Version 1 RAT-STATS Companion Manual TABLE OF CONTENTS Page Random Numbers 1-1 Single-Stage Random Numbers ........................................ 1-2 Sets of Two Numbers................................................ 1-3 Sets of Three Numbers............................................... 1-4 Sets of Four Numbers ............................................... 1-5 Frames - Single Stage ............................................... 1-6 Frames - Sets of Two ................................................ 1-8 RHC Sample Selection ............................................. 1-11 Procedure for Two-Stage RHC Sampling ............................ 1-11 Procedure for Three-Stage RHC Sampling ........................... 1-18 Summary of Input for RHC Sample Selection ........................ 1-20 Generating Spares for RHC Sampling ............................... 1-24 Comparison of RHC and Multistage SRS ............................ 1-29 Attribute Appraisals .................................................. 2-1 Unrestricted Attribute Appraisal ....................................... 2-2 Stratified Attribute Appraisal.......................................... 2-6 Two-Stage Unrestricted ............................................. 2-11 Three-Stage Unrestricted ............................................ 2-19 RHC Two Stage ................................................... 2-28 RHC Three Stage .................................................. 2-36 Stratified Cluster .................................................. 2-62 Stratified Multistage ............................................... 2-68 Variable Appraisals ................................................... 3-1 Unrestricted Variable Appraisal ....................................... 3-2 Stratified Variable Appraisal .......................................... 3-8 Using a Stratified Sample ......................................... 3-8 Strata Formation ............................................... 3-11 Two-Stage Unrestricted Variable Appraisal ............................. 3-17 Three-Stage Unrestricted Variable Appraisal ............................ 3-24 RHC Two Stage Variable Sampling ................................... 3-35 RHC Three Stage Variable Sampling .................................. 3-48 Stratified Cluster .................................................. 3-77 Stratified Multistage ............................................... 3-82 Post Stratification.................................................. 3-91 Unknown Universe Size ............................................ 3-99 Sample Size Determination ............................................. 4-1 Variable Sample Size Determination.................................... 4-2 Unrestricted Using a Probe Sample ... .............................. 4-2 Unrestricted Using Estimated Error Rate ............................. 4-7 Stratified...................................................... 4-11 Total Sample Size Unknown ................................... 4-11 Total Sample Size Known ..................................... 4-16 Attribute Sample Size Determination .................................. 4-22 PREFACE The purpose of this manual is to provide: # an overview of each program in the Windows version of RAT-STATS, # examples illustrating the application of the software, # snapshots of data sets used by the programs, # some discussion regarding the program output, and # formulas used within the software. The intent is for the auditor/specialist to use as much of this discussion as he/she finds helpful. While the RAT-STATS Users Guide gives descriptions of program input and output, this Companion Manual should provide insight as to how to better use the software and exactly how the program derives the results. The formulas are provided so that OAS has a single source for all formulas in the event that a question is raised as to exactly how a particular result was obtained. We hope you find that the manual makes the OAS software easier to understand and easier to apply. Please pass on any suggestions or corrections to Office of Inspector General, Office of Public Affairs at [email protected]. RAT-STATS Companion Manual Random Numbers RANDOM NUMBERS Whatever statistical sampling design you end up using (including stratified and/or multistage), at some point in the data collection you will need one or more random samples. The next section, dealing with Unrestricted Random Sampling, will examine the mechanics and estimation procedures using such a sample in detail, but first it is necessary to discuss procedures for generating a random sample. A number of programs exist for such purposes; namely: # Single-Stage Random Numbers # Sets of Two Numbers # Sets of Three Numbers # Sets of Four Numbers # Frames - Single Stage # Frames - Sets of Two # RHC Sample Selection Page 1-1 Single-Stage Random Numbers RAT-STATS Companion Manual Single-Stage Random Numbers This program generates an unduplicated quantity of random numbers. Random values (the output from this program) can be output in sequential order, random order, or a mixture of both. For values in sequential order, you will see the values printed in sequential order, beginning with the smallest selected item number and proceeding to the largest item number. For values in random order, the items are printed in the order in which they were selected by the program. Example 1. A universe contains 1,000 payments and a simple random sample of 10 payments (with four spares) is needed. What items should be selected? Solution: Using this program and a seed value of 12345, the sampled payments are those numbered as follows: 9, 236, 337, 340, 346, 497, 556, 641, 658, and 884 The four spares are payments 404, 624, 927, and 947. Page 1-2 RAT-STATS Companion Manual Sets of Two Numbers Sets of Two Numbers This program will generate unduplicated pairs of random numbers. This is useful when sample items are selected through a two-step process (e.g., page number and line number). Random values (the output from this program) can be output in sequential order, random order, or a mixture of both. Values in sequential order will be printed in sequential order, beginning with the smallest selected item number and proceeding to the largest item number. For values in random order, the items are printed in the order in which they were selected by the program. Example 2. Items are selected from a computer printout that had pages numbered 1 through 658 and had 66 lines on each page. A simple random sample of 10 items (with four spares) is needed. Which items should be selected? Solution: Using this program and a seed value of 12345, the sampled payments are: PAGE: 224 258 266 327 366 400 422 433 561 610 ITEM: 23 37 42 1 16 7 23 59 40 63 The spares are: PAGE: 579 109 188 330 ITEM:61 54 50 49 Page 1-3 Sets of Three Numbers RAT-STATS Companion Manual Sets of Three Numbers This program will generate unduplicated sets of three random numbers. This should be used when sample items are selected through a three-step process (e.g., month, page, and line number). Random values (the output from this program) can be output in sequential order, random order, or a mixture of both. Values in sequential order will be printed in sequential order, beginning with the smallest selected item number and proceeding to the largest item number. For values in random order, the items are printed in the order in which they were selected by the program. Example 3. Same as Example 2, where the pages are numbered 1 through 658 for each month. Here the universe consists of 1 year's worth (12 months) of computer printouts. We need four sample items and two spares. Solution: Using this program and a seed value of 12345, the sampled items are: MONTH: 3 5 6 8 PAGE: 224 266 6 582 ITEM: 23 42 37 43 The two spares are: MONTH: 12 8 PAGE: 623 258 ITEM: 57 37 Page 1-4 RAT-STATS Companion Manual Sets of Four Numbers Sets of Four Numbers This program will generate unduplicated sets of four random numbers. This should be used when sample items are selected through a four-step process (e.g., year, month, page, and line number). Random values (the output from this program) can be output in sequential order, random order, or a mixture of both. For values in sequential order, you will see the values printed in sequential order, beginning with the smallest selected item number and proceeding to the largest item number. For values in random order, the items are printed in the order in which they were selected by the program. Example 4. Same as Example 3, where the pages are numbered 1 through 658 for each month and year (total of 5 years). We need three sample items and two spares. Solution: Using this program and a seed value of 12345, the sampled items are: YEAR: 2 3 4 MONTH: 5 1 5 PAGE: 433 366 266 ITEM: 59 16 42 The two spares are: YEAR: 5 2 MONTH: 12 7 PAGE: 561 400 ITEM: 40 7 Page 1-5 Frames - Single Stage RAT-STATS Companion Manual Frames - Single Stage This program will generate an unduplicated set of random numbers which is useful when the universe of sampling items either (1) contains gaps of numbers or (2) the numbering system repeats within the universe. For instance, the universe of items consists of two frames, numbered 1 through 1,050 and 8,405 through 9,565. Random values (the output from this program) can be output in sequential order, random order, or a mixture of both. For values in sequential order, you will see the values printed in sequential order, beginning with the smallest selected item number in the first frame (if