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Analyzing Data Sets with Missing Data: An Empirical Evaluation of Imputation Methods and Likelihood-Based Methods Ingunn Myrtveit, Erik Stensrud, Member, IEEE, and Ulf H. Olsson AbstractÐMissing data are often encountered in data sets used to construct effort prediction models. Thus far, the common practice has been to ignore observations with missing data. This may result in biased prediction models. In this paper, we evaluate four missing data techniques (MDTs) in the context of software cost modeling: listwise deletion (LD), mean imputation (MI), similar response pattern imputation (SRPI), and full information maximum likelihood (FIML). We apply the MDTs to an ERP data set, and thereafter construct regression-based prediction models using the resulting data sets. The evaluation suggests that only FIML is appropriate when the data are not missing completely at random (MCAR). Unlike FIML, prediction models constructed on LD, MI and SRPI data sets will be biased unless the data are MCAR. Furthermore, compared to LD, MI and SRPI seem appropriate only if the resulting LD data set is too small to enable the construction of a meaningful regression-based prediction model. Index TermsÐSoftware effort prediction, cost estimation, missing data, imputation methods, listwise deletion, mean imputation, similar response pattern imputation, full information maximum likelihood, log-log regression, ERP. æ 1 INTRODUCTION M ISSING data are often encountered in software engi- neering data sets that are used to construct effort prediction models [37], [13]. The International Software Benchmarking Standards Group (ISBSG) database is no exception. It has a large fraction of missing data, in some variables more than 40 percent [4], [16]. The ERP 1 data set presented in this paper also includes several observations with missing data in one or more variables. There are several reasons why observations may have missing values. High data collection cost may cause missing values. The cost of gathering and reporting data from software projects is nonnegligible. DeMarco estimates it would constitute a 5 to 10 percent of total cost [11]. From our personal experience as an ERP project manager, we know that some data indeed cost more to collect. For example, it is more difficult, and, therefore, costly to collect data on Interfaces and Effort than on Users, Sites and Modules. Therefore, we expected, and did indeed find, that there are more missing values in Interfaces and Effort than in Users, Sites, and Modules (see Table 1). Another reason for missing values is that some values are so called wild values. A wild value is a value we know is untrue. For example, if a reported Effort is negative, we know it must be untrue. Typically, a wild value is due to a punching error. Also, it may be due to someone who did not know how to measure and report that variable correctly. A common data screening procedure is to replace wild values by ªmissingº thereby creating additional missing values. It should be observed that a wild value is not synonymous to an outlier, an outlying observation. Yet another reason why observations may have missing values is that some respondents just do not report some of the variables for some reason. There are basically three ways to handle missing data. One option is to remove incomplete observations by listwise deletion (LD). Alternatively, one may fill in the holes by some imputation method. A third option is to use a model- based method. In the two first cases, complete-case analysis methods may be applied to the resulting complete data set whereas a model-based method like the full information maximum likelihood method (FIML) is able to analyse incomplete data sets directly. That is, it is an incomplete- case analysis method. Thus far, the common practice when constructing regression-type effort prediction models has been to apply a two-stage procedure, consisting of ignoring missing data (using LD) before applying regression analysis to construct the effort prediction model. LD is routinely used since most statistical packages use LD as default. Also, some effort prediction models based on machine learning use LD as default. An example is estimation by analogy as imple- mented in ANGEL [34]. Unfortunately, LD has several drawbacks. The most obvious drawback is that it discards a considerable amount of information. This is especially unfortunate in empirical IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. 27, NO. 11, NOVEMBER 2001 999 . The authors are with the Norwegian School of Management, PO Box 580, N-1301 Sandvika, Norway. E-mail: {ingunn.myrtveit, erik.stensrud, ulf.olsson}@bi.no. Manuscript received 01 Jan. 2001; revised 01 May 2001; accepted 01 Aug. 2001. Recommended for acceptance by S. Pfleeger. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number 114703. 1. Actually, package-enabled reengineering (PER) projects implement enterprise resource planning (ERP) systems. Therefore, the term ªPER projectº is more appropriate than the term ªERP project.º Still, we have chosen to use the latter since the term ERP recently has become an established term in the research community with the April 2000, vol. 43, no. 4, issue of the Communications of the ACM. 0098-5589/01/$10.00 ß 2001 IEEE

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d+-()')N (+ !"= #LM" 1**71 *]/'88N 2*88 1/0(*& (+ E+(56'((*)-1 +, 70110-3 &'('? d+-10&*) ,0)1( (5* 6'((*)- 0-@'E8* >R? Z*( /1 '11/7* (5'( 2* 2'-( (+ 076/(* '" ,0)1(?"- (501 .'1*= 2* 5'G* (5* .5+0.* +, 1*8*.(0-3 '-N 1/E1*( +,(5* G')0'E8*1 '+ (+ '"# '1 7'(.50-3 G')0'E8*1? @5* 1'7*3+*1 ,+) 076/(0-3 '% 0- @'E8* >J? @5'( 01= 076/(0-3 '%&+*1 -+( .'/1* '-N 7+)* ()+/E8* (5'- 076/(0-3 (5* +(5*)G')0'E8*1? @5* 7'0- +EP*.(0+- '3'0-1( #LM" 01 (5'( 0()*]/0)*1 ' (5+)+/35 B-+28*&3* +, (5* &'(' 20(5 )*3')& (+1*8*.(0-3 (5* 7'(.50-3 G')0'E8*1 ,+) *'.5 G')0'E8* (+ E*076/(*&? U81+= (5* 7+)* G')0'E8*1 2* 1*8*.( '1 7'(.50-3G')0'E8*1= (5* ,*2*) (5* 1/E1*( +, 6+(*-(0'88N 10708') .'1*1?d+-10&*) ,+) *F'768* 076/(0-3 '" /10-3 '+ (+ '"# '17'(.50-3 G')0'E8*1? "- (501 .'1*= (5*)* 2+/8& E* -+7'(.50-3 .'1*1? @5'( 01= (5*)* ')* -+ )+21 0- @'E8* >J

(5'( ')* .+768*(* 0- '+ (+ '"# '1 2*88 '1 0- '"? U( (5*+(5*) *F()*7*= 2* .+/8& 1*8*.( +-8N '$ (+ '"# '1 7'(.50-3G')0'E8*1 (+ 076/(* '" 0- )+2 >? "- (501 .'1*= '88 )+21 [(5)+/35 f 7'N E* .+-10&*)*& ,+) 10708') .'1* .'8./8'(0+-1?

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\* +E1*)G* (5'( 0- (5* 16*.0'8 .'1*1 &*60.(*& 0- @'E8* >R'-& @'E8* >J= (5* ZQ 7*(5+& 2+/8& 0-./) ' >II 6*).*-( 8+11+, 0-,+)7'(0+- 6)+G0&*& 2* 2'-( (+ /1* '88 (5* >I G')0'E8*1?

@5* 6'((*)- +, 70110-3 &'(' 0- (5* KLM &'(' 1*( 7+1(8N)*1*7E8*1 (5'( 0- @'E8* >J? #LM" 1**71= (5*)*,+)*= ' 7+)*'66)+6)0'(* 7*(5+& (5'- !" 0- (501 .'1*? \* '81+ E*80*G*(5'( #LM" 3*-*)'88N 01 (+ E* 6)*,*))*& +G*) !" E*.'/1* 0(80B*8N 0-()+&/.*1 8*11 E0'1 (5'- !"?

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Y-* 7'N 1/16*.( (5'( 8*11 1/..*11,/8 6)+P*.(1 )*6+)( (+ '8*11*) *F(*-( (5'- 7+)* 1/..*11,/8 6)+P*.(1? S+88+20-3 (50180-* +, (5+/35(= +-* 2+/8& '11/7* (5'( 8')3*= .+768*F6)+P*.(1 )*6+)( (+ ' 8*11*) *F(*-( (5'- 17'88= -+( 1+ .+768*F=6)+P*.(1 E*.'/1* (5*N 5'G* ' 5035*) 6)+E'E080(N +, -+( E*0-31/..*11,/8? \*)* (501 (5* .'1*= (5*)* 2+/8& E* ' E0'1 .'/1*&EN (5* 70110-3 &'('= '-& (5* &'(' 2+/8& -+( E* !dUL?S+)(/-'(*8N= 8')3* 6)+P*.(1 ')* 7'-'3*& EN (5* 7+1(*F6*)0*-.*& 6)+P*.( 7'-'3*)1 25*)*'1 (5* 17'88*) 6)+P*.(17+)* +,(*- ')* 5*'&*& EN P/-0+) 7'-'3*)1? "( 01 (5*)*,+)*-+( +EG0+/1 25*(5*) 17'88 +) 8')3* 6)+P*.(1 5'G* (5* 5035*)1/..*11 )'(* '-&= (5/1= 25*(5*) (5*)* ')* 7+)* 70110-3 &'('0- 17'88 +) 0- 8')3* 6)+P*.(1? \* E*80*G* (5*)*,+)* CE'1*& +-*F6*)( B-+28*&3* +, (5* &'('D (5'( 70110-3 &'(' ')* *]/'88N,+/-& '7+-3 17'88 '-& 8')3* 6)+P*.(1 '-& (5'( (5* &'('(5*)*,+)* ')* !dUL? O+2*G*)= +/) *760)0.'8 )*1/8(1 C0-#*.(0+- J?>D 1**7 (+ 1/33*1( ' 18035( E0'1 C-+-)'-&+770110-3 &'('D?

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