Sugi-89-265 Bretheim.pdf - sasCommunity

COMPUTING THE STANDARD DEVIATION FROM A GEOMETRlC MEANDaniel R. Bretlteim, KPMG Peat MarwickINTRODUCTIONTIle requirement to compute the standard deviationfrom fI. geometric (rather than arithmetic) mean hasbeen prompted by the desire to medel " portion of theFederal government's methodology for reimbursinghospitals for servK"" provided to Medicare recipients.The Health Care Financing Administralion (HCF A) h ..used this approach at; the mecllanism to identifyunusually higil-L"()st cases, known as outliers. Thegeomett'ic mean cluu'ge for each diagllosis-l"elatedgroup (DRG) serves as .. benchmark to which multiplesof the DRG's alandard devialion are added. Theresulting threshold mual be exceeded for an individualcase to achieve llouUieI.'lI status. These obseITationsare deleted from the data base prior to computing theaverage charge for each DRG.Procedures available within the SAS* System forcomputing averages are based pdmariJs on thearithmetic mean. The arithmetic mean is distinguishedfrom other seldom-used statistics sucb as thegeometric mean, which averages the iOg'al'ithms ofnumbers and the harmonic mean, wWch averagesreciproe~ls. F orttmately. ti,e SAS* System providestile flexibility Bnd data manipolaUon power to satisfytltis computing requirement.mean. The seven-step process outlined below was usedfor tllese purposes:1. Compute the natur91logar

EldUbit 2SAS* System statementsoptions nodate:libname in 'd;\sas':* compute logarithm for each value *:data one:set in. raw:logchg",logt charge.l ;proc sort data~one:by drg;'* compute mean of log values for each group *:proc means data=one mean noprint:var logchg~by drg:output out~stats m€an~mchg;* merge each observation with its group mean *:data twa:merge one stats:by drg;if first.drg then do:sum_sd""O :end:* compute geometric mean *:geo_mean ~ exp(mchg):* accumulate squared deviations ~:squardev ~ (charge - geo_mean)**2:sum_sd + squardev;* ,compute the standard deviation and threshold *.if last.drg an~ _freq_ > 1 then do:std_dev '" sqrt(sum_sd/Lfreq_ - 1)):thres = geo_mean ~ !strl_dev * 3}:output:end:procvarfQrmatlabeltitleltitle2run~print data~twolabel:drg _freq_ geo_mean std_dev thres:geo_mean std_dev thres dollarl0.2:_freq_'~'geo mean ~ 'GEO. MEAN'std~dev 'STD. DEV.'thres ' THRESHOLD' ;'OUTLIER iHRESHOLDS BASED ON'~'THREE STANDARD DEVIATIONS FROM THE GEOMETRIC MEA~;:1431

Exhibit 3Sample OutpntOUTLIER THRESHOLDS BASED ONTHREE STANDARD DEVIATIONS FROM THE GEOMETRIC MEANQ!lSllil.G.NGFA).MEANSTD.DEV.THRESHOLD123105201379433$18,623.466,162.691,356.76$2,729.59812.30157.90$26,812.238,599.591,830.48•1432