, 46:172-80. 1975 "Trend studies with available survey data: opportunitiesand pitfalls," in J. Southwick (ed.), Survey Data for Trend Analysis. Williams College Center, Roper Public Opinion Research. Glenn, Norval, and J. L. Simmons 1967 "Are regional cultural differences diminishing?" Public Opinion Quarterly31:176-93.

D GARTH TAYLOR

100

Goodman, Leo 1963 "On methods for comparing contingency tables," Journal of the Royal Statistical Society (A) 126:94-108. 1976 "The relationshipbetween modified and usual multiple regression approachesto the analysis of dichotomous variables," Sociological Methodology. San Francisco: Jossey Bass. Grizzle, James E., C. Frank Starmer, and G. Koch 1969 "Analysis of categorical data by linear models," Biometrics 25:489-504.

Hansen, Morris, W. Hurwitz, E. Marks, and W. Mauldin 1951 "Response errors in surveys," Journalof the American Statistical Association 46:147-90. Hyman, H. H., J. Wright, and J. Reed 1975 The EnduringEffects of Education.Chicago:University of Chicago. Kish, Leslie 1957 "Confidence intervals for clustered samples," American Sociological Review 22:154-65. 1965 Survey Sampling. New York: Wiley. Mayer, K. 1959 "Diminishing class differentials in the United States," Kyklos XIII:605-28. Newman, Karen 1976 "The use of AIPO surveys: to weight or not to weight," in James A. Davis (ed.), Studies of Social ChangeSince 1948Volume 1. Chicago: National Opinion Research Center. Nie, N., S. Verba, and J. Petrocik 1976 The Changing American Voter. Cambridge: Harvard University Press. Reed, John Shelton 1972 The EnduringSouth. Lexington, Mass.: D.C. Heath. Smith, Tom W., D Garth Taylor, and Nancy Mathiowetz 1979 "Public opinion and public regardfor the federalgovernment,"in A. Barton and C. Weiss (eds.), Making BureaucraciesWork. Beverly Hills, California:Sage. Sullivan, John L., James E. Pierson, and George Marcus 1979 "Political intolerance:the illusion of progress," Psychology Today 12:87-91. Taylor, D Garth forth- "Analyzingqualitativedata," in P. Rossi, J. Wright,and A. Andercom- son (eds.), The Handbook of Survey Research. New York: Acaing demic Press. Tukey, John 1962 "The future of data analysis," Annals of MathematicalStatistics 33:1-67. Wonnacot, R. J., and T. H. Wonnacot 1970 Econometrics. New York: Wiley.