RANKED SET SAMPLING FOR ENVIRONMENTAL STUDIES

Transcription

1 . RANKED SET SAMPLING FOR ENVIRONMENTAL STUDIES 000 JAYANT V. DESHPANDE Indian Institute of Science Education and Research, Pune , India Talk Delivered at the INDO-US Workshop on Environmental Statistics held at Duke University, NC, U.S.A. from March 4 to March 6,

2 ABSTRACT Here the fact that Ranked Set Sampling (RSS) procedures were initially developed for environmental studies is emphasized. These procedures envisage ranking a large number of observations (in small groups) and actually measuring only a few of them. The basic procedure to estimate the mean and its higher efficiency over the SRS procedure is demonstrated. Then we discuss the modification to be carried out for developing nonparametric confidence intervals for quantiles. Lastly we discuss some tests for perfect judgment ranking of observations which should be used as preliminary procedures before the RSS methodology is adopted. 2

3 Outline Examples from Environmental Studies Introduction to Ranked Set Sampling Estimation of Mean Confidence Intervals for Quantiles Tests for Perfect Ranking Conclusions. 3

4 1. Introduction and Summary This paper reviews the use of ranked set sampling methodology for statistical problems arising in environmental studies. These studies typically require high cost observations. Hence it is important that the inferences be based on the smallest number of observations. Ranked set sampling is a methodology which can improve the efficiency of techniques such as estimation and confidence intervals without increasing the number of substantial observations; if additional information on their relative ranking is available (or can be obtained without significant expense). 4

5 In the second section we discuss the modes of obtaining environmental data and the reasons why this can be very expensive, bringing out the need for exercising parsimony in the number of observations. We introduce ranked set sampling (RSS) and explain why this leads to greater efficiency of the statistical procedures compared to those based on the usual simple random sampling (SRS) procedures. We exhibit the gain in efficiency of the mean based on RSS over SRS. We also explicitly bring out the increase in coverage probabilities of distribution free confidence intervals based on order statistics in similar circumstances. 5

6 In the operation of RSS procedures one needs to rank a larger number of observations than those measured for the characteristic of interest. This ordering is often done on the basis of some surrogate variable or covariate whose values are available practically free of cost. For choice of optimal procedures we must have perfect rankings. But the question of perfection of ranking remains unanswered. We therefore discuss tests available for testing perfect ranking against the extreme alternative of random rankings. We see that reasonable test procedures allow us to either decide in favour of perfect ranking or against it. 6

7 In the last section we discuss some aspects of the large body of work of RSS methodology giving references to a recent book and some review papers. It may be observed that after introducing the well known properties of the RSS estimation of the mean, we discuss some newer work, not yet reported in say the book of Chen, Bai and Sinha (2003). 7

8 2. Problems Arising in Environmental Studies where Ranked Set Sampling is Useful Observations in environmental studies are notoriously difficult and/or expensive to obtain. Studies after studies have confirmed this. Hence the RSS methodology which has higher efficiency for the same number of measured (as opposed to ranked) observations is the preferred one. In fact the origins and much of its development has been with respect to the domain of environmental studies. 8

9 McIntyre (1952) who initially proposed this methodology did so in the context of estimating mean pasture and forage yields. In order to obtain observations one must harvest the forage. It would involve moving and clipping the browse and then weighing it after drying it. All this is a labourious, expensive and time consuming process. So all avenues towards reduction of number of observations are explored. Now one can rank the observational units (called quadrats) more or less by just imputing them visually. Thus ranking can be accomplished in a large number of quadrats without much expense and included in the RSS procedures. 9

10 Another example often quoted is that of assessing the status of hazard waste sites. The radiochemical analysis of the soil samples is expensive as well as hazardous. But the samples can be ranked according to the value of a surrogate variable, viz., the Field Instrument for the Determination of Low Energy Radiation (FIDLER) counts per minute taken at the location from where the soil samples for the radiochemical analysis would be obtained. These can be obtained at relatively low cost and then used for ranking a large number of soil samples. The actual measurements are on plutonium concentration, per square metre of surface soil (Yu and Lam (1997)), which are expensive. 10

11 Another experiment described in details in Murray, Ridout and Cross (2000). Leaves on standing trees were sprayed with a fluorescent, water soluble tracer at 2% concentration in water. Then a large number of leaves were harvested and were ranked according to the surface area covered with the chemical. Once ranking was complete, the deposits on fewer number of selected ranked leaves were washed and the flow directed in a test tube. The relevant concentration of the tracer was then measured using a spectrometer. Thus the data was collected. It may be noted that the ranking stage is much less laborious, time consuming and expensive than the measuring stage. 11

12 Another example concerns estimation of mercury contamination in fish as discussed in Murff and Sager (2006). Ten appropriate live or freshly dead fish were selected from a catch. Their length was measured for ranking. Then in the laboratory a fillet was removed from each fish, homogenized with a blender and analyzed for mercury concentration (in mg/kg) with a gas chromatograph. This is an expensive process, besides resulting in destruction of the sample. RSS methodology was suggested for use in such an experiment. However, in this case it was found that although more efficient, the gain was not very much for ordinary least squares regression technique. Hence it is of relevance to see how much gain in the efficiency is actually made by the RSS methodology over the SRS methodology. 12

13 3. Ranked Set Sampling Methodology and Estimation of the Mean The methodology of Ranked Set Sampling was introduced in 1952 by McIntyre. For a while it did not attract the attention it deserved, but there has been a surge in the interest in it over the last twenty years or so. Patil (1994), Barnett (1999), Chen, Bai and Sinha (2003), Wolfe (2004) may be seen as some of the landmark contributions. Let us first describe the basic framework of this methodology. It consists of the following stages. Suppose we are interested in a characteristic represented by the values of a random variable X taken by each of the units in the population. Let X have continuous probability distribution with c.d.f. F and p.d.f. f. Let µ be the expectation of X and σ 2 its variance. 13

14 (i) First choose k units from the population as a simple random sample. By some (hopefully inexpensive) procedure select the unit with the smallest value of X. Let it be X [1]. (ii) A second independent SRS of size k is chosen and the unit with the second smallest value X [2] of the character is selected. Similarly, continue this process until one has X [1], X [2],, X [k], a collection of independent order statistics from k disjoint collections of k simple random samples. 14

15 This constitutes the basic balanced Ranked Set Sample. These are independently distributed with respective p.d.f. s f (i) (x [i] ) = k! (i 1)!(k i)! (F (x [i])) i 1 (1 F (x [i] )) k i f(x [i] ), < x [i] <. 15

16 Note that we obtained k 2 SRS samples from the population, but retained only one (with the appropriate rank) from each group of k samples. So the total effort has been to rank k random samples of size k each, and retaining only k, the i-th (i = 1, 2,, k) order statistic from the i-th group respectively, which are to be actually measured. It is expected that ranking is cheap and measuring is expensive. Ranking k 2 observations and using k of them after measurement is thus a sampling method of boosting the efficiency of the statistical procedures from that based merely in groups of k (SRS) measurements. 16

17 In order to retain the reliability of the rankings, it is usually suggested that it be carried out in small groups, say upto 4, 5, 6 in size. However, this would limit both the versatility and efficiency of the procedures. Hence recourse is taken to replicating the whole process m times. Thus mk 2 units are examined, each group of k is ranked within itself and the i-th order statistic is obtained from m groups, leading to the data X [i]j, i = 1,, k, j = 1,, m. This constituted balanced RSS sampling. In unbalanced sampling instead of m replications for each order statistic, one will have m i replications of the i-th order statistic. 17

18 4. Estimation of Mean Let us consider the estimation of µ, the mean of F now. If we use k SRS observations then X, its sample mean is an unbiased estimator with variance σ 2 /k. The mean X = 1 k X [i] k i=1 of the k RSS observation is also unbiased for µ as seen below. E(X ) = 1 k E(X [i] ) k = 1 k = = i=1 k i=1 k! x (i 1)!(k 1)! (F (x)) i 1 (1 σ(x)) k i f(x)dx [ k ( ) k 1 x i 1 i=1 (F (x)) i 1 (1 F (x)) k i] f(x)dx xf(x)dx = µ. 18

19 Further we see that V (X ) = 1 k V (X k 2 [i] ) i=1 = 1 k E(X k 2 [i] E(X [i] ) 2 i=1 = 1 k 2[kσ2 ( µ (i) µ)2 ] = σ2 k 1 k 2 k (µ (i) µ)2 i=1 where µ (i) = E(X [i]) = V (X) 1 k 2 (µ (i) µ) 2 V (X). Hence, if the rankings are perfect then the RSS estimator of m has a smaller variance than the SRS estimator. This increase in the efficiency comes at the cost (if any) of ranking the k 2 observations, in addition to measuring the k selected order statistics. 19

20 5. Confidence Intervals for Quantiles If F is a continuous c.d.f. with the quantile of order p, q p = inf{x : F (x) p} then a standard nonparametric confidence interval for q p is provided by the order statistics. Let X 1,, X n be a random sample (SRS) of size n and X 1:n X n:n the order statistics from it then if we set P [X r:n q p X s:n ] s 1 ( ) x = p j (1 p) n j = 1 α, j j=r [X r:n, X s:n ], r < s, provides a (1 α)100% confidence interval for q p. 20

21 Let us now adopt the RSS methodology. We select n 2 independent observations. These are ranked in groups are n each. Then let X [1], X [2],, X [n] be the 1-st, 2-nd, n-th order statistics from these disjoint groups. Hence, although the marginal distribution of X [i] is the same as that of X [i:n], they further are independent. Then one interprets the interval [X [r], X [s] ], r < s, as a confidence interval for q p. Due to independence of X [r] and X [s], the confidence coefficient is P [X [r] q p X [s] ] P [X [r] q p ][1 P (X [s] q p )] n ( ) n = p j (1 p) n j j j=r n ( ) n 1 p j (1 p) n j j. j=1 21

22 One notes two properties here (i) E(X s:n X r:n ) = E(X [s] X [r] ). Hence both the SRS based and the RSS based confidence intervals have the same expected lengths. (ii) If r and s are so chosen that the confidence coefficient for the traditional SRS confidence interval is 1 α with probability α/2 in each tail then the confidence coefficient of the RSS based confidence interval is ( 1 α ) 2 α 2 = 1 α + 2 4, giving an increase of α2 4 in coverage property. 22

23 It is recognized that X [r] and X [s] being order statistics from independent SRS do not necessarily obey X [r] < X [s], so in some cases we may not have a proper interval at all. It is therefore suggested that we order the two statistics X [r] and X [s] as X (rs1), X (rs2) and use [X (rs1), X (rs2) ] as the confidence interval. The confidence coefficient of this modified interval is P [X (rs1) q p X (rs2) ] P [{X [r] q p X [s] } or {X [s] q p X [r] }] = P [X [r] q p X [s] ] + P [X [s] q p X [r] ] P (X [r] q p )P (q p X [s] ) +P (X [s] q p )P (q p X [r] ). 23

24 Again, if r and s are so chosen that the SRS confidence intervals leave probability α/2 in the tails then the probability of coverage of [X (rs1) X (rs2) ] is ( 1 α ) 2 ( α ) 2 α 2 + = 1 α This C.I. thus adds a further α 2 /4 to the confidence coefficient of the SRS C.I. However, this comes at the cost of some increase in the expected length of the new C.I.. It can be seen that E(X (rs2) X (rs1) ) = E F(ss) {2XF [r] (X) X} +E F(rr) {2XF [s] (X) X} and can be calculated for specific distributions. 24

25 To increase the flexibility of the confidence intervals, it is suggested that m independent groups of n 2 observations be obtained for ranking purposes. From these X [i]j, j = 1, 2,, m, i = 1,, n replicates of the i-th order statistic be obtained. These N independent order statistics be ordered from lowest to highest as X 1:N X N:N. Then 1 r < s N can be appropriately chosen to obtain nonparametric confidence intervals with confidence coefficient (1 α)100%. Fo details see Ozturk and Deshpande (2004). Further similar RSS based confidence intervals for quantiles of finite populations have been discussed in Deshpande, Frey and Ozturk (2006). 25

26 6. Judgment Rankings and Tests for Perfect Judgment The ranked set sampling protocol depends heavily on the ability to rank observations which are not measured. As the groups in which ranking is to be made become large such rankings become more difficult and thus more unreliable. So it is always preferred to actually rank observations only in small groups (usually restricted to 4, 5 or 6 units). There are other practical considerations as well. If two quadrats are adjacent then it is easy to visually compare them for their prospective yields. But if the two or more quadrats are far flung them to make comparative judgments is far more difficult. 26

27 The extensive ranked set sampling literature includes optimal procedures which rely on the assumption of perfect rankings and also those which are more robust with respect to the violation of this assumption. So an applied statistician who assumes that the rankings are perfect and chooses the appropriately optimal procedure faces the possibility that the procedures are not optimal and perhaps may not even be valid, if the rankings are imperfect. So as a preliminary procedure we suggest that a test for the perfectness of rankings may be used. 27

28 In Frey, Deshpande and Ozturk (2007) the following approach has been suggested. Let X [i]j, i = 1,, m, j = 1,, n j, be a ranked set sample. Here m observations are ranked and X [i]j is the i-th order statistic to be measured. This is done n i times j = 1, 2,, n i, giving us a full set of N = n 1 + n n m observations. Let R [i]j be the rank of X [i]j among these N observations. In case of perfect rankings one can find the probabilities of each of the N! possible rank vectors which are the permutations of {1, 2,, N}. The null hypothesis says that the distributions of the ranks follow the distribution of order statistics ranks. 28

29 This distribution although theoretically known, is not the usual equal probability for all possible rankings. In a very small example with m = 3, n 1 = n 2 = n 3 = 1 one can obtain the probabilities under the null hypothesis, of the vector R = (R [1] 1, R [2] 1, R [3] 1) as follows. R Prob / / / / / /105 A possible way to test the H 0 is to form the critical region as the union of the least likely (under H 0 ) outcomes. For example, the union {(2 3 1), (3 1 2), (3 2 1)} will provide a critical region with almost.05 as the probability of first type of error. This is in consonance with Fisher s approach of constructing tests of significance. 29

30 Another approach due to Neyman-Pearson requires, first of all, an alternative hypothesis. Heuristically, one may say that the alternative to perfect rankings is totally random ranking, i.e., those providing equal probability to each rank vector. This would indicate that the rankings are arbitrary and devoid of any information regarding the sizes of the observations. The Neyman-Pearson approach will reject the H 0 when the ratio of the probabilities of R under H 1 and under H 0 exceeds a threshold value so chosen that the probability of type I error is the specified α. It is quickly seen that this approach too leads to exactly the same critical region as the Fisherian approach described above. Also see Frey and Wang (2013). 30

31 However, the distribution theory of the ranks becomes too complicated for even moderate sized m and n i. Frey (2007) has provided a recursion formula for this purpose. But as is usual in nonparametric tests one investigates functions (linear or otherwise) of ranks whose exact distributions may be tabulated for small samples and asymptotic distributions may be obtained by appropriate versions of the central limit theorem. Two such statistics have been proposed by Frey, Ozturk and Deshpande (2007). Let R [i]. = 1 n i n i j=1 R [i]j, T i = 1 N (R [i]. E(R [i]. ) K = T QT where T = (T 1,, T m ), and and Q is the Moore-Penrose inverse of the asymptotic covariance matrix of T. The expectations and the covariances are under H 0. Then K has asymptotically, under H 0, the chi-squared distri- 31

32 bution with m 1 degree of freedom. Large values, indicating large departures of the observed R from its null expectation, will indicate evidence against H 0 and for some alternative hypothesis. 32

33 Another statistic proposed in the same paper is m n W = ir [i]j, i=1 j=1 the test rejecting for small values of the statistic. After standardization, its asymptotic distribution is the standard normal. We consider performance of these tests through simulated power for alternatives which are convex combinations of probabilities under H 0 and under the extreme random rankings. We find the test based on W consistently out-performing the one based on K. Similar work was undertaken by Vock and Balakrishnan (2011, 2013). In the first paper they propose a Jonkheere-Terpstra type test for perfect rankings and in the second paper they find that it is essentially the Frey, Ozturk, Deshpande (2007) test, the test statistics being linear functions of each other. 33

34 Example : One of the examples introduced earlier was about ranking and measuring the percent cover of leaves under various sprayer settings. The experiment had m = n = 5 giving a total of 25 observations. The ranks observed in the 5 groups of order statistics were (1, 5, 4, 6, 3), (2, 11,10, 8, 15), (22, 12, 14, 18, 13), (7, 9, 20, 19, 23), and (16, 25, 24, 17, 21). We find W = which has (simulated) p- value > Hence it may be concluded that there is insufficient evidence to reject the null hypothesis of perfect orderings. 34

35 7. Conclusions In this review we have only provided an introduction to the ranked set sampling methodology as applicable to environmental studies. Since its introduction in 1952 it has taken great strides in analysis of parametric and nonparametric models as introduced here. Further, problems like optimal estimation in the context of lognormal extreme value and other distributions are discussed by Barnett (1999). An easy to read introduction is available in Patil (2002). Chen, Bai and Sinha (2003) have an entire book on ranked set sampling. Ozturk and Deshapnde (2004) have proposed a new test for the nonparametric two sample scale problem. Newer contributions include more detailed power studies. Murff and Sagar (2006) have shown that the use of this methodology in ordinary least squares regression does improve the efficiency, but only marginally. 35

36 However, the basic result which states that if inexpensive (or cost free) ranking is incorporated along with measurements which are expensive, then RSS does provide some increase in efficiency of the procedure over its SRS version. 36

37 References [1] Barnett V., (1999), Ranked set sample design for environmental investigations, Env. Eco. Statist., 6, [2] Chen Z., Bai Z., Sinha B. K. (2003), Ranked Set Sampling, Springer. [3] Deshpande J. V., Frey H., Ozturk O, (2006), Nonparametric rank set sampling confidence intervals for quantiles of a finite population, Environ Eco. Statist., 13, [4] Frey J., Ozturk O., Deshpande J. V., (2007), Nonparametric tests for perfect judgment rankings, J.Am. Statisti. Assoc., 102, [5] Frey J., (2007), A note on probability involving independent order statistics, J. Statist. Comp. Sim., 77, [6] Frey J., Wang L., (2013), Most powerful tests to perfect rankings, Comp. Statist. Data An., 60, [7] McIntyre G. A., (1952), A method for 37

38 unbiased selective sampling, using ranked sets, Aust. J. Agr. Res., 3, [8] Murff E., Sager T., (2006), The relative efficiency of ranked set sampling in ordinary least squares regression, Environ Eco. Statist., 13, [9] Murray J. A., Ridout M. S., Cross J. V., (2000), The use of ranked set sampling in spray deposit assessment, Aspects of App. Bio., 57, [10] Ozturk O., Deshpande J.V., (2004), A new nonparametric test using ranked set data for a two sample scale problem, Sankhya, 66, [11] Ozturk O., Deshpande J. V., (2006), Ranked set sample nonparametric quantile confidence intervals, J. Statist. Planning Inf., 136, [12] Patil G. P., Sinha A. K., Taillie C., (1994), Ranked set sampling, in Handbook of Statistics, 12, Ed. G. P. Patil and C. R. Rao,

39 [13] Patil G. P., (2002), Ranked set sampling, Ency. Environmentrics, Ed. A. H. El-Sharawi, W. W. Piegorsch, 3, [14] Vock M., Balakrishnan N., (2011), A Jonkheere-Terpstra type test for perfect ranking in balanced ranked set sampling, J. Statist. Planing Inf., 141, [15] Vock M., Balakrishnan N., (2013), A connection between two nonparametric tests for perfect ranking in balanced ranked set sampling, Comm. statist.(th. Methods), 42, [16] Wolfe D. A., (2004), Ranked set sampling : An approach to more efficient data collection, Statist. Sc., 19, [17] Yu P.L.H., Lam K., (1997), Regression estimator in ranked set sampling, Biometrics, 53,