# Calculation of Sampling Weights

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3 lected. The calculaton of such weghtng factors s smlar to the frst stage weght snce samplng geographcal regons was also done wth probablty proportonal to sze (PPS). The resultng frst stage weght s smply the product of the "regon" weght and the frst stage weght as descrbed earler. In some countres, schools were selected wth equal probabltes. Ths generally occurred when no relable measure of school sze s avalable. In ths case, the basc frst stage weght for the th sampled school was defned as N ---- n where n s the number of sampled schools and N s the total number of schools n the stratum. It should be noted that n ths case the basc weght for all sampled schools s dentcal School-Level Response Rate (Partcpaton Rate) A school-level response rate, weghted and unweghted, was calculated to measure the proporton of orgnally selected schools that ultmately partcpated n the assessment. Snce replacement schools were used to mantan the sample sze, school-level response rates have been reported both wth and wthout the use of replacement schools. The calculaton of the response rate used the followng terms, derved from the data collecton: n ex number of sampled schools that should have been excluded number of orgnally sampled schools that partcpated n rp number of replacement schools that partcpated n nr number of non-respondng schools (nether the orgnally selected schools nor ther replacements partcpatng.) Note that the followng equaton holds: n ex + + n rp + n nr n The unweghted school-level response rate s defned as the rato of orgnally sampled schools that partcpated to the total number of sampled schools mnus any excluded schools. It was calculated by the followng equaton: sc R unw n rp + n nr 73

4 The weghted school-level response rate s defned n a smlar manner. The weght assgned to the th sampled school for ths purpose s the samplng nterval used to select t,. The weghted school-level response rate, based solely on orgnally selected schools, s therefore the rato of the weghted sum of orgnally sampled schools that partcpated to the weghted sum of all sampled schools less any excluded schools. It was calculated by the followng equaton: R w sc n rp n nr 1 The weghted school-level response rate, ncludng replacement schools, was calculated by the followng equaton: sc R wrp, n rp 1 n rp n nr School-Level Nonresponse Adjustment Frst stage weghts were calculated for orgnally sampled schools and replacement schools that partcpated. Any sampled schools that were no longer elgble were removed from the calculaton of ths nonresponse adjustment. Examples are secondary schools ncluded n the samplng frame by mstake and schools that no longer exsted. The school-level nonresponse adjustment was calculated separately for each desgn doman and explct stratum. The school-level nonresponse adjustment was calculated as follows: n n A ex sc n rp and the fnal frst stage weght for the th school thus becomes FW sc A sc * 74

6 where c g s the number of sampled classrooms as defned earler and c g K g, k j 1 Agan, as a rule, c g takes the value 1 or 2 and wll reman fxed for all sampled schools. In cases where c g has the value 2, and only one of the sampled classrooms partcpated, a classroom-level nonresponse adjustment was appled to the second stage weght by multplyng t by the factor Thrd-Stage (Student) Weght The thrd stage weght represents the nverse of the thrd stage selecton probablty attached to a sampled student. If ntact classrooms were sampled as specfed n Foy, Rust, and Schlecher (1996), then the basc thrd stage weght for the jth grade g classroom n the th school was BW st 1.0 If, on the other hand, subsamplng of students was requred wthn sampled classrooms, then the basc thrd stage weght for the jth grade g classroom n the th school was gj BW,, k st s g where k g,,j s the sze of the jth grade g classroom n the th school, as defned earler, and s g s the number of sampled students per sampled classroom. The latter number usually remans constant for all sampled classrooms n a grade Student-Level Response Rate (Partcpaton Rate) and Adjustment The basc thrd stage weght requres an adjustment to reflect the outcome of the data collecton efforts. The followng terms were derved from the data collecton for each sampled classroom: s ex number of sampled students that should have been excluded s rs number of sampled students that partcpated s nr number of sampled students that dd not partcpate. 76

7 Note that the followng equaton holds: s ex s rs s nr + + s where s g,,j s the number of sampled students per sampled classroom. Ths number should be constant f subsamplng of students s done wthn each sampled classroom and represents the classroom sze, k g,,j, when ntact classrooms are tested. The student-level response rate, for a gven classroom, was calculated as follows: R st s rs s rs + s nr Excluded students (.e., those meetng the gudelnes for student-level exclusons specfed n Foy, Rust, and Schlecher, 1996) were not ncluded n the calculaton of the response rate. The student-level nonresponse adjustment was calculated as follows: s A rs st s nr s rs Note that the student-level nonresponse adjustment s smply the nverse of the student-level response rate. The fnal thrd stage weght for the jth grade g classroom n the th school thus becomes FW st A st * BW st The weghted overall student-level response rate was computed as follows: R w st rs 1 rs + nr 1 BW cl1 * * BW cl1 * * BW st BW st where the numerator s the summaton of the basc weghts over all respondng students, and the denomnator s the summaton of the basc weghts over all respondng and nonrespondng students. Weghted student response rates were reported separately by grade n the TIMSS nternatonal reports. 77

8 4.2.4 Overall Samplng Weghts The overall samplng weght s smply the product of the fnal frst stage weght, the approprate fnal second stage weght, and the approprate fnal thrd stage weght. If ntact classrooms were tested, then the overall samplng weght was W FW sc FW sc * * FW st If subsamplng wthn classrooms was done, then the overall samplng weght was W FW sc FW cl2 * * FW st It s mportant to note that samplng weghts vared by school, grade, and classroom. However, students wthn the same classroom have the same samplng weghts. The use of samplng weghts s crtcal to obtanng proper survey estmates when samplng technques other than smple random samplng are used. TIMSS has produced a samplng weght for each student sampled for the TIMSS man (wrtten) assessment and subsampled for the performance assessment. Secondary analysts usng the TIMSS data wll need to be aware of ths and use the proper weghts when conductng analyses and reportng results. 78

9 REFERENCES Foy, P., Rust, K., and Schlecher, A. (1996). Sample desgn. In M.O. Martn and D.L. Kelly (Eds.), TIMSS techncal report, volume I: Desgn and development. Chestnut Hll, MA: Boston College. Harmon, M. and Kelly, D.L. (1996) Performance assessment. In M.O. Martn and D.L. Kelly (Eds.), TIMSS techncal report, volume I: Desgn and development. Chestnut Hll, MA: Boston College. Schlecher, A. and Snscalco, M.T. (1996). Feld operatons. In M.O. Martn and D.L. Kelly (Eds.), TIMSS techncal report, volume I: Desgn and development. Chestnut Hll, MA: Boston College. 79

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