HYPERGEOMETRIC SAMPLING TOOL a BACKGROUND OF CALCULATION AND VALIDATION

Transcription

1 DRUGS WORKIG GROU HYERGEOETRIC SALIG TOOL a (versio ) BACKGROUD OF CALCULATIO AD VALIDATIO DOCUET TYE : REF. CODE: ISSUE O: ISSUE DATE: Guidelie - Validatio DWG-SGL- 7 DECEBER a The hypergeometric samplig tool (sample size calculator) is a module withi a excel based EFSI DWG Calculator for Qualitative Samplig of seized drugs, where some other tools are also available. Ref code: DWG-SGL- Issue o. age: /34

2 TABLE OF COTETS. Itroductio Defiitios Why ew versio of the hypergeometric tool Theory How to fid (the highest iteger lower tha K for H test) i practice Samplig based o the umber of expected positives Calculatio of for iteger K Sample size ad correspodig actual proportio of positives Samplig strategy based o the predefied proportio (k) of expected positives Calculatig of for itegers or o itegers K Calculated sample size ad actual proportio of positives Software: geeral explaatio EXCEL hypergeometric fuctio - basic Calculatio based o umber of positives Calculatios based o proportio k of positives: Truc or RoudUp? Truc fuctio Roud Up fuctio Software: EFSI hypergeometric tool for sample size calculatio Data required for calculatio Results Dyamic graph acro buttos Formulas applied Calculatio based o the umber positives (itegers) Calculatio based o proportio Restrictios limitatios Calculatio based o the umber of positives Calculatio based o the proportio of positives rotectio of the software Validatio of the hypergeometric samplig tool (versio ) Correctess of the sample size () calculatio whe the proportio of positives k is specified (iteger ad o iteger Ks) Criteria Validatio procedure Results Criteria fulfilled? Does the calculated sample size»guaratee«a 'at least' requested proportio of positives?... 5 Ref code: DWG-SGL- Issue o. age: /34

3 8... Criteria Validatio procedure Results Criteria fulfilled? Calculatio based o the umber positives - validatio Some additioal tests Compariso of sample sizes calculated by EFSI hypergeometric tool ad with HyperBay calculator Idepedat validatio obtaied from HSA Testig performed by the author 5 of the»hyperbay«sample size calculator Coclusios Software Other Appedix Calculatios by had Details o calculatio of S, S, S Biomial coefficiet ad calculatios»by had« Resposible for errors Refereces Ref code: DWG-SGL- Issue o. age: 3/34

4 . ITRODUCTIO A represetative samplig procedure ca be performed o a populatio of uits with sufficiet similar exteral characteristics (e.g. size, colour). Differet samplig approaches, i.e. arbitrary or statistical, may be applied. Samplig strategies for appropriate sample size calculatio may be supported by computerized tools. The first versio of Excel based EFSI DWG Calculator for Qualitative Samplig of seized drugs (from here o: EFSI Samplig Calculator ) has bee published i 3 ad validated i 9. The calculator offers applicatios usig Hypergeometric, Bayesia or Biomial based fuctios for sample size calculatios. Further it provides a calculatio for the Estimatio of weight or the Estimatio of umber of tablets i bigger or multi package seizures. The EFSI Samplig Calculator foud good acceptace i the foresic commuity ad has bee used almost worldwide. However, from the users DWG received suggestios for improvemet, especially o the hypergeometric tool. The argumets are described i Chapter 3. Therefore, DWG decided to improve this tool ad make it more user-friedly. This documet presets the ew versio () of the hypergeometric tool. It briefly explais the backgroud of the calculatio ad reports the validatio of the adjusted tool. The EFSI Samplig Calculator was updated with the ew hypergeometric tool (versio ), while other calculatios (i. e. Bayesia, biomial, etc ) remaied uchaged. The validatio report o the uchaged calculatios is also available i the documet Validatio of the guidelies o represetative samplig. 3 We like to thak all who cotributed basically to the improvemet of the software ad its validatio: Dr. Soja Klemec (atioal Foresic Laboratory, Sloveia). Without her steerig, coordiatig ad leadig effort combied with her great ethusiasm i improvig the calculator ad its validatio, this project could ot be realized. Tomislav Houra, Dr. aja Jelea etek ad Dr. Ies Gmajički (Foresic Sciece Cetre, Croatia) We ackowledge their cotributio ad support i the checkigs of the draft documet ad software. Ref code: DWG-SGL- Issue o. age: 4/34

5 Dr. Laurece Dujourdy (iistere de l'iterieur, Istitut atioal de olice Scietifique, Frace) for commets ad correctios of draft documet. Dr. Agelie Yap Tiog Whei (Health Scieces Authority, Sigapore) iflueced the project essetially. Her valuable ad costructive suggestios helped to make the calculator more userfriedly ad fit for purpose. Dr. Cheag Wai Kwog (atioal Istitute of Educatio, Sigapore). His tremedous, i depth review ad commets to the draft documet ad hypergeometric part of software esured the quality ad validity of the calculator. Joh Gerlits (Utah Bureau Of Foresic Services, USA) With his profoud mathematical ad aalytical skills he tested the software ad suggested may helpful hits ad practical solutios for calculatio improvemet ad flexibility of the calculator. Dr. ichael Boves EFSI DWG Chairma 7- Drugs Workig Group Hypergeometric Samplig Subcommittee wishes to thak also to: Dr. ichael Boves (Foresic Sciece Istitute Zurich, Switzerlad) for his multi-level support ad always helpig had, critical reviews, correctios ad valuable suggestios, which made the fial versio of this documet ad calculator better. Dr. Soja Klemec Ref code: DWG-SGL- Issue o. age: 5/34

6 . DEFIITIOS Some defiitios ad labels as applied i this documet: populatio size umber of similar samples K threshold umber of positives (drugs) guarateed i the populatio k K/ threshold proportio of positives (drugs) guarateed i the populatio sample size () umber of samples to be aalyzed x the value of umber of positives i the sample r x the value of the umber of egatives i the sample H ull hypothesis H hypothesis alterative (opposite) to H i α umber of positives i the populatio (ote: i is the iteger lower tha K if H is true) highest iteger lower tha K at which H is tested probability of rejectig H whe H is true, i.e., α (Type I error) - α probability of acceptig H whe H is true TRUC excel fuctio - cuttig the decimals off (for example: TRUC (8.9) 8), equals to ROUDDOW to zero decimals ROUDU excel fuctio - roudig umber up (for example: RoudUp (to zero decimals). ) Other as defied i text. 3. WHY EW VERSIO OF THE HYERGEOETRIC TOOL Foresic laboratories workig with statistical samplig for qualitative aalysis usually set a miimum requiremet of the expected proportio (k) or umber (K) positives i populatio () ad cofidece level (- α). Therefore the resultig/calculated sample size (umber of samples for aalysis) has to be as such that laboratory requiremets o umber/proportio of positives are met exactly or are slightly higher. The software should guaratee this. The ew versio of the tool replaces the hypergeometric part of the EFSI Samplig Calculator from 9 which did ot fulfill requiremet stated above, as i some situatios sample sizes were uder estimated. I the ew versio error from the previous oe is corrected. Beside this, two types of hypergeometric calculatios are offered ow: Hypg_roportio is based o threshold proportio of positives k specified by the laboratory, while Hypg_umber is based o the umber of expected positives K specified. Ref code: DWG-SGL- Issue o. age: 6/34

7 I summary, the backgroud of the ew hypergeometric calculatio is as follows: By testig H at K if K (a iteger) is specified, or at RoudUp (K) if k is specified (K k, eed ot to be a iteger), the calculatio will theoretically (see explaatios i the followig sectios) give a sample size that guaratees at least k proportio/ K umber of positives i populatio, at a cofidece level of at least α. Some ew fields with back calculatios (actual proportio of positives for calculated sample size, cofidece level) were added ad the graphical presetatio has bee improved. See more i chapter THEORY b The purpose of samplig is to fid the lowest sample size such that miimum laboratory requiremets o umber or proportio of positives are met exactly or are slightly higher (see above - chapter 3). Guarateeig with (-α)% cofidece that at least proportio k % (or correspodig umber K) of populatios are drugs is the same as guarateeig that the probability o fidig oly (or mostly) drugs i the sample will be less tha α whe the proportio of positives i the populatio is less tha k (or umber of positives less tha K). Determiatio of the miimum required sample size () for at least requested proportio of positives is based o a test of the ull hypothesis that umber of positives i the populatio is less tha K agaist the alterative hypothesis that the umber of positives is at least K:,4 H : i < K agaist H : i K The hypotheses are tested with the umber of positives i the sample, X, as the test statistic. The ullhypothesis is rejected whe X is larger tha a certai umber. If this umber is take as the umber of positives expected i the sample, x, the, should be selected such that (X x, i < K) α Equatio Ituitively, (Reject H i ) icreases as the umber of positives i the populatio i icreases. (H is i < K.) Therefore, to fid the smallest sample size () which guaratees at least proportio of positives k we cocetrate oly o the highest possible iteger (from here o labeled as ) smaller tha K. b Some passuses of the text were adopted from the documet Validatio of the guidelies o represetative samplig, DWG- SGL- documet, versio, 9. However, geeralizatio of the theory, equatios correctios ad further explaatios of calculatio were performed by the author of this documet ad the ew versio of software. Ref code: DWG-SGL- Issue o. age: 7/34

8 I other words: the ull hypothesis (H ) is tested at the highest possible (iteger) which is lower tha K. If H is rejected (H is accepted), the the calculated sample size will give the smallest umber of samples for aalyses, which guaratees at least k proportio (or correspodig K) of positives i the populatio, at a cofidece level at least equal or greater tha (- α). Equatio may be rewritte as: (X x, < K) α So give that < K the required miimal sample size () is the smallest value for which (X x < K) α. Whe all sampled drug uits are expected to cotai drugs (i.e. x which is equivalet to r ), X follows a hypergeometric distributio: X ~ HY(,, ) Resultig i: ( X x / < K, x ) zero S Equatio Whe at most oe sampled drug uit is expected ot to cotai drugs (i.e. x - which meas that x - or x are possible; hece, the umber of egatives ca be at most, i.e.: r or r are possible), X is distributed as a mixture of two hypergeometric radom variables: ( X x / < K, x ) oe S S S Equatio 3 Whe at most two sampled drug uits are expected ot to cotai drugs (i.e. x - which meas the umber of egatives ca be at most two, i.e.: r or r or r are possible), X is distributed as a mixture of three hypergeometric radom variables: Ref code: DWG-SGL- Issue o. age: 8/34

9 Ref code: DWG-SGL- Issue o. age: 9/34 < x K x X S S S S S two ), / ( Equatio 4 Ad so o for higher umber of egatives at most allowed. Smallest populatio size is actually calculated by the cosecutive use of appropriate equatio above, i.e. cumulative hypergeometric probability is calculated (see example i chapter.).

10 5. HOW TO FID (THE HIGHEST ITEGER LOWER THA K FOR H TEST) I RACTICE 5.. Samplig based o the umber of expected positives Data etered ito calculatio (based o # of positive samples) are: (populatio size) ad K (umber of positives). The two umbers are always itegers Calculatio of for iteger K For H test: K- is the highest iteger lower tha K (which is iteger too). See Figure. Iteger K H test at K- H : K- K Δ Figure : Itegers K how to fid the highest iteger lower tha K (for H test) 5... Sample size ad correspodig actual proportio of positives If H is rejected, H is accepted ad calculated sample size will correspod to proportio of positives k K/, which match requested proportio exactly. 5.. Samplig strategy based o the predefied proportio (k) of expected positives 5... Calculatig of for itegers or o itegers K If the samplig strategy is based o defied miimum proportio of positives k the umber of expected positives is calculated as: K k ad ca result i iteger or o iteger K. See example i the table below. Ref code: DWG-SGL- Issue o. age: /34

11 Table : Example calculated K for k.9 ad differet populatio sizes () opulatio size Calculated umber of expected positives K k x opulatio size Calculated umber of expected positives K k x , , , , , , , , , If we follow the theory, for H test, we will fid the highest iteger lower tha K as show i the table below (Table ). Table : Formulas for calculatio Descriptio calculatio (formula) Iteger K (as described i 5.) K- o iteger K: (see Figure ) Truc (K) RoudUp (K)- For o iteger umbers the highest iteger lower tha K is actually trucated K. The same value ca be obtaied by roudig K up to the earest higher iteger ad subtractig (see Figure ). Ref code: DWG-SGL- Issue o. age: /34

12 for o itegers K & Truc ad Roudup for H test TRUC (K) ROUDU K K- K K Iteger umbers Δ o iteger umbers H is tested at a umber of positives TRUC(K) ROUDU(K) - (red marked iteger) sice this is the highest possible iteger ( ) lower tha K (red lie). If H is rejected, the actual proportio of positives which calculated sample size guaratees is equal k` (ROUDU(K))/, where is the populatio size. The actual proportio of positives will be above our request k which correspods to the umber of samples K (o iteger). Figure : o itegers K how to fid the highest iteger lower tha K (for H test) Table 3: Example - How to fid (the highest iteger < K) for itegers ad o itegers K? EXALE: How to fid (highest iteger < K) for itegers ad o itegers K arameters defied by laboratory (costat, regardless of populatio size received) : - proportio of positives k.9 - cofidece level -α.95 - umber of egatives r Case work (material as received ito the lab): Case A: populatio size A A Case B: populatio size B B Calculatios K ad label K k x equatio applied Case A 8. 7 K- Case B TRUC(K) RoudUp(K)- I geeral, to calculate the sample size (), we have first to calculate K (umber of expected positives correspodig to proportio k) ad the where we test the H. Ref code: DWG-SGL- Issue o. age: /34

13 After K ad are determied we ca calculate the lowest sample size (which guaratees at least k proportio of positives) with the cosecutive use of Equatio or Equatio 3 or Equatio 4 (depedet o the umber of egatives at most allowed). Cumulative hypergeometric probability is calculated by icreasig, util actual cofidece level laboratory request is fulfilled or at maximum to. See calculatios by had i chapter Calculated sample size ad actual proportio of positives For iteger Ks requested ad calculated proportio of positives match exactly (see 5..). For o iteger Ks: o iteger parts of K are kid of chameleos. As samples i reality are itegers, the laboratory has to decide what to do with chameleos. By promotig them (i.e. roudig up) to the earest higher iteger, the origial laboratory request o proportio of positives (k) will be pushed to a bit higher level (k`), i.e. above request. I such case the laboratory ca describe its geeral samplig strategy as: The aalyzed sample size guaratees at least k proportio of positives i the populatio. So, for at least, chameleos are first promoted, the H is tested at Truc (K) RoudUp (K) -, ad if rejected, the calculated sample size () will guaratee the proportio of positives a bit higher tha requested (k`> k). Calculatios of the actual proportio are show i the table below. Table 4: Actual proportio of positives descriptio Actual proportio of positives for o itegers K (always slightly above requested k) Actual proportio of positives for itegers K (always match requested proportio exactly) calculatio k` RoudUp (K)/ k K/ Opposite, if the laboratory (or software) degrades the chameleo to the earest lower iteger by trucatig K ad the test H at i Truc (K) - (which is always lower tha ), the geeral samplig strategy will fit for at most requested proportio (which might ot be a very useful statemet for the court). Ref code: DWG-SGL- Issue o. age: 3/34

14 6. SOFTWARE: GEERAL EXLAATIO 6.. EXCEL hypergeometric fuctio - basic EXCEL hypergeometric fuctio has four argumets ad is defied as: HYGEODIST (A, B, C, D), where C stads for the umber of positives as take ito accout for the ull hypothesis test (H test). Label A is the umber of successes i sample, B stads for the sample size ad D for the populatio size. Oe should be aware that HYGEODIST fuctio is discrete, which meas, that it processes oly ITEGER (WHOLE) umbers. Hece, if the argumet C is ot iteger (this may happe if the calculatio is based o the proportio of positives ad the product betwee kx is ot iteger) it will be trasformed to iteger by software default fuctio (trucatig cuttig decimals off) or alog our istructios (for example Roud U). Roudig or trucatig has o effect o itegers. 6.. Calculatio based o umber of positives All umbers for calculatio are itegers so K- is iteger ad sample size is calculated alog: HYGEODIST (A, B, K-, D), 6.3. Calculatios based o proportio k of positives: Truc or RoudUp? Actually both fuctios may be applied. The differece is: if we apply Truc (excel hypergeometric default) we will eed two differet equatios for the sample size calculatio: oe for itegers K ad a differet oe for o itegers. If we apply RoudUp oe equatio fits for all situatios Truc fuctio To refresh: K k x, where k is a predefied value alog the laboratory samplig strategy ad the populatio size is flexible (differet from case to case)! Data k ad are etered ito the excel calculatio by the user. is calculated by the software from K k x. Geeral form: HYGEODIST(A, B,, D) Ref code: DWG-SGL- Issue o. age: 4/34

15 Calculatio of the third hypergeometric argumet ad excel formula: descriptio excel hypergeometric fuctio Itegers K*: K- Truc (K)- HYGEODIST (A, B, Truc (K) -, D) o itegers K: Truc (K) HYGEODIST (A, B, Truc (K), D) * trucatig ad roudig do ot chage iteger umbers This meas i other words: if trucatig (software default) is applied, the the TEST O ITEGERS shall be icluded ito the calculatio. Such test will istruct the software to do the followig: Is K k x iteger? Yes calculate alog formula: HYGEODIST(A,B,Truc (K) -,D) o calculate alog formula: HYGEODIST(A,B,Truc (K),D) Figure 3: Test o itegers, if truc (Excel default) is applied Roud Up fuctio The same effect as with the test o itegers ca be achieved if the software is istructed to ROUD decimals U to zero decimals (this solutio is more elegat ad has additioally some logical backgroud see chapter 5..). RoudUp(K)- works fie for itegers (roudig actually has o effect o itegers) ad for o itegers roudig up actually auls the effect of - (from trucatig), ad the at least requiremet is achieved. excel hypergeometric fuctio descriptio for itegers ad o itegers Itegers K: K- RoudUp(K)- HYGEODIST(A, B, RoudUp(K)-, D) o itegers K: RoudUp(K)- Truc(K) RoudUp(K)-, see Figure. Ref code: DWG-SGL- Issue o. age: 5/34

16 7. SOFTWARE: EFSI HYERGEOETRIC TOOL FOR SALE SIZE CALCULATIO The Hypergeometric tool (versio ) was origially desiged ad validated by the icrosoft Excel 3 software. Excel_3 file was the saved as the Excel 7 acro-eabled format (.xlsm) file ad basic fuctioality of applicatio was retested. Icosistecies were ot detected.. I the curret () versio of EFSI Samplig Calculator two types of hypergeometric sample size calculatios are eabled: For calculatios based o umber of positives oe will select the Hypg_umber tab ad the Hypg_roportio tab for calculatios based o the proportio of positives. See Figure 4 ad Figure 6 for data iput ad results widows. 7.. Data required for calculatio Data are etered i steps to 4 (cell B to B4). op-up messages (see example o Figure 5) will appear if the user eters values out of rage ad additioally some forbidde etries may also be show as red labeled strikethrough umbers. 7.. Results Results are show i steps 5 to 7. umbers appear i red colour (see Figure 4 ad Figure 6). The sample size is calculated i cell B5. Calculatios of the actual proportio of positives for a calculated sample size ad a actual cofidece level, oe ca see i cells C ad C4, respectively Dyamic graph I this plot the calculated cofidece level versus calculated sample size, for umber of egatives from to is show. The plot rage is updated automatically. If the umber of egatives r is too high for the give criteria (, k, CL) ad the sample size does ot fulfill the criteria, the curve appears as a lie with CL. See example o Figure 4 ad ote the lie i aqua colour for r acro buttos The aximum populatio size ( max ) is adjustable by the user. To keep the file size relatively small, max is set as default. The curret value of max oe ca see i the cell C. A click o the appropriate macro butto (see Figure 4) will chage, i. e. exted/ reduce the populatio size ad the file will be saved. Remark: ote that with higher values of max the file size will icrease sigificatly, calculatios are becomig slower ad the idle time for file savig ad opeig will icrease! It is recommeded to reduce the max. populatio size to the default value before you close the applicatio. Ref code: DWG-SGL- Issue o. age: 6/34

17 The Uhide/Hide butto (oly available for calculatios based o proportio see Figure 6) will ope/ close two additioal colums with side calculatios (useful for better uderstadig of the calculatio). Figure 4: Data iput ad results widow for calculatio based o the umber of positives Ref code: DWG-SGL- Issue o. age: 7/34

18 Figure 5: opup message example (the populatio size over rage) Figure 6: Data iput ad results widow for calculatio based o the proportio of positives Figure 7: Side calculatios (colum D ad E) ca be made visible by click o the macro butto UHIDE Ref code: DWG-SGL- Issue o. age: 8/34

19 7.5. Formulas applied The formulas used i the software (Hypg_umber tab ad the Hypg_roportio tab) are based o Equatio, Equatio 3 ad Equatio 4 from chapter 4. The hypergeometric probabilities i these equatios are calculated usig the Excel fuctio HYGEODIST. Equatios for calculatig the sample size are show bellow ad reffer to the row 7 of the hypergeometric calculatios sheets. ote, that i cosecutive rows of the applicatio (i.e. row 8, row 9, etc ) relative parts of the cell umberig are chaged. Beside the hypergeometric part of the equatio (labeled i bold fots), which has already bee explaied extesively, some additioal excel logical fuctios (Boolea: OR, AD ad coditioal IF statemets) were applied i the calculatio. The first coditio IF(A7"","", is icluded for display purposes oly ad does ot ifluece the sample size calculatio. Fuctios of the other coditioals (uderlied italic) are briefly explaied below Calculatio based o the umber positives (itegers) Calculatio of (S) for zero egatives (r ) : IF(A7"","",IF(A7<($B$),(HYGEODIST($A7,$A7,($B$)-,$B$)),)) To see why (S) is calculated oly whe A7 < $B$ ( < K), ote that the hypergeometric distributio is valid oly if -r K-, which is equivalet to K- whe r. To see why the coditio -r K- is ecessary, ote that K- is the umber of positives i the populatio (whe H is true), while -r is the umber of positives i the sample. Calculatio of (S) for at most oe egative (r ): IF(A7"","",IF(A7<($B$),HYGEODIST($A7-,$A7,($B$)-,$B$))) To see why (S) is calculated oly whe A7 < $B$ ( K), ote that the coditio -r K- is equivalet to K whe r. Calculatio of (S) for at most two egatives (r ) : Ref code: DWG-SGL- Issue o. age: 9/34

20 IF(A7"";"";IF(OR(A7<;B$B$);"";IF(A7<B$;HYGEODIST(A7-;A7;B$-;B$)))) To see why (S) is calculated oly whe A7 < $B$ ( < K), ote that the coditio -r K- is equivalet to K whe r. To see why (S) is ot calculated whe $B$ $B$ ( K), ote that the hypergeometric distributio is valid oly if -K r (see remark c ), which is equivalet to K - whe r. Calculatio of the actual proportio (k`) Actual proportio of positives for calculated sample size is calculated i cell C.) $B$/$B$, which equals K/. c ote that K is the umber of egatives i the populatio (whe H is true), while r is the umber of egatives i the sample Ref code: DWG-SGL- Issue o. age: /34

21 7.5.. Calculatio based o proportio The ROUDU fuctio is applied for the calculatio i the third hypergeometric argumet (blue part of formula). To see why the IF statemets (uderlied italic fots ) are used, replace $B$ i chapter 7.5. by ROUDU($B$ x $B$), i.e., replace K by ROUDU( x k). Calculatio of (S) for zero egatives (r ) : IF($A7"","",IF($A7<ROUDU($B$*$B$,)-, (HYGEODIST($A7,$A7,ROUDU($B$*$B$,)-, $B$)))) Calculatio of (S) for at most oe egative (r ): IF($A7"","",IF($A7<ROUDU($B$*$B$,), HYGEODIST($A7-,$A7, ROUDU($B$*$B$,)-,$B$))) Calculatio of (S) for at most two egatives (r ) : IF(A7"";"";IF(OR($A7<;$B$ROUDU($B$*$B$;));"";IF($A7<ROUDU($B$* $B$;);HYGEODIST($A7-;$A7;ROUDU($B$*$B$;)-;$B$)))) Back calculated (actual) proportio (k`) of positives for calculated sample size (cell C). k` ROUDU($B$*$B$)/$B$, which equals ROUDU(K)/ Restrictios limitatios For particular data oe ca see limitatios by choosig DATA from Excel meu bar followed by the commad Validatio (visible oly whe the sheet protectio is off see chapter 7.6.3). I settig up the limitatios we had i mid the reasoable use of the software, i. e. the software shall cover realistic laboratory situatios. Ref code: DWG-SGL- Issue o. age: /34

22 7.6.. Calculatio based o the umber of positives populatio size : max, where max is adjustable (by macro buttos) up to 65 umber of positives K: K Remark: Zero positives i the populatio (K) is ot allowed (as this is ot a realistic laboratory assumptio ad results for such example would be wrog also due to theoretical reasos). For example, whe K, H : - does ot make sese, if < mathematical expressio udefied. max. umber of egatives (r):,,, util the coditio -K r is fulfilled To see why the coditio -K r is ecessary, ote that -K is the umber of egatives i the populatio (whe H is true), while r is the umber of egatives i the sample. cofidece level (CL) rage:. CL.9999 (alog the survey performed i withi EFSI laboratories typical reported values of this parameter were:.95 ad.99) is Calculatio based o the proportio of positives populatio size : max, where max is adjustable (by macro buttos) up to 65 proportio of positives (k):. k Remark: Zero proportio of positives (k) is ot allowed (see explaatio i poit (alog the survey (EFSI ) typical rages applied were betwee:.5 ad.9). max. umber of egatives (r):,,, util coditio -RoudUp(k x ) r is fulfilled. To see why the coditio is ecessary, ote that -RoudUp(k x ) is the umber of egatives i the populatio (whe H is true). cofidece level (CL) rage:. CL.9999 (see remark from poit 7.6.) rotectio of the software The protectio optio (without a password) is eabled so that users may oly eter data i specific required cells. This protectio ca be disabled if you wish to experimet/ or chage the package Choose: Tools/Uprotect sheet. To uhide colums choose: Format/Colum/Uhide. Ref code: DWG-SGL- Issue o. age: /34

23 8. VALIDATIO OF THE HYERGEOETRIC SALIG TOOL (VERSIO ) 8.. Correctess of the sample size () calculatio whe the proportio of positives k is specified (iteger ad o iteger Ks) The validatio was performed for, or egatives (at most allowed), respectively Criteria Calculated sample sizes ad calculated actual cofidece levels shall match whe calculated by the software ad by had. Calculated sample sizes, cofidece levels ad actual proportios obtaied with calculatios based o the umber of samples shall match with the calculatio based o proportio Validatio procedure For two examples (case A ad B) from Table 3 calculatios were made by had ad by software ad the results have bee compared. CASE A: ; k.9; (-α).95; for r, ad K.9x 8 (iteger) CASE B: ; k.9; (-α).95; for r, ad K.9x8.9 (o iteger) Results obtaied by the Hypg_roportio excel sheet were compared with the results obtaied with Hypg_umber, o such a way that for a o iteger K from example B, K was rouded up ad applied i the calculatio with the umber of positives Results Table 5: Compariso of results calculated by had (see i appedix.) versus calculatio by the software (summary) Kkx RoudUp(K)- calculated by had calculated by software CL calculated by had CL calculated by software Criteria fit? r, k.9, CL yes yes r, k.9, CL yes yes r, k.9, CL yes yes Ref code: DWG-SGL- Issue o. age: 3/34

24 Table 6: Compariso of results calculated with Hypg_roportio versus calculated with Hypg_umber (hypgeom. proportio of positives) k.9, CL.95 (hypgeom. umber positives) CL.95 sample size actual Sample K* actual actual CL proportio actual CL size populatio size requested proportio calculated k` r r r *see descriptio of validatio procedure poit Criteria fulfilled? Yes. Ref code: DWG-SGL- Issue o. age: 4/34

25 8.. Does the calculated sample size»guaratee«a 'at least' requested proportio of positives? 8... Criteria Resultig/ calculated sample size (umber of samples for aalysis) has to be such that the laboratory requiremets o the umber/ proportio of positives are met exactly or are slightly higher Validatio procedure Sample sizes are calculated, for populatio sizes () from to 5, at a cofidece level CL ( α).95 for k.9 ad r. Actual proportios are back calculated. Results are show below Results See Figure 8 ad Table actual proportio of positives actual proportio of positives requested umber of positives (Kk*) requested proportio (k.9) Figure 8: Actual proportio of positives k` for itegers ad o itegers K. Whe K is a iteger (ote the umbers i blue rectagles) actual ad requested proportio match exactly. For o itegers K the actual proportio k` is higher tha the requested proportio k. ote that for the give example the requested k.9 (red lie) Ref code: DWG-SGL- Issue o. age: 5/34

26 Table 7: Actual proportio k` of positives ad actual CL for calculated sample size (for itegers ad o itegers K) populatio size umber of positives requested Kk* umber positives for H test RoudUp(k*)- calculated sample size Actual CL actual k` RoudUp(k*)/ 9 8, 8,9778,9 9,9 9, 9,988,99,8, 9,9545,967 3,7,,965,93 4,6,,967, ,5 3,,974, ,4 4,,95, ,3 5, 3,9559,94 8 6, 6, 4,968, , 7, 5,9649, ,,959,9 8,9 8, 3,9579,948 9,8 9, 4,9636,99 3,7, 4,956,93 4,6, 5,9585,967 5,5, 6,9635,9 6 3,4 3, 6,9538,93 7 4,3 4, 7,959, , 5, 8,9634, , 6, 8,9548, , 5,95,9 3 7,9 7, 6,9566,93 3 8,8 8, 7,96, ,7 9, 7,9555, ,6 3, 8,968, ,5 3, 8,9545, ,4 3, 9,9596, ,3 33, 9,9537, , 34,,9585, , 35,,959, , 8,96,9 4 36,9 36, 8,955, ,8 37, 8,95, ,7 38, 9,9558, ,6 39, 9,95, ,5 4,,9565,9 46 4,4 4,,95, ,3 4,,957, , 43,,958, , 44,,9577, , 9,9537, Criteria fulfilled? Yes. Ref code: DWG-SGL- Issue o. age: 6/34

27 8.3. Calculatio based o the umber positives - validatio Validated through poit Some additioal tests Compariso of sample sizes calculated by EFSI hypergeometric tool ad with HyperBay calculator Values obtaied by the ew hypergeometric tool (Hypg_roportio sheet) were compared by results obtaied with HyperBay calculator 5 (Hg sheet) published o SWGDRUG web pages: Results match. Table 8: Sample sizes calculated by EFSI Hypg_roportio (results agree with correspodig results obtaied by the HyperBay calculator) r CL.95 CL.99 k. k.5 k.7 k.9 k. k.5 k.7 k r CL.95 CL.99 k. k.5 k.7 k.9 k. k.5 k.7 k r CL.95 CL.99 k. k.5 k.7 k.9 k. k.5 k.7 k / / 4 7 / 4 8 / Ref code: DWG-SGL- Issue o. age: 7/34

28 8.4.. Idepedat validatio obtaied from HSA 6 Idepedet validatio data d, which were kidly provided to the DWG form the reviewers 6 of the draft versio of this documet ad software, cofirmed correspodig results published i this documet (Table 5, Table 7 ad Table 8). For results published i Table 8 idepedet validatio has bee performed oly for the followig set of parameters: CL.99, k.9 ad r, r ad r, respectively Testig performed by the author 5 of the»hyperbay«sample size calculator Testigs of the draft versio of the EFSI hypergeometric sample size calculatio were kidly performed also by the author of HyperBay sample size calculator. He ru (with EFSI calculator) the examples icluded i the»readme file«of the published HyperBay calculator (see ) ad did ot fid ay icosistecies. e 9. COCLUSIOS 9.. Software The hypergeometric sample size calculatio tool (versio ) is validated ad fit for purpose. Other tools of the EFSI DWG Calculator for Qualitative Samplig of seized drugs (versio ) remaied uchaged with respect to the former versio (9) ad have bee validated previously. Therefore, we may coclude that the software package versio is validated ad fit for purpose. 9.. Other The validatio report (Validatio of the Guidelies O Represetative Samplig, DWG-SGL-, versio, 9) has bee revised (cocerig the hypergeometric calculatio), sectio withdraw ad the ew versio of the validatio report has bee released. 3 The documet Guidelies o Represetative Drug Samplig, UODC & EFSI DWG, ST/AR/38, April 9 is suggested to be reviewed (oly cocerig the hypergeometric samplig part) ad revised appropriately, if ecessary. d As a idepedet validatio, a program writte usig the R software (R available at project.org/, is a free software uder the GU roject.) was used. The program code applied was a part of the reviewer report. e private commuicatio: J. Gerlits S.Klemec, e mail 6 ov Ref code: DWG-SGL- Issue o. age: 8/34

29 . AEDIX.. Calculatios by had Table 9: Calculatio by had for zero egatives (r) alog Equatio. If H is true, H is rejected with red marked probability α (i.e., H is accepted with red marked probability -α). CASE A ; k.9; (-α).95; r K k* 8; H test at K -7 sample size 3 cosecutive calculatios (α)(sample positives -r) (-α) (Sample egatives > r) CASE B ; k.9; (-α).95; r K k* 8.9; H test at TRUC(K)8 sample size cosecutive calculatios (α)(sampl e positives -r) (Sample egatives > r) ad so o ad so o ad so o to to Ref code: DWG-SGL- Issue o. age: 9/34

30 Table : Calculatio by had for at maximum oe egative allowed alog Equatio 3. If H is true, H is rejected with red marked probability α (i.e., H is accepted with red marked probability -α). CASE A (Kiteger) ; k.9; (-α).95; r K k* 8; H test at K -7 sample size cosecutive calculatios oe zero S zero 7 7 (α)(sample positives -r).85.5 (-α) (Sample egatives > r) CASE B (K iteger) ; k.9; (-α).95; r K k* 8.9; H test at TRUC(K)8 sample size cosecutive calculatios oe zero S zero 8 8 (α)(sample positives -r) (-α) (Sample egatives > r) zero zero to 6 ad so o ad so o (ot calculated by had) 3 to 7 ad so o ad so o (ot calculated by had). 7 zero zero ad so o Ref code: DWG-SGL- Issue o. age: 3/34

31 Table : Calculatio by had for at maximum two egatives allowed alog Equatio 4. If H is true, H is rejected with red marked probability α (i.e., H is accepted with red marked probability -α). CASE A (Kiteger) ; k.9; (-α).95; r K k* 8; H test at K -7 sample size cosecutive calculatios two oe S (α) (Sample positives - r) (-α) (Sample egatives > r) CASE B (K iteger) ; k.9; (-α).95; r K k* 8.9; H test at TRUC(K)8 sample size cosecutive calculatios two oe S (α) (Sample positives - r) (-α) (Sample egatives > r) oe.5.85 oe oe oe to 8 ad so o. ad so o (ot calculated by had) 3 to 9 ad so o ad so o (ot calculated by had) 9 oe oe () oe oe () *see ote *see ote *if x>, take x, i.e.: 8 7 ad 8 9 Ref code: DWG-SGL- Issue o. age: 3/34

32 .. Details o calculatio of S, S, S A geeral equatio ca be writte as: ( X x / < K ) x x () ad S, S, S i Equatio to Equatio 4 are calculated: For S x (i.e. r), which gives: x x () () ( X x / < K, x ) zero S () ote, that x -r, therefore for r, S above may be rewritte also as: S r r () () () For, take x - (i.e. r),ad equatio is follows: S S x x () (), which is equivalet to S r r () (). For S take x - (i.e. r) ad equatio is as follows: S x x () (), which is also equivalet to S r r () (). Ref code: DWG-SGL- Issue o. age: 3/34

33 .3. Biomial coefficiet ad calculatios»by had«for ay set cotaiig elemets, the umber of distict k-elemet subsets of it that ca be formed (the k- combiatios of its elemets) is give by the biomial coefficiet, where k ad are positive itegers. k For easier uderstadig of»calculatios by had«(chapter.), ote that (geeral otatio is applied here): k! k!( k)!, wheever k, ad which is zero whe k >. ad ad! as well. Have i mid also: Factorial of egative umbers are ot defied, it is therefore ot possible to calculate. Ref code: DWG-SGL- Issue o. age: 33/34

34 . RESOSIBLE FOR ERRORS lease address questios, report errors ad/or bugs fidigs i the hypergeometric part software or withi this documet to the soja.klemec@policija.si ad/or to DWG cotact perso through the cotact form (for the latest updates about curret cotact perso please see at EFSI ublic ope area: ). Dr. Soja Klemec soja.klemec@policija.si Head of Chemistry Departmet atioal Foresic Laboratory, Vodovoda 95 Ljubljaa Sloveia. REFERECES UODC, Guidelies o Represetative Drug Samplig, UODC & EFSI DWG, ST/AR/38, April 9, ISB , U, 9 Validatio of the guidelies o represetative samplig, DWG-SGL-, versio, 9 3 Validatio of the guidelies o represetative samplig, DWG-SGL-, versio, 4 Frak, R.S., Hikley, S.W. ad Hoffma, C.G., Represetative Samplig of Drug Seizures i ultiple Cotaiers, Joural of Foresic Scieces, JFSCA, 99, 36 (), Joh Gerlits, Utah Bureau Of Foresic Services, USA, author of a excel based hypergeometric samplig probability calculator: HyperBay. Software available at: 6 Agelie Yap Tiog Whei, Health Scieces Authority, Sigapore ad Dr. Cheag Wai Kwog, atioal Istitute of Educatio, Sigapore, i:»reviewer report o draft documet: EFSI Hypergeometric Software vers. backgroud of calculatio ad validatio report«, pp 6-8, 9 ov. (report was kidly provided to DWG by Dr. Agelie YA Tiog Whei). Ref code: DWG-SGL- Issue o. age: 34/34