Weighting Methods in Survey Sampling


 Georgiana Holt
 2 years ago
 Views:
Transcription
1 Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiaohih Chang Ferry Butar Butar Abstrat It is said that a welldesigned survey an best prevent nonresponse. However, no matter how well a survey is designed, in pratie, nonresponse almost always ours. The easiest way to deal with nonresponse is to ignore it, but frequently, ignoring nonresponse results in poor survey quality. Item nonresponse and unit nonresponse are two types of nonresponse. Imputation proedures are popular remedies for the former while weighting methods are ommonly used to ompensate for the latter. This paper fouses on weighting methods that help redue nonresponse bias. Key Words: nonresponse, adjustment, alibration, missing data 1. Introdution In survey sampling, a good sample is desirable for making good inferenes about a population. The bias of a properly alulated estimate results from sampling error and nonsampling errors. Pratially, sampling error always exists beause of sampletosample variation. The only sure way to avoid sampling error is to study the entire population, i.e., ensus, and that is impratial for a huge population. Seletion bias is an example of nonsampling error, and it inludes nonresponse whih might greatly bias estimates alulated without adjustments. If there is nonresponse together with underoverage in survey sampling, missing data will result. Underoverage ours when not all of the elements in the target population are inluded in the sampling frame. Nonresponse an be divided into item nonresponse and unit nonresponse. In item nonresponse, at least one but not all of the measurements of interest are obtained from the sampled unit. In unit nonresponse, the sampled unit provides nothing. However, some information usually is available by other means. Example 1.1. Suppose a telephone survey is onduted to estimate household eletriity onsumption in a ertain region, and a sample of n individuals is drawn from the residential phone diretory. Therefore, the target population inludes all households in the region and the sampling frame is the list of people in the telephone diretory. Eah of the n individuals is asked the type of housing unit, the number of bedrooms, and the average monthly eletrial use see Table 1. Lohr 009 stated that missing data and haphazard mistakes in data olletion are often the biggest auses of error in a survey. Designing a good survey and questionnaire, and being areful in olleting data an prevent poor response rates and mistakes in data olletion. Followup and allbak are ommon proedures used to inrease response rate. In the telephone survey example, it is possible that the seleted person unavailable at the first all is ontated and there is a response at the kth allbak. k is a positive integer. Sine making phone alls may be expensive and timeonsuming, it is impratial to try to ontat someone a number of times. Furthermore, it is usually the ase that some people refuse to be interviewed no matter how many times an interviewer tries. Department of Mathematis and Statistis Sam Houston State University, Huntsville, TX Department of Mathematis and Statistis Sam Houston State University, Huntsville, TX
2 Setion on Survey Researh Methods JSM 01 Table 1: Hypothetial data for example Housing Unit Number of Average Monthly Individual Type Bedrooms Eletri Usage kw h a House 3,000 b House ,000 d Apartment f g  1,000 h Other  i Unit nonresponse: f. Item nonresponse: b,, d, g, h, i... Sine nonresponse almost always ours in a survey, one should be autioned if the nonrespondents differ ritially from the respondents, espeially when the response rate is low. Due to the presene of nonresponse and the plausible differene between respondents and nonrespondents, the sample is onsidered to be representative for the population of people who will answer survey questions but not for the target population. Therefore, inferenes based only on the respondents do not seem to be valid. When nonresponse is inevitable and nonignorable, there are ways to ompensate.. Simple Random Sampling SRS Simple random sampling is the most ommon probability sampling proedure. Although it is referred to as simple random sampling, this proedure is very important beause fully understanding simple random sampling is a prerequisite for further studies in other sampling tehniques. In simple random sampling, every element in the population has an equal probability of being seleted in the sample. Intuitively, eah subset of a fixed number of elements in the population has an equal probability of being seleted as a sample..1 Estimation in Simple Random Sampling In many sampling studies, the most ommon objetive is to estimate the population total, mean, or proportion. The estimation of population total is emphasized here beause estimated population total divided by the population size yields the estimated population mean and estimating proportion is a speial form of estimating mean. To derive the estimator of the population total, let U = {1,,..., N} and let S be a set of n elements hosen from U. Let U y = {y i i U} denote the population of size N, then y i is the survey harateristi of the ith unit and the simple random sample is denoted by S y = {y i i S}. Let t and t s be population total and sample total, respetively, then t = N y i and t s = y i. The i=1 i S 4769
3 Setion on Survey Researh Methods JSM 01 unbiased estimator of the population mean, Y = t N, 1 is simply the sample mean, y = t s. The population total is estimated by substituting y for n Y in 1, so ˆt = N n t s. Note that t and Y are unknown sine, in a survey, a sample is a small portion of the population, i.e., n < N. In fat, n is muh smaller than N, and this is the idea of sampling using n units to represent N units, i.e, one unit in the sample represents N/n units in the population. Suppose that the perfet representative sample is given, then a logial relation between the sample total and the population total should be t s : t = n : N. From this relation, one an see that the estimation of the population total is Let w i = N/n for all i S, then may be rewritten as ˆt = N n t s. ˆt = i S w i y i, 3 where w i is alled the sampling weight. Sampling weights are alulated to help simplify the alulation in many sample surveys. More importantly, making adjustments on sampling weights may help redue nonresponse bias. Example.1. A simple random sample of size 40 drawn from a population of size 10 is illustrated in Figure.1. In this sample design, w i = for all i S, i.e., one seleted unit represents 3 units inluding itself in the population. Cornfield 1944 introdued a useful method of deriving the expeted value and the variane of an estimator in sampling without replaement from a finite population. In order to use the method the researher must show that ˆt is an unbiased estimator of t and to obtain the variane of ˆt. Let Z i be a random variable suh that { 1, i S Z i = 0, i S. Note that E Z i = n/n, and E Z i Z j = n N N 1. By definition, given n and N, the variane of ˆt is V ar ˆt n, N [ = E ˆt t ] n, N, and it an be shown V ar ˆt n, N = N 1 n N S n 1 n, where S = N 1 1 N y i Y, where the unknown S an be i=1 4770
4 Setion on Survey Researh Methods JSM 01 estimated by the sample variane, s = n 1 1 y i y. Thus, i S V ar ˆt n, N = N 1 n s N n 4 is an unbiased estimator of V ar ˆt n, N.. Nonresponse in Simple Random Sampling Two widely disussed remedies for nonresponse are imputation and weighting adjustment. In imputation, the researher may fill in missing values by plausible values whih are generated from observed variables, hene imputation is the ommon remedy for item nonresponse. There are many imputation proedures used suh as hotdek imputation, olddek imputation, regression imputation, multiple imputation, and frational imputation. However, our researh fouses on weighting adjustment, in whih sampled units are lassified into lasses by auxiliary information, and then eah responding sampled unit is assigned an adjustment weight alulated by taking the inverse of the response rate of the orresponding lass. Responding sampled units in a lass with low response rate are assigned higher weights than those in a lass with a high response rate. Weighting adjustment is the ommon remedy for unit nonresponse, and it does not require filling in gaps in the data. A weighted estimator is unbiased under the assumption the probability of nonresponse is the same for all units within a lass. Although the assumption is usually not satisfied in pratie, it is more reasonable than to assume that the probability of nonresponse is the same for all units in the sample, hene nonresponse is ignorable. As a result, weighting adjustment might not eliminate nonresponse bias but it is useful for reduing nonresponse bias. Example.. The small data set in Table. is reated to mimi the situation in Example 1.1 in whih one an see both item nonresponse and unit nonresponse. This sample of size, n = 40, is drawn from the population of size, N = 10. The goal is to estimate t, the total average monthly eletriity onsumption in the region of interest. Y is the study variable; X 1, X, and X 3 are the auxiliary variables. X 3 is obtained from other available soures but not from the sampled units, hene it is always observed. Y, X 1, and X are subjet to nonresponse. Suppose that nonresponse an be totally ignored and only observed Y s are onsidered in the estimation, then ˆt and V ar ˆt n, N an be alulated by using 3 and 4, respetively. Although ˆt may not be affeted by the nonresponse, V ar ˆt n, N beomes large beause n is 6 instead of 40. Consequently, the onfidene interval may be too wide and does not provide muh information. Ignoring nonresponse is usually not a good idea. The following are some possible approahes to handling this data set. Fill in every gap in the data set by some imputation tehniques. Form lasses by X 3 and perform weighting adjustment. Fill in the gaps in auxiliary information and form lasses by the best possible set of auxiliary variables. Then, perform weighting adjustment. Fill in the gaps for the data points with suffiient auxiliary information. Weighting adjustment an then ompensate for the remaining nonresponse. Note that item nonresponse may be treated as unit nonresponse when a respondent answers too few questions. 4771
5 Setion on Survey Researh Methods JSM 01 Table : Hypothetial sample of 40 from a population of 10 ID X 1 X X 3 Y ID X 1 X X 3 Y Note: X 1 =1 house, = mobile home, =3 apartment,=4 other; X =1 one bedroom, = two bedrooms,= 3 three bedrooms,=4 at least four bedrooms; X 3 =1 the ommunity was establishd after 1950, = otherwise; Y denotes average monthly eletrial usage in kw h. Lohr 009 noted that response rates an be manipulated by defining them differently and presented several formulas that are used in surveys for alulating response rates. Here, for demonstration purposes, assume that only one variable is to be observed in a survey. Therefore, the alulation of response rates is straightforward. Let Y be the study variable subjet to nonresponse, and the response rates are alulated by dividing the number of observed Y s by the number of units in sample. Let m be the number of observations obtained from the simple random sample of size n drawn from the population of size N and let r 100% denote the response rate, where r = m n. Figure.1 displays the sample with 100%, a response rate that is nearly impossible to ahieve in surveys. The following example is an altered Example.1 in whih nonresponse ours. It shows the effet of ignoring nonresponse. Example.3. Consider the situation in Example 1.1 again. The goal is to estimate the total average monthly eletriity onsumption in the region of interest. Let Y denote the average monthly eletrial usage. Suppose that the sample is shown in Figure.3 in whih and represent seleted persons who do not give information about their average monthly eletrial usage while and represent seleted persons who answer the question about their average monthly eletrial usage. The response rate is 65% beause m = 6 and n = 40. Sine the sample is drawn from a telephone diretory, seleted persons addresses are usually available notwithstanding nonresponse. This additional information enables the researher to find out in what ommunity a person lives. Suppose it is observed that housing units in old ommunities are generally larger than those in new ommunities. Also, it is reasonable to state that people who live in large properties normally have higher eletrial 477
6 Setion on Survey Researh Methods JSM 01 usage than those who live in small properties. Ideally, one would like to form lasses by some harateristis and then view the group of respondents in eah lass as a simple random sample of the orresponding lass. Consider only Y and X 3, then the data set in Table. an be lassified into two groups by the age of the ommunity, X 3. One an visually tell the differene in response rate between the two lasses in Figure.4. Class I is omprised of n 1 = 30 people and m 1 = 4 out of 30 people are respondents. Thus, the response rate in lass I is 80%, i.e., r 1 = 0.8. Class II is omprised of n = 10 people and only m = out of 10 people are respondents. Thus, the response rate in lass II is 0%, i.e., r = 0.. Suppose that people in lass II onsume more eletriity than people in lass I, then ignoring nonresponse an seriously bias the estimate. To show that, use the data of the 6 respondents to estimate the total average monthly eletriity onsumption, t. This an be thought of as taking a simple random sample of size, m = 6, drawn from the population of size, N = 10, so here, S is a set ontaining 6 respondents IDs. The sampling weight is N/m. Use 3 to find ˆt 49, 31. As an example, assume that the sample is representative and the lassifiation is perfet, then the group of respondents in eah lass an be viewed as a simple random sample obtained from a orresponding subpopulation. Let N 1 and N be the number of members in subpopulation I and the number of members in subpopulation II, respetively. Sine the sample is representative, then n N = n N holds for = 1,. Therefore, N 1 = 90 and N = 30. Let y 1 and y be the mean average monthly eletrial usage in lass I and II, respetively. So, y 1 =, 000 and y = 3, 000. Let Y 1 and Y be the mean average monthly eletrial usage in subpopulation I and II, respetively, then t = N 1 Y 1 + N Y. The reasonable estimate of t should be: ˆt adj = N 1 y 1 + N y = 70, 000. If y is lose to Y for = 1,, then the differene between ˆt adj and t will be small. Assume that y 1 = Y 1 and y = Y, then ˆt adj = t. Compare ˆt with ˆt adj, one may onlude that ˆt underestimates t, and ˆt t is large if: for a fixed r r 1 0, Y Y 1 is large. for a fixed Y Y 1 0, r r 1 is large. If r = r 1 or Y = Y 1, then ˆt = t..3 Weighting Adjustment in Simple Random Sampling In general, suppose that the lassifiation forms C lasses. The set, S, ontaining the identifiation numbers of sampled units an be partitioned so that S = C S. Let U = 4773
7 Setion on Survey Researh Methods JSM 01 Table 3: The omposition of sample response and nonresponse 1... C Note m m 1 m m C m = C n m n 1 m 1 n m n C m C n m n n 1 n n C n = C m n N N 1 N N C N = C r m 1 n 1 m n m C n C m n N S S. There exist partitions U 1, U,, U C, suh that S U for = 1,,, C. Let A be the set that ontains the identifiation numbers of responding units in lass, so A S for = 1,,, C. Let N, n, and m denote the number of elements in U, S, and A, respetively, and assume that m > 1 for = 1,,, C. So, m = C m, n = C n, and N = C N. The overall response rate and response rate for eah lass an be alulated and ompared by onstruting a table similar to Table 3. Suppose that eah of the n units has the same probability of responding, then m responding units may be interpreted as a simple random sample drawn from the n sampled units. To interpret in terms of weighting, the m units is to represent the n units, so eah of the m units has a weight of n. For = 1,,, C, let y i be the observation of the study having the m identifiation number, i, and let t s = y i. Let t = C t and t = y i, then t i S i U an be estimated by N t s. Sine not every y i where i S is available, t s should be n estimated. Let t a = y i, then ˆt s = n t a. If N is known for = 1,,, C, i A m then ˆt = N ˆt s = N n t a = N y i. n n m m i A It an be regarded as using a simple random sample of size m to represent the population of size N. Therefore, the adjusted estimator of t is ˆt adj = C i A w i y i 5 where w i = N /m, and it an be expressed as w i = w bi w adji where w bi = N /n is alled the base weight in lass and w adji = n /m is alled the nonresponse adjustment weight in lass. In 5, N should be known for all = 1,,, C, and ˆt adj is also alled the poststratified estimator of t that may also be employed in adjusting underoverage. Generally, ˆt adj should provide a better estimate than ˆt = N i A m y i where nonresponse is totally ignored. To show that ˆt is an unbiased estimator of t, follow the method proposed by Cornfield 4774
8 Setion on Survey Researh Methods JSM and define: Z i = { 1, i S 0, i S { and A i = 1, i A 0, i A. Thus, E Z i = n /N, E A i = m /n, and E Z i A i = m /N. For = 1,,, C, let u = m, n, N, then it an be shown that E ˆt u = t. Sine the m responding units an be regarded as a simple random sample of size m representing the population of size N for = 1,,, C, then N V ar ˆt u = N 1 m N S m 6 where S = N 1 1 y i t /N. Note that 6 is just the analogous to the i U variane of t seen in Setion.1. Therefore, E ˆt adj u C N = E i A y i u = m C E ˆt C u = t = t, and V ar ˆt adj u C = V ar ˆt u = C V ar ˆt u = C 1 m N S m, where u = m 1,, m C, n 1,, n C, N 1,, N C. In the poststratified estimator, the estimated variane 5 is where s = m 1 1 V ar ˆt adj u = C N 1 m s 7 N m = S. y i t a /m ; E s i A So far, the estimation an be done when N is available for = 1,,, C. However, if this is not the ase, onsider the example of the perfet simple random sample in Example.3 in whih the ratio of eah sub sample size to its orresponding subpopulation size is equal to the ratio of the sample size to the population size. Although this ondition is almost impossible to satisfy in a single sampling ativity, the sampling distribution of n /N should be entered at n/n. That is En /N = n/n and EN /n = N/n. Therefore, t an be estimated by t adj = C i A w iy i 8 where wi = N n and t adj is alled the weighting lass estimator of t. n m Let u = m 1,, m C, n 1,, n C ; therefore, it an be shown that the expeted value of t adj given u, that is E t adj u = t + C n N t t t. The variane n N of t adj given u an be estimated by substituting n n N for N in 7. Thus, V ar t adj u = C N n n 1 n N m s. 9 n m Consider the weighting lass estimator. Oh and Sheuren 1983 proposed that an approximate 1001 α% onfidene interval for t may be onstruted by t adj ± z α/ MSE t adj u
9 Setion on Survey Researh Methods JSM 01 where z α/ is the 1 α/th perentile of the standard normal distribution and MSE t adj u = V ar t adj u [ + Bias t adj u ] C = N n 1 n n N m s + N n n m N 1 3. Stratified Random Sampling ST R C n n N ta t adj. m In simple random sampling, it is possible that some ertain groups of sampling units in the population are underrepresented and hene, will result a bad sample. Stratified random sampling enables one to avoid having a bad sample and to improve the preision of estimates for a fixed sample size or to ahieve the same level of preision with a sample size not as large as in simple random sampling. In stratified random sampling, the population of size N is divided into H subpopulations alled strata. Let stratum h have size N h for h = 1,,, H. The sample is obtained by randomly seleting n h sampling units from stratum h for h = 1,,, H. Therefore, H simple random samples are obtained in the stratified random sampling. Stratified random sampling works well when the variation in the measurement of interest is low within the strata but is high between the strata. Suppose that the total sample size is n so n = H n h. To alloate the sample among the strata may depend on the sample design or some onstraints suh as osts. Disussions of the alloation of sampling units among the strata an be found in Cohran 1977 and Lohr Estimation in Stratified Random Sampling Sine a proedure of simple random sampling is implemented in eah stratum, the results of the preeding setions an be applied diretly to eah stratum. Let index set, U = {1,,, N}, denote the population of size N. Let U h denote stratum h of size N h for h = 1,,, H suh that U = H U h, N = H N h, and U h U k = for h k. Let S h denote the simple random sample of size n h drawn form U h for h = 1,,, H and let S = H S h. Therefore the total sample size is n = H n h. Let y hi be the measurement of interest in stratum h, then t h = y hi and t str = H t h are the total in stratum h and the i U h population total, respetively. The unbiased estimator of t h is ˆt h = i Sh w strhi y hi 11 where w strhi = N h /n h, and the variane of ˆt h given n h and N h is V ar ˆt h n h, N h = Nh 1 n h/n h Sh /n h where Sh = N h 1 1 yhi Y h and Y h = t h /N h. i U h The sample variane obtained from stratum h is s h = n h 1 1 y hi y h where i S h y h = n 1 h y hi and E s h = S h. Therefore, i S h V ar ˆt h n h, N h = N h 1 n h s h. 1 N h n h 4776
10 Setion on Survey Researh Methods JSM 01 Sine the population total is t str = H t h, then it an be estimated by ˆt str = H ˆt h. Thus, ˆt str = H w strhi y hi, and E ˆt str q = H E ˆt H h n h, N h = t h = t str, where i S h q = n 1,, n H, N 1,, N H. The variane of ˆt str given q an be estimated by V ar ˆt str q = H beause V ar ˆt str q H = V ar ˆt h q N h = H 1 n h s h 13 N h n h Nh 3. Nonresponse in Stratified Random Sampling 1 n h S h. N h n h In stratified random sampling, the researher deals with multiple simple random samples. Suppose that there are H strata and a simple random sample of size n h is obtained from eah of the H strata having size N h. Thus, there are H simple random samples. Consider only unit nonresponse and for h = 1,,, H, let m h denote the number of the observations obtained from the n h sampled units. If m h < n h whih is almost always the ase in surveys then nonresponse ours in the sample obtained from stratum h and estimation without adjustments might not produe the best results for making inferenes about the subpopulation as well as the population. 3.3 Weighting Adjustment in Stratified Random Sampling Suppose that a stratified random sampling proedure is arried out and there exists nonresponse. Assume that it is possible to divide the sample seleted from stratum h into C h lasses by some available auxiliary information for h = 1,,, H suh that every sampled units in the same lass has an equal probability of responding. Let set, S h, denote lass having size n h in the sample seleted from stratum h for = 1,,, C h and h = 1,,, H, then S h = C h S h where S h S hk = for h k and n h = C h n h. For h = 1,,, H, let U h1, U h,, and U hch be the substrata of U h and eah of them has size N h1, N h,, and N hch, respetively, so that N h = C h N h and S h U h. Let A h denote the group of responding units in lass of the sample drawn from stratum h and let m h be the size of A h. For h = 1,,, H, r h denotes the response rate in S h. In poststratified approah, the base weight, w bhi = N h /n h, indiates that one unit in S h is to represent N h /n h units in substratum h. The nonresponse adjustment weight, w adjhi = n h /m h, indiates that one unit in A h is to represent n h /m h units in S h. The final weight, w hi = N h /m h is w hi = w bhi w adjhi. 14 This indiates that one unit in A h represents N h /m h units in substratum h. Using the tehniques in the preeding setions, one an find an unbiased estimator, given u h = m h, n h, N h, for t h, the total of substratum h: ˆt h = i A h w hi y hi
11 Setion on Survey Researh Methods JSM 01 The variane of ˆt h given u h may be estimated by V ar ˆt h u h = N h 1 m h s h N h m h where s h = 1 y hi t a h and t m h 1 i A h m = ah y hi. Moreover, for h = h i A h 1,,, H, the poststratified estimator of the total of stratum h is C h ˆt h adj = i A h w hi y hi 16 and E [ ] ˆt h adj u h = th where u h = m h1,, m hch, n h1,, n hch, N h1,, N hch sine E ˆt h u h = th. Thus, for h = 1,,, H, the variane of ˆt h adj given u h [ ] an be estimated by V ar ˆt h adj u h, by summing V ar ˆt h u h from = 1 to C. Therefore, using a poststratified approah, the population total an be estimated by C h ˆt str adj H = i A h w hi y hi 17 [ ] and it is unbiased given u = u 1 ; u ; ; u H sine E ˆt h adj u h = t h. The variane of ˆt str given u may be estimated by adj [ ˆt V ar adj ] str u = H C h N h 1 m h s h. 18 N h m h Suppose that the N h is not available for = 1,,, C h and h = 1,,, H, then onsider the weighting lass estimator: t h = i A h w hi y hi where w hi = w strhi w adjhi. That is replaing w bhi, = 1,,, C h with w strhi t h adj = C h t h and t str adj = H t h the population total may be estimated by adj = N h n h in 14. Furthermore,. Therefore, using weighting lass approah, C h t str adj H = i A h w hi y hi 19 Let u h [ = m h1,, ] m hch, n h1,, n hch and let u = u 1 ; u ; ; u H. To show that E t str adj u t str, define: Z hi = { 1, i S h 0, i S h { and A hi = 1, i A h 0, i A. h It an be shown that E Z hi = n h, E A hi = m h, E Z hi A hi = m h N h n h [ 1,,, C h and h = 1,,, H, and E t str adj u = t str + H C h n h n h for = N h Nh t h t h N h. 4778
12 Setion on Survey Researh Methods JSM 01 The variane of t str in 18. That is [ t V ar adj ] str u = adj given u may be estimated by replaing N h with n h n h H C h N h N h nh 1 n h m h s h. 0 n h N h n h m h If proportional alloation is used in the sample design, then n h = n for all h = 1,,, H. N h N In 19, w hi beomes N n w adj hi and 0 an be written as [ t V ar adj ] str u = N n H C h n h 1 n N [ t Oh and Sheuren 1983 proposed MSE adj ] str u as [ t MSE adj ] [ t str u = V ar adj ] str u + { [ ]} and Bias t str adj H u = N h n h N h 1 C h n h n h { Bias 4. TwoStage Cluster Sampling m h s h. n h m h [ ]}, t str adj u [ N h ta h t ] h. m adj h A multistage random sample is onstruted by seleting a sample in at least two stages. Sampling in stages enables one to redue the population and to ombine sampling proedures. Suppose the population an be divided into a number of subpopulations. Contrary to stratifiation, it is expeted that the variane with respet to the measurements of interest is high within eah subpopulation, whereas the variane with respet to the measurements of interest is low between subpopulations. The subpopulations are alled lusters here, and they are usually naturally formed from, for example, ommunities, ity bloks, and shools. In the first stage, a random sample of lusters is obtained, and a number of sublusters are formed within eah seleted luster for seletion in the next stage. The subgroups formed for seletion in eah stage prior to the final stage an be alled the population units in general, and the proess of seleting population units within eah population unit obtained in the previous stage an be repeated until the sizes of the population units meet the requirements. In the last stage, a random sample of sampling units is obtained from eah seleted population unit. 4.1 Estimation in TwoStage Cluster Sampling Suppose that the population onsists of L lusters, then in the first stage, let a simple random sample of l lusters be obtained. Let τ be the population total and let t g be the total of luster g for g = 1,, L. Let index set U = {1,,, L} denote the population, then τ = t g. Let S denote the simple random sample of size l drawn from U, then an g U unbiased estimator of τ is ˆτ = w g t g
13 Setion on Survey Researh Methods JSM 01 where w g = L/l. The variane of ˆτ given l and L an be estimated by V ar ˆτ l, L = L 1 l/l s t where s t = l 1 1 tg t and t = l 1 t g. These estimators l are usable in onestage luster sampling, but additional work is required in twostage luster sampling. Sine, in the seond stage, simple random sample g is to be drawn from luster g for all g S, then t g in 1 is not observed but should be estimated. Let U g denote luster g having size N g, and let S g denote simple random sample g having size n g drawn from U g for g S. Let y gi be the measurement of interest obtained from sampling unit i in luster g, then for g S, an unbiased estimator of t g is ˆt g = i Sg w gi y gi where w gi = N g. The variane of ˆt g given n g and N g an be estimated by V ar ˆt g n g, N g n g Combining 1 and, the unbiased estimator of τ for twostage luster sampling is given by ˆτ = w g w gi y gi. 3 i S g Lohr 009 proved the variane of ˆτ has two omponents: the variability between lusters and the variability of sampling units within eah luster. The variane of ˆτ an be estimated by V ar ˆτ l, L, q = L 1 L l s ˆt l + L l N g 1 n g s g 4 N g n g where q = { n g, N g g S }, s = l 1 1 ˆt ˆt, ˆt g and ˆt = l 1 ˆt g. 4. Nonresponse in TwoStage Cluster Sampling Nonresponse appears in the seond stage of twostage luster sampling. If a single luster is seleted in the first stage, then the nonresponse problem in the sample drawn from this luster is idential to that in a regular simple random sampling. However, here one deals with l simple random samples with nonresponse. For g S, suppose that there are m g responding units in simple random sample g, then the response rate for sample g is r g = m g n g. Nonresponse appears if r g < Weighting Adjustment in TwoStage Cluster Sampling Assume that it is possible to divide sample g into C g lasses by some available auxiliary information for all g S so that eah unit in a lass has an equal probability of responding. Let S g denote lass of sample g onsisting of n g units and let A g S g denote a set of m g responding units in lass of sample g for = 1,,, C g within eah g S, C g C g then n g = n g and m g = m g. For g S, let U g1, U g,, and U gcg be the subluster of U g having size N g1, N g,, and N gcg, respetively, so that N g = C g N g 4780
14 Setion on Survey Researh Methods JSM 01 and S g U g. Suppose that N g are known for all g S and = 1,,, C g. The poststratified estimator for the total of luster g where g S is ˆt g adj = Cg i A g w gi y gi 5 where w gi = w bgi w adjgi, w bgi = N g, and w adjgi = n g. Thus, w gi = N g. The n g m g m g variane of ˆt g adj given u g = m g1,, m gcg, n g1,, n gcg, N g1,, N gcg is estimated by V ar [ ] C g ˆt g adj u g = Ng 1 m g s g where s g = 1 ygi y N g m g m g g i A g and y g = 1 y gi. m g i A g Sine ˆτ = w gˆt g, then ˆτ adj = w g ˆt g. When using poststratified estimators adj in the seond stage to ompensate for nonresponse, the unbiased estimator of the population total is ˆτ adj = C g w g w gi y gi. 6 i A g The variane of ˆτ adj given u = {u g g S} may be estimated by V ar ˆτ adj l, L, u = L 1 l s ˆt L l + L l C g N g where s ˆt = l 1 1 [ ˆt g adj ˆt adj ], and ˆt adj = l 1 ˆt g 1 m g s g N g m g If using weighting lass adjustment, then one an replae w gi with w gi = N g n g w adjgi in 5. Thus, the weighting lass estimator of t g for g S is t g adj = Cg adj. i A g w giy gi. 7 Let u g =. m g1,, m gcg, n g1,, n gcg Given u g, for g S, the variane of t g adj and squared bias of t g may be estimated by adj and [ V ar t g { [ Bias t g adj u g adj u g ] C g = N g ng 1 n g m g s g n g N g n g m g ]} = N g n g N g 1 C g n g n g [N g y g t g adj], respetively. When using weighting lass estimators in the seond stage to ompensate for nonresponse, the estimator of τ is τ adj = C g w g i A g w giy gi
15 Setion on Survey Researh Methods JSM 01 Given u = { u g g S }, the variane of τ adj and squared bias of τ adj may be estimated respetively by V ar τ adj l, L, u = L 1 l s t + L L l l and C g [ Bias τ adj l, L, u ] L N g n g = l N g 1 N g ng 1 n g m g s g n g N g n g m g C g where s t = l 1 1 [ t g adj t adj ], and t adj = l 1 t g n g n g [N g y g t g An approximate α % onfidene interval for τ may be onstruted by τ adj ± z α/ MSE τ adj l, L, u 9 where MSE τ adj l, L, u = V ar τ adj l, L, u [ + Bias τ adj l, L, u ] and z α/ is the 1 α/th perentile of the standard normal distribution. 5. Conlusion Two very ommon weighting adjustments for unit nonresponse, poststratifiation and weighting lass methods, are desribed in simple random sampling, stratified random sampling, and twostage luster sampling. It is expeted that these methods an be used to redue nonresponse bias. It is inappropriate to fully rely on these weighting adjustments beause the assumption of forming a group in whih the nonresponse is ompletely negligible is rarely satisfied in the real world. A researher should always try to avoid low response rates. Furthermore, item nonresponse and unit nonresponse normally appear together; thus, some imputation tehniques may be neessary as omplementary remedies sine filling in missing values by using some known information ould be effiient and useful for obtaining desirable lassifiation in weighting adjustments. adj. adj ] REFERENCES Cohran, W. G. 1977, Sampling Tehniques, third edition. Wiley, New York. Cornfield, J On Samples from Finite Populations, Journal of the Amerian Statistial Assoiation, 39: 6, Holt, D., and Elliot, D. 1991, Methods of Weighting for Unit NonResponse, Journal of the Royal Statistial Soiety. Series D The Statistiian, 40: 3, Holt, D., and Smith, T. M. F. 1979, Post Stratifiation, Journal of the Royal Statistial Soiety. Series A General, 14: 1, Little, R. J. A., and Rubin, D. B. 00, Statistial Analysis with Missing Data, seond edition. Wiley, New York. Lohr, S. L. 009, Sampling: Design and Analysis, seond edition. Brooks/Cole, Cengage Learning, Boston, MA. Oh, H. L., and Sheuren, F. J. 1983, Weighting Adjustment for Unit Nonresponse, In W. G. Madow, I. Olkin, and D. B. Rubin Eds., Inomplete Data in Sample Surveys, Volume : Theory and Bibliographies pp Aademi Press, New York. Smith, T. M. F. 1991, PostStratifiation Journal of the Royal Statistial Soiety. Series D The Statistiian, 40: 3,
Static Fairness Criteria in Telecommunications
Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma 92002 Mat208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, email: vesatimonen@hutfi
More informationChapter 1 Microeconomics of Consumer Theory
Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deisionmakers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationCHAPTER 9: FUNDAMENTALS OF HYPOTHESIS TESTING: ONESAMPLE TESTS
228 CHAPTER 9: FUNDAMENTALS OF HYPOTHESIS TESTING: ONESAMPLE TESTS 1. Whih of the following would e an appropriate null hypothesis? a) The mean of a population is equal to 55. ) The mean of a sample is
More informationA Holistic Method for Selecting Web Services in Design of Composite Applications
A Holisti Method for Seleting Web Servies in Design of Composite Appliations Mārtiņš Bonders, Jānis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,
More informationSet Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos)
A.1 Set Theory and Logi: Fundamental Conepts (Notes by Dr. J. Santos) A.1. Primitive Conepts. In mathematis, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existene
More informationComputer Networks Framing
Computer Networks Framing Saad Mneimneh Computer Siene Hunter College of CUNY New York Introdution Who framed Roger rabbit? A detetive, a woman, and a rabbit in a network of trouble We will skip the physial
More informationAn Iterated Beam Search Algorithm for Scheduling Television Commercials. Mesut Yavuz. Shenandoah University
0080569 An Iterated Beam Searh Algorithm for Sheduling Television Commerials Mesut Yavuz Shenandoah University The Harry F. Byrd, Jr. Shool of Business Winhester, Virginia, U.S.A. myavuz@su.edu POMS 19
More informationComputational Analysis of Two Arrangements of a Central GroundSource Heat Pump System for Residential Buildings
Computational Analysis of Two Arrangements of a Central GroundSoure Heat Pump System for Residential Buildings Abstrat Ehab Foda, Ala Hasan, Kai Sirén Helsinki University of Tehnology, HVAC Tehnology,
More information1.3 Complex Numbers; Quadratic Equations in the Complex Number System*
04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x  2x  22 x  2 2 ; x
More informationA Comparison of Service Quality between Private and Public Hospitals in Thailand
International Journal of Business and Soial Siene Vol. 4 No. 11; September 2013 A Comparison of Servie Quality between Private and Hospitals in Thailand Khanhitpol Yousapronpaiboon, D.B.A. Assistant Professor
More informationAUDITING COST OVERRUN CLAIMS *
AUDITING COST OVERRUN CLAIMS * David PérezCastrillo # University of Copenhagen & Universitat Autònoma de Barelona Niolas Riedinger ENSAE, Paris Abstrat: We onsider a ostreimbursement or a ostsharing
More informationCapacity at Unsignalized TwoStage Priority Intersections
Capaity at Unsignalized TwoStage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minorstreet traffi movements aross major divided fourlane roadways
More informationChemical Equilibrium. Chemical Equilibrium. Chemical Equilibrium. Chemical Equilibriu m. Chapter 14
Chapter 14 Chemial Equilibrium Chemial Equilibriu m Muh like water in a Ushaped tube, there is onstant mixing bak and forth through the lower portion of the tube. reatants produts It s as if the forward
More information5. Hypothesis Tests About the Mean of a Normal Population When σ 2 is Not Known
Statistial Inferene II: he Priniples of Interval Estimation and Hypothesis esting 5. Hypothesis ests About the Mean of a Normal Population When σ is Not Known In Setion 3 we used the tdistribution as
More informationChapter 5 Single Phase Systems
Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude
More informationSoftEdge Flipflops for Improved Timing Yield: Design and Optimization
SoftEdge Flipflops for Improved Timing Yield: Design and Optimization Abstrat Parameter variations ause high yield losses due to their large impat on iruit delay. In this paper, we propose the use of
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationThe Optimal Deterrence of Tax Evasion: The Tradeoff Between Information Reporting and Audits
The Optimal Deterrene of Tax Evasion: The Tradeoff Between Information Reporting and Audits Yulia Paramonova Department of Eonomis, University of Mihigan Otober 30, 2014 Abstrat Despite the widespread
More informationOptimal Sales Force Compensation
Optimal Sales Fore Compensation Matthias Kräkel Anja Shöttner Abstrat We analyze a dynami moralhazard model to derive optimal sales fore ompensation plans without imposing any ad ho restritions on the
More informationClassical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk
Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples
More informationTrigonometry & Pythagoras Theorem
Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises
More informationIsaac Newton. Translated into English by
THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY (BOOK 1, SECTION 1) By Isaa Newton Translated into English by Andrew Motte Edited by David R. Wilkins 2002 NOTE ON THE TEXT Setion I in Book I of Isaa
More informationMagnetic Materials and Magnetic Circuit Analysis
Chapter 7. Magneti Materials and Magneti Ciruit Analysis Topis to over: 1) Core Losses 2) Ciruit Model of Magneti Cores 3) A Simple Magneti Ciruit 4) Magneti Ciruital Laws 5) Ciruit Model of Permanent
More informationSymmetric Subgame Perfect Equilibria in Resource Allocation
Proeedings of the TwentySixth AAAI Conferene on Artifiial Intelligene Symmetri Subgame Perfet Equilibria in Resoure Alloation Ludek Cigler Eole Polytehnique Fédérale de Lausanne Artifiial Intelligene
More informationElectrician'sMathand BasicElectricalFormulas
Eletriian'sMathand BasiEletrialFormulas MikeHoltEnterprises,In. 1.888.NEC.CODE www.mikeholt.om Introdution Introdution This PDF is a free resoure from Mike Holt Enterprises, In. It s Unit 1 from the Eletrial
More informationHierarchical Clustering and Sampling Techniques for Network Monitoring
S. Sindhuja Hierarhial Clustering and Sampling Tehniques for etwork Monitoring S. Sindhuja ME ABSTRACT: etwork monitoring appliations are used to monitor network traffi flows. Clustering tehniques are
More informationSupply chain coordination; A Game Theory approach
aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 upply hain oordination; A Game Theory approah JeanClaude Hennet x and Yasemin Arda xx x LI CNRUMR 668 Université
More informationTEACHING FUNDAMENTALS OF ELECTRICAL ENGINEERING: NODAL ANALYSIS
TACHING FUNDAMNTALS OF LCTICAL NGINING: NODAL ANALYSIS Đorđević, A..; Božilović, G.N.; Olćan, D.I.; Sh. of letr. ng., Univ. of Belgrade, Belgrade, Serbia 00 I. Personal use of this material is permitted.
More informationRELEASING MICRODATA: DISCLOSURE RISK ESTIMATION, DATA MASKING AND ASSESSING UTILITY
Setion on Survey Researh Methods JSM 008 RELEASING MICRODATA: DISCLOSURE RISK ESTIMATION, DATA MASKING AND ASSESSING UTILITY Natalie Shlomo 1 1 Southampton Statistial Sienes Researh Institute, University
More informationchapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374
CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems
More informationA Keyword Filters Method for Spam via Maximum Independent Sets
Vol. 7, No. 3, May, 213 A Keyword Filters Method for Spam via Maximum Independent Sets HaiLong Wang 1, FanJun Meng 1, HaiPeng Jia 2, JinHong Cheng 3 and Jiong Xie 3 1 Inner Mongolia Normal University 2
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More informationTheory of linear elasticity. I assume you have learned the elements of linear
Physial Metallurgy Frature mehanis leture 1 In the next two letures (Ot.16, Ot.18), we will disuss some basis of frature mehanis using ontinuum theories. The method of ontinuum mehanis is to view a solid
More informationFOOD FOR THOUGHT Topical Insights from our Subject Matter Experts
FOOD FOR THOUGHT Topial Insights from our Sujet Matter Experts DEGREE OF DIFFERENCE TESTING: AN ALTERNATIVE TO TRADITIONAL APPROACHES The NFL White Paper Series Volume 14, June 2014 Overview Differene
More informationChapter 5 lecture Notes
Lebanese Amerian University From the SeletedWorks of Ghassan Dibeh 04 Chapter 5 leture Notes Ghassan Dibeh, Lebanese Amerian University Available at: http://works.bepress.om/ghassan_dibeh/00/ Chapter
More informationTELECOMMUNICATION SYSTEMS AND TECHNOLOGIES Vol. I  Analog and Digital Transmission of Data  Simon Haykin ANALOG AND DIGITAL TRANSMISSION OF DATA
ANALOG AND DIGITAL TRANSMISSION OF DATA Simon MMaster University, Canada Keywords: Modulation and demodulation, Coherent detetion, Amplitude modulation, Double sidebandsuppressed arrier modulation, Single
More informationIntelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds
Intelligent Measurement Proesses in 3D Optial Metrology: Produing More Aurate Point Clouds Charles Mony, Ph.D. 1 President Creaform in. mony@reaform3d.om Daniel Brown, Eng. 1 Produt Manager Creaform in.
More informationUse of Track Geometry Measurements for Maintenance Planning
84 TRANSPORTATION RESEARCH RECORD 147 Use of Trak Geometry Measurements for Maintenane Planning WILLEM EBERSOHN AND ERNEST T. SELIG Trak geometry measurements are disussed as a means of evaluating the
More informationDeadlinebased Escalation in ProcessAware Information Systems
Deadlinebased Esalation in ProessAware Information Systems Wil M.P. van der Aalst 1,2, Mihael Rosemann 2, Marlon Dumas 2 1 Department of Tehnology Management Eindhoven University of Tehnology, The Netherlands
More informationTHE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES
Proeedings of FEDSM 98 998 ASME Fluids Engineering Division Summer Meeting June 225, 998 Washington DC FEDSM98529 THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES John D. Wright Proess
More informationAn Efficient Network Traffic Classification Based on Unknown and Anomaly Flow Detection Mechanism
An Effiient Network Traffi Classifiation Based on Unknown and Anomaly Flow Detetion Mehanism G.Suganya.M.s.,B.Ed 1 1 Mphil.Sholar, Department of Computer Siene, KG College of Arts and Siene,Coimbatore.
More informationAn integrated optimization model of a Closed Loop Supply Chain under uncertainty
ISSN 18166075 (Print), 18180523 (Online) Journal of System and Management Sienes Vol. 2 (2012) No. 3, pp. 917 An integrated optimization model of a Closed Loop Supply Chain under unertainty Xiaoxia
More informationGranular Problem Solving and Software Engineering
Granular Problem Solving and Software Engineering Haibin Zhu, Senior Member, IEEE Department of Computer Siene and Mathematis, Nipissing University, 100 College Drive, North Bay, Ontario, P1B 8L7, Canada
More informationCHAPTER 14 Chemical Equilibrium: Equal but Opposite Reaction Rates
CHATER 14 Chemial Equilibrium: Equal but Opposite Reation Rates 14.1. Collet and Organize For two reversible reations, we are given the reation profiles (Figure 14.1). The profile for the onversion of
More informationSpecial Relativity and Linear Algebra
peial Relativity and Linear Algebra Corey Adams May 7, Introdution Before Einstein s publiation in 95 of his theory of speial relativity, the mathematial manipulations that were a produt of his theory
More informationOpen and Extensible Business Process Simulator
UNIVERSITY OF TARTU FACULTY OF MATHEMATICS AND COMPUTER SCIENCE Institute of Computer Siene Karl Blum Open and Extensible Business Proess Simulator Master Thesis (30 EAP) Supervisors: Luiano GaríaBañuelos,
More informationprotection p1ann1ng report
f1re~~ protetion p1ann1ng report BUILDING CONSTRUCTION INFORMATION FROM THE CONCRETE AND MASONRY INDUSTRIES Signifiane of Fire Ratings for Building Constrution NO. 3 OF A SERIES The use of fireresistive
More informationJEFFREY ALLAN ROBBINS. Bachelor of Science. Blacksburg, Virginia
A PROGRAM FOR SOLUtiON OF LARGE SCALE VEHICLE ROUTING PROBLEMS By JEFFREY ALLAN ROBBINS Bahelor of Siene Virginia Polytehni Institute and State University Blaksburg, Virginia 1974 II Submitted to the Faulty
More informationRISKBASED IN SITU BIOREMEDIATION DESIGN JENNINGS BRYAN SMALLEY. A.B., Washington University, 1992 THESIS. Urbana, Illinois
RISKBASED IN SITU BIOREMEDIATION DESIGN BY JENNINGS BRYAN SMALLEY A.B., Washington University, 1992 THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Siene in Environmental
More informationREDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD SECTION
C I E 17 th International Conferene on Eletriity istriution Barelona, 115 May 003 EUCTION FACTO OF FEEING LINES THAT HAVE A CABLE AN AN OVEHEA SECTION Ljuivoje opovi J.. Elektrodistriuija  Belgrade 
More informationPricebased versus quantitybased approaches for stimulating the development of renewable electricity: new insights in an old debate
Priebased versus based approahes for stimulating the development of renewable eletriity: new insights in an old debate uthors: Dominique FINON, Philippe MENNTEU, MarieLaure LMY, Institut d Eonomie et
More informationSebastián Bravo López
Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More informationDSPI DSPI DSPI DSPI
DSPI DSPI DSPI DSPI Digital Signal Proessing I (879) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:
More informationSuggested Answers, Problem Set 5 Health Economics
Suggested Answers, Problem Set 5 Health Eonomis Bill Evans Spring 2013 1. The graph is at the end of the handout. Fluoridated water strengthens teeth and redues inidene of avities. As a result, at all
More informationCustomer Efficiency, Channel Usage and Firm Performance in Retail Banking
Customer Effiieny, Channel Usage and Firm Performane in Retail Banking Mei Xue Operations and Strategi Management Department The Wallae E. Carroll Shool of Management Boston College 350 Fulton Hall, 140
More informationProcurement auctions are sometimes plagued with a chosen supplier s failing to accomplish a project successfully.
Deision Analysis Vol. 7, No. 1, Marh 2010, pp. 23 39 issn 15458490 eissn 15458504 10 0701 0023 informs doi 10.1287/dea.1090.0155 2010 INFORMS Managing Projet Failure Risk Through Contingent Contrats
More informationChapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112
More informationHEAT EXCHANGERS2. Associate Professor. IIT Delhi Email: prabal@mech.iitd.ac.in. P.Talukdar/ MechIITD
HEA EXHANGERS2 Prabal alukdar Assoiate Professor Department of Mehanial Engineering II Delhi Email: prabal@meh.iitd.a.in Multipass and rossflow he subsripts 1 and 2 represent the inlet and outlet, respetively..
More information5.2 The Master Theorem
170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended
More informationExploiting Relative Addressing and Virtual Overlays in Ad Hoc Networks with Bandwidth and Processing Constraints
Exploiting Relative Addressing and Virtual Overlays in Ad Ho Networks with Bandwidth and Proessing Constraints Maro Aurélio Spohn Computer Siene Department University of California at Santa Cruz Santa
More informationExperimental demonstration of Doppler spectral broadening using the PC sound card
APPARATUS AND DEMONSTRATION NOTES Jeffrey S. Dunham, Editor Department of Physis, Middlebury College, Middlebury, Vermont 05753 This department welomes brief ommuniations reporting new demonstrations,
More informationA Survey of Usability Evaluation in Virtual Environments: Classi cation and Comparison of Methods
Doug A. Bowman bowman@vt.edu Department of Computer Siene Virginia Teh Joseph L. Gabbard Deborah Hix [ jgabbard, hix]@vt.edu Systems Researh Center Virginia Teh A Survey of Usability Evaluation in Virtual
More informationRATING SCALES FOR NEUROLOGISTS
RATING SCALES FOR NEUROLOGISTS J Hobart iv22 WHY Correspondene to: Dr Jeremy Hobart, Department of Clinial Neurosienes, Peninsula Medial Shool, Derriford Hospital, Plymouth PL6 8DH, UK; Jeremy.Hobart@
More informationFindings and Recommendations
Contrating Methods and Administration Findings and Reommendations Finding 91 ESD did not utilize a formal written prequalifiations proess for seleting experiened design onsultants. ESD hose onsultants
More informationarxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA
LBNL52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrene Berkeley National Laboratory
More informationPattern Recognition Techniques in Microarray Data Analysis
Pattern Reognition Tehniques in Miroarray Data Analysis Miao Li, Biao Wang, Zohreh Momeni, and Faramarz Valafar Department of Computer Siene San Diego State University San Diego, California, USA faramarz@sienes.sdsu.edu
More informationSampling: Design and Procedures
Chapter Eleven Sampling: Design and Procedures 111 Brand name change and reposition Coke to New Coke Legend to Lenovo (2003) Panasonic, National, Technic all to Panasonic Vegemite to isnack 2.0 (social
More informationHenley Business School at Univ of Reading. PreExperience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD)
MS in International Human Resoure Management For students entering in 2012/3 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date of speifiation:
More informationMethod for characterizing single photon detectors in saturation regime by cw laser
Method for haraterizing single oton detetors in saturation regime by w laser Jungmi Oh, Cristian Antonelli, 2 Moshe Tur, 3 and Misha Brodsky,* AT&T Labs, 2 S. Laurel Ave., Middletown, New Jersey 7748,
More informationWORKFLOW CONTROLFLOW PATTERNS A Revised View
WORKFLOW CONTROLFLOW PATTERNS A Revised View Nik Russell 1, Arthur H.M. ter Hofstede 1, 1 BPM Group, Queensland University of Tehnology GPO Box 2434, Brisbane QLD 4001, Australia {n.russell,a.terhofstede}@qut.edu.au
More informationImproved Vehicle Classification in Long Traffic Video by Cooperating Tracker and Classifier Modules
Improved Vehile Classifiation in Long Traffi Video by Cooperating Traker and Classifier Modules Brendan Morris and Mohan Trivedi University of California, San Diego San Diego, CA 92093 {b1morris, trivedi}@usd.edu
More informationDeduplication with BlockLevel ContentAware Chunking for Solid State Drives (SSDs)
23 IEEE International Conferene on High Performane Computing and Communiations & 23 IEEE International Conferene on Embedded and Ubiquitous Computing Dedupliation with BlokLevel ContentAware Chunking
More informationImpact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities
Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary
More informationEffects of Oven Drying Temperature and Drying Time on Rough Rice Moisture Content Determination
Effets of Oven Drying Temperature and Drying Time on Rough Rie Moisture ontent Determination V. K. Jindal, T. J. Siebenmorgen MEMBER ASAE ASSO. MEMBER ASAE ABSTRAT HE effets of oven drying temperature
More informationBig Data Analysis and Reporting with Decision Tree Induction
Big Data Analysis and Reporting with Deision Tree Indution PETRA PERNER Institute of Computer Vision and Applied Computer Sienes, IBaI Postbox 30 11 14, 04251 Leipzig GERMANY pperner@ibaiinstitut.de,
More informationMirror plane (of molecule) 2. the Coulomb integrals for all the carbon atoms are assumed to be identical. . = 0 : if atoms i and j are nonbonded.
6 Hükel Theory This theory was originally introdued to permit qualitative study of the πeletron systems in planar, onjugated hydroarbon moleules (i.e. in "flat" hydroarbon moleules whih possess a mirror
More informationA ContextAware Preference Database System
J. PERVASIVE COMPUT. & COMM. (), MARCH 005. TROUBADOR PUBLISHING LTD) A ContextAware Preferene Database System Kostas Stefanidis Department of Computer Siene, University of Ioannina,, kstef@s.uoi.gr Evaggelia
More informationEffectiveness of a law to reduce alcoholimpaired driving in Japan
Effetiveness of a law to redue aloholimpaired driving in Japan T Nagata, 1,2 S Setoguhi, 3 D Hemenway, 4 M J Perry 5 Original artile 1 Takemi Program, Department of International Health, Harvard Shool
More information10.1 The Lorentz force law
Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been
More information1) Overview 2) Sample or Census 3) The Sampling Design Process i. Define the Target Population ii. Determine the Sampling Frame iii.
1) Overview 2) Sample or Census 3) The Sampling Design Process i. Define the Target Population ii. Determine the Sampling Frame iii. Select a Sampling Technique iv. Determine the Sample Size v. Execute
More informationEconomic Analysis of the Effects of Winter Cover Crops on NoTillage Corn Yield Response to Fertilizer Nitrogen
Eonomi Analysis of the Effets of Winter Cover Crops on NoTillage Corn Yield Response to Fertilizer Nitrogen By Roland K. Roberts 1 James A. Larson 1 Donald D. Tyler 2 Bob N. Duk 3 and Kim D. Dillivan
More informationUser s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities
File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear LeastSquares Fits with a Common Interept: Determination
More informationRESEARCH SEMINAR IN INTERNATIONAL ECONOMICS. Discussion Paper No. 475. The Evolution and Utilization of the GATT/WTO Dispute Settlement Mechanism
RESEARCH SEMINAR IN INTERNATIONAL ECONOMICS Shool of Publi Poliy The University of Mihigan Ann Arbor, Mihigan 481091220 Disussion Paper No. 475 The Evolution and Utilization of the GATT/WTO Dispute Settlement
More informationRobust Classification and Tracking of Vehicles in Traffic Video Streams
Proeedings of the IEEE ITSC 2006 2006 IEEE Intelligent Transportation Systems Conferene Toronto, Canada, September 1720, 2006 TC1.4 Robust Classifiation and Traking of Vehiles in Traffi Video Streams
More informationRecovering Articulated Motion with a Hierarchical Factorization Method
Reovering Artiulated Motion with a Hierarhial Fatorization Method Hanning Zhou and Thomas S Huang University of Illinois at UrbanaChampaign, 405 North Mathews Avenue, Urbana, IL 680, USA {hzhou, huang}@ifpuiuedu
More informationBasic Properties of Probability
Basi Properties of Probability Definitions: A random experiment is a proedure or an operation whose outome is unertain and annot be predited with ertainty in advane. The olletion of all possible outomes
More informationThe Quantum Theory of Radiation
A. Einstein, Physikalishe Zeitshrift 18, 121 1917 The Quantum Theory of Radiation A. Einstein (Reeived Marh, 1917) The formal similarity of the spetral distribution urve of temperature radiation to Maxwell
More informationBayes Bluff: Opponent Modelling in Poker
Bayes Bluff: Opponent Modelling in Poker Finnegan Southey, Mihael Bowling, Brye Larson, Carmelo Piione, Neil Burh, Darse Billings, Chris Rayner Department of Computing Siene University of Alberta Edmonton,
More informationSinglePhase Transformers
SinglePhase Transformers SinglePhase Transformers Introdution Figure shows the transformer hemati symbol and the orresponding ommonly used Steinmetz model is shown in Figure. Therein, the model voltage
More informationState of Maryland Participation Agreement for PreTax and Roth Retirement Savings Accounts
State of Maryland Partiipation Agreement for PreTax and Roth Retirement Savings Aounts DC4531 (08/2015) For help, please all 18009666355 www.marylandd.om 1 Things to Remember Complete all of the setions
More informationOPTIMAL TAXATION AND SOCIAL INSURANCE IN A LIFETIME PERSPECTIVE
January 26 OPTIMAL TAXATION AND SOCIAL INSURANCE IN A LIFETIME PERSPECTIVE A. Lans Bovenberg, Tilburg University, CEPR and CESifo ) Peter Birh Sørensen, University of Copenhagen, EPRU and CESifo ) Abstrat
More informationChapter 1: Introduction
Chapter 1: Introdution 1.1 Pratial olumn base details in steel strutures 1.1.1 Pratial olumn base details Every struture must transfer vertial and lateral loads to the supports. In some ases, beams or
More informationSUPPLEMENT TO BUBBLES AND SELFENFORCING DEBT (Econometrica, Vol. 77, No. 4, July 2009, )
Eonometria Supplementary Material SUPPLEMENT TO BUBBLES AND SELFENFORCING DEBT (Eonometria, Vol. 77, No. 4, July 2009, 1137 1164) BY CHRISTIAN HELLWIG AND GUIDO LORENZONI In this doument, we present three
More informationScalable Hierarchical Multitask Learning Algorithms for Conversion Optimization in Display Advertising
Salable Hierarhial Multitask Learning Algorithms for Conversion Optimization in Display Advertising Amr Ahmed Google amra@google.om Abhimanyu Das Mirosoft Researh abhidas@mirosoft.om Alexander J. Smola
More informationA ThreeHybrid Treatment Method of the Compressor's Characteristic Line in Performance Prediction of Power Systems
A ThreeHybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition of Power Systems A ThreeHybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition
More informationImproved SOMBased HighDimensional Data Visualization Algorithm
Computer and Information Siene; Vol. 5, No. 4; 2012 ISSN 19138989 EISSN 19138997 Published by Canadian Center of Siene and Eduation Improved SOMBased HighDimensional Data Visualization Algorithm Wang
More informationDesign Implications for Enterprise Storage Systems via MultiDimensional Trace Analysis
Design Impliations for Enterprise Storage Systems via MultiDimensional Trae Analysis Yanpei Chen, Kiran Srinivasan, Garth Goodson, Randy Katz University of California, Berkeley, NetApp In. {yhen2, randy}@ees.berkeley.edu,
More informationProduct Warranties and Double Adverse Selection
rodut Warranties and Double Adverse eletion David A. oberman Assistant rofessor of Marketing INEAD Boulevard de Constane 77305 Fontainebleau Cede, Frane The author thanks rofessors Andy Mithell, Jak Mintz,
More informationAn Enhanced Critical Path Method for Multiple Resource Constraints
An Enhaned Critial Path Method for Multiple Resoure Constraints ChangPin Lin, HungLin Tai, and ShihYan Hu Abstrat Traditional Critial Path Method onsiders only logial dependenies between related ativities
More information