Studying categorical data over time

SOCIAL

SCIENCE

RESEARCH

Studying

7, IsI-179(1978)

Categorical

Data over Time

JAMES A. DAVIS Harvard

University

A review of recently developed simple techniques for analyzing data in the form of proportions from replicated, cross-sectional, sample surveys. (A) One variable over time-tests for homogeneity, pooling homogeneous proportions, tests for linear trends with as few as three time points, tests for departures from linearity; (B) Two variables with a constant percentage difference-four fold tables as rudimentary causal models, decomposing change with linear flow graphs; (C) Extension of the flow graph techniques to-three or more variables and changing coefficients; (D) Comments implying panel designs are somewhat over-rated and successive cross-sections somewhat under-rated.

In an era when there is rampant cynicism, sociologists still have faith that longitudinal research is a good thing. The faith bums stronger among those who have never done such research, but I have never heard anyone fail to agree things would be much better if we could study it over time. Why is a longitudinal design good? If we had interval scale data and many time points we could carry out time-series analysis. In an extremely important article Land and Felson (1976) review the possibilities with interval measures and opt for “dynamic macro structural-equation models” as the most promising approach if one lacks the 30 to 50 time points required by spectral analysis (Mayer and Arney, 1974). But we are sociologists with nominal or dichotomous measures and only a handful of time points, typically just two. Full-blown time-series analysis is seldom appropriate. In the last two years the NORC Social Change project has developed a set of statistical techniques for handling such primitive longitudinal data (Davis, 1975a, 1975b; Taylor, 1975). The procedures are logically and statistically elementary, but they provide a consistent package of tools The research reported here was supported by National Science Foundation Grant No. GS-38534. I am indebted to D. Garth Taylor for many of the ideas here and to Paul R. Abramson for comments and corrections on the first draft. I am indebted to Dartmouth College for secretarial and drafting help. Paper prepared for The Conference on Strategies of Longitudinal Research on Drug Use, San Juan, Puerto Rico, April 7-9, 1976. Address reprint requests to Library, NORC, 6030 S. Ellis, Chicago, Illinois 60637. 151 0049-089X/78/0072-0151$02.00/0 Copyright @ 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

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JAMES A. DAVIS

organized around the notions of linear systems (Heise, 1975) and “flow graphs” (Huggins and Entwisle, 1968; Stinchcombe, 1968). The aim of this paper is to explain and illustrate these tools. ONE VARIABLE:

TWO TIMES

Roizen, Cahalan, and Shanks (1976) report panel data on a probability sample of white, noninstitutionalized males ages 21-59 years in San Francisco, measured in 1967- 1968 and in early 1972. Respondents who had ever been treated for alcoholism (N = 55) and those reporting themselves as teetotalers both times (N = 39) were excluded from the analysis, leaving 521 cases.’ On the basis of questionnaires the respondents were classified each time as having one or more alcohol-related problems or having none, with the results given in Table 1 (adapted from Roizen et al., 1976, Table 2). Proportion

TABLE 1 with One or More Alcohol-Related

Year 1972 1967-1968

Proportion .443 ,393

Problems, 1967-1968 and 1972 N

(260) wa

Adjusted .95 confidence limits” 2.087 + ,086

(1 See footnote 1.

In the sample the proportion with a problem increased .050 (.443 - .393 = .050). Applying the standard test for a difference in proportions, (1) we conclude the change is not significant since the difference, d, .050, is not as large as its two (T confidence limits, &. 122. We are unable to detect any significant change in the proportion of San Francisco males with alcohol problems between 1967-1968 and 1972. The point is hardly profound-though in their elaborate analysis of the data Roizen et al. never seem to ask whether the proportion with problems is changing-but it has a nonobvious implication. If two (or more) samples give homogeneous estimates of a particular proportion we can pool them to get a better estimate of the common universe value. (The actual calculations are explained in Davis 1975a, pp. 1 Although these data come from a panel-a longitudinal study where the same cases are measured each time-the techniques explained here do not require panel designs. Therefore, the data in Table I have been treated as if the design involved 520 cases, 260 measured at each time. Furthermore, I correct all estimated sampling errors for multistage or cluster effects. As a rule of thumb, cluster designs have sampling variances twice as large as the Simple Random Sample (SRS) formulas in textbooks. I routinely multiply estimated (TSby 21 = 1.414 as an adjustment.

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DATA OVER TIME

153

126-128.) The pooled value turns out to be .418, with two u confidence limits of +- .061. Here the pooled value is the arithmetic mean of the two estimates because the two Ns are identical and the proportions are close. In general, however, the pooled estimate will be closer to the more reliable results since the technique weights each proportion inversely to its estimated sampling variance. We can now say, “In the period 1967-1972 the proportion of San Francisco males with alcohol problems is quite likely between .479 and .357.” Confidence limits for the pooled proportion are substantially smaller than for the two separate estimates (.061 vs .087 and .086). The confidence limits for the pooled proportion are virtually the same as those we would obtain by applying a single N of 520 to the average proportion, .418 (.061168 vs .061115). That is, we obtain no magic improvement or deterioration in reliability by dividing a total sample into replicates for pooling.2 But the point can be turned around like this: Dividing N into subsamples replicated at two or more times: (a) Costs nothing in terms of confidence in the pooled estimate of an unchanging proportion, and (b) Allows one to estimate changes in the proportion. Whether the execution of two samplings size .5N costs more than one sample size N is not entirely clear. Offhand, it sounds more expensive, but I have not been impressed by the economies of increased scale in surveys unless (a) the survey is really huge, as in the Current Population Survey or (b) interviewer assignments (cluster sizes) are increased. The homely example in Table I provides fundamental support for our faith in longitudinal research. In the case of a proportion, at least, longitudinal designs can give us useful information about stability and change and are not a technical extravagance when the proportion turns out to be stable. Obviously, the test for differences in proportions and the pooling techniques can be extended directly to variables consisting of more than two categories, though the purist would leave one category unanalyzed since its results will be redundant. (If you know how much the other categories went up or down, you can find the change in the last one by subtraction.) Warning. When the item is ordered (e.g., No Problems, Minor Problems, Major Problems, Overwhelming Problems) it is not obvious whether the trend results will tell you whether the thing, if changing, is going “up” or “down.” Consider the example from a panel study of University of California men given in Table 2 (Mellinger, Somers, Davidson, and Manheimer, 1975, Table 2). 2 The principle is well known in the form, “Stratification in a sample design can’t hurt you,” even if the strata turn out to be unrelated to the variable you are estimating.

154

JAMES A. DAVIS TABLE 2 Drug Use in a Sample of University of California Men, 1970 and 1973 Year

Level

1970

1973

Shift

Confidence limits”

Multiple drug use Marijuana only Never used

.16 .43 .41

.24 .53 .22

+.08 +.10 -.I9

2.078 + .097 -c .089

1.000 (4171

.99 (417)

N” (1 See footnote 1.

There are significant changes in all three categories, but did “level of drug use” go up or down? An arbitrary but plausible rule says: “An ordinal variable increases (decreases) if one finds a positive (negative) net shift for any possible dichotomy.” (If, no matter where you draw the line, the proportion above it increased and the proportion below decreased, the level of the whatchamacallit has increased.) Drug use in these data increased. If we dichotomize as Multiple vs Other we get an increase of +.08; if we dichotomize as Ever vs Never used, we get an increase of + .18. But such consistent results are not inevitable. If Multiple had decreased and Never Used had also decreased we could produce an increase or a decrease depending on the cut. With more categories, the possibilities for ambiguity are enhanced. ONE VARIABLE:

K TIMES

Testing differences between two proportions is old hat, but examining differences among a set of K proportions is less familiar. The statistical principles were worked out by Goodman (1963) and extented to trend research by Taylor (1975) and Davis (1975a). We will not present the detailed procedures, which are given in the cited papers. Instead, we will concentrate on the logic of the system. For example, consider answers to the question, “Would you favor or oppose a law which would require a person to obtain a police permit before he or she could buy a gun? (Favor, Oppose, Don’t Know),” a Gallup item appearing in the 1972-3-4-5 NORC General Social Surveys.3 Table 3 gives the proportions answering “Favor” each year. First, we assume (temporarily) there have been no changes, that is, each proportion is an independent estimate of some common universe :’ The General Social Survey is an annual sampling of United States adults stressing replicated measures to study social change. Copies of the data are put into the public domain immediately after the questionnaires are coded and cleaned via the Roper Public Opinion Research Center, Yale University. and the Inter-University Consortium for Political Research. The project is funded by NSF.

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Proportions

TABLE 3 Favorable to Gun Permits, 1972-1975O Proportion

Year

.737 .753 .735 .702

1975 1974 1973 1972 Pooled value

DATA OVER TIME

N

Adjusted .95 confidence limits

(1488) (1477) (1495) (1610)

.032 ,032 .032 .032 .016

.732

0 x2 = 5.3, p = .147.

proportion. If so, we can estimate the true value by pooling our various estimates, weighting each inversely to its estimated sampling variance, i.e., giving greater weight to the more reliable estimates. In Table 2 the pooled value is .732, and we observe again how pooling improves our confidence intervals. Second, we compare each value with the pooled result, for example, .737 - ,732 = .005. A x2 test can be made for the discrepancy between observed and expected proportions and the x2s summed over all proportions. In Table 2 we get a total x2 of 5.3 over the 4 years. Our computer program4 tells us that this x2 has a probability level of .147. Since this is well above the conventional .05 level, we infer the proportions to be homogeneous-we can fit the estimates very comfortably with the common value. Thus, there is no trend in Attitude to Gun Permits for the years 1972-1975. Proportions Year 1975 1974 1973 Pooled value

TABLE 4 Saying Drug Control Spending is “Too Little”” Proportion .55 I A00 .659

N

Adjusted .95 confidence limits

(1482) (1473) (1493)

.036 ,036 .035

A05

0 Homogeneity x2 = 18.5,~ = C.001.

Table 4, however, tells a different story. The item, again from the General Social Survey (in abbreviated form), is “are we spending too much, too little, or about the right amount on Dealing with drug addiction? (Too little, About right, Too much, Don’t Know).” Table 4 gives the proportions responding “Too Little.” 4 Copies of the program, TRENDLET, written in Dartmouth BASIC by John Fry and Garth Taylor, are available from the writer.

156

JAMES

A. DAVIS

The significant homogeneity x2 at the bottom of Table 4 says we can not fit the three data points comfortably with the pooled proportion. Attitudes toward Drug Control Spending are not homogeneous for the years 19731975. (the item was not used in 1972). Longitudinal design has detected a change, for a change. Third, we can go farther than that. Even though we are working with categorical data, Time is a classic interval variable. Thus, we can ask whether the changes-if significant-show a linear trend. To do so, we run a weighted regression, as usual weighing each proportion inversely to its estimated sampling variance. Table 5 shows the results. Table 5 suggests we can fit the data very well by a linear equation, whose coefficient, - .0541, says support for Drug Control Spending decreases about .054 each year, at least for the 3 years 1973-1975. Two x2 tests at the bottom of Table 4 help us interpret such results. The test for Improvement asks whether the fit from the regression model is significantly better than the fit using the constant pooled value. The result in Table 4 is highly significant @ < .OOl). This may be thought of as a significance test for the regression. We are not used to situations where regressions based on three points are significant at the .OOl level. The difference, of course, is that here each “point” is based on almost 1500 cases. In this approach, rather like Tukey’s exploratory data analysis, the analyst should pay careful attention to individual points in his regression plot. The second test, Linear Fit, asks whether the points show significant scatter around the regression line. Here they don’t, asp = .815. We have a nice clean result: Introduction of the trend line not only improves fit significantly, but the trend line describes the data well since residual variation is insignificant. A regression model for decreasing endorsement of Drug Control Spending is necessary (the pooled constant Linear

Trends

Year

Regression prediction = ?

197.5 1974 1973

..549 .603 ,657

Regression Tests Improvement Linear Fit

equation

? = 4.61 - 0.0541

TABLE 5 for Data

in Table

Value

= Y

3

Error

= Y -

.551 .600 ,659 * (Year

-

+.002 -.003 +.002

1900)

x’

P

18.4 0.1

<.OOl .815

?

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157

DATA OVER TIME

doesn’t fit) and sufficient (departures from linear fit are not significant). It is quite possible for data to show significant improvement along with significant discrepancies for Linear Fit. When examining Subjective Social Class Identification in 21 United States national surveys from 1948 to 1974, I found highly significant Improvement (the proportion claiming to be Middle or Upper Class increases +.0044 per year) along with highly significant departures from Linear Fit (Davis, 197%). Whether the departures represent outlyers (errors in measurement) or substantive blips cannot be determined on statistical grounds alone. The statistics wili tell which points will differ significantly from the regression prediction, but they will not tell you why. Conversely, it is possible for data to show significant departures from homogeneity without linear trends. Seven United States Gallup pools for the years 194% 1952- 1956- 1957- 1963- 1966 were used to examine marginal changes in subjective Happiness (“In general, how happy would you say you are-Very Happy? Fairly Happy? Not Very?“). The Results (Davis, 1975d) showed significant departure from homogeneity, but no significant improvement for fitting a linear trend. Inspection of individual x’s showed the 1956 and the 1957 samples with values significantly above the pooled value and the 1948 significantly below, as shown in Table 6. In these data happiness shows significant fluctuations but no trend. TABLE

6

Happiness in Seven Gallup Polls, 1948-I%6 Year 1966 1963 1957 1956 1952 1948

Gallup study No. 736 735 675 580 570 508 425K

Proportion “very happy”

Versus pool value = .494

.463 S16 .472 ,534 .538 .474 .443

NS” NS NS +&to +.044 NS -.051

’ NS = Not statistically significant by x2 test. Numerical values are significant at the .05 level.

These tests can be arranged to give an algorithm for analyzing longitudinal data where a set of proportions are measured three or more times. Thus: (1) Test the proportions (ps) for homogeneity (a) If x2 is not significant, infer: “P (the universe proportion) is a constant, pooled value.” (b) If significant,

estimated

by the

158

JAMES A. DAVIS

(2) Test for Linear Improvement (a) If x2 is not significant, infer: “P shows trendless fluctuation.” (b) If x2 is significant, go to (3). (3) Test for Linear Fit (a) If x2 is not significant, infer: “P is described by a linear function estimated by the regression equation.” (b) If x2 is significant, infer: “P shows a rough linear trend.” . . . and test for outlyers. After removing the case with the largest x2 for the difference between p and the regression prediction, return to step 1. In sum, with three or more readings on a proportion one can apply a nested set of tests to sort the data into one of the following cases (Case I will be explained later): (II) P is a constant (III) P is described by a linear trend (IIIb) P shows a rough linear trend (IV) P shows trendless fluctuation. TWO OR MORE VARIABLES:

CONSTANT COEFFICIENTS

With two dichotomous variables, our old friend the fourfold table, we can begin viewing longitudinal data as a system of variables changing over time. Consider, for example, the association between Marijuana use and Depressive Mood in a probability sample of New York State high school students in Fall 1971 (Kessler, 1976). There is a significant difference. Marijuana users are more likely to be Depressed. For the present we assume Marijuana affects Depression, not vice versa. The matter will be discussed at length later. TABLE 7 Marijuana Use and Depressive Mood among New York High School Students, Fall 1971” Uses marijuana Yes No d

Proportion depressed ,542 .442 +.100

” Adapted from Kessler (1976, Table 5). Cf. footnote 1.

(450)

(1942) (2.074)

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DATA

Although fourfold tables are rudimentary, linear equations (Davis, 1975). Thus:

OVER

TIME

they can be translated

159

into

Y=(daX)+Ky (2) where Y = the Marginal proportion for the nonbase category of the dependent variable, X = the Marginal proportion for the nonbase category of the independent variable, d = the difference in proportions, and KY = the constant or intercept value, the proportion in the nonbase category of Y for cases in the base category of X. To express data this way one merely chooses a “base” and “nonbase” category for each variable (rather like assigning + and - to the categories of a dichotomy). Using “No” as the base category for both variables in Table 7, we get: x

= $00

+

1942

=

.188

d = +.100 KY = .442 Y = (+.I00

* .188) + .442 = .461

(3)

Most of us would find Eq. (3) harder to read than Table 7, although they have identical information save for N. (With three or more variables, however, the contrast is less. Multivariate percentage tables are notoriously ambiguous.) However, any set of linear equations can be translated into a simple, useful pictograph called a “linear flow graph” (Davis 1975a; Heise, 1975; Huggins and Entwistle, 1968; Stinchcombe, 1968). This tool, appropriated from electrical engineering where it is used to analyze the flow of current in an electronic device, is similar to the “path diagrams” of regression analysis although the rules are not quite identical. Figure 1 represents the flow graph for Eq. (3). To make the translation (Fig. 1): (a) Nonbase categories become nodes (circles). (b) The independent and dependent variables are connected by an arrow, whose value is d. (c) One draws an arrow from the constant, KY, to the dependent variable, Y, assigning the value + 1BOO to the arrow. The flow graph rule: The value of a dependent variable is found by multiplying the arrow coefficient for each incoming arrow times its origin value and summing gives the same answer as Eq. (3) (+. 100 * .188)

+(l.OO * .422) = .461. Now, let’s measure our two variables again at some later time, Time,. If

160

JAMES

A. DAVIS

0-z KY

= .442

1.000

.I68 I

+.100

x

FIG.

I.

Flow

graph

Y

for Eqs.

(2-3).

we define the difference, T, - T,, for any term as A, we can rewrite Eq. (2) to give Y at Time, like this (numerical subscripts indicate times): Y, = (d, + Ad) * (X, + Ax) + (Ky + AK).

(4)

Equation (4) says we get the Time, values by adding its own A to each Time, term in Eq. (2). Next, after multiplying out: Y, = [d, *Ax,] + (d, * Ax) + (Ad * X,) + (5) (M * Am + KY1 + (Au. Observing the terms in brackets (as opposed to parentheses) in Eq. (5) are the terms in Eq. (2): to find AY, the change in Y from Time, to Time,, we remove the bracketed terms. Thus: AY=(d,*~)+(M*X,)+(M*hX)+(~).

(6)

Now let us assume that the arrow coefficient, d, remains the same at both times, i.e., Ad = 0. If so, two terms in Eq. (6) drop out and we are left with: AY = (d *AX) + (AK).

(7)

Comparing Eq. (7) with Eq. (2) we see they are almost the same. The only difference is that Eq. (7) has As and Eq. (2) has cross-sectional values. Equations (4-7) illustrate a powerful principle in linear systems (Heise, 1975, pp. 23-25): Zf the coefJicients for relationships between pairs of variables do not change, one may USC the flow graph of the system to analyze change by substituting As for source (initial variable) value and KS.

Thus we have the change graph in Fig. 2. Translating Fig. 2 into English: The total change in Y can be found by multiplying and adding change in the constant term.

d by the change in X

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DATA OVER TIME

FIG. 2. Change graph for system in Fig. 1 assuming d is constant.

In point of fact, we are all familiar with this idea since we often interpret a regression coefficient by saying “If X were to change one unit, Y would increase (decrease) b units,” but we seldom add the crucial assumption “given stability in b .” Jumping back to Table 7, all this high school mathematics says: If the relationship between Marijuana and Depression stays the same . . . . Change in the level of Depressive Affect can be broken down into two parts: (1) Change due to the increase or decrease in Marijuana use (d * Change in Marijuana marginals). (2) Residual change, not accounted for by the model. Since the data in Table 7 are from a panel study with a follow-up in Spring 1972, we can estimate these effects. It is convenient to arrange such data as if they were a three-variable system with Time as the source or exogenous variable (Table 8). First, we estimate AX, change in the marginal proportion of Marijuana users. This is easily found by collapsing Table 8 (see Table 9). The change, +.038, is small but significant, since d exceeds its two u TABLE 8 Longitudinal Data for Marijuana Use and Depression among New York High School Students, 1971-1972” Depressive mood Time Fall 1971

Marijuana No Yes

Spring 1972

No Yes

0 See footnote 1. N = 4784.

No

Yes

1083 206 1095 273

859 244 756 268

162

JAMES A. DAVIS TABLE 9 Time and Marijuana Use in Table 8” Time Spring 1972 Fall 1971 d

Proportion .226 .I88 +.038

users

(2392) (2392) (k.034)

’ See footnote 1.

confidence level. More High School students were Marijuana users in the Spring follow-up. Second, we examine stability in the coefficient for Marijuana and Depression. Breaking Table 8 into two fourfold tables for Marijuana by Depression within Times (the first two rows and the last two rows) we find that d, = .lOO (2.074) and d, = + .087 (k-+.069), close but not identical. To test for a significant shift in d we may use the classical test for a difference in the difference between proportions. (For more general methods that apply to systems with more than three variables, see Davis, 1975a, pp. 126- 129.) The difference in ds is not significant. Therefore, we may consider both as estimates of a common d, estimated by their weighted average, weights being inverse to the estimated sampling variances of the ds. The pooled value, d^ = + .093 (k.050). (When the pooled value is not significantly different from zero and there are no significant fluctuations over time, we have a Type I case. Since the notion does not apply to single proportions, the cases in the previous section began with II.) Third, we can estimate AKy. Algebraically this is an amount added (or subtracted) across the board, regardless of scores on other variables, so we may think of it as the net change in Y within categories of the prior variable. In terms of data arranged as in Table 8, AKy is the pooled d for Time by Y within categories of X, i.e., ds in the fourfold table formed by the first and third rows and the table formed by the second and fourth row. In Table 8 we get ad of - .034 (+ .045) among students in the No category for Marijuana and - .046 (+ .090) in the Yes group. There is no significant difference between the two, so we calculate the pooled value, obtaining - .037 (k.040). The residual is not strictly significant, being only 1.85 times its estimated sampling standard deviation. The associated twotailed probability, .064, is so close to the conventional .05 level, however, I’m willing to keep it for our purposes. These estimates enable us to fill in our change graph, as shown in Fig. 3. Following the flow graph rule, the total change in level of Depression is: From increase in Marijuana (.093 * .038) = + .003534 Residual, not accounted for by the model =-.037 - .033466

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DATA OVER TIME

Q KY

163

:

FIG. 3. Change graph for data in Table 8. Solid lines, positive values; dashed lines, negative values.

According to the model, increased Marijuana smoking produced a significant, albeit small, increase in the level of Depressed Mood, but this was more than offset by a “secular decline” in Depression so that, all in all, the proportion Depressed dropped - .033. The decline of -.033 is exactly what one would obtain by comparing the Time, and Time, marginal proportions for the dependent variable. When the data are perfectly free from interactions (conditional ds are perfectly homogeneous) the modeled change and raw marginal change must be identical. Even with minor interaction effects the modeled results are generally within one or two units in the second decimal, as in our example. Such analyses can be extended to three or more variables but, before explaining how, it may be useful to discuss causal direction and magnitudes . We have assumed throughout that Marijuana influences Mood while Mood has no influence on Marijuana, i.e., the two variables are connected by a one-way causal arrow. (Whether this is obvious, debatable, or preposterous is unknown. I am a methodologist and untroubled by such problems.) Why make such a strong assumption? Because the central statistic, d, is asymmetrical. Dy,y is not in general equal to D.yy although in my experience they are pretty close unless the variables are both highly skewed in their marginals. There are, of course, contingency table statistics that are symmetrical, the odds ratio and Yule’s Q being so fashionable these days as to be almost suspicious. However, odds ratio-based statistics do not follow the elegant principles of flow graphs (Davis and Schooler, 1974). The practical researcher thus faces a dilemma. Shall one opt for conservative (ambiguous) causal direction or for a powerful calculus? Anyone who has wrestled with nonexperimental data involving more than two variables knows the answer.

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JAMES

A. DAVIS

Warning. Sociological folk wisdom to the contrary notwithstanding, mere replication cannot establish one-way causal direction. (For a “proof,” construct a set of fake data in which the two variables have a mutual or loop effect, derive results for as many times as you like, and then try to figure out which way the one-way arrow “really” runs.) Whether shifting to a panel design changes things much is discussed later. The matter of magnitudes also deserves comment. The effects derived from Fig. 3, + .003534 and - .037, are scarcely heroic in size and, paradoxically, the former is statistically significant but the latter is borderline. This is not a quirk of drug research. I have been working with such data for several years now and can assure you that the results are not unusual (see, for example, the Subtitle in Davis, 1975d). In social research data, mine at least, significant ds seem to average no larger than .20 and changes in source proportions or KS seldom exceed .05 per year. Since (.20 * .05) = .Ol , you can see short-run massive effects are hard to come by, especially in larger systems where variables are connected by indirect paths whose values are products of decimals. In sum, my experience in change analysis has been this: By and large the cross-sectional insights of social scientists are supported by change analysis’ but the magnitudes are so small as to be of little practical importance. The logic and statistics are easily extended to three or more variables and to polytomous measurement (for specifics, see Davis, 1975a, 1975b). A detailed explanation would add considerably to length without adding new ideas, aside from that of “path product.” Instead, let us briefly consider what might happen if a third variable, let’s call it Hanging Around With Alienated Insightful Intellectuals (HAWAII), were introduced into the system in Fig. 3. Assuming all coefficients are constant and there are no interactions, the change graph would be as seen in Fig. 4. Given data at two times,” AMarijuana, AK2, and “a” would be estimated as explained above; “b” and “c.” would be weighted averages of the partial ds for prior variables and Depression summed across times and categories of the outside variable, and AK3 would be the weighted average of the Time by Depression d within categories of the two prior variables. By extension of the principles preceding Fig. 2 and Eq. (4), we can decompose the net change in Depression into these components: Residual change unexplained mode1

by variables in the AK3

a An exception: My experience and that of an increasing literature is that age differences in cross-sectional data generally turn out to be Cohort effects when subjected to longitudinal analysis. 5 These procedures can be used for studies involving more than two times, but it will be simpler to postpone this until the next section.

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DATA OVER TIME

Effect of Change in HAWAII, net of change in Marijuana Effect of Change in Marijuana Direct Via effect of Changing Levels of Marijuana on Changing HAWAII

165

b *AK2 c * AMarijuana b * a * AMarijuana

Total

Q AK2

A

AK3 FIG. 4. Hypothetical

three-variable

change graph.

The example is a bit fanciful but, in Davis (1975b), change in Tolerance of Communists and Atheists from 1954 to 1972-1973 is decomposed into (a) a substantial residual, (b) an effect of educational change within a Cohort Group, and (c) effects of Cohort replacement, direct and via effects on Educational attainment. In sum, without invoking levels of measurement beyond nominal or statistics beyond those for sampling proportions, it is possible to handle longitudinal data from as few as two times (no panel design required) in terms of linear models that allow one to ask whether changes in the system of prior variables account for changes in a dependent variable.

TWO OR MORE VARIABLES:

CHANGING

COEFFICIENTS

Constant coefficients make for felicitous flow graphs, but there is no guarantee the real world behaves so neatly. Consider, for example, a fascinating set of data analyzed by Abramson (I 975, pp. 15- 18). He gives the cross tabulation of SES (Head of Household White Collar vs Blue Collar) by Major Party Presidential Vote Among nonfarm Whites for the years 1948-52-56-60-64-68-72 from University of Michigan Survey Research surveys. Since the printed table excludes Farmers and Blacks and I have excluded Third-party voters and nonvoters, the results should not

166

JAMES A. DAVIS

be taken at face value. Table 10 gives Abramson’s results adapted for our purposes. This system may be specified as represented in Fig. 5. The left hand columns in Tables 10 and 11 tell us the source variable, SES, has changed. The tests show (a) significant heterogeneity, (b) significant improvement when fitting a linear trend, and (c) insignificant residual variation around the trend line. The proportion White Collar increased steadily at the rate of +.00.52 (the regression coefficient) per year, as shown in Table 12. Since the proportion White Collar is increasing and White Collar Voters are less inclined to vote Democratic, one might expect a steady decrease in the Democratic vote from 1948 to 1972. But, the data are not that simple. The same tests we apply to a proportion can be applied to d (the difference between two proportions). Table 1 I assures us that d is not constant. There is significant heterogeniety, significant improvement TABLE IO SES and Major Party Vote among Nonfarm Whites, 1948-1972” Proportion democratic Year

Proportion white collar

Blue collar

1948 1952 1956 l%O 1964 1968 1972

,506 ,486 ,472 .492 ,516 .566 ,592

.754 ,513 .429 563 ,746 .471 .3lO

(271) (899) (1006) ( 1140) (879) (680) (1262)

(134) (462) (531) (579) (425) (295) (587)

White collar ,314 .31 I ,347 .447 .555 .371 .293

d

(137) (437) (475) (561) (454) (385) (675)

- A40 - .202 - .082 -.116 -.19l -.I00 -.017

Total proportion democratic .531 .415 ,391 ,506 ,647 .415 ,301

a Adapted from Abramson (1975, pp. 15-17). TABLE 11 Statistical Effects for Data in Table 10 Proportion white collar Homogeneity probability Linear trend Improvement probability Linear fit Regression Constant Coefficient R2 Type

Proportion democratic among blue collar

d

.ool

.OOl

.OOl

A01 .329

301 .OOl

.OOl .006

.21 + .0052 .649

1.08 -.oo9o ,270

-.76 +.0100 ,588

III

IIIb

IIIb

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d FIG. 5. Flow graph for data in Table 10.

when fitting a linear trend, and significant variation around that trend line. Far from stable, d shows a secular decline along with significant electionto-election “blips.” For the period 1948-1972 the correlation between SES and Vote has declined, but particular elections show nonrandom waxing and waning of the relationship. This analysis manages to support both those who argue for trendless fluctuations (e.g., Converse’s earliest interpretation, 1958) and those who argue for secular decline (Abramson, 1975; Glenn, 1973) but not those who argue for constancy (Alford, 1967). Table 13 gives the results in detail. The 1948 (Truman-Dewey) election has a stronger SES effect and the 1956 (Eisenhower-Stevenson) election has a weaker effect, but the other five elections are not significantly off the trend line. (Remember in samples of 1000, ds have two (T confidence intervals of roughly 2.09, after correction for multistage sampling.) When d is not stable, the constant term, K, cannot be estimated by the pooled within-category changes. If the value of d changes from Time, to Time,, the groups in question show unequal shifts in the dependent variable of mathematical necessity. When d is not homogeneous we estimate K by the proportions “high” on the dependent variable for cases TABLE 12 Linear Trend in Proportion White Collar in Table 10 Year

Regression estimate

1948 1952 1956 1960 1964 1%8 1972 1976

.456 .477 .498 .519 ,539 ,560 .581 .605

Error (proportion minus estimate) + .050 +.009 - .026 -.027 -.023 +.006 +.011 ?

168

JAMES A. DAVIS TABLE 13 Linear Trend and Departures from It for the Association between SES and Vote in Table IO Deviation

Year

Regression estimate

1948 1952 1956 I%0 1964 1968 1912 1976

- .284 - .244 -.204 -.164 -.I24 - .084 -.044 +.ooo

Significantly weaker

Other

Significantly stronger -.I56

+.042 +.122

?

+.048 -.067 -.016 + .027 ?

?

in the “base” category-here we look at the proportion Democratic among Blue Collar voters. Table 11 shows it has the same pattern as d, a secular decrease along with significant year-to-year fluctuations. Table 14 gives the details. We now have the basic facts: (a) The proportion White Collar among White Two-Party voters has increased steadily since 1948. (b) Among Blue Collar voters, Democratic proportions declined steadily since 1948, but this is masked by significant departures from trend in five of the seven elections in the series. (c) White Collar voters are generally less apt to vote Democratic, but the difference has declined steadily and the 1948 and 1956 elections show significant departures from the trend. TABLE 14 Linear Trend and Departures from It for the Proportion Democratic among Blue Collar Voters in Table IO Deviation Year

Regression estimate

1948 1952 1956 I%0 1964 1968 1972 1976

.649 .613 ,577 ,541 ,505 ,469 ,433 ,396

Significantly low

Other

Significantly high +.105

-.I00 -.I48 + ,022 +.241 +.002 p.123 ?

?

7

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How do we build these facts into a linear flow graph model? Jumping back to Eq. (6) we notice: (d, * AX) + (Ad * AX) = (d, * AX).

(8)

Therefore: Y=(Ad*X,)+(d,*Ax)+(AK),

(9)

can be graphed as seen in Fig. 6.

FIG. 6. Linear flow graph for Eqs. (6-8-9).

Translating Fig. 6 into English: Change in the marginal proportion for the dependent category, Y, can be decomposed into three parts7 (1) Change in Y among cases in the base category for X. This is a residual and is the amount of change not accounted for by properties in the system. (2) The Time, coefficient, d,, times the marginal shift in the proportion X. This may be viewed as “demographic change” (Stinchcombe, 1968, chap. 3) since it estimates the contribution coming from increases or decreases in the level of prior variable, i.e., changes in population composition. (3) X,, the initial marginal proportion for the prior category, times Ad, change in the coefficient. This is the contribution of shifts in the degree of association for the two varables. X, * Ad may also be interpreted as the contribution of differential rates of change. To see why, consider the 1960 and 1964 data plucked out of Table 10 and shown in Table IS. The results at the bottom of Table 15 show that Ad for 1964 vs 1960 is -.075. Now, however, let us look across the rows. Comparing 1964 and 1960 results for White Collar we find a f. 108 shift toward the Democrats. For Blue Collar the shift is +. 183. Now if we compare the two shifts we get a difference of -.075, exactly the same as Ad. This is no coincidence 7 This decomposition differs from and is better (I think) than the one reported in Davis (1975b, pp. 496-499). Both are mathematically correct.

JAMES A. DAVIS

170

TABLE I5 SES and Vote, I%0 and 1964 (Taken from Table IO): Proportion SES

1960

1964

Shift

White collar

.447 = (I

,555 = b

+.I08

Blue collar

.563 = c

,746 = d

+.I83

Democratic Difference

(+.I08 - ,183 = -.075) -.I91

d, d,

-.I16 (Ad = -.075)

as shown by the following four cell proportions:

algebra, using the letters a, 6, c, d, and for the

Ad = (b - d) - (a - c) = (b + c) - (d + a)

Relative

(10)

Shift = (b - a) - (d - c) = (b + c) - (a + d).

(11)

Thus (M * X,) can be interpreted as a contribution to change coming from differential “rates” of change in the two prior categories. When the node category shows a greater shift to the dependent category, the term will be positive; when the node category shows a lesser shift, it will be negative; when the changes are equal, Ad will be zero and the branch will disappear. To illustrate, let us analyze the election-to-election values in Table 10 (see Tables 16 and 17). Reading Table 17 from left to right: (1) The column headed “Demographic” tests the so-called “bourgeoisification” hypothesis that socioeconomic upgrading generates recruits for conservative parties. The hypothesis appears correct in that all the signs are negative. The increasing proportion White Collar and the White Collar reluctance to vote Democratic have indeed eroded the Democratic vote. TABLE 16 Modeled Results for Data in Table 10 Year

K

Lw

X,

Ax

d

M

1948 1952 1956 1960 1964 I%8 1972

.754” .513a .429” .541 ,746” .469 .310”

-.241 - ,084 +.112 +.205 -.277 -.159

.456 .477 .498 ,519 ,539 .560 ,581

+.021 +.021 +.021 +.021 +.021 +.021

- 440” -.244 - .082” -.I64 -.I24 -.084 -.04l

+.I96 +.162 -.082 +.040 +.040 +.040

o Values are raw data and significant outlyers from regression predictions. All others are regression estimates as explained in text.

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TABLE 17 Modeled Change in Democratic Vote, Election to Election

Period 1948-52 1952-56 1956-60 l%O-64 1964-68 1968-72

Demographic W*dJ - .005 -.002 -.003 -.003 -.002 -.OOl

AY

Relative shift w, * Ml

Residual AK

Total

Raw data”

+ ,089 +.007 -.041 +.021 +.022 + .022

-.241 -.084 +.112 + .205 -.277 -.I59

-.I57 - .009 + .068 +.223 -.257 -.138

-.116 - .024 +.115 +.141 -.232 -.114

a Discrepancies between total and raw data are due to discrepancies between regression estimates and raw data results.

However, the magnitudes are small-the total effect for 1948-1972 being just .016 (but see the comments just prior to footnote 5)-and the values are declining as the SES association withers. (2) The second column, relative shift, shows larger effects, positive in sign except for one election. Thus, except for 1956-1960 (EisenhowerStevenson vs Kennedy-Nixon), the differential rates of change for White and Blue Collar Voters have favored the Democrats. In the four elections where the Blue Collar Voters have moved toward the Republicans, the White Collar voters have gone less far. In one of the two elections where the Blue Collar Vote shifted toward the Democrats, the White Collar vote moved even farther. Of course, as Eqs. (10 and 11) tell us, this is logically equivalent to saying, in five of the six adjacent elections, the association between SES and Vote declined. Here the later interpretation “feels better,” but frequently the analyst will find the equivalent “relative shift” view useful. (3) The third column, Residual, shows relatively large values. SES differences do not go very far in accounting for trends in presidential voting. [What would happen if we reversed our base category from Blue Collar to White? Not much: AX, dZ, and M would remain the same, while X, would shift to (1.00 - XJ. IfX has a split very far from 50-50, choosing the larger category as the node will decrease the residual a bit (if d is changing, not otherwise). Since there are other good reasons for doing so, a good conservative rule is: Assign the larger of the two categories ofX as the base, the smaller as the node.] The approach is easily extended to three or more variables, as illustrated in Fig. 7. Following conventional tlow graph principles, the total change in 2 equals:

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Component

Type

Value

Demographic Changing level of Y, net of X Changing level of X Direct Via Y Differential

LN, * h, Ax * c2 AX*u,*b,

shift in Y Categories of Y Categories of X

Mixed Residual not explained by model

Y, * Lib x, * AC X, * h

* b,

a2

Figure 7 can also be generalized to polytomous variables using principles explained in the technical references above. In sum, when coefficients change over time, the system becomes more complex (and often more interesting), but the same flow graph and statistical principles may be applied so that change in a dependent variable can be decomposed into causally plausible components and a residual term indicating the extent to which variables in the system account for the changes.

FIG. 7. Flow graph for three variables with for cases in base categories at X and Y. Lower

changing coefficients. AKz is the shift in 2 case letters are d coefficients.

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A NOTE ON PANEL ANALYSIS

If the sociologist’s attitude toward longitudinal research is one of religious faith, our feeling toward panels-longitudinal studies in which the same respondents are studied in all waves-approaches superstitious reverence. If longitudinal studies are Good, panel studies must be Wonderful. Why? The question has been argued for 30 years (Lazarsfeld, 1946; Heise, 1970; Kessler, 1976) and has generated a vast, if not always intelligible, literature. Such a hardy topic is impossible to cover briefly, but it may be useful to consider panel designs in the light of the approach presented above. Let me begin with a bit of iconoclasm. First, I am not much impressed by the utility of panel analysis for establishing causal direction. Here the definitive paper seems to be Heise’s (1970). Casual reading (if the term applies) might suggest Heise is optimistic about establishing directions from panel designs. However, his argument requires that one know the causal lug exactly. Causal lag (Heise, 1975, chap. 6) is the time required for influence to occur. If weather influences one’s mood, the influence probably occurs the same day; but if a college degree influences income, the effect cannot be detected the evening of commencement. Heise does indeed show that if we know the lag and if the panel waves occur close to “one lag apart” we can establish direction from panel data by comparing the two “crosslagged correlations.” But how likely is it we would know lag but not causal direction (e.g., not know whether Marijuana usage produces Depression or Depression produces Marijuana usage, but knowing whichever it is, it takes 10 days to 2 weeks)? Extremely unlikely, in my opinion. Thus, to use panel design to discover causal direction requires information that, for all practical purposes, depends on knowing causal direction. And the assumption is not merely technical. Consider, for example, the Kessler study in Table 8. His original data (Kessler, 1976, Table 5) are in panel form (see Table 18). TABLE 18 Panel Data on Marijuana and Depression” Spring 1972 Marijuana Fall 1971 Marijuana No No Yes Yes a N = 4785.

Depression No Yes No Yes

No

Yes

No

Yes

No

Yes

1586 482 77 45

401 1019 15 78

135 58 243 110

44 159 78 255

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This is the classic “16-fold table” whose cabalistic properties have obsessed a generation of survey methodologists. When Table 18 is analyzed as a d system, we get the flow graph in Fig. 8. (Numbers in parentheses are two (T confidence intervals. Footnote 1 explains why confidence intervals for Marijuana, and Depression, differ from previous tables .) Both cross-lagged associations (Marijuana, to Depression, and Depression, to Marijuana,) are small and insignificant. Does this mean there is no causal relationship between the variables? I doubt it, since we get significant associations between them in both waves. More likely, in my opinion, the causal lag is very short and the effect wears off fast. Presumably with a shorter lag something would turn up, but how short? A month? A week? A day? Six hours? It would take an elaborate program of research to nail it down, and we can hardly begin the research unless we already know quite a bit about the relationship. (Thus, for example, I assumed the lag was too long, not too short, because I have some ideas about the causal relationship already.) The lag problem is equally serious in routine, cross-sectional designs. Consider two hypothetical items, “Are you drinking (an alcoholic beverage) at this instant?” Do you have a hangover?” In a cross-sectional design the two measures would probably show a very strong negative correlation which would undoubtedly lead policy makers to offer free drinks as a method to cut down the number of hangovers. Despite the truly frightening possibilities for error, social scientists have paid virtually no attention to lag problems except as assumptions to justify simpler mathematics (but see Heise, 1975, chap. 6). If you want the testimony of a reformed sinner, I had never given the problem 10 min thought until I was assigned this essay.

FIG.

8.

Flow

graph

for data in Table

18. K values

are eliminated

for simplicity

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In sum, correct assumptions about causal lags are necessary whether or not one is attempting to determine causal direction. Before interpreting a cross-sectional or panel association between X and Y as an estimate of causal effect, one should go beyond the problem of specifying direction to ask: (1) Might the effect have so long a lag (incubation period) the Y scores have not yet come into line? (e.g., Education and Earnings in a sample of young people). (2) Might the effect, once established, be so persistent the Y scores are lined up with earlier values ofX, not the current ones? (e.g., Parental Income and respondent’s earnings in a sample of mature adults). Consider, for example, Mellinger, Somers, Davidson, and Manheimer (1975)) who use drug measures in 1970 to predict college students’s grades and vocational decisions in 1973. This perfectly sensible design actually invokes three rather strong assumptions, which cannot be tested with the data. First, the investigators assume Drug use affects Grades and Decisions, not vice versa or both. Otherwise, any causal (as contrasted to forecasting) interpretation is unwarranted. To me, the assumption is not obvious. Second, they assume (reasonably, in my opinion) one doesn’t have to wait a decade to see the havoc wreaked by marijuana. Third, they assume 1973 Grades and Decisions are not reacting to (lined up with) levels of drug use coming after and different in amount from those of 1970. (If individual levels of drug use do not change from 1970 to 1973, this would not be a problem.) Assumption three, like assumption one, does not seem irresistable to me. Moving on to the second heretical opinion: Of logical necessity, panel design is of no special help when one is interested in the problems discussed in this paper. If you merely want to know (a) whether a variable has changed, (b) what the relationship between variables is at various times, (c) whether the relationship is stable, and (d) whether changes in a prior variable account for changes in a dependent variable-longitudinal studies of samples from the same universe are required, but panel design is not mandatory. Third, I think the statistical power of panel designs may be overestimated. We are frequently told panels are more efficient (more likely to detect significant differences) since cases serve as their own controls. That is, for a nonchanging prior variable (e.g., ethnicity) which is correlated with a dependent variable (e.g., drinking) AX will be zero. When such variables do not change, the residual variation in the dependent variable will be lessened. True indeed, but (a) the improvement will only be detected by significance tests using residual variance, i.e., F tests. “Nonparametric” tests involving proportions, binomials, likelihood ratios, etc. commonly used by sociologists simply will not notice the

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improvement, (b) in causal models involving “demographic characteristics,” one cannot estimate AX, and (c) N is cut in half. If we measure 500 cases twice, N is 500, not 1000. (Consider how reliable our estimate of the United States sex ratio would be if we randomly select one case and measure its sex every day for 500 days.) Since all statistical significance tests depend on N in one way or another, the last point can be telling. Sampling design is so technical an expert must be consulted before a plan is chosen, and it may well be that a panel design will be optimal, but the investigator should not blithely assume panel design has miraculous powers to produce statistical significance when compared with successive independent samples. It is time to ask what panel designs are good for. One answer has been given already. When investigating variables with a considerable lag (e.g., drug use in the freshman year and career plans during the senior year in Mellinger et al., 1975) panel design eliminates memory distortion. If data had been collected only during the freshman year, senior outcomes would be unknown; if data had been collected only during the senior year, memories of earlier drug usage might be distorted; if data had been collected on different samples at two times, one could not examine the relationship between drug usage and later outcomes. Here a panel design is an obvious advantage. A second advantage of panel design is that it allows one to study respondent change, which is not exactly the same thing as studying changes among the variables. The two “stability coefficients” in Fig. 8 (+ .475 for Depression and + .658 for Marijuana) illustrate. Such numbers simply cannot be estimated from successive crosssections. A more revealing specification for studying respondent change is to treat change as a dependent variable. Returning to Marijuana and Depression, suppose one is interested in the hypothesis that Marijuana usage promotes emotional instability, not Depression per se. Table 18 can be easily rearranged so that the second wave measures are defined as change (i.e., a No who stays No and a Yes who stays Yes are nonchangers; Noes who shift to Yes and Yeses who switch to No are Changers). Table 19 illustrates. The data can be treated as four ordered variables and analyzed as a d system (Davis, 1975a). Since none of the interactions are statistically significant, the results can be presented in the flow graph in Fig. 9. Reading Fig. 9 from left to right: The + .I00 for Marijuana and Depression is the same old Time, association analyzed above. 1 The +. 133 from Marijuana, to Marijuana change says Marijuana users were more likely to change (stop) than nonusers were to change (start), whether or not they were Depressed at Time,.

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TABLE 19 Values in Table 18 Rearranged to Treat Change as Dependent” Time, Marijuana No No No No Yes Yes Yes Yes

Change in depression Change in marijuana

Depression No No Yes Yes No No Yes Yes

No Yes No Yes No Yes No Yes

No

Yes

1586 135 1019 159 243 77 255 78

401 44 482 58 78 15 110 45

n N = 4785.

(Why then did Marijuana usage go up? Because there were many more nonusers at Time,.) The +.042 from Depression, to Marijuana change says there is a significant tendency for Depressed students to change Marijuana usage (go from No to Yes or Yes to No). The tests here show no interaction (p = .735), which is important in such analyses. If significant interactions were present we would conclude the effect varied with initial Marijuana usage (e.g., hypothetically, “Depressives are more likely to start smoking than nondepressives, but not more likely to stop.“)

FIG. 9. Flow graph for data in Table 19. K values are deleted for simplicity. Values in parentheses are two (T confidence intervals.

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JAMES A. DAVIS

The +. 106 from Depression to Depression Change says that Depressives were more likely to cheer up than Nondepressives to go into a blue funk. Again interactions are important. Sincep = .201, we can say the differential stability of the two mental states is about the same in each possible Marijuana situation. Finally, we see there is virtually no association between Time, Marijuana and Change in Depression or between Marijuana Change and Depression Change. Our hypothesis came out badly. Marijuana smoking does not seem to promote emotional instability-but panel data are required to test the proposition. Data in a panel design (16-fold table) may be specified in a third way, taking Time, values as dependent on changes, e.g., is change in mood correlated with marijuana usage at Time,? This specification is just a rescrambling of Table 19, so we will not discuss it. Such specifications, while fascinating, are very sensitive to spurious results stemming from measurement error, i.e., categorical analogs to the “regression fallacy.” Assuming some true and unchanging proportion, P, and the same probability of misclassification for cases in each category of P, it is easy to show that the category with the smaller true proportion will appear to have a larger proportion of changes strictly as a function of independent, successive errors in classification. In sum, panel data (1) are required when one is studying respondent stability or to avoid recall in lagged relationships, (2) have certain technical advantages in design but extract a terrible price in terms of N, and (3) are unlikely in most research to be helpful in establishing causal directions.

REFERENCES Abramson, P. R. (1975), Generational Change in American Politics, Heath, Lexington, Mass. Alford, R. R. (1%7), “Class voting in the Anglo-American political systems,” in Party Systems and Voter Alignments: Cross-National Perspectives (S. M. Lipset and S. Rokkan, Ed,..), pp. 67-93, Free Press, New York. Converse, P. E. (1958), “The shifting role of class in political attitudes and behavior,” in Readings in Social Psychology (E. E. Maccoby, T. M. Newcomb, and E. L. Hartley, ’ Eds.), pp. 388-399, Holt, Rinehart & Winston, New York. Davis, J. A. (1975a), “Analysing contingency tables with linear flow graphs: D Systems,” in Sociological Methodology 1976, (D. Heise, Ed.), pp. 11 I-145, Jossey-Bass, San Francisco. Davis, J. A. (1975b), “Communism, conformity, cohorts, and categories: American tolerance in 1954 and 1972-3,” American Journal of Sociology 81, 491-513. Davis, J. A. (1975c), “Subjective social class, party identification and presidential vote, 1952-1974,” British Sociological Association Quantitative Sociology Newsletter No. 15. 8-37.

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Davis, J. A. (1975d), Does Economic Growth Improve the Human Lot? Yes Indeed, About .0005 Per Year, paper presented at The International Conference on Subjective Indicators of Quality of Life, Fitzwilliam College, Cambridge, England, September 8- 11, 1975, NORC (litho). Davis, J. A., and Schooler, S. R. (1974), “Nonparametric path analysis-The multivariate structure of dichotomous data when using the odds ratio or Yule’s Q,” Social Science Research 3, 267-297. Glenn, N. D. (1973) “Class and party support in the United States: Recent and emerging trends,” Public Opinion Quarterly 37, l-20. Goodman, L. (1%3), “On methods for comparing contingency tables,” Journal Royal Statistical Society, Series A, 126, 97. Heise, D. R. (1970), “Causal influence from panel data, ” in Sociological Methodology 1970 (E. F. Borgatta and G. W. Bohmstedt, Eds.), pp. 3-27, Jossey-Bass, San Francisco. Heise, D. R. (1975) Causal Analysis, Wiley, New York. Huggins, W. H., and Entwisle, D. (1968), Introductory Systems and Design, Blaisdell, Waltham, Mass. Kessler, R. C. (1976), Rethinking the 16 Fold Table Problem, paper presented at the 1976 Eastern Sociological Meetings, Boston, March 27, 1975. Land, K. C., and Felson, M. (In press), “A general framework for building dynamic macro social indicator models.” American Journal of Sociology. 82, 565-620. Lazarsfeld, P. F. (1946), “The use of panels in social research,” Proceedings of the American Philosophical Society 92, 405-410. Mayer, T. F., and Amey, W. R. (1974), “Spectral analysis and the study of social change,” in Sociological Methodology 1973-1974 (H. L. Costner, Ed.), pp. 309-355, JosseyBass, San Francisco. Mellinger, G. D., Somers, R. H., Davidson, S. T., and Manheimer, D. I. (1975), Drug Use, Academic Performance and Career Indecision: Longitudinal Data in Search of a Model, Institute for Research in Social Behavior, Berkeley, California (litho). Roizen, R., Cahalan, D., and Shanks, P. (1976), “Spontaneous remission” among untreated problem drinkers, paper prepared for presentation at Conference on Strategies of Longitudinal Research on Drug Abuse, San Juan, Puerto Rico, April 7-9, 1976 (litho). Stinchcombe, A. L. (1968), Constructing Social Theories, Harcourt, Brace, Jovanovich, New York. Taylor, D. G. (1975), Procedures for Evaluating Trends in Qualitative Indicators, NORC (litho).