Notes on the Case-Study and the Unique Case
University of Chicago
A perennial controversy in the social and psychological sciences is that between advocates of statistics and advocates of the case-study method. Although nobody disputes the importance of the case-study as a procedure for getting new ideas, some statisticians unfortunately fail to recognize adequately the importance of the case-study as a procedure for making direct predictions about the behavior of an individual. This lack of recognition may be due partly to the fact that those who have undergone a rigorous discipline in quantitative methods develop a bias against the loose and sometimes pretentious vocabulary so often used by exponents of the case-study, as well as against the tendency of some of these exponents to lay claims to certainty of prediction which are unsupported by either logic or empirical evidence. The same charge of making extravagant claims is laid at the door of the statisticians by experts in the use of the case-method.
Perhaps we can get to the heart of the difficulty if we examine the concept of the unique case. The case-study procedure makes a prediction, it is said, by "analyzing the unique dynamic configuration of traits within the individual."
WHAT IS A CONFIGURATION?
Suppose that we want to predict success in some activity for a given individual named Smith. For simplicity, let us assume that we can rate Smith as of two separate time periods, with respect to three behavior items, each of which is described in four mutually exclusive categories, qualitative or quantitative, as follows:
|Time t||Time t’|
Let the underlined symbols indicate the ratings for Smith. They tell us that at time t his trait configuration was a2b1c4; at time t' it was a'3b'1c'3.
Although a2b1c4 are reported as of a single time interval t, the pattern actually may represent sequences within that time interval, e.g., a2b1 may be followed by c4. Similarly, within the time interval t', a'1b'1 may be followed by c'3. The notation here is intended to be quite general. For example, a2b1 may represent Smith's report that b1 is an "effect" of a2, or may represent the investigator's inference that b1 is an "effect" of a2. It must be remembered that such a report or recorded inference, even if subject to error, may be a datum which can be treated either by the statistician or the case history investigator just like any other item for predicting Smith's success in an activity. An excellent discussion of this point, with many practical suggestions for research, appears in a manuscript by Paul F. Lazarsfeld not yet published, entitled, "The Art of Asking Why."
Consider a single cross-section in time, for example, time t. The number of possible static configurations of the kind a2b1c4 is 43 = 64. If ten-traits were to be considered simultaneously the number of configurations would be 410= 1, 048, 576.
Next, consider an individual's rating on a single trait in the two time periods as constituting a simple dynamic configuration. Smith's simple dynamic configuration with respect to trait a would be a2a'3. The number of possible simple dynamic configurations with respect to any specified trait is 42 = 16.
Finally, let us consider Smith's ratings on all three traits in both time periods as constituting a single complex dynamic configuration. Smith's complex dynamic configuration is a2a'3b3b'1c4c'3. Such a configuration might be illustrated by the fragment of a case history. Before Smith was married he got into fast company (a2) and drank heavily (b1) and, perhaps because of drink, had great difficulty holding a job (c4). Since marriage he has given up the fast company (a'3) although he still drinks too much (b'1), and, consequently perhaps, has been having some, though less, difficulty in holding his job (c3). The number of possible complex dynamic configurations of this type (43)2 = 642 = 4096. If there were ten traits, the number of possible complex dynamic configurations would be (410)2 = (1,048,576)2 = 1,099,511,627,776.
With such an astronomically large number of different complex dynamic configurations possible from a relatively
( 351) small number of trait categories and with only two time periods, it is evident how easy it is to make classifications which put every individual in the world in a different configuration. (This is the general principle, of course, by which a small number of traits are used, in configurational analysis, to identify an individual from his fingerprints.)
Now, our problem is, knowing Smith's pattern a2a'3, b1b'1, c4c'3, etc., how can we predict his success or failure in a given activity?
It will be instructive to compare and contrast the approaches by the statistician and case-study investigator, respectively, to this problem. Consider, first, the statistician:
HOW THE STATISTICIAN PREDICTS
If the number of possible configurations is small and the sample of individuals is very large, such that numerous individuals are characterized by a common configuration, the statistician's task is simple. All he needs to do is to observe the proportion of successes among those characterized by each given configuration and to make a direct actuarial prediction. For example, for trait a there are only 16 possible simple dynamic configurations involving one subcategory in each of the two time periods. Let us assume that out of 100 individuals with a2a'3, 80 succeed in a given activity and 20 fail. Then the best actuarial prediction which can be made for Smith on the basis of trait a alone is that he will succeed.. The statistician would expect to be correct about 80 percent of the time on such predictions and wrong about 20 percent of the time, the exact percentage of correct predictions being subject to sampling error.
Direct applications of this method breaks down, however, when the number of possible configurations becomes so large that no sample is large enough to provide an experience table. Thus, even a very large sample may not yield a single example of an individual who is characterized by a particular complex dynamic configuration, out of a possible 4096, such as a2a'3b1b'1c4c'3. Hence, when confronted with Smith, who is characterized by this pattern, the statistician ordinarily is helpless unless he can recombine various configurations in such a way as to get a small number of groups in which a sufficient number of individuals will fall.
There are a great many methods by which the statistician can make these recombinations. Only a few examples will be mentioned.
One method might be to make a direct typological reduction. Thus he might
reduce the number of configurations by cutting each trait to two categories:
|Time t||Time t’|
This yields the investigator only 64 types of complex dynamic configurations with which to work. Or he might simplify still further by treating B1C1 and B2C1 as members of the same class (BC), and B1C2 and B2C2 as members of another class (BC)2 in time t, and by using similar groupings for time t'. Then he gets
|Time t||Time t’|
or only 16 types of complex dynamic configurations. It will be noted that the relationship- of the type A1A'1 (BC)1(BC)'1 has precisely the same general structure as the relationship from which it was derived, except for a sacrifice of detail. With only 16 types, the statistician is able to observe enough individuals in each type to acquire data for direct actuarial prediction. Much information about Smith was lost in order to make this prediction possible. A factor analysis of some sort might provide a method of typological reduction with a minimum loss of information.
A different procedure, relatively new to most statisticians, is the use of the discriminant function, by which a frequency distribution of prediction scores is worked out for the successful individuals in the trial sample and another frequency distribution of prediction scores is worked out for the unsuccessful individuals. The method would assign weights to the subcategories of traits at each time period in such a way as to maximize the difference between the means of the two frequency distributions. By referring to these two distributions, the likelihood that a person who possesses a given prediction score will be successful can be determined. If Smith has that score, an actuarial forecast can be made for him on the basis of the sample experience.
Much more common is the statistical procedure of assigning arbitrary or item-analysis weights to each subcategory
( 353) and computing by simple addition a prediction score for each individual in the trial sample. All individuals whose prediction scores fall in the same class interval are treated alike, regardless of the different configurations which they may possess. (The time factor might be handled in various ways, for example, by treating the direction of change in each item as a separate item with its own set of weights.) If 75 out of 100 whose prediction scores fall in the same class interval succeeded, and if Smith's prediction score falls into that class interval, the usual actuarial prediction can be made. This is the method used in parole prediction.
These statistical procedures have in common the operation involving a sacrifice of information about the individual configurations. This sacrifice is made in order to obtain broad enough classes to assure an adequate number of cases from the trial sample in each class. It is an artifact to treat thousands of different configurations as belonging to a single class, supposedly homogeneous. If the classes are too broad and too heterogeneous, the statistician will make many bad guesses.
HOW THE CASE-STUDY INVESTIGATOR PREDICTS
Let us turn now to the case-study investigator who is confronted with a dynamic configuration a2a'3b1b'1c4c'3 for Smith. Like the statistician, he may possess no experience about the success and failure of others in this one out of thousands of possible configurations.
He may, and frequently actually does, operate much as the statistician operates, by making typological reduction; or by mentally assigning informal predictive weights on various traits separately, which weights he combines in some informal way. If he lacks the numerical information from the trial sample which the statistician requires, he makes up for the lack by assuming some value out of his general experience. The outcome is an actuarial prediction, of necessity less accurate on the average than would have been possible if numerical experience tables were available.
While this description perhaps characterizes fairly well certain of the operations performed by the case-study investigator, it is far from adequate. Statisticians who criticize the case method because of pretentious claims made for its accuracy are likely to be thinking of such a description or are overlooking another operation which is also frequently performed.
The additional operation is one which the case-study investigator can perform because of the extreme flexibility available to him as contrasted with the rigidity of the statistical framework. This flexibility enables the case-study investigator to economize by seeking out a limited number of traits which seem important for Smith and by analyzing them intensively while ignoring other traits. True, the statistician can and does ignore traits (one way he does it is to give the traits zero values for Smith). But the case-study investigator can do what the statistician cannot do, namely, concentrate on an intensive, detailed, free-flowing analysis of the configuration of the limited number of traits which he thinks is important in Smith. Moreover, he is not bound by any prior list of traits but can add others freely. Thus, he might decide to ignore traits b and b' and c and c' entirely for Smith, but consider d and d'. Or he might get a detailed developmental history of Smith, introducing observations on additional time periods, represented by a" and d" in time t" and by a"' and d"' in time t"'. Thus Smith's complex dynamic configuration may be represented by his behavior over several time intervals. In the first time interval the contribution to the configuration may be a2d1, in the second a'3d'2, in the third a''3d''2, and in the fourth a'''3d'''3.
The investigator must now predict Smith's success. If he is familiar with other configurations somewhat like Smith's and knows how they turned out, he will apply this knowledge to Smith. Such a prediction is, of course, not only implicitly actuarial but is likely to be subject to gross error. If the investigator is not familiar with any other configurations remotely resembling Smith's he probably is as helpless as the statistician in the same situation, unless--note the reservation--unless the time series on Smith alone is of a very special kind. Specifically, the sequence of observations on Smith must contain information on Smith's success at one or more points of time in activities resembling that for which a prediction is to be made. Thus, if at time t, when Smith's behavior pattern or trait pattern was a2d 1, he was a success in an activity similar to that being predicted, and if at time t' , when his pattern was a'3d'2, he also was a success--in fact, if he always seemed to be a success--there ordinarily would be considerable confidence that he would succeed again. It should be carefully noted, however, that, even in this favorable circumstance, prediction can not be made with complete certainty, for the following reasons:
(1) The assumption that the new situation is analogous is subject to various kinds of error.
(2) The assumption (often not explicit, but necessarily present) that it is the general rule that people who succeed in past situations will continue to succeed, does not refer to an invariable sociological law, because none has been demonstrated. It refers rather to a hypothesis, which, if carefully studied, would be found to have exceptions--hence, to be a "law" of an actuarial character.
If the second point above be questioned, and there may be some who would question it, one can make the point clearer, perhaps, by another example. Suppose, when Smith had configuration a1d1 at time t, that he failed in an activity, as also when he had the configuration a'3d'2 at time t'. But suppose he was successful at t" and t'''. The prediction may now be that he will succeed, on the implied, if not specified, assumption that people who are improving will succeed in the future. The language of common sense is full of such usages, sometimes making an explicit recognition of the general relationship, as when we say "I think he'll grow out of those bad habits as he gets older; folks usually do."
One reason why some case-study investigators have difficulty recognizing the point that a projection of an intraindividual trend implies a reference to a general relationship is that they sometimes phrase their forecast in more complicated language than that used above. For example, they do not say simply, "Smith is improving; therefore he will succeed," but they say, "Smith has been improving, and he now is characterized by a behavior pattern a3'''d3''', which makes for success, as compared with his earlier behavior pattern a2d1, which made for failure." But how does the investigator know that "a3" d3" makes for success" and that "ad, makes for failure"? He can not know, unless he has had experience with persons whose configuration was similar to a3''' d3''' and to a2d1. If he is using that experience, he is making an explicit actuarial reference, and without adequate data if these configurations are rare.
Even when the prediction for Smith is made solely on his individual time sequence, the ultimate test is a statistical one and can be made empirically. The test is not for Smith as an individual but as a random member of a class of individuals concerning whom the investigator makes predictions with about the same feeling of confidence. If the investigator has felt about the same confidence in 50 previous cases as in Smith's case and has been correct 40 times, the best actuarial forecast is that he is correct about Smith, but, of course, Smith may become one of his mistakes and this can not be known in advance.
Let us summarize the operation on Smith of the case-study investigator. Many times his operations are the same as those of the statistician--for example, he may use typological reduction or he may relate certain traits of Smith to his past experience with other cases. Unlike the statistician, he is free to concentrate intensively on variates or configurations he thinks important in Smith and to explore, at will, especially taking advantage of cues from repetitions or sequences in Smith's success and failure experience over time. If Smith's dynamic configuration can not be compared in whole or in part with that of other individuals whose success or failure is known, the investigator must obtain this special kind of time sequence-record of success and failure--or he is helpless. Even if the investigator secures this unique time sequence of success and failure of Smith, the ultimate forecast depends in part on the correctness of whatever general theory is implicit in the projection of the trend.
Although the trend is to replace many case-study operations by quantitative techniques easy to administer, especially when prediction must be made quickly for a large number of individuals, the case-study is likely to continue to be a useful-often indispensable--supplement to the work of the statistician, even in situations where the value of the statistician's methods is most obvious. If the case method were not effective, life insurance companies hardly would use it as they do in supplementing their actuarial tables by a medical examination of the applicant in order to narrow their risks. Its great virtue in direct prediction is its flexibility, permitting an intensive study of the configuration of selected factors in a time setting.
No detailed comments are needed about the important contribution of the case-study procedure as a fruitful source of new ideas, which can be eventually set up as hypotheses for explicit statistical checking. Few statisticians, if any, will dispute that function of the case-study. In conclusion, however, at least one point should be made to indicate how one of the principal present advantages of the case method could also be better utilized by the statistician. The case method, as has been emphasized, often relies heavily on information about an individual's record of success or failure in situations analogous to that about which a prediction is to be made. There is no reason, of course, why such a record should not be an explicit
( 357) entry as one of the predictive items in a formal statistical analysis. Sometimes it is. In parole studies, past record of recidivism becomes one of the most powerful predictive items. The case-study has no monopoly on the use of such time trends, and still greater use of such information, together with that of typologies which are dynamic in that they indicate sequences or connections in time, should improve statistical prediction.
The statistician and the case-study investigator can make mutual gains if they will quit quarreling with each other and begin borrowing from each other.