When the court has a party, how many "friends" show up? A note on the statistical distribution of amicus brief filings.

• Author: Farber, Daniel A.
• Position: Symposium: The Rehnquist Court in Empirical and Statistical Retrospective

1. INTRODUCTION

   Amicus briefs have been a significant subject of empirical research. (1) Researchers have also used the number of amicus briefs filed in a case as a measure of the case's importance. (2) Lawyers also have reason to be interested in amicus briefs, which have gone from being exceptional to being the norm in Supreme Court cases. (3) This study addresses some basic questions about amicus filings: what is the distribution of this measure of case importance? How does it relate to other measures of importance such as a case's citation rates?

   Because this study is an outgrowth of a previous one, (4) it may be helpful to begin with a summary of the previous re search. In that study, using citation data for the Supreme Court's 1984 and 1990 Terms, I examined patterns of citation frequency in order to test three models:

   * Under the first model, the extent of an opinion's contribution to the law (and thereby its influence) is determined by a host of independent factors. This model produces a bell-shaped distribution of "step lengths," ranging from baby to giant steps. The data did not support this model.

   * Under the second model, judges have bounded rationality and strong attachments to existing rules, leading them to take "baby steps" most of the time but occasional "giant steps" when continued adherence to an existing norm proves untenable. In empirical studies by various social scientists, this kind of model has been found to produce frequency distributions that are roughly normal but have a characteristic known as "leptokurtosis." The data also failed to support this model: the degree of leptokurtosis was much too high, so that the curves were far from the normal distribution.

   * The third model stems from complexity theory (also known as chaos theory or fractal geometry.) This type of model applies to many dynamic processes--for example, it fits the frequency distribution of earthquakes. This model was supported by the data, explaining most of the variance in the data (with [R.sup.2] over .80 for both of the Terms I studied) (5)

   The current study tests these models (along with one additional one) in the context of amicus brief filings.

   During the earlier study I became intrigued by the apparent divergences between the number of times a case was cited in later court rulings versus other citations (primarily law reviews). A regression analysis (6) showed that my impression was correct. Although an increase in the number of case citations did predict a higher average number of noncase citations, almost none of the variance was explained (with an [R.sup.2] of only .07). (7) When I divided the ten most heavily cited cases into two groups, based on the proportion of judicial versus non-judicial citations, the difference between the groups was striking. Of the five cases most frequently cited by courts, all but one dealt with a procedural issue, and the exception dealt with ERISA preemption. (8) The five cases most frequently cited in law reviewers involved more socially salient issues such as discrimination. (9) In short, the courts seemed most keenly interested in procedure, while the commentators were drawn to cases with quasi-constitutional overtones. (10)

   The two primary results of the current study regarding the frequency distribution of amicus briefs are as follows. First, a power law distribution does provide improved fit (over linear regression), but less strikingly than for citation frequencies. An exponential distribution is a slightly better fit and might well be preferred. Both distributions leave significant unexplained variance. The broader implication is that the number of briefs filed in a case probably depends on a fairly complex set of frictions and feedback loops. It is plausible to assume that the same probably holds true of other types of efforts to influence government decisionmakers (i.e., lobbying).

   Second, amicus brief filings are unrelated to the number of federal appellate citations received by an opinion, but are modestly related to the number of law review citations. We should be cautious about using any one of these measures as the sole gauge of an opinion's importance. There seem to be distinct dimensions of case importance, an issue that deserves further investigation in its own right.

1. POSSIBLE MODELS

   We do not have a systematic understanding of the process that leads to the filing of amicus briefs. (11) A 1993 study revealed several important factors. First, many amici are repeat players. (12) Second, surveys of amici suggest that the most important factor is the perceived relevance of the case to the organization's goals, followed by the quality of the case as a legal vehicle. (13) Third, filing an amicus brief is costly--ranging from $8,000 to about $20,000. The average organization surveyed had an interest in about sixteen cases; for half the organizations, participating in that number of cases would have exceeded the organization's entire litigation budget. (14) Fourth, economic interest groups filed an increasing share of the amicus briefs. (15)

   We also know that individual organizations do not make filing decisions in a vacuum. State governments have formed a network that results in concerted filing activities:

   During the 1990 Term, for example, in cases in which at least one state filed a friend-of-the-court brief, the average number of other states participating was 15.7. In only three cases did a state participate as amicus curiae without the support of others. (16) Or consider the following advice to lawyers with cases before the Supreme Court: "[i]n today's world, effective representation of your client requires that you at least seriously explore the possibility of enlisting persuasive amicus support on your client's behalf." (17)

   One obvious motive for filing an amicus brief is to influence the result in a case. It is unclear how effective briefs actually are in this regard, particularly if we exclude those filed by the Solicitor General on behalf of the United States. The authors of the most recent and thorough empirical study report that "amicus briefs supporting respondents enjoy higher success rates than do amicus briefs supporting petitioners; that small disparities of one or two briefs for one side with no briefs on the other side may translate into higher success rates but larger disparities do not...." (18) Notably, the number of cases filed tends to be similar on both sides of the case. (19) A final relevant fact: the Court itself does not serve a gatekeeper function; it routinely approves filing of briefs in cases where the parties themselves fail to consent.

   In the absence of a strong theory for predicting filing, we may turn to more general models as a source of guidance. One possibility is that the number of amicus briefs filed in a given case is more or less random--that is, that it is the product of unrelated factors operating in different directions, which happen to balance out one way or another in a particular case. Trying to identify and measure these various factors would be difficult. But, it turns out, we may be able to identify this kind of randomness without specifying the causal links. A basic theorem of mathematical statistics links this form of randomness with the famous bell-shaped, normal distribution. More precisely, the central limit theorem states that "the sum of a large number of independent random variables will be approximately normally distributed almost regardless of their individual distributions; any random variable which can be regarded as the sum of a large number of small, independent contributions is thus likely to follow the normal distribution approximately." (20)

   We could not expect an exact correspondence between citation data and the normal distribution, if only because the normal distribution requires an infinite domain in both directions while the number of citations to an opinion cannot be a negative number. In assessing deviations from normality, a few parameters are especially useful. For later reference, here is a list:

   Central The mean, the median, and the mode of a Tendency normal distribution are the same. Skew A normal curve is symmetrical rather than being skewed in either direction. Symmetry is measured by the skew parameter, which is zero for the normal distribution. Kurtosis Kurtosis measures whether a curve is flattened out or unusually peaked, compared with the normal distribution. Kurtosis for the normal distribution is sometimes given as 3. (21) However, my software used a different formula, for which the normal distribution comes out at zero. Leptokurtosis in data has an important implication for decisionmaking. (22) Change data from human institutions have, in comparison to the Gaussian (normal) distribution, an excess of cases in the central peak, an excess of cases in the tails of the distribution, but a paucity of cases in the "shoulders," the area between the central peak and the tails. In terms of amicus briefs, the idea would be that most briefs get some average amount of attention from interested groups that results in filings, but there may be a tendency for attention to snowball once a case begins getting attention. The snowballing effect can lead to distribution tails that follow the third model (discussed below).

   Another variant of the second model leads to an exponential distribution. In this model, cognitive and institutional factors function only as a source of friction, essentially impeding the reaction of decisionmakers to relevant information. (23) The hypothesis is that the number of cases with N + 1 amicus briefs is a fixed fraction of the number of cases with N briefs. This results in a distribution having the form y = [be.sup.-ax] . Apart from friction, this might also reflect a diffusion process. For example, there could be a core group whose members tend to be the first to decide to file in a case, information leaks out from this group to a larger group...