Confidence Interval for the Ratio of Two Normal Variables (an Application to Value of Time)

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1 Interdisciplinary Information Sciences Vol. 5, No. (009) 37 3 #Graduate School of Information Sciences, Tohoku University ISSN print/ online DOI 0.036/iis Confidence Interval for the Ratio of Two Normal Variables (an Application to Value of Time) Hisa MORISUGI, Jane ROMERO, and Takayuki MORIGUCHI Graduate School of Information Sciences, Tohoku University, Aramaki-aza-aoba 0, Aoba-ku, Sendai , Japan morisugi@plan.civil.tohoku.ac.jp, jane@plan.civil.tohoku.ac.jp Received January 5, 007; final version accepted July 3, 008 The problem of making inferences about the ratio of two normal variables occurs in many fields such as bioassay, bioequivalence and ecology. In this paper the theory of the ratio of two normal variables is applied in evaluating the value of time, which is a major component in cost-benefit analysis of transport projects. The subjective value of time is the ratio of the marginal rate of substitution between travel time and travel cost. This paper explores the building of confidence intervals for the subjective value of time, applying methods that make inferences about the mean for the ratio of normal variates: direct substitution of Fieller s pdf and t-test method. In addition, we also propose a method to minimize the length of the confidence interval. KEYWORDS: ratio of normal means, confidence interval, Fieller s pdf, t-test method, minimizing length of confidence interval, value of time. Introduction Let X ¼ðX ;...; X n Þ and Y ¼ðY ;...; Y n Þ be independent random samples such that X and Y are two normal variables with means and, variances and, respectively, and correlation coefficient. The aim of the problem is to establish the method of inference on the mean of ratio X=Y by producing inferences about ¼ =, which describes the relative magnitude of the mean. This problem of making inferences on the ratio arises in different scenarios. It has drawn considerable interest not only in the field of statistics (Maragsalia, 965, Malley, 98, Rukhin, 985, Srivastava, 986, Juola, 993) but also in numerous fields like bioassay (Finney, 978), bioequivalence (Chow and Liu, 99, Hsu, Hwang, Lui and Ruberg, 99), cost data (Laska, Meisner and Siegel, 997, O Hagan and Stevens, 003), biomedical data (Heitjan, 000) and ecology (Raftery and Schweder, 993). Its application also varies from analyzing the relative potency of a new drug to that of a standard drug, in the estimation of red cell life span to ratio of the weight of a component of the plant to that of the whole plant among others. In this paper, the inferences on the ratio will be applied to derive the subjective value of time. It is commonly referred to as marginal willingness to pay of an individual for marginal savings in travel time. The estimates of the coefficients of marginal travel cost and travel time can be asymptotically regarded as components of a general multivariate normal population. However, the ratio of two normal variables does not follow a normal distribution. In the context of value of time analysis for transportation, the current methods of deriving the value of time are lengthily discussed in Armstrong et al. (00) (see also Garrido and Ortúzar, 993). What is common among those methods discussed in that paper is that they did not utilize the associated probability density function of the ratio. Further, it was considered as unknown a priori. Here we propose two methods that has been long established but not yet applied in VOT analysis based on the inferences on the ratio of two normal variables, that is, by direct substitution of Fieller s pdf and t-test method. Other than discussing its application in the value of time (VOT) context, we introduce an approach on how to minimize the confidence interval. As the resulting distribution of the ratio is more likely a skewed distribution, symmetric endpoints do not result to optimal values. That is why we propose to minimize the length of the confidence interval. The organization of this paper is as follows. Section will define the value of time. In Section 3 is the discussion on the methods of building the confidence interval by direct substitution of Fieller s pdf and t-test method as well as the method on how to minimize the length of the confidence interval. The application to value of time is discussed in Section. Section 5 summarizes the conclusions.. Definition of Value of Time Value of time is defined as the change in travel cost relative to change in travel time with the utility level kept

2 38 MORISUGI et al. constant. The subjective value of time is given as ¼ dp i dt i U¼const P i is travel cost, T i is the travel time and U is a given utility level Travel cost and travel time are assumed as variables of a general multivariate normal population. Assuming an aggregate Logit model, and for simplification of the model, two mode choices are given and an individual has the following choice of modes and given utility function. The choice function is given as, j y j ¼ ln ; j ¼...n; j y j ¼ 0 0 þ ðp j p j Þþ ðt j t j Þþ" j ð:þ : j : share of mode of sample number j y j : deference in utility of mode from mode for sample number j 0i,, : parameters p ij ¼ travel cost of mode i for sample number j t ij ¼ travel time of mode i for sample number j " j : random error term for sample number j Let 0 ¼ð 0 0 Þ, P j ¼ p j p j, T j ¼ t j t j and assuming normality of the " i term then, y j ¼ 0 þ P j þ T j þ " j ð:3þ The subjective value of time or the willingness to pay is ¼. If the parameters a, a are denoted as the estimates of the parameters,, then, a ¼ a Nð; Þ a 0 P ¼ T ; ¼ ðx 0 XÞ... ; X ¼ B... A P n T n ð:þ 3. Proposed Methods: Direct Substitution of Fieller s Pdf and t-test Method In statistical literature, Fieller s theorem is already a well-established method in analyzing ratio inferences. However, its application in value of time analysis has not yet been tapped. Borrowing the concept from bioassay, this section will discuss how this method could be applied in deriving the subjective value of time. In addition, we propose the t-test method as a simplified version of Fieller s and could be more appropriate in the application to the value of time. Direct Substitution of Fieller s Pdf The simplest form of the ratio can be stated as follows. Given two normal variables X and Y with means and, variances and, respectively, and correlation coefficient, their ratio does not necessarily follow the normal distribution. Fieller (93) and Hinkley (969) derived the density function of the ratio ^ ¼ X=Y as (! ) pffiffiffiffiffiffiffiffiffiffiffiffiffi hl h f ð ^Þ ¼pffiffiffiffiffi p g 3 g ffiffiffiffiffiffiffiffiffiffiffiffiffi k þ exp ð3:þ g ð Þ : g ¼ ^ ^ þ! = k ¼ ^ þ

3 Confidence Interval for the Ratio of Two Normal Variables (an Application to Value of Time) 39 h ¼ ^ l ¼ exp ð Þ! þ h g k Where ðþ: cdf Nð0; Þ To derive the confidence interval for value of time, estimated values of,, and from actual data are plugged into Eq. 3. ^ ¼ a. The other notations are as follows: a Eða Þ¼ Eða Þ¼ Vða Þ¼ Vða Þ¼ covða ; a Þ¼ ; and ¼ From the resulting graph of the distribution, the confidence interval is computed given a 95% confidence limit, Pr½ <0:95 Š¼0:95 ð3:þ such that, 0:05 < ^ < 0:975 ð3:3þ t-test Method The t-test method is an instance of what statisticians call as the method of pivots, in a pivot is a function of the data and the parameters whose distribution is independent of the value of the true parameter. The t-test method is slightly sophisticated as it uses Student s t-distribution to account for population variances, which need to be estimated by sample variances as in the case of value of time data. Specifically what is called Fieller s method is helpful for our problem (Fieller (9), Maddala (977)). Our approach is as follows. Suppose the linear statistics a and a are jointly normally distributed with expectations E½a Š¼, and E½a Š¼, variances Vða Þ¼ and Vða Þ¼, and covariance covða a Þ¼. Then define Whatever is the true value of, ¼ Z ¼ a a Nð0; z Þ z ¼ þ ð3:þ ð3:5þ Let s z be the unbiased estimator of z, then the t-statistic is t ¼ a a s z and s z ¼ s s þ s s, s and s be the unbiased estimator of, and, respectively. Setting the confidence level as 95%, Prðt < t0 Þ¼0:95 t 0 is the value of t-statistics for 95% interval. Substituting (3.6) and (3.7) into (3.8), ða a Þ Pr ðs s þ s Þ < t 0 ¼ 0:95 The inequality in the parenthesis can be transformed as ða s t 0 Þ ða a s t0 Þþða s t 0 Þ < 0 ð3:6þ ð3:7þ ð3:8þ ð3:9þ ð3:0þ

4 0 MORISUGI et al. Supposing that a s t 0 is positive, solving for leads to the following confidential interval. a a s t0 p ffiffiffiffiffiffiffiffi D= a << a a s t0 þ p ffiffiffiffiffiffiffiffi D= s t 0 a s t 0 ð3:þ D= ¼ðs s s Þt 0 þða s a a s þ a s Þt 0 Behavior of the Confidence Interval Considering that the values of the interval are derived from a quadratic function of left hand side of (3.0), its results could either be real or imaginary numbers by the resulting interval could be either finite or infinite. The value of a s t 0 might be positive for most cases because a =s is the t-value of estimate a which is usually more than 5% significant, therefore, greater than s. Minimization of Length of the Confidence Interval If the sampling distribution is symmetric, the symmetric interval is the best one in a sense that the length of interval is minimal. If the sampling distribution is not symmetric, however, as in the skewed case of value of time, this procedure is not optimal. The obvious criterion is to minimize the length of the confidence interval (Greene, 003, Section.8). The measure of desirability of a confidence interval is its expected length; thereby we propose minimizing the length of confidence interval. However, short intervals are desirable only when they cover the true parameter value but not necessarily otherwise (Lehmann, 959). If a short confidence interval is taken to indicate accurate information about the parameter, then it may be preferable that the interval be long when it is far from the true parameter value (Pratt, 96). This leads to a condition considering both expected length and the probability of covering false values conditional on the true value being covered. To minimize the length of the confidence interval from the t-test method, set Prð = < ^ < 0:95þ= Þ¼0:95 ð3:þ for a given value of 0 <<0:05. Then, assuming again that a s t 0 is positive as before and following the same way as (3.0) and (3.), the resulting value of for the t-test method is solved by sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a s t= D = a a s t0:05 = þ D 0:05 = a s t = a ð3:3þ s t 0:05 = : a a s t 0:05 = D = a s t 0:05 = sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 0:05 = a a s t = þ a s t = sffiffiffiffiffiffiffiffiffi D = ¼ðs s s Þt = þða s a a s þ a s Þt = D 0:05 = ¼ðs s s Þt 0:05 = þða s a a s þ a s Þt 0:05 = Noting that on the critical value of t-statistic, as t0:05 = < t = for 0 <<0:05, it can easily shown that the length of confidential interval of (3.) is smaller than (3.3), and for otherwise (3.3) is smaller than (3.). Therefore the confidential interval with the minimal length can be obtained by varying the value of. Similar method of minimization of length could be applied to the results of the Fieller s pdf. But this case needs direct calculation of and 0:95þ from the FIeller s pdf by varying the value of.. Application to Value of Time ð3:þ As benefits accrued from time savings consists of almost 70 to 80% of all benefits derived from improvement of transport facilities, the evaluation of travel time savings is an important factor in the cost benefit analyses of transport projects. To derive the monetary value of time savings, it is necessary to derive the marginal rate of substitution between travel time and travel cost in the discrete choice model (Ben-Akiva and Lerman, 985).

5 Confidence Interval for the Ratio of Two Normal Variables (an Application to Value of Time) f ( θˆ ) Value of time θˆ (yen /minutes) Fig.. The distribution of the value of time (yen/min). Table. Comparison of direct substitution of Fieller s pdf and t-test method. :5 3 L U L U L U :5 Fieller s t-test Fieller s t-test Fieller s t-test Building of Confidence Interval for VOT by Fieller s Pdf The data was taken from the long-distance travel survey in Japan (Ministry of Land, Infrastructure and transportation 003) and the resulting estimated values of and are the following: ¼ 5:50 (util/yen) ¼ :0 0 3 (util/min) ¼ 6:50 0 (util /yen min) ¼ 3:9 0 (util /yen ) ¼ 9:9 0 (util /min ) The distribution of the ratio =, i.e., the estimated value of time is plotted by solving the direct substitution method and is shown in Fig.. The graph is a skewed unimodal case (single peak). This is not a normal distribution but is well behaved enough to define a pseudo mean and pseudo standard deviation. This applies to large data in the assumption can be justified by the Central Limit Theorem in statistics, which says the mean of a sufficiently large sample will be approximately normally distributed no matter what the underlying population distribution. The confidence intervals (lower bound and upper bound), computed by Fieller s pdf and the estimation by t-test method, are shown in Table. The characteristic of the result from the Fieller s pdf is that it is leaning on the left-side of the curve as compared to the values from the t-test method that is more on the right-side of the curve. Comparison of Fieller s Method and t-test Method Comparison was also done by varying the values of and while the covariance value is fixed as 6:50 0 as shown in Table. The left side values are the lower bound (L) while the right side values are the upper bound (U) of the confidence interval. From Table, the confidence interval derived from the Fieller s pdf is lower and narrower as compared to the results from the estimation by the t-test method. Further, Fig. shows the divergence of the values with the varying of the values of (results of Fieller s pdf are designated as direct while t-test as t-test ). The same figure is obtained by varying. Minimization of Length of the Confidence Interval Table shows the results of the minimization of length of intervals as applied to both methods. Note that with that minimization process, the discrepancy of values as shown graphically in Fig. was minimized if not eliminated.

6 MORISUGI et al. Value of time E+0 5.0E.0E.5E σ direct U direct L t-test U t-test L Fig.. Confidence interval of value of time with varying values of. Table. Comparison of results from Fieller s theorem and t-test method with minimized length of interval. :5 3 L U L U L U Fieller s :5 t-test Fieller s t-test Fieller s t-test Value of time E+0 5.0E.0E.5E.0E σ direct U direct L t test U t test L Fig. 3. Confidence interval of value of time with varying values of after minimization of length of interval. Figure 3 shows graphically the results of the comparison between the Fieller s theorem and the t-test method after minimizing the length of interval. 5. Conclusions The principal virtue of using the direct substitution of Fieller s pdf for obtaining a confidence interval for the ratio is that it could be applied regardless of the probability distributions of the underlying random variables. However, considering the complexity of the pdf, it is worthwhile to look for an easier way of estimating the confidence interval given a certain probability level. The method of pivots like the t-test method is a good alternative to come up with a valid estimation of the confidence interval for the ratio. Both methods are classified to be normal-theory methods in it invokes the Central Limit Theorem, which says that the mean of a sufficiently large sample will be approximately normally distributed no matter what the underlying population distribution is.

7 Confidence Interval for the Ratio of Two Normal Variables (an Application to Value of Time) 3 The resulting confidence intervals derived from the two methods as indicated in Table gave differing values. However, by introducing minimization of length of the confidence interval to both results, the discrepancy was minimized if not eliminated. This minimized interval indicates the optimum interval for the corresponding skewed distribution. Thus it can be recommended that the confidential interval can be better estimated by the t-test with minimized length. The ease of using t-test with minimized length is notable for applications that require quick results with minimal mathematical knowledge as t-test is included in most spreadsheet applications. With t-test and the modification minimizing the length of the interval, it is capable of producing estimation with higher degree of accuracy. This is most helpful in widening the usage and applications of confidence interval not only in estimating the value of time but for other similar problems. REFERENCES Armstrong, P., Garrido, R., and Ortúzar, J., (00), Confidence interval to bound the value of time, Transportation Research Part E, 37: 3 6. Ben-Akiva, M., and Lerman, S. R., (985), Discrete Choice Analysis: Theory and Application to Travel Demand, MIT Press, Cambridge, MA. Chow, S. C., and Liu, J. P., (99), Design and Analysis of Bioavailability and Bioequivalence Studies, Marcel Dekker, Inc., New York. Fieller, E. C., (93), The distribution of the index in a normal bivariate population, Biometrika, : 8 0. Fieller, E. C., (9), A fundamental formula in the statistics of biological assay and some applications, Quarterly Journal of Pharmacy, 7 3. Fieller, E. C., (95), Some problems in interval estimation, Journal of the Royal Statistical Society, Series B, 6: Finney, D. J., (978), Statistical Method in Biological Assay 3rd ed. Macmillan, New York. Garrido, R., and Ortúzar, J., (993), The Chilean value of time study: methodological developments, Proceedings of the st PTRC Summer Annual Meeting, University of Manchester Institute of Science and Technology, England. Greene, W. H., (003), Econometric Analysis th ed. Prentice Hall, Upper Saddle River, NJ. Heitjan, D. F., (000), Fieller s method and net health benefits, Health Economics, 9: Hinckley, D. V., (969), On the ratio of two correlated normal random variables, Biometrika, 56: Hsu, J. C., Hwang, J. T., Lui, H. K., and Ruberg, S., (99), Confidence intervals associated with tests for bioequivalence, Biometrika, 8: 03. Juola, R. C., (993), More on the confidence interval, The American Statistician, 7: 7 9. Laska, E. M., Meisner, M., and Siegel, C., (997), Statistical inference for cost-effectiveness ratios, Health Economics, 6: 9. Lehmann, E. L., (959), Testing Statistical Hypotheses, New York, John Wiley & Sons. Maddala, G. S., (977), Econometrics, International student edition, McGraw-Hill, pp Malley, J., (98), Simultaneous confidence intervals for ratios of normal means, J. Am. Stat. Assoc., 77: Maragsalia, G., (965), Ratios of normal variables and ratios of sums of uniform variables, J. Am. Stat. Assoc., 60: Ministry of Land, Infrastructure and transportation, (003), Long-distance Travel Survey. O Hagan, A., and Stevens, J., (003), Assessing and comparing costs: how robust are the bootstrap and methods based on asymptotic normality?, Health Economics, : Pratt, J. W., (96), Length of confidence intervals, J. Am. Stat. Assoc., 56: Raftery, A. E., and Schweder, T., (993), Inference about the ratio of two parameters, with application to whale censusing, The American Statistician, 7: Rukhin, A., (985), Estimating the ratio of normal parameters, Ann. Stat., 3: Srivastava, M. S., (986), Multivariate bioassay, combination of bioassay s and Fieller s theorem, Biometrics, : 3.