Using Statistics in Patent Liability: A Case Study

Using Statistics in Patent Liability: A Case Study Chapter 13 A Primer on Economic Damages in Intellectual Property Matters Esfandiar Maasoumi, Ph.D. and Matthew G. Mercurio, Ph.D. SMU, Dallas, Texas and FTI Consulting, Inc. August 12, 2005 ABSTRACT While the use of statistics (particularly survey methods) in copyright and trademark matters continues to grow, statistics has seen far less use in patent cases. However, elementary statistics can be a powerful tool in investigation of patent liability as well. Of course, as in other fields where applied statistics are used, statistics are just as often misused. Our analysis illustrates how statistics can be used, as well as some pitfalls and potential misuses of statistics, in conceptualizing the similarity of two products, and possible solutions. I. Background The determination of whether the intellectual property owned by one party has been infringed upon by another party is usually the realm of patent lawyers and subject matter experts. One approach that has been used is to attempt to determine the degree of similarity between two products that resulted from two (or more) purportedly independent processes. There are numerous metrics and methods in mathematics and statistics for defining similarity or closeness. 1 Care must be exercised in the choice of appropriate metrics and concepts, as well as in the practical execution of measurements and tests. Otherwise, nonsensical outcomes and inferences can result, camouflaged 1 Cf., for example, Cressie, N. and Read, T. R. C. (1984). Multinomial goodness-of-fit tests. J. Roy. Statist. Soc. Ser. B, 46(3), 440-464. 1

by sophisticated technology and/or presentations. In the case study discussed in this chapter, the authors are aware of work on a matter which involved a more purely mathematical approach to these issues, namely physical measures of sameness. The product at hand is golf putters. 2 The plaintiff in this case alleged that the design for his putter (which we will hereafter refer to as the Duff Putter ) had been copied by another manufacturer and was being sold under a brand name which we will hereafter refer to as the Cramer Putter. The plaintiff s expert, Dr. M, conducted a statistical analysis of the allegedly infringing Cramer Putter as well as a sample of other putters, and concluded that it was exceedingly unlikely that the two putters in question would have appeared so similar by chance alone. The expert for the defense responded with a different statistical analysis of Dr. M s work using the same data. 3 The basic ideas may be summarized as follows: Any entity may be sufficiently well represented by a multitude of measurements on as many distinct characteristics of that entity as possible. A notion of fundamentalism in economics suggests that an object or individual may be arbitrarily well represented or summarized more precisely as the number of characteristics by which it is analyzed increases indefinitely. 4 This is intuitively appealing since, in the limit, one could include all the attributes that fully identify and distinguish an entity. For example, this notion is implicit and central to concepts and implementation of such techniques as hedonic regressions in hedonic pricing and demand analysis. In hedonic regression, a computer, for instance, is but a collection of its parts which are sold in markets and generally experience technical improvement over time. This way a true price for computers is arrived at, derived from the prices of its components. In practice only a finite number of characteristics can be measured. Assuming all agree to the adequacy of a necessarily finite number of attributes, the important task becomes one of assessing similarity of two or more objects characterized in multiple dimensions. This is a daunting task, subject to many pitfalls and assumptions. Once it is understood that measurements are typically realizations of a number of random variable (e.g., the statistical measurement of a series of average measurements), the task of assessment of equality or similarity can be assessed through multivariate statistical 2 Neither of the authors is a golfer of any measurable skill whatsoever. 3 Both of the present authors consulted on the matter discussed here. 4 Kolm, S. CH., Multi-dimensional Egalitarianism, Quarterly Journal of Economics, 1977, 91, 1-13. 2

distributions and testing. It will be seen that formal testing is both necessary and challenging for scientific implementation and to provide reliable and probative evidence. Two products or objects may be quite similar in some aspects, but quite different in others. In rare cases, similarity in a few dimensions may be sufficient evidence if the allegations have been successfully addressed to those few characteristics. II. Dr. M s Methodology Dr. M presented a set of physical measurements on three different characteristics of a sample of putters to measure his concept of similarity. These three characteristics were: Face Width Front-to-Back Length Alignment Lines By way of explanation, the Face Width of a putter is the distance across the part of the putter which is perpendicular to the line of swing and actually strikes the ball. In the picture at left, the face width is clearly shown on the upper view. Face Width FTB Length The Front-to-Back Length of a putter is the distance from the face of the putter to the back and of the putter, parallel to the line of swing. 5 In the picture at left, the front-to-back length is clearly shown on the lower overhead view of the putter head. Alignment Lines are physical or colored markings, usually parallel along the line of swing, which help the golfer to align the putt properly. In the picture below are several examples of alignment lines. 5 On many putters, particularly older models, this distance is quite small, being only the width of the putter head. On newer models, particularly those with alignment lines, this distance can be several inches. Cf. the left two putters in the alignment lines photos above. 3

Dr. M s concept of similarity is based on the probability that a randomly drawn putter s characteristics would be close to those of the allegedly infringed putter. If the estimated probability is negligibly small, the plaintiff would argue that a competing putter with similar characteristics is adjudged to be an infringing putter. Dr. M made at least the following critical assumptions about these three putter characteristics: (1) These characteristics each follow a known distribution, and that in fact that the relevant distribution is the Normal (or bell curve ) distribution across his sample of putters and (2) These three characteristics are independent of one another. As we discuss below, both of these assumptions, while expedient, are inappropriate and unsupported. Dr. M used sample statistics from these measurements across his sample of putters to compute what he referred to as a test of the probability that the two putters at issue, the Cramer Putter and the Duff Putter, would demonstrate similarity with regard to these characteristics. Specifically, Dr. M computed the probability that a randomly selected putter would exhibit measurements for these three characteristics within a small interval around the corresponding measurements for the Duff Putter. Before further detail is given that will help in practical refutations of this approach, it is helpful to note a fundamental conceptual problem with Dr. M s metric. Unlike the number of balls in a box, a\the measurement of the face width of a putter is continuous, i.e., it could have infinitely many 4

values from the shortest putter to the longest putter. In statistics, for all such continuously distributed measurements (including Normal variables) the probability of the selection of any single value in the distribution is zero. It is only intervals over which there is a probability. However, by extension, the probability that a particular measurement would fall into an arbitrarily small interval can be made to be arbitrarily small, by simply shrinking the measurement interval. Thus, a priori, Dr. M s metric is tautological under his own assumptions: A priori, no two putters would have a significant probability of having very close measurements, in any dimension/attribute! Dr. M s calculations were presented in his expert report, and are reproduced below: Table 1, M Expert Report Face Width Length Alignment Lines Duff Putter 3.4017 3.296 1.882 ± 0.1125 Cramer Putter 3.4153 3.373 1.8415 Table 1 presents measurements of the three putter characteristics selected by Dr. M to examine for the two putters at issue in this matter: Tables 2-4, Dr. M Expert Report Table 2 - Face Widths Table 3 - Front-to-Back Length Table 4 - Alignment Lines Mean 4.1761 Mean 2.0891 Mean 1.1400 Variance 0.22 Variance 0.20 Variance 0.82 Standard Deviation 0.47 Standard Deviation 0.44 Standard Deviation 0.91 Skew 0.40 Skew 0.50 Skew 0.92 Kurtosis 2.39 Kurtosis -0.47 Kurtosis 0.002 P=Probability (of) P=Probability (of) P=Probability (of) P(<3.401) 0.04834 P(<3.25) 0.994572 P(<1.7695) 0.995472 P(<3.416) 0.05166 P(<3.40) 0.998397 P(<1.9945) 0.998397 P(3.401 W 3.416) 0.00331 P(3.25 W 3.40) 0.002924 P(1.7695 W 1.9945) 0.07074 5

Tables 2, 3 and 4 present summary statistical measures on the three characteristics for the sample of putters chosen by Dr. M for his analysis. 6 Each table presents the mean and the variance (a measure of the dispersion of the data around the mean) for each characteristic. In addition, the skew (a measure of excess length in either tail of the distribution) and kurtosis (a measure of the heaviness of the tails) are also provided. The last row in Tables 2-4 provides the probability calculations alluded to above. These figures purport to show the probability that a randomly selected putter would have measurements of the characteristic in question within the particular interval Dr. M selected. It is important to note that Dr. M s choice of interval length is completely ad hoc. Dr. M offers no explanation or justification of how the selection of these particular intervals was derived. In the hypothetical distribution above, the probability that some measured quantity x falls between 40 and 60 is given by the blue shaded area. In practice, it is easier to calculate the probability that x is less than 60 and the probability that x is less than 40, and then subtract the latter from the former. Using the figures given in Tables 2-4 above, Dr. M calculates what he refers to as the probability that a randomly selected putter would exhibit measurements for all three characteristics which fall within the selected intervals as follows: Pr{Randomly selected putter has a Face Width x where 3.401 x 3.416} = (0.04834-0.05166) = 0.00331 6 Although, as we discuss later, the samples for the three characteristics are not the same. 6

Pr{Randomly selected putter has a Front-to-Back Length y where 3.25 y 3.40} = (0.994572-0.998397) = 0.002924 Pr{Randomly selected putter has alignment lines z where 1.7695 z 1.9945} = (0.995472-0.998397) = 0.070744 Pr{All three are true} = 0.00331 * 0.002924 * 0.0707=6.85 x 10-7 or one chance in 1.46 million The crux of Dr. M s report is that he believes that this result, based on his assumptions, provides sufficient evidence for one to reasonably conclude that the two putters at issue could not be so similar by chance alone, and that indeed the Cramer putter must therefore infringe on the Duff putter. Note that the present authors do not maintain that the use of elementary probability theory is inappropriate here. Elementary applied statistics, when used properly, can make a powerful impression on the triers of fact in a patent matter. But Dr. M s report is not properly grounded in the correct statistical fundamentals, and thus much of the work in his report is rendered speculative. III. The General Similarity Among Putters Dr. M states explicitly in his report that he is attempting to rebut the notion that all putters are relatively similar. To demonstrate in a simple way why Dr. M s methodology does not allow one (even in principle) to conclude putters can t be similar by chance alone or to draw any meaningful statistical inference whatsoever from the data on his ad hoc choice of putter characteristics, we utilized the data reported in the appendices to Dr. M s report. Those tables present the individual measurements of the three characteristics selected by Dr. M across the entire sample of putters. Dr. M measures the Face Width for a sample of 172 putters, the Front-to-Back Length for a sample of 23 putters, and the Alignment Lines for a sample of 13 putters. This, unfortunately, leaves a sample of only 8 putters (in addition to the Duff Putter and The Cramer putter) for which Dr. M provides all three measurements. While it is a flawed methodology, for illustrative purposes only we performed the same analysis as Dr. M except that instead of measuring the similarity between the two putters at issue in this case, 7

we applied Dr. M s approach to two other putters in the sample. Following the methodology implicit in Dr. M s analysis, we selected intervals based on the relative differences between the two putters we selected for analysis. With regard to the Face Width, we calculate the probability that these two putters would both fall into the interval between 3.6030 and 3.3845 as P = 0.064789. With regard to Front-to-Back Length, we calculate the probability that these two putters would both fall into the interval between 2.1655 and 2.2110 as P = 0.039792. With regard to Alignment Lines, we calculate the probability that these two putters would both fall into the interval between 0.3405 and 0.3030 as P = 0.010979. Using these figures, Dr. M s approach indicates that the probability that a randomly selected putter would exhibit measurements for all three characteristics which fall within the selected intervals as: Pr{All three are true} = 0.064789 * 0.039792 * 0.010979=2.83 x 10-5 or one chance in 35,330 This exercise simply shows how vacuous Dr. M s analysis is as an indication of statistical evidence of theft of intellectual property. That the two putters we just examined should show such unlikely similarity in a sample of only eight putters provides strong evidence that many putters are exceedingly similar with regard to the characteristics identified by Dr. M. Another point can be made with reference to examination of the three characteristics separately. Again, using Dr. M s methodologies underlying assumptions and based on the measurement of Face Width, it should be extremely unlikely that any two putters would have the exact same measurement for Face Width. The basis for Dr. M s underlying assumption and methodology is questionable given that golf has been played since the 1400 s and while technology has changed, we have seen no evidence, nor has Dr. M offered any, that putters Face Width have changed significantly. 7 Dr. M is assuming that no two putters should be the same in the categories he has chosen and that putters that are not copied should not have similar measurements in the categories he has chosen. In effect, Dr. M is assuming his own conclusion, that is, he assumes that in order for the dimensions of any two putters to be close one must have been copied from the other or it is a very rare statistical event to have occurred by chance. 7 Anecdotally, we understand that most putter Face Widths are approximately the same as the diameter of the cup, or about 4.5 inches. 8

But in the sample of 172 putters for which Dr. M measured Face Width, there are five pairs of putters with precisely the same Face Width, measured to one ten thousandth of an inch. In addition there is one group of three putters with exactly the same Face Width. Thus, regardless of the measurements for the other two characteristics (which Dr. M provides for only 8 putters), the fact that two putters have the exact same Face Width means that the probability of arriving at these similarities by chance approaches zero. Once again, this conclusion is predicated on Dr. M s underlying maintained hypothesis that if any two putters have similar measurements for a given attribute, it must be due to copying or be due to a very rare statistical event occurring. If Dr. M s approach is a statistically sound one, it should be infinitely unlikely that any two putters would have the exact same measurement for Face Width. Using his data, we have just demonstrated that is not the case. The data demonstrate conclusively that many putters exhibit similar measurements for these characteristics, which were chosen by the plaintiffs own expert. The test of sameness upon which Dr. M s entire report relies is in fact common to numerous putters in the sample he provides. Thus, we find Dr. M s approach completely devoid of probative value in this matter. IV. Statistical Tests As further evidence of the general similarity among many putters with no insidious motives, we used the data in Tables 1-4 of Dr. M s report to examine the profile of the average or representative putter according to each of the three characteristics Dr. M chose for his analysis. For example, beginning with Face Width, the average putter has a Face Width of 4.1761 inches. Chart 1 below graphs the distribution of Face Widths around the mean of 4.1761 inches: 9

Distribution of Putters According to Face Width 60 Mean 4.1761 in. 50 40 Frequency 30 20 10 0 3.09 3.35 3.61 3.87 4.13 4.39 4.65 4.91 5.17 5.42 5.68 5.94 6.20 More Inches We then used the data provided by Dr. M to calculate some basic statistical ratios, or t-tests. The purpose of these tests is to determine whether the measured differences across putters for these various elements are more likely due to chance, or rather represent meaningful differences from a statistical perspective. These tests demonstrate clearly that neither the Duff Putter nor the Cramer putter differ from the average or representative putter to any customary degree of statistical significance. These t-statistics are obtained from first taking the difference between the Face Width of the Cramer Putter (3.4153 in., or that of Duff Putter, 3.4017 in.) and the average of the 10

distribution, 4.1761 in., and then dividing this difference by the standard deviation of.47 found in Table 2. Both of these t-statistics are less than 1.96, the cutoff for significance at the 95% level. Thus, the Cramer Putter and the Duff Putter are statistically no different than the average putter. As a result, there is no valid statistical reason to assert that their similarity is probative here. Using the raw data on Face Widths in the appendix, we then calculated that out of the 172 putters for which Dr. M measured Face Width, only 9 are statistically distinct from the average putter at the 95% confidence level. Furthermore, we calculate that 91 putters are statistically similar to the Cramer Putter and 93 putters are statistically similar to the Duff Putter. In other words, virtually all putters in this sample are statistically indifferent from the average putter based on Face Width, and in most cases are not statistically different from either the Cramer Putter or the Duff Putter! 8 We then analyzed the data on Front-to-Back Length for each putter. Analysis of the data in Table 3 demonstrates that while both the Cramer and Duff putters are statistically different from the average putter length using similar t-statistics to those computed above, the data also demonstrate that several other putters are of statistically similar length to the Cramer Putter and the Duff Putter. To demonstrate this, we computed a t-statistic based on the difference between the Cramer Putter (3.3730 in., or the Duff Putter, 3.2960 in.) and the remaining putters in the appendix for which Dr. M measured the Front-to-Back length. In four cases, the value of this t-statistic is less than 1.96, meaning that the length of these putters is statistically indistinguishable from the Cramer Putter and the Duff Putter. The analysis of Table 4 and the underlying data in the Appendix indicate that, again, there is no statistically significant difference between Cramer and the average putter based on this attribute, the characteristic Alignment Lines. The relevant t-statistic here is the difference between either the Cramer Putter measurement (1.8415 or the Duff Putter measurement, 1.882) and the mean (1.1400), divided by the standard deviation of 0.91. These t-statistics are far less than 1.96. Furthermore, the data demonstrate that only one putter is statistically different from the average putter and that no putter is statistically different from either the Cramer Putter or the Duff Putter. In other words, virtually all other putters in this sample are not statistically different from the average putter, the 8 Note that implicit in this calculation is Dr. M s assumption that the underlying distribution is Normal. But as we discuss below in detail, because certain values produced by this distribution do not match those of the Normal distribution, this assertion is dubious. As such, the significance values for the Normal or T-density are conservatively high here. 11

Cramer Putter, or the Duff Putter based on this characteristic. We must reiterate all of these conclusions are based on giving a similarity test interpretation to a simple statistical exercise, which we don t believe is appropriate in the first place. V. Non-Independence of Putter Characteristics Dr. M s probability calculations are also flawed because they rely crucially on the assumption of independence, i.e., the idea that each of the three attributes chosen by Dr. M is unrelated to the other attributes for any given putter. But these measurements are not in fact independent from one another; they are interrelated as part of the overall design parameters which govern modern putter design. In fact, the USGA rules for putters and irons states When the clubhead is in its normal address position, the dimensions of the head must be such that the distance from the heel to the toe is greater than the distance from the face to the back. 9 In other words, it must be the case that the Face Width is greater than the Front-to-Back Length. Thus, as a matter of USGA rules, these two measurements are NOT independent. As such, the multiplication of three already artificially small probability numbers is incorrect as a threshold matter, and compounds the prior errors. Indeed, given a large enough number of attributes or characteristics to compare and small enough intervals over which to measure differences, every putter will appear unique. Consequently, the findings of these small probabilities of falling within small intervals is meaningless, and can be arbitrarily reproduced for numerous other pairs of putters in the expert s own data set as shown above. The correct approach would be to have a joint test of equality of the selected attributes, based on the joint distribution of those attributes. There are several methods available for estimating such distributions, the chief among them being the nonparametric kernel methods, requiring very large samples (not available here, or generally). Alternatively, approximate asymptotic methods (such as the Kolmogorov-Smirnov metric 10 or maximum likelihood techniques) can be used to fit parametric distributions and/or to test the fit of the data to any distribution. The latter also will generally 9 USGA Rules of Golf, Section 4.1a, Appendix II, Section 4.b.ii. 10 Hogg, Robert V. and Allen T. Craig. Introduction to Mathematical Statistics, Prentice Hall, fifth edition. 12

require large samples for reliability. Mechanically, however, such approximations can be implemented, and are indeed the most common, albeit implicit, justification for applications of statistics in law and many other fields. For instance, central limit theorems can be invoked that suggest, for large samples, the joint distribution of the measured attributes around some desired value (true values, means, averages, other objects ) is approximately Normal. Once this is accepted, a number of multivariate test statistics are available for testing equality of two vectors (of attributes). The most commonly used is a Chi-Squared statistic, which is merely a generalization of the t- statistics we reported above, but taking into account the dependence between the attributes. This would be an example of a joint test of the hypothesis of similarity in several dimensions. To be more specific, the distances between pairs of corresponding characteristics are measured and arranged in a vector W which would have an approximate normal distribution. Given the known Normal distribution, it is possible to estimate the variance (covariance matrix) of this vector of measurements. Under the hypothesis that objects are equal), the computed value of the statistic Q=W [Cov(W)] -1 W is then compared with the significance level of a Chi-Squared distribution with appropriate degrees of freedom. If Q is larger than the critical level, the hypothesis of similarity or equality is rejected at a desired level of statistical confidence. One says that similarity is not supported to a degree of statistical confidence (e.g., 95%). An alternative, sequential application of single dimension tests, such as t-tests, can be considered but require care and sophistication in controlling for test size (such as the Bonferroni method). This latter alternative falls under the topic of multiple comparisons and testing, and would be difficult to justify in situations like the present case, where various attributes do not have any natural order or logical nesting. It is also possible that in some applications, one may wish to test whether one object s attributes are all larger (or smaller) than another object s. Such ranking tests, or of inequality restrictions, are available, for instance by using Chi-bar-Squared tests or extended Kolmogorov-Smirnov tests. See Maasoumi (1998, Companion to Econometric Theory, edited by Badi Baltagi, Blackwell) for a review. VI. The Presentation of Statistical Evidence 13

Besides the usual hurdles in preparing a statistical analysis, the adversarial process of litigation raises further issues which the expert must address: Absence of Measures of Statistical Confidence Dr. M s probability computations are not accompanied by statistical indications of significance, and, thus, lack statistical validity. This is important in legal applications because not only the results but in particular the reliability of those results dictate the admissibility and weight offered to statistical testimony. Dr. M failed to indicate how the putters were measured, which obviously is the starting point for all of his analysis and potentially the most critical. Errors in this first step will render his entire analysis speculative. Because of Dr. M s failure to provide standard errors for his estimates, he in fact offers statisticalsounding assertions that are not actually based upon reliable statistics. Data Reliability Issues In particular with regard to litigation, it is important that all data relied on by the expert is clearly described. Apparently Dr. M was provided access to 248 putters. There is no mention of how these putters were selected or if the selection was random. What was eventually revealed is that all 248 putters were provided by the plaintiff, the owner of the intellectual property in question (i.e., the inventor of the Duff Putter). These putters were assembled by the plaintiff and kept in a shack in his backyard long before the initiation of the litigation in question. One might speculate that the plaintiff selected these putters because he liked them, and thus that his own putter design is likely to be closer to these putters than a true random sample. In any case the 248 putters are not likely to be a random sample in any meaningful sense. That in and of itself is a major issue when calculating statistical significance, as the simple mean and variance calculations are invalid without random sampling. Assumptions Regarding the Distribution of the Data As is obvious from Chart I above, the distribution of Face Widths for this sample of putters is hardly Normal. Even the expert s own computations of skew and kurtosis are different 14

from those of the Normal distribution (for which the correct theoretical values are 0 and 3, respectively). 11 Furthermore, the expert report does not provide any of the standard tests for Normality extant in the statistical literature and text. Indeed, the report is totally silent as to the method of estimation of the data distributions, and therefore the method of estimating the probabilities that are the center piece of the unsupported conclusions of similarity between putter measurements. Failing this, Dr. M could have relied on nonparametric tests, i.e., tests which make no specific assumption about the distribution in the underlying data. Ad Hoc Choice of Putter Attributes In any statistical analysis, it is important that the expert clearly state the rationale for the choice of attributes to be measured as well as the sample population and any sampling plans utilized. Ad Hoc choices may suffer from biases which render the results of any statistical analysis meaningless. Dr. M offers no arguments or evidence to support the decision to characterize putters with the particular attributes/measurements which he chose to analyze. There is no statistical or industry evidence provided. In particular, a bit of basic research on golf club design indicates that the following parameters are important to the performance of a putter: Materials of construction Shaft axis location Shaft entry point Type of insertion Center of gravity measurements Face to back width None of these attributes are measured in the expert report of Dr. M, nor does he discuss what characteristics might be appropriate or why. On the surface, we seem to have been presented with a conundrum. On the one hand, statistics can be manipulated to show the apparent uniqueness of every putter. On the other hand, statistical analysis is capable of demonstrating that most putters are extremely similar as a general matter. In 11 The skew measures the amount of asymmetry in a Normal distribution, the kurtosis measures excess weight in the tails of the distribution. 15

that sense, this case study certainly demonstrates that statistics can be a powerful tool, here in rebuttal. As for the original affirmative decision to use statistics in this case, there may be no general agreement about which putter characteristics are actually important to the function of the putter, or against what type of metric any measure of sameness should reasonably be compared. It may simply be that applied statistics are inadequate to the task here, and that other forms of evidence will have to be used to argue the liability question here. 16