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{"sheet":"TOC","index":0,"n_rows":25,"n_cols":3,"rows_kept":25,"cut":false,"hidden":false,"merged":[[7,0,7,11],[13,2,13,11]]}
["Aldo Benini for OkularAnalytics"]
["An Excel-based method to generate ordinal random variates (such as for simulated ratings)"]
["with desired number of variables, levels, distributions and correlation patterns"]
["Version 19 June 2019"]
[]
["[Table of contents at the bottom of this sheet]"]
[]
["Ratings are frequently used in humanitarian needs assessments. They produce ordinal variables. For didactic or testing purposes, one may want to resort to simulated rating data, particularly when real data are not available or are incomplete, or are distributed such that they frustrate the purpose. Ratings of various items, topics, aspects, etc. should reflect the close associations in reality through an appropiate correlation pattern. For each rating-type indicator a specific distribution over its levels may also be decided and and then generated.\n\nIn Excel, the correlation pattern can be generated with the help of a mathematical device known as the Cholesky decomposition. https://en.wikipedia.org/wiki/Cholesky_decomposition gives background, but we rather give a small demo. We import a community-contributed function, stored in a VBA module, to do the actual calculation of the Cholesky matrix.\n\nThe need to create such an artificial data structure arose in a project that developed a methodology to assess the conditions of affected camps and village communities (\"sites\") in several humanitarian sectors. For each sector, five so-called \"screening questions\" were devised. To these key informants would respond by rating the situation of their sites on a scale with five levels. \n\nYet the procedure is of wider interest. For this demo we retain the screening questions as mere placeholders for any kind of ordinal rating variable with five levels. We assume that responses are highly positively correlated for screening questions within a given sector, but moderately only for responses to questions from different sectors. For simplicity, the demo works with two sectors only.\n\nWe demonstrate the generation of such variates step by step. We also demonstrated how, for training purposes mainly, causal factors can be incorporated into the simulated data. Participants can then be challenged to figure out, with normal statistical methods, which factors show up with significant effects in the response pattern.\n\nWe assume that the reader is familiar with the standard Excel functions used in the formulas. To access the two VBA modules re: Cholesky and correlation matrices, type Alt + F11. To update calculations in manual mode, type F9."]
[]
["Table of Contents"]
["Foundation:"]
[null,"CholeskyDemo","Mathematical foundation, can be skipped. Workbook must be saved with .xlsm extension"]
["Step-by-step:"]
[null,"CorrMatrix_Cholesky","Desired correlations among normal random variates for 10 rating variables ; Cholesky decomposition of the correlation matrix"]
[null,"CorrelatedNormal","Generation of correlated normal random variates for 10 rating variables and 100 observations"]
[null,"CorrVerified","Achieved correlation matrix"]
[null,"Distrib_NoCovariates","Desired distribution of the 10 rating variables"]
[null,"SimulatedResponses","The desired result: 10 ordinal variables with five levels"]
[null,"CorrObserved","Achieved correlations [see cautionary note]"]
["Incorporating causal effects into the simulated variables:"]
[null,"FactorInfluence","Defining desired effect sizes"]
[null,"ResponsesWithFactors","Simulated data that incorporated the causal effects"]
[null,"DistrictwiseResults","One possible analysis type for a workshop"]
["Auxiliary table:"]
[null,"NamedRanges","Convenience listing of named ranges"]

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["Using the Cholesky decomposition to generate correlated standard normal random variates"]
["Press F9 to get a new set of values in col 10 - 12 and recalculate col 14 - 15 and the achieved correlations!",null,null,null,null,null,null,null,null,"3. The starting point:",null,null,null,"4. The desired result:"]
["1. The plan:",null,null,null,"2. The tool:",null,null,null,null,"Uncorrelated standard normal random variates [obtained from =NORM.S.INV(RAND()) ]",null,null,null,"Correlated standard normal random variates"]
["Desired correlation matrix",null,null,null,"Its Cholesky decomposition",null,null,null,null,"Var1","Var2","Var3",null,"Var1","Var2","Var3"]
[1,0.8,0.4,null,{"$f":"CholeskyDecompose(corr)","v":1},0,0,null,null,{"$f":"_xlfn.NORM.S.INV(RAND())","v":-1.8376887911198481},{"$f":"_xlfn.NORM.S.INV(RAND())","v":8.3437990582623786E-2},0.20660206216500843,null,{"$f":"MMULT(J5:L25, TRANSPOSE($E$5:$G$7))","v":-1.8376887911198481},-1.4200882385463043,-0.53112148496400302]
[0.8,1,0.5,null,0.8,0.59999999999999987,0,null,null,{"$f":"_xlfn.NORM.S.INV(RAND())","v":0.3174226180494657},0.6339945169725093,0.39661866833506371,null,0.3174226180494657,0.63433480462307812,0.66064924470485886]
[0.4,0.5,1,null,0.4,0.3,0.8660254037844386,null,null,1.8003221796499114,-0.1320346575271551,-1.2865836375686104,null,1.8003221796499114,1.361036949203636,-0.43369563962598967]
["Must be symmetrical",null,null,null,"[user-written VBA function]",null,null,null,null,-0.65932583824762314,0.75537322996613521,-1.271941922049753,null,-0.65932583824762314,-7.4236732618417511E-2,-1.1386523829427009]
["because corr(X, Y) = corr(Y, X).",null,null,null,"Elements of upper triangle all = 0.",null,null,null,null,2.2265152605507068,-1.1296539573802018,-0.86756269664147012,null,2.2265152605507068,1.1034198340124446,-0.19962141766102326]
[null,null,null,null,null,null,null,null,null,0.20461094939039676,0.36293957290804651,-0.31249427645259537,null,0.20461094939039676,0.3814525032571453,-7.9901730316612241E-2]
["5. Checking the result in col 14-16:",null,null,null,null,null,null,null,null,1.121017431438764,0.612082523570187,-0.35572590394842429,null,1.121017431438764,1.2640634592931232,0.32396406004304318]
["Achieved correlations",null,null,null,null,null,null,null,null,-0.3674745183144093,0.75547726035009566,-0.61426923804376987,null,-0.3674745183144093,0.15930674155852986,-0.45231939412995026]
[{"$f":"CorrelationMatrix(DemoCorrelatedVars)","v":0.99999999999999978},0.86659073506636053,0.47740593151571381,null,null,null,null,null,null,0.5957558376315446,1.018733348789302,0.71970294629983078,null,0.5957558376315446,1.0878446793788168,1.1672033743635695]
[0.86659073506636053,1,0.54442716430294724,null,null,null,null,null,null,-0.35915035095399123,-0.44126729243489432,0.36295423103110847,null,-0.35915035095399123,-0.55208065622412961,3.8287256371921319E-2]
[0.47740593151571381,0.54442716430294724,1,null,null,null,null,null,null,-1.888266295229569,-9.6385758695869114E-2,-0.75104284815054911,null,-1.888266295229569,-1.5684444914011768,-1.4346444315295825]
["The achieved correlations deviate from the desired ones because of the sample variance.",null,null,null,null,null,null,null,null,0.51881858815525139,-0.37334517408070295,0.59067317374867823,null,0.51881858815525139,0.19104776607577942,0.60706185683822467]
[null,null,null,null,null,null,null,null,null,-0.7188033546482393,-2.0283349188214617,-0.85023671411612578,null,-0.7188033546482393,-1.7920436350114683,-1.6323484111605064]
[null,null,null,null,null,null,null,null,null,0.22110966242355742,-1.4390606237894299,1.4758026398091424,null,0.22110966242355742,-0.68654864433481178,0.93480825487944696]
[null,null,null,null,null,null,null,null,null,-0.65590521057720086,-0.4908954478687651,-1.7003978761113978,null,-0.65590521057720086,-0.8192614371830198,-1.8822184758450851]
[null,null,null,null,null,null,null,null,null,1.2980281421340809,0.38338203726373676,-0.8270609024499791,null,1.2980281421340809,1.2684517360655068,-8.2029883965811901E-2]
[null,null,null,null,null,null,null,null,null,-7.5636008437097182E-2,0.70991470056941119,-0.49571429867015321,null,-7.5636008437097182E-2,0.36544001359196887,-0.24658116887155473]
[null,null,null,null,null,null,null,null,null,0.58793156919616307,1.1928809113412808,-0.68423706630183945,null,0.58793156919616307,1.1860738021616988,4.7021945251934927E-4]
[null,null,null,null,null,null,null,null,null,-1.8628150552927896,0.62697443597139169,0.33799525992805896,null,-1.8628150552927896,-1.1140673826513967,-0.26432120986927482]
[null,null,null,null,null,null,null,null,null,1.664021158120224,-0.48149812104639156,-0.45852562406629233,null,1.664021158120224,1.0423180538683443,0.12406418820664966]
[null,null,null,null,null,null,null,null,null,0.75665293844896175,-0.85135817443000417,0.18310445607092263,null,0.75665293844896175,9.4507446101167014E-2,0.20582683355413423]

{"sheet":"CorrMatrix_Cholesky","index":2,"n_rows":27,"n_cols":14,"rows_kept":27,"cut":false,"hidden":false,"merged":[]}
["Two sectors, five screening questions each. Desired correlations among the responses to all ten questions"]
["Within a given sector, high positive correlations are plausible (green), across sectors low or moderate positive ones"]
[null,null,"Sector1","Sector1","Sector1","Sector1","Sector1","Sector2","Sector2","Sector2","Sector2","Sector2",null,"The matrix can be changed,"]
[null,null,"Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10",null,"but it has to stay symmetrical."]
["Sector1","Screen1",1,0.8,0.8,0.8,0.8,0.4,0.4,0.4,0.4,0.4]
["Sector1","Screen2",0.8,1,0.8,0.8,0.8,0.4,0.4,0.4,0.4,0.4]
["Sector1","Screen3",0.8,0.8,1,0.8,0.8,0.4,0.4,0.4,0.4,0.4]
["Sector1","Screen4",0.8,0.8,0.8,1,0.8,0.4,0.4,0.4,0.4,0.4]
["Sector1","Screen5",0.8,0.8,0.8,0.8,1,0.4,0.4,0.4,0.4,0.4]
["Sector2","Screen6",0.4,0.4,0.4,0.4,0.4,1,0.8,0.8,0.8,0.8]
["Sector2","Screen7",0.4,0.4,0.4,0.4,0.4,0.8,1,0.8,0.8,0.8]
["Sector2","Screen8",0.4,0.4,0.4,0.4,0.4,0.8,0.8,1,0.8,0.8]
["Sector2","Screen9",0.4,0.4,0.4,0.4,0.4,0.8,0.8,0.8,1,0.8]
["Sector2","Screen10",0.4,0.4,0.4,0.4,0.4,0.8,0.8,0.8,0.8,1]
[]
[]
[null,null,"Cholesky decomposition of the above matrix:"]
[null,null,{"$f":"CholeskyDecompose(CorrMatrix)","v":1},0,0,0,0,0,0,0,0,0]
[null,null,0.8,0.59999999999999987,0,0,0,0,0,0,0,0]
[null,null,0.8,0.26666666666666661,0.53748384988656983,0,0,0,0,0,0,0]
[null,null,0.8,0.26666666666666661,0.16537964611894448,0.51140831195675873,0,0,0,0,0,0]
[null,null,0.8,0.26666666666666661,0.16537964611894448,0.12033136751923718,0.49705012174770841,0,0,0,0,0]
[null,null,0.4,0.1333333333333333,8.2689823059472242E-2,6.0165683759618588E-2,4.7338106833115033E-2,0.89973541084243736,0,0,0,0]
[null,null,0.4,0.1333333333333333,8.2689823059472242E-2,6.0165683759618588E-2,4.7338106833115033E-2,0.67744783875195291,0.59210491916054664,0,0,0]
[null,null,0.4,0.1333333333333333,8.2689823059472242E-2,6.0165683759618588E-2,4.7338106833115033E-2,0.67744783875195291,0.25432694514278514,0.53470182370032771,0,0]
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[null,null,"Uncorrelated standard normal random variates",null,null,null,null,null,null,null,null,null,null,"Standard normal random variates with desired correlations"]
[null,"Site ID","Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10","ID","Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10"]
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{"sheet":"CorrVerified","index":4,"n_rows":13,"n_cols":14,"rows_kept":13,"cut":false,"hidden":false,"merged":[]}
["Achieved correlation pattern:"]
[]
[null,null,"Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10",null,"The proof is in the pudding!"]
[null,"Screen1",{"$f":"CorrelationMatrix(CorrNormalVariates)","v":1},0.83361566817825128,0.85274104158137809,0.84620090464719466,0.84231582987307552,0.40254935364300348,0.44184900844562786,0.35381994455227295,0.40355768971955119,0.33480164854220373,null,"The pattern of moderate vs. high correlations"]
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[null,"Screen3",0.85274104158137809,0.87221449460666767,1,0.87733855318722198,0.82578687179220966,0.31353625065055224,0.34448058583949598,0.24603850652551826,0.33345929267209196,0.23955306464685963]
[null,"Screen4",0.84620090464719466,0.84103857652688763,0.87733855318722198,1,0.80326004765677883,0.43032302011286383,0.43500609701857645,0.37498649186498456,0.4242119776222667,0.37971041698507718]
[null,"Screen5",0.84231582987307552,0.81530998297954871,0.82578687179220966,0.80326004765677883,1,0.3914431194515785,0.39672172191963856,0.32451323777995839,0.34261473317619012,0.32191135172430241]
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{"sheet":"Distrib_NoCovariates","index":5,"n_rows":16,"n_cols":27,"rows_kept":16,"cut":false,"hidden":false,"merged":[]}
[null,"Desired distribution of responses to screening questions",null,null,null,null,null,null,null,null,null,null,null,null,null,"Actual distribution of the simulated responses to screening questions"]
[null,null,null,"Sector1","Sector1","Sector1","Sector1","Sector1","Sector2","Sector2","Sector2","Sector2","Sector2",null,null,null,null,"Sector1","Sector1","Sector1","Sector1","Sector1","Sector2","Sector2","Sector2","Sector2","Sector2"]
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[null,"Levels of the response to screening question",1,0.2,0.4,0.4,0.4,0.4,0.2,0.4,0.4,0.4,0.4,null,null,"Levels of the response to screening question",1,{"$f":"COUNTIF(SimulatedResponses!B$3:B$102,$Q4) / COUNT(SimulatedResponses!B$3:B$102)","v":0.14000000000000001},{"$f":"COUNTIF(SimulatedResponses!C$3:C$102,$Q4) / COUNT(SimulatedResponses!C$3:C$102)","v":0.38},{"$f":"COUNTIF(SimulatedResponses!D$3:D$102,$Q4) / COUNT(SimulatedResponses!D$3:D$102)","v":0.34},{"$f":"COUNTIF(SimulatedResponses!E$3:E$102,$Q4) / COUNT(SimulatedResponses!E$3:E$102)","v":0.37},{"$f":"COUNTIF(SimulatedResponses!F$3:F$102,$Q4) / COUNT(SimulatedResponses!F$3:F$102)","v":0.35},{"$f":"COUNTIF(SimulatedResponses!G$3:G$102,$Q4) / COUNT(SimulatedResponses!G$3:G$102)","v":0.2},{"$f":"COUNTIF(SimulatedResponses!H$3:H$102,$Q4) / COUNT(SimulatedResponses!H$3:H$102)","v":0.38},{"$f":"COUNTIF(SimulatedResponses!I$3:I$102,$Q4) / COUNT(SimulatedResponses!I$3:I$102)","v":0.43},{"$f":"COUNTIF(SimulatedResponses!J$3:J$102,$Q4) / COUNT(SimulatedResponses!J$3:J$102)","v":0.38},{"$f":"COUNTIF(SimulatedResponses!K$3:K$102,$Q4) / COUNT(SimulatedResponses!K$3:K$102)","v":0.44}]
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[null,null,"Total",{"$f":"SUM(D4:D8)","v":1},0.99999999999999989,0.99999999999999989,0.99999999999999989,0.99999999999999989,1,0.99999999999999989,0.99999999999999989,0.99999999999999989,0.99999999999999989,null,null,null,"Total",{"$f":"SUM(R4:R8)","v":1},1,1,0.99999999999999989,0.99999999999999989,1,0.99999999999999989,1,1,1]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Scale values to look up"]
[null,"Cut-off points of the normal distribution",null,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,1]
[null,"for VLOOKUP",null,{"$f":"_xlfn.NORM.S.INV(SUM(D$4:D4))","v":-0.84162123357291452},{"$f":"_xlfn.NORM.S.INV(SUM(E$4:E4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(F$4:F4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(G$4:G4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(H$4:H4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(I$4:I4))","v":-0.84162123357291452},{"$f":"_xlfn.NORM.S.INV(SUM(J$4:J4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(K$4:K4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(L$4:L4))","v":-0.25334710313579978},{"$f":"_xlfn.NORM.S.INV(SUM(M$4:M4))","v":-0.25334710313579978},2]
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{"sheet":"SimulatedResponses","index":6,"n_rows":102,"n_cols":11,"rows_kept":102,"cut":false,"hidden":false,"merged":[]}
[null,"Ordinal random variates, with five levels and approximating the desired correlation pattern"]
["Site ID","Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10"]
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[100,{"$f":"VLOOKUP(CorrelatedNormal!N102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(B102)-2, 5, 13-COLUMN(B102)),13-COLUMN(B102),TRUE)","v":3},{"$f":"VLOOKUP(CorrelatedNormal!O102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(C102)-2, 5, 13-COLUMN(C102)),13-COLUMN(C102),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!P102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(D102)-2, 5, 13-COLUMN(D102)),13-COLUMN(D102),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!Q102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(E102)-2, 5, 13-COLUMN(E102)),13-COLUMN(E102),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!R102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(F102)-2, 5, 13-COLUMN(F102)),13-COLUMN(F102),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!S102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(G102)-2, 5, 13-COLUMN(G102)),13-COLUMN(G102),TRUE)","v":5},{"$f":"VLOOKUP(CorrelatedNormal!T102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(H102)-2, 5, 13-COLUMN(H102)),13-COLUMN(H102),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!U102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(I102)-2, 5, 13-COLUMN(I102)),13-COLUMN(I102),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!V102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(J102)-2, 5, 13-COLUMN(J102)),13-COLUMN(J102),TRUE)","v":3},{"$f":"VLOOKUP(CorrelatedNormal!W102,OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(K102)-2, 5, 13-COLUMN(K102)),13-COLUMN(K102),TRUE)","v":5}]

{"sheet":"CorrObserved","index":7,"n_rows":15,"n_cols":12,"rows_kept":15,"cut":false,"hidden":false,"merged":[]}
["Achieved correlation pattern:"]
[]
[null,null,"Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10"]
[null,"Screen1",{"$f":"CorrelationMatrix(SimulatedResponses)","v":1},0.70591855317963714,0.728951956322753,0.73722879932351359,0.71857201405819449,0.38871343493440808,0.36157295220892116,0.33603178088053803,0.40699890399955463,0.31117714975721916]
[null,"Screen2",0.70591855317963714,0.99999999999999989,0.79674539387893495,0.70652783070542624,0.73991998792490266,0.32000253837003817,0.32521254182254677,0.32348521493783938,0.30277009520396614,0.29804090977461195]
[null,"Screen3",0.728951956322753,0.79674539387893495,1,0.77153014708757872,0.75106152886245525,0.19256254205509615,0.23411992617788407,0.14943498300342342,0.24894876059083793,0.15474574589524884]
[null,"Screen4",0.73722879932351359,0.70652783070542624,0.77153014708757872,0.99999999999999989,0.75753154931963318,0.33533064935039453,0.3741495510841395,0.24432597843300036,0.35749420821866734,0.34246649319659539]
[null,"Screen5",0.71857201405819449,0.73991998792490266,0.75106152886245525,0.75753154931963318,0.99999999999999989,0.35802796732972558,0.42328009185204285,0.2976463523181927,0.35972168699614337,0.31365702367837228]
[null,"Screen6",0.38871343493440808,0.32000253837003817,0.19256254205509615,0.33533064935039453,0.35802796732972558,1.0000000000000002,0.71563773208674097,0.66402011842043873,0.73219896442416332,0.66996300656460739]
[null,"Screen7",0.36157295220892116,0.32521254182254677,0.23411992617788407,0.3741495510841395,0.42328009185204285,0.71563773208674097,1,0.69151781302335102,0.73291618717403673,0.71368805068906316]
[null,"Screen8",0.33603178088053803,0.32348521493783938,0.14943498300342342,0.24432597843300036,0.2976463523181927,0.66402011842043873,0.69151781302335102,1.0000000000000002,0.76070781726403902,0.76630121325528944]
[null,"Screen9",0.40699890399955463,0.30277009520396614,0.24894876059083793,0.35749420821866734,0.35972168699614337,0.73219896442416332,0.73291618717403673,0.76070781726403902,1,0.72817731684318288]
[null,"Screen10",0.31117714975721916,0.29804090977461195,0.15474574589524884,0.34246649319659539,0.31365702367837228,0.66996300656460739,0.71368805068906316,0.76630121325528944,0.72817731684318288,1.0000000000000002]
[]
["These are Pearson-moment correlations. This is not entirely correct. Polychoric correlations are appropriate, but are not available in Excel."]

{"sheet":"FactorInfluence","index":8,"n_rows":16,"n_cols":6,"rows_kept":16,"cut":false,"hidden":false,"merged":[]}
["Creation of correlated ordinal variates into which we simulate the effect of a causal variable of interest, e.g. for a data analysis workshop. Fictitious example:"]
[null,"Affected region is composed of two districts."]
[null,"These report different levels of problems."]
[]
[null,"District","Situation","Effect on underlying driver"]
[null,"X","Stronger pre-war economy; limited displacements since war began -> less intense problems",-0.5,null,"You can experiment with these values,"]
[null,"Y","Weaker pre-war economy; high level of displacements since war began -> more intense problems",0.5,null,"e.g. -1 and +1 would lead to much more"]
[null,null,null,null,null,"drastic differences in the screening responsed"]
[null,null,"This sheet will hold the coefficients for the values of the factors",null,null,"between Districts X and Y."]
[null,null,"by which we want to differentiate the simulated levels of problems."]
[null,null,null,null,null,"[See effect in sheet \"Districtwise results\"]"]
[null,null,"Here is only one example, about the effects that the different ",null,null,"The effect is incorporated by adding the values in the 6th column [red]:"]
[null,null,"situations in two geographical areas will have.",null,null,"=VLOOKUP(CorrelatedNormal!RC[7] + RC6, OFFSET(Distrib_NoCovariates!R12C4, 0, COLUMN(RC)-7, 5, 18-COLUMN(RC)),18-COLUMN(RC),TRUE)"]
[]
[null,null,"Other variables and their effects (e.g., social group-wise)"]
[null,null,"can be modeled analogously."]

{"sheet":"ResponsesWithFactors","index":9,"n_rows":102,"n_cols":16,"rows_kept":102,"cut":false,"hidden":false,"merged":[]}
[null,"Factors impacting on problem levels",null,null,null,null,"Ordinal random variates, incorporating the effects of underlying factors"]
["Site ID","District","TBD","TBD","TBD","EffectSum","Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10"]
[1,"X",null,null,null,{"$f":"VLOOKUP(B3, DistrictEffect, 3, FALSE)","v":-0.5},{"$f":"VLOOKUP(CorrelatedNormal!N3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(G3)-7, 5, 18-COLUMN(G3)),18-COLUMN(G3),TRUE)","v":3},{"$f":"VLOOKUP(CorrelatedNormal!O3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(H3)-7, 5, 18-COLUMN(H3)),18-COLUMN(H3),TRUE)","v":3},{"$f":"VLOOKUP(CorrelatedNormal!P3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(I3)-7, 5, 18-COLUMN(I3)),18-COLUMN(I3),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!Q3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(J3)-7, 5, 18-COLUMN(J3)),18-COLUMN(J3),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!R3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(K3)-7, 5, 18-COLUMN(K3)),18-COLUMN(K3),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!S3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(L3)-7, 5, 18-COLUMN(L3)),18-COLUMN(L3),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!T3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(M3)-7, 5, 18-COLUMN(M3)),18-COLUMN(M3),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!U3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(N3)-7, 5, 18-COLUMN(N3)),18-COLUMN(N3),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!V3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(O3)-7, 5, 18-COLUMN(O3)),18-COLUMN(O3),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!W3 + $F3, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(P3)-7, 5, 18-COLUMN(P3)),18-COLUMN(P3),TRUE)","v":1}]
[2,"X",null,null,null,-0.5,{"$f":"VLOOKUP(CorrelatedNormal!N4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(G4)-7, 5, 18-COLUMN(G4)),18-COLUMN(G4),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!O4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(H4)-7, 5, 18-COLUMN(H4)),18-COLUMN(H4),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!P4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(I4)-7, 5, 18-COLUMN(I4)),18-COLUMN(I4),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!Q4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(J4)-7, 5, 18-COLUMN(J4)),18-COLUMN(J4),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!R4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(K4)-7, 5, 18-COLUMN(K4)),18-COLUMN(K4),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!S4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(L4)-7, 5, 18-COLUMN(L4)),18-COLUMN(L4),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!T4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(M4)-7, 5, 18-COLUMN(M4)),18-COLUMN(M4),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!U4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(N4)-7, 5, 18-COLUMN(N4)),18-COLUMN(N4),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!V4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(O4)-7, 5, 18-COLUMN(O4)),18-COLUMN(O4),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!W4 + $F4, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(P4)-7, 5, 18-COLUMN(P4)),18-COLUMN(P4),TRUE)","v":1}]
[3,"X",null,null,null,-0.5,{"$f":"VLOOKUP(CorrelatedNormal!N5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(G5)-7, 5, 18-COLUMN(G5)),18-COLUMN(G5),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!O5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(H5)-7, 5, 18-COLUMN(H5)),18-COLUMN(H5),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!P5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(I5)-7, 5, 18-COLUMN(I5)),18-COLUMN(I5),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!Q5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(J5)-7, 5, 18-COLUMN(J5)),18-COLUMN(J5),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!R5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(K5)-7, 5, 18-COLUMN(K5)),18-COLUMN(K5),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!S5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(L5)-7, 5, 18-COLUMN(L5)),18-COLUMN(L5),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!T5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(M5)-7, 5, 18-COLUMN(M5)),18-COLUMN(M5),TRUE)","v":1},{"$f":"VLOOKUP(CorrelatedNormal!U5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(N5)-7, 5, 18-COLUMN(N5)),18-COLUMN(N5),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!V5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(O5)-7, 5, 18-COLUMN(O5)),18-COLUMN(O5),TRUE)","v":2},{"$f":"VLOOKUP(CorrelatedNormal!W5 + $F5, OFFSET(Distrib_NoCovariates!$D$12, 0, COLUMN(P5)-7, 5, 18-COLUMN(P5)),18-COLUMN(P5),TRUE)","v":2}]
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{"sheet":"DistrictwiseResults","index":10,"n_rows":18,"n_cols":12,"rows_kept":18,"cut":false,"hidden":false,"merged":[]}
["Actual distribution of the simulated responses to screening questions"]
["District","X","Sector1","Sector1","Sector1","Sector1","Sector1","Sector2","Sector2","Sector2","Sector2","Sector2"]
[null,null,"Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10"]
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[null,"Total",{"$f":"SUM(C4:C8)","v":1},1,1,1,1.0000000000000002,1,1.0000000000000002,1,1,1]
[]
["District","Y","Sector1","Sector1","Sector1","Sector1","Sector1","Sector2","Sector2","Sector2","Sector2","Sector2"]
[null,null,"Screen1","Screen2","Screen3","Screen4","Screen5","Screen6","Screen7","Screen8","Screen9","Screen10"]
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[null,5,{"$f":"COUNTIFS(ResponsesWithFactors!G$3:G$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.36},{"$f":"COUNTIFS(ResponsesWithFactors!H$3:H$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.2},{"$f":"COUNTIFS(ResponsesWithFactors!I$3:I$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.18},{"$f":"COUNTIFS(ResponsesWithFactors!J$3:J$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.16},{"$f":"COUNTIFS(ResponsesWithFactors!K$3:K$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.2},{"$f":"COUNTIFS(ResponsesWithFactors!L$3:L$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.46},{"$f":"COUNTIFS(ResponsesWithFactors!M$3:M$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.22},{"$f":"COUNTIFS(ResponsesWithFactors!N$3:N$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.28000000000000003},{"$f":"COUNTIFS(ResponsesWithFactors!O$3:O$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.2},{"$f":"COUNTIFS(ResponsesWithFactors!P$3:P$102, $B17, ResponsesWithFactors!$B$3:$B$102,\"=Y\") / COUNTIF(ResponsesWithFactors!$B$3:$B$102,\"=Y\")","v":0.22}]
[null,"Total",{"$f":"SUM(C13:C17)","v":1},1,1,1,1.0000000000000002,1,0.99999999999999989,1,1,1]

{"sheet":"NamedRanges","index":11,"n_rows":10,"n_cols":3,"rows_kept":10,"cut":false,"hidden":false,"merged":[]}
["Name","RefersToLocal","Cells"]
["Cholesky","=CorrMatrix_Cholesky!R18C3:R27C12",100]
["corr","=CholeskyDemo!R3C1:R5C3",9]
["CorrMatrix","=CorrMatrix_Cholesky!R5C3:R14C12",100]
["CorrNormalVariates","=CorrelatedNormal!R3C14:R102C23",1000]
["DemoCorrelatedVars","=CholeskyDemo!R3C14:R23C16",63]
["DistrictEffect","=FactorInfluence!R5C2:R7C4",9]
["NormalVariates","=CorrelatedNormal!R3C3:R102C12",1000]
["ResponsesWithFactors","=ResponsesWithFactors!R3C7:R102C16",1000]
["SimulatedResponses","=SimulatedResponses!R3C2:R102C11",1000]