As a result, a civil suit was filed by the City of New York alleging that Brink's had failed to honor its contract and acted negligently. The city was seeking monetary compensation from Brink's for losses incurred by the criminal activities of its employees. Brink's was subsequently found guilty of negligence and breach of contract. The question remained as to what the actual dollar amount of the damages were.
The purpose of this paper is to examine the statistical methods by which the City of New York's expert witness arrived at the amount of damages, and to contrast that with the defense's expert witness' criticisms. We suggest an alterhate approach as well.
In order to investigate the existence of any trend Fairley fit piecewise regressions to the data over the 22 month period. As Fairley points out himself, piecewise regression is a nice idea since it will estimate changes in level (intercept shift) and trend (slope shift) over the 22 month period, while a single regression line will not. He used seasonally adjusted monthly average per meter day as the dependent variable and month as the independent variable ( the data is seasonally adjusted by dividing the city average for the month by the predicted revenue for the month, and then multiplying this by the period average for the city). It seems somewhat strange that he seasonally adjusts the data given his previous comments about how choosing the same 10 calendar months over each period made such adjustment unnecessary. Fairley argued that although there exists no evidence for a general upward trend (main point of contention for the defense, see next section), there does exist significant evidence of a change in level (intercept shift) over the two periods. Since we don't have access to the results of this regression, we estimated similar models ourselves. We fit many models, for both unadjusted and seasonally adjusted data. We formed averages as revenue per collection day, while Fairley forms averages as revenue per meter day. For instance, we compute the monthly average by dividing monthly revenue by the total number of times Brink's collected for the month, while Fairley forms averages by dividing the monthly revenue by the number of days of the month that parking meters were in effect. Since we neither have his regression results or the data concerning collection days, we are unable to examine whether or not these different procedures produce different results. For most of the models, while there exists strong evidence for an intercept shift, there isn't much evidence for a slope shift. However, upon deleting the last CDC observation (which has a Cook's distance twice as big as any other observation), the following regression results are obtained.
Model: MODEL1 Dependent Variable: Total Monthly Revenue (including area 1A) Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 72094721798 24031573933 2.852 0.0725 Error 15 126391276555 8426085103.7 C Total 18 198485998353 Root MSE 91793.70950 R-square 0.3632 Dep Mean 1765510.21053 Adj R-sq 0.2359 C.V. 5.19927 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 1720364 62707.041458 27.435 0.0001 DUMMY* 1 558901 213199.54966 2.621 0.0193 MONTH 1 1977.127273 10106.154803 0.196 0.8475 INT** 1 -29958 15574.630545 -1.924 0.0736 * Dummy = 0-1 indicactor variable, 1 = CDC, 0 = Brink's. ** Int = Month*DummyThe overall fit of the model is improved, and the slope shift approaches statistical significance. Regarding the substantive issue of an upward trend in revenue, however, this shift is in the wrong direction. Thus, it seems that Fairley's piecewise regressions demonstrate that he has accounted for the possible confounding to causal interpretation; namely, trend and Seasonality.
Fairley goes on to introduce a more elaborate nonlinear model based on Borough level data, arguing that such a model provides more degrees of freedom, etc. Based on this model, he estimates an intercontractor difference and uses this to come up with an estimate of the damages that Brink's is liable. Unfortunately, we currently do not have borough level data and thus cannot verify his results. In any event, we will argue in section 5 that a simpler model based solely on city-wide totals is sufficient and also easier for a jury to understand.
With regard to (2), Levin argues that the difference between the two periods may (emphasis our own) be attributable to factors other than theft. He presents evidence of a gasoline shortage that led to ``marked drops'' in revenues for automobile toll bridges and tunnels. This happened during the Brink's period, June-September 1979.
Levin's analysis was based on data for the entire Brink's period for two sets of data. Since an area of the city designated as area 1A was not under the jurisdiction of Brink`s, Levin examined the given time period for city data excluding 1A and for area 1A alone. He searched for a trend in each area.
Levin plotted five-month moving averages over the Brink's and CDC (May '78 to Mar '81) contract for the city as without 1A and for 1A alone. He argued that the increase in revenue collected in area 1A during the CDC period showed that other factors besides theft were accounting for the increase in revenue across the city as a whole. In addition, he argued that these five-month moving averages exhibited an upward trend operating over the Brink's period.
Levin also fitted regression lines separately to the monthly city revenue excluding area 1A and to area 1A alone over the entire period of the Brink's contract (May '78 to Mar '80). He found a positive slope for the city-wide data, and argued that if this regression line was extrapolated over the CDC period, the predicted revenue difference for the two 10-month periods is $2,155,000, making the observed $1,000,000 difference less surprising. With regard to the line fitted to the area 1A data, he also found a positive slope. Thus, the conclusions he reached through the regression analysis are similar to the ones he made from the moving average analysis, i.e., the difference in revenue could be accounted for by factors besides theft.
Total Revenues Brink's CDC Shortfall Jun 1685938 1941688 -255750 Jul 1644110 1889106 -244996 Aug 1746709 1741465 5244 Sep 1754081 1832510 -78429 Oct 1853363 1926233 -72870 Nov 1754081 1670136 83945 Dec 1692441 1948290 -255849 Jan 1801019 1627594 173425 Feb 1702335 1655290 47045 Mar 1678305 1844604 -166299 ======================================================= -764,534.0 Total -76,453.4 Monthly Average =======================================================It is quite easy for a jury to relate to the shortfall in revenue as they are formed by taking simple differences. Regression methods are not as convincing here for two reasons. First, it is easier to attack regression models. For instance, although our models indicate significant intercept shifts, the fit of these models is very poor (F = 2.04; prob > F = .15), an easy point for the defense to attack. Second, regression models are often inaccessible to a jury. In addition, the method of controlling for seasonality is easy to understand. The data above yield shortfall for ten corresponding months. A scatterplot and boxplot are shown in Figure 1a and Figure 1b respectively.
Given this data, what is a reasonable way of forming an estimate of the total amount of damages to be awarded to the city? There are several simple ways to proceed. On average, Brink's collected $76,453 less than CDC in any given month for months in which we have comparative information. In light of this fact, a quick and dirty damage estimate is:
However, all of the above methods are susceptible to Levin's criticism that there exists an upward trend over the entire Brink's period. Figure 2 is a plot of total revenue per month over the entire Brink's period. Indeed, there appears to be an increasing trend, with two specific periods (the 13 months prior to the 10-month comparison period, and the 10-month comparison period.) If we look at Figure 2 carefully, we see that there is an upward linear trend that appears to end at the beginning of the ten-month comparison period. A regression over the entire 23 month period exhibited both a significant intercept and slope shift, with the slope of the 10-month comparison period not being significantly different than zero (F=22.3; Prob > F=.0001). Note that month 22 was not included in the analysis since it was a highly influential observation and appeared to be an outlier (Cook's Distance=.29, four times larger than the next highest value).
If we assume that theft was occurring throughout the entire Brink's period, then factors other than theft most likely contributed to the increasing revenue over time. Possible explanations are perhaps an increase in the number of cars, the building of shopping malls, or an increase in the fare per meter. Although the city's parking meter theft expert refuted most of these possibilities, the trend is still apparent in the graph. Therefore, in trying to be fair to Brink's, we attempted to control for this upward trend by downweighting the months prior to the ten-month comparison period. Our argument goes as follows:
The average Brink's/CDC shortfall was calculated based on data for this flat period. If we divide the total revenue per month during this 13 month period by the average revenue per month for the ensuing 10 month period, we would have a rough estimate of how much weight each of these earlier months should be given. We then multiply each of these ``weights'' by the expected monthly shortfall calculated above and obtain an estimate of the expected shortfall over the entire period. (All months during the 10-month comparison period are given a weight of one.) Using this downweighting procedure we produce a more conservative damage estimate of 1.6 million dollars (approximately $200,000 less than our original raw estimate. See Appendix A for calculations).
In summary, we believe the method we used to calculate the average monthly Brink's/CDC revenue shortfall is reasonable for two reasons. It controls for Seasonality in a straightforward manner, and it is easy for the jury to understand. We also feel that the general upward trend prior to the 10-month comparison period need to be taken into consideration when forming the damage estimate. Although any method of downweighting is susceptible to criticism, our method employs a simple approach which we consider fair to Brink's and the City and reasonable given the data at hand.
Month Total 10-Month Weight Shortfall Revenue Brink's Avg 13 1337159 1731238.2 0.77237 59050.43 14 1532810 1731238.2 0.88538 67690.59 15 1318521 1731238.2 0.76161 58227.35 16 1502054 1731238.2 0.86762 66332.37 17 1393093 1731238.2 0.80468 61520.53 18 1564212 1731238.2 0.90352 69077.34 19 1474861 1731238.2 0.85191 65131.50 20 1554116 1731238.2 0.89769 68631.49 21 1572284 1731238.2 0.90818 69433.81 22 1129834 1731238.2 0.65262 49894.72 23 1781470 1731238.2 1.02901 78671.69 24 1659206 1731238.2 0.95839 73272.38 25 1752172 1731238.2 1.01209 77377.86 26 1685938 1731238.2 1.00000 76453.40 27 1644110 1731238.2 1.00000 76453.40 28 1746709 1731238.2 1.00000 76453.40 29 1754081 1731238.2 1.00000 76453.40 30 1853363 1731238.2 1.00000 76453.40 31 1754081 1731238.2 1.00000 76453.40 32 1692441 1731238.2 1.00000 76453.40 33 1801019 1731238.2 1.00000 76453.40 34 1702335 1731238.2 1.00000 76453.40 35 1678305 1731238.2 1.00000 76453.40 ========== 1628846.05