Imputation Methods for Longitudinal Data: A Comparative Study

Transcription

1 Internatonal Journal of Statstcal Dstrbutons and Applcatons 017; 3(4): do: /j.sd ISSN: (Prnt); ISSN: (Onlne) Imputaton Methods for Longtudnal Data: A Comparatve Study Ahmed Mahmoud Gad 1, Rana Hassan Mohamed Abdelkhalek 1 Statstcs Department, Faculty of Economcs and Poltcal Scence, Caro Unversty, Caro, Egypt Department Statstcs, Mathematcs and Insurance, Faculty of Commerce, Benha Unversty, Benha, Egypt Emal address: Ahmed.gad@feps.edu.eg (A. M. Gad), gendy176@yahoo.com (R. H. M. Abdelkhalek) To cte ths artcle: Ahmed Mahmoud Gad, Rana Hassan Mohamed Abdelkhalek. Imputaton Methods for Longtudnal Data: A Comparatve Study. Internatonal Journal of Statstcal Dstrbutons and Applcatons. Vol. 3, No. 4, 017, pp do: /j.sd Receved: March 5, 017; Accepted: March 8, 017; Publshed: November 10, 017 Abstract: Longtudnal studes play an mportant role n scentfc researches. The defnng characterstc of the longtudnal studes s that observatons are collected from each subject repeatedly over tme, or under dfferent condtons. Mssng values are common n longtudnal studes. The presence of mssng values s always a fundamental challenge snce t produces potental bas, even n well controlled condtons. Three dfferent mssng data mechansms are defned; mssng completely at random (MCAR), mssng at random (MAR) and mssng not at random (MNAR). Several mputaton methods have been developed n lterature to handle mssng values n longtudnal data. The most commonly used mputaton methods nclude complete case analyss (CCA), mean mputaton (Mean), last observaton carred forward (LOCF), hot deck (HOT), regresson mputaton (Regress), K-nearest neghbor (KNN), The expectaton maxmzaton (EM) algorthm, and multple mputaton (MI). In ths artcle, a comparatve study s conducted to nvestgate the effcency of these eght mputaton methods under dfferent mssng data mechansms. The comparson s conducted through smulaton study. It s concluded that the MI method s the most effectve method as t has the least standard errors. The EM algorthm has the largest relatve bas. The dfferent methods are also compared va real data applcaton. Keywords: Dropout Mssng, Longtudnal Data, Mssng Data, Multple Imputatons, Sngle Imputaton 1. Introducton Longtudnal studes become an ncreasngly common research area especally n the feld of publc health and medcal scences. Such studes are desgned to nvestgate changes n a specfc varable, whch s measured repeatedly ether at dfferent tmes or under dfferent condtons. Mssng values are common n longtudnal studes because some ndvduals may mss a planned vst. There are many possble causes leadng to mssng values ncludng falure of measurement, accdents, errors resulted from collectng or enterng data, refusal to contnue, or other admnstratve reasons. Whenever there are mssng values, there s loss of nformaton, whch causes reducton n effcency. Also, under certan crcumstances, mssng data can ntroduce bas and thereby lead to msleadng nferences about the parameters. Mssng data can be classfed, based on the occurrence n tme, nto two patterns: ntermttent pattern and dropout pattern. The ntermttent mssngness, also termed as nonmonotone, means mssng values due to occasonally omsson, wth observed values afterwards. The dropout pattern, also termed monotone, where mssng values due to premature wthdrawal, wth no observed values afterwards (Gad and Ahmed [8]). Methods that handle mssng values depend upon the mechansm of mssngness. Mssng data mechansm refers to the underlyng process of generatng mssng data. Rubn [1] ntroduced three mssng data mechansms: mssng completely at random, mssng at random and mssng not at random. The mechansm s mssng completely at random (MCAR) f the mssngness s not related to any observed or unobserved responses. The mechansm s mssng at random (MAR) f the mssngness s ndependent of the unobserved data condtonal on observed responses, whereas the mechansm s mssng not at random (MNAR) f the mssngness depends on unobserved as well as some observed responses.

2 Internatonal Journal of Statstcal Dstrbutons and Applcatons 017; 3(4): Several statstcal approaches have been appled to the analyss of longtudnal data wth mssng values. These approaches should be selected based on the amount of mssngness and the mssngness mechansm. Some statstcal methods are vald only under certan stuatons wth specfed mssng rates. In other words, there s no unque best method avalable for all stuatons. Lttle and Rubn [1] revewed many tradtonal approaches for dealng wth mssng data and concluded that these methods are only approprate under the strong assumpton of MCAR mechansm. However, n practce the MAR mechansm s much more common than the MCAR mechansm. When the sze of the dataset s large enough, analyss can be conducted usng deleton methods such as the complete case analyss (CCA) method. The CCA can be used for any statstcal analyss and does not need specal computatons snce t s a default method n most statstcal computer packages. However, gnorng mssng values even n ths stuaton leads to loss of nformaton and reducton of statstcal power, whch may result n ncorrect statstcal nference. Imputaton methods are consdered as alternatves to the deleton methods. The term mputaton means replacng mssng values by other observed values or estmated values (Rubn [1]). However, even when an mputed value s closer to an deal predcted observaton; t s stll consdered as mputed data, not real data. The rule of thumb suggests that 0% or less of mssng data s acceptable rate to use mputaton methods (Lttle and Rubn [1]). The mputaton technques can be classfed accordng to the number of mputed values as sngle mputaton (SI) and multple mputatons (MI) methods. In SI technques each mssng observaton s replaced by a sngle value. In MI technques each mssng value s substtuted by two or more acceptable values to account for the uncertanty nherent n the mputaton process (Rubn [1]). Many smulaton studes have been conducted n lterature to evaluate the effcency of dfferent mputaton methods, see for example Engels and Dehr [6], Mshra and Khare [14], Naka [16], Naka and Ke [18], Naka et al. [17] and Zhu [8]. The man purpose of ths artcle s to compare the performance of eght mputaton methods. Ths s accomplshed by a smulaton study usng dfferent mssngness mechansms (MCAR, MAR, and MNAR) wth varous mssngness rates. For smplcty, and wthout loss of generalty, a monotone pattern of mssng data s assumed. The performance of mputaton methods s evaluated usng two crtera; the relatve bas (RB) and the mean squared error (MSE). The rest of the artcle s organzed as follows. In Secton, the basc notatons are descrbed. In Secton 3, dfferent mputaton approaches of handlng mssng data are revewed. In Secton 4, a smulaton study s presented to compare the eght mputaton methods. In Secton 5, the selected mputaton methods are appled to a data set concernng qualty of lfe among breast cancer patents n a clncal tral controlled by the Internatonal Breast Cancer Study Group. Fnally, Secton 6 s devoted to conclusons and dscussons.. Notaton For a longtudnal dataset wth balanced desgn all subjects have complete measurements and are measured on the same tme ponts. The man nterest s on the relatonshp between the response varable and some covarates. Unbalanced longtudnal data are possble when some values are ntermttently mssng or drop out from the data. The th repeated measures are potentally observed on the subject th at t = 1,, m; j = 1,, n and the total j tme ponts ( ) number of observatons s N= n. m Y represent the repeated response varables of subject, and (,, 1 X ) X = X are covarates or explanatory varables. p The y denotes the value of the varable Y and x k denotes the value of X k recorded at tme ( = = = ) t 1, m; j 1,, n ; k 1,, p. The (, T Y = y 1 yn ) k n p s a vector of values for the repeated measures and X = x s a matrx of values of tme-varyng or tmendependent covarates on the 3. Imputaton Methods th subject. Imputaton methods are used to compensate for the unt wth mssng values. Imputaton methods become mportant n statstcal analyss of ncomplete data. Some methods use only nformaton that belongs to the subject whose data were mssng, whle some used the values of other subjects. Imputaton methods are classfed based on the number of mputed values n place of mssng values nto sngle or multple mputatons. In sngle mputaton, each mssng value s mputed wth a sngle value whle n multple each mssng value s substtuted wth multple values producng several dfferent complete datasets. The eght mputaton methods are revewed n ths secton The Complete Case Analyss Method (CCA) The CCA s easy and straghtforward technque to handle mssng data. Ths method excludes all subjects, n the dataset, wth one or more mssng values at any measurement occason. Only cases havng complete observatons are consdered. The CCA method can be used for any type of statstcal analyss and does not need specal computatons snce t s the default method n most statstcal computer packages. If the mssng data mechansm s MCAR, the remanng sample of subjects can be consdered as a random sample from the orgnal sample. Ths mples that, for any parameter of nterest, f the estmates would be unbased for the full dataset wthout mssng data, they wll also be unbased for

3 74 Ahmed Mahmoud Gad1 and Rana Hassan Mohamed Abdelkhalek: Imputaton Methods for Longtudnal Data: A Comparatve Study the complete case dataset (Naka [15]). When the mssng data mechansm s not MCAR, the results from the CCA method may be based because the complete cases become unrepresentatve to the full populaton (Naka et al. [17]). Therefore, when the data s MCAR and only a small proporton of unts are excluded, ths method can be a sensble choce. 3.. The Mean Substtuton Method (MS) Based on mean mputaton method, the mean of the varable s consdered the best estmate of any subject who has mssng value for that varable. The mean value of non-mssng observatons s used to fll n mssng values for all observatons. Although mean substtuton mantans the same sample sze from reducton, t has some challenges. When data contans farly large mssngness rate, the mean mputaton method can dstort the dstrbuton of the varable because the possble extreme values are shfted to the mddle of the dstrbuton whch may complcate the analyss and results n underestmaton of the varance whch may cause large kurtoss (Lttle and Rubn [1]). The covarance also s underestmated because the mean mputaton for the mssng subjects has zero varance. In addton, ths mputaton method smlar to the CCA; t requres MCAR assumpton to obtan unbased and effcent estmates but ths assumpton s very restrctve The Last Observaton Carred Forward Method (LOCF) The LOCF method s a very common approach for handlng mssng data especally n dropout mssngness (Saha and Jones []). Ths method mputes the unobserved value by the last observed value for the same subject. For dropout mssngness, t s assumed that the last observed value s carred forward to the end of the study. Ths mples that the last observaton remans the same after dropout. The LOCF can also be appled to longtudnal data where the subjects are observed at several occasons, and some subjects are lost-to-follow up or have ntermttent mssng values. Ths stuaton could be consdered as unrealstc n many settngs. The LOCF method tends to underestmate the true varablty of the data. It s shown that LOCF method does not gve vald analyses f the mssngness mechansm s not MCAR (Lane [11]). However, t creates bas even f the strong MCAR assumpton s satsfed. The LOCF can gve satsfactory results, f the observatons n the dataset are approxmately close to each other. When the measurements occasons are short to some extent, ths ensures the effectveness of the LOCF method The Hot Deck (HOT) Method Ths method proposed by Madow et al. [13] n whch any mssng value of unt s replaced by a smlar respondng unt n the same sample. The respondng unt s chosen randomly or selected on the bass of smlarty crtera. In the case of more than one smlar subject to the subject whch contan the mssng values n the sample, the most smlar subject s selected and replace the mssng values from hs or her measurements. Also, n ths method the mssng value may be flled based on the correlaton among the varable contanng mssng data and the other varables whch has no mssng. Ths method performs well when the varable used to sort the data s hghly predctve of the varable wth the mssng values and when there s a large sample to ensure easly dentfyng a smlar case (Strener [5]). The hot deck method does not dstort the dstrbuton of the sampled values besdes the conceptual smplcty of applyng t. In addton, usng a smlarty crteron s a realstc matter and preserves some of the measurement error that would lkely be found f the value had been completed by the respondent. Based on the hot deck method, the standard devaton of the varable wth the nserted values s a better approxmate to the standard devaton value for the varable wthout the substtuted values. However, standard devatons are stll lkely to be lower (Strener [5]). It has some cautons lke dstortng of both correlatons and covarance because the mssng values are replaced wth values that already exst n the dstrbuton of scores. The smaller standard errors lead to greater lkelhood of a Type I error (Naka and Ke [18]) The K-Nearest Neghbors (KNN) Method Accordng to the KNN method, each mputed value s selected from the respondent who s the nearest to the subject wth mssng value based on the dstance between them. The dstance s computed usng the nformaton from the observed data. The KNN mputaton method s approprate only when the mssngness mechansm s MCAR. If MCAR assumpton s volated, ths leads to based results. Also, Rancourt et al. [0] stated that the mean estmates are unbased usng the KNN assumng the gnorable mssngness mechansm. Ths method has some nce features (Chen and Shao [3]). Frst, t s a hot deck method n the sense that donors are substtuted by a value from the same varable for a respondent of the same sample. The mputed values are actually occurrng values, and they may not be perfect substtutes, but are unlkely to be nonsenscal values. Second, the KNN method may be more effcent than the mean mputaton method, snce t makes use of auxlary nformaton provded by the x values and t s a nonrandom mputaton method. However, t does not use an explct model relatng y and x, hence, t s expected to be more robust aganst model volatons than other methods whch are based on explct models. Fnally, the KNN method provdes asymptotcally vald dstrbuton. Rancourt et al [0] stated that the KNN mputaton yelds normally pont estmates wth small or neglgble bas, assumng that a lnear relatonshp exsts between the varable of nterest y and the concomtant varable x used for nearest neghbor dentfed. But ths clam was not supported by any theoretcal result n general The Regresson Imputaton (Regress) Method Regresson mputaton method s sometmes dentfed as a

4 Internatonal Journal of Statstcal Dstrbutons and Applcatons 017; 3(4): condtonal mean mputaton. The basc dea behnd regresson method s dentfyng several predctors for the varable wth mssng values usng a correlaton matrx. The best predctors (the hghest correlatons) are selected and used as ndependent varables n a regresson equaton. The varable wth mssng data s used as a dependent varable. Ths varable s regressed on all other varables to produce a regresson equaton on the bass of the subjects wth complete data for the predctor varables. The regresson equaton s then used to replace mssng values for ncomplete subjects wth the predcted values. In an teratve process, the values for the mssng varable are nserted and then all subjects are used to predct the dependent varable. These steps are repeated untl there s lttle dfference between the predcted values from one step to the next, that s, they converge. The predctors from the last round are the ones whch are used to replace the mssng values (Saunders et al. [3]). Regresson assgns the subject s predcted value to the mssng value but subjects wth the same covarates wll exactly have the same mputed value (Engel and Dehr [6]). Ths method can yeld consstent estmates for the mean under normalty and MCAR assumpton for the mssng mechansm but, the sample covarance s underestmated (Lttle and Rubn [1]). Also, Allson [1] ponted out that regresson parameter estmates based on regresson mputaton under MCAR are relatvely unbased n large samples. However, t has two problems stemmng from the fact that the mputed values were perfectly predcted from other varables, they tend to ft a regresson lne together too well. Frst, they do not reflect the random error or varance so, the varance of the mputed value of the data set s underestmated whch lead to small standard errors and p- values at the tme of analyss. Second, the correlatons wth the mputed varables are overestmated because the underestmated varance of the mputed varable s n the denomnator of the correlaton formula (Allson [1]) The Expectaton Maxmzaton (EM) Algorthm The EM algorthm s ntroduced by Dempster et al. [4] and mplemented for many mssng data problems. It s an teratve algorthm that fnds the parameters whch maxmze the log-lkelhood functon when there are mssng values n the dataset. The EM algorthm s carred out through two steps: the expectaton step (E-step) and the maxmzaton step (M-step). Gven the current parameter estmates, the E-step calculates the condtonal expectaton of the complete data log-lkelhood gven the observed data and the current set of parameter estmates. The E-Step can be expressed symbolcally (Naka [15]) as follows: ( ˆ) ( ) Q θ θ = E g θ Y Y, ˆ obs θ = θ ( θ ) (, ˆ ms obs θ θ ) = g Y f Y Y = where s an estmate for θ and g ( Y ) dy ms θ s the complete data log-lkelhood. Gven the complete data log-lkelhood, the M-step fnds the parameter estmates that maxmze the complete data loglkelhood produced from the E-step to obtan updated parameter estmates. The M- step can be expressed (Naka [15]) as follows: Q t + 1 ( θ ˆ θ ) Q ( θ ˆ θ ) for all θ The teraton between M-steps and E-steps are contnued untl some convergence s met, that s untl values that are reestmated by the second step approxmate the prevous estmated values. Many advantages have been reported to the EM algorthm. Frst, the observed data lkelhood ncreases at every step. Second, the EM algorthm s preferred to regresson mputaton because the estmated parameter values that maxmze the observed data log-lkelhood functon are consstent, effcent under MAR condton and tend to be approxmately unbased n large samples and normally dstrbuted (Fchman and Cummngs [7]). Thrd, the obtaned varances are close to what s theoretcally desrable (Dragset [5]). However, the convergence of the teratons can be very slow n case of large fractons of mssng data (Naka and Ke [18]) The Multple Imputaton (MI) Method Multple mputatons method s consdered as a contnuaton to sngle mputaton method from the condtonal dstrbuton. The MI approach nvolves mputng each mssng value by two or more acceptable values to produce several dfferent complete datasets. Then each dataset s analyzed to produce dfferent parameter estmates. The sets of parameter estmates from each mputaton are then combned usng a specal rule (maybe by takng the average) to gve an overall (sngle) estmate of the complete data parameters as well as reasonable estmates of standard errors that ncorporate the varablty n results between the mputed datasets. A key feature of the MI method s that the uncertanty about the parameters n the mputaton model s taken nto account when mputng the unobserved values. In addton, the mputaton phase of the MI s operatonally dstnct from subsequent analyss. Applyng the MI typcally results n effectve estmates that are less based compared wth the estmates obtaned from sngle mputaton methods. Also, the MI provdes more correct standard errors, P-values, and confdence ntervals as opposed to sngle mputaton methods, whch gves too small standard errors (Van der Heden et al. [7]). It s also effcent, even f the number of mputatons s relatvely small and when between-mputaton varance s not too large (Naka and Ke [18]). However there are some dsadvantages of the MI method. Frst, snce some values are mputed nto the mssng value, mssng value ndvduals are allowed to have varyng probablty thus ndvdual varaton s gnored. Second, the uncertanty nherent n mssng values s gnored because the analyss doesn t dstngush between the observed and mputed values. Thrd, the MI procedure takes more work both to create the mputatons and to analyze the results. For

5 76 Ahmed Mahmoud Gad1 and Rana Hassan Mohamed Abdelkhalek: Imputaton Methods for Longtudnal Data: A Comparatve Study example, mputng 5 to 10 datasets cost tme, cause computatonal dffculty, and need testng models for each data set separately (Sheh [4]) Fourth, the MI does not satsfy normalty test n most stuatons (Naka [16]). Fnally, although each of the mputatons used n the MI procedure based on regresson parameters from the observed data and t s assumed that these regresson mputaton parameters are the true populaton parameters, but n fact they are only sample estmates from a sample dstrbuton. Therefore, when multple mputaton methods are mplemented, t s preferable to use new parameters drawn randomly for each mputaton from a Bayesan posteror dstrbuton of regresson mputaton parameters rather than usng the sample regresson parameters for each mputaton (Newman [10]). 4. Smulaton Study 4.1. Smulaton Settng The am of ths smulaton s to evaluate the behavor of eght mputaton methods under the three mssng data mechansms. It s based on a dataset for n subjects wth fve measurement tmes. The sample sze n, s chosen to range from small to large. We consder the sample szes n = 10, n = 50, and =100 to represent small, moderate and, large sample szes respectvely. It s assumed that there are two covarates; the tme TIME and the treatment group Grp. Hence the data are smulated accordng to the followng model y = + Tme + Grp + ε 0 1 j where Tme j was coded 0, 1,, 3, 4 for the fve tme ponts, and Grp s a dummy varable takes the value 0 for placebo group and value 1 for treatment group. The smple lnear regresson model for the mean profles of the repeated measurements Ε ( y ) = µ, j=0,1,, 3, 4 s used. The j varance-covarance structure s assumed as frst-order autoregressve AR (1). The parameters are fxed at 0 = 1, 1 = 0.5, and = 1. The ε ' s were generated from a multvarate normal wth zero mean and V( ε ) = σ = 1. The mean response s ( ) 0 1 Ε y = + Tme + Grp and the (co) varance for j j j tme ponts j and j equals σ ρ, for ρ 0 and σ s the error varance. The data are smulated to satsfy the multvarate normal dstrbuton and the correlatons between two varables Y and Y. The correlaton coeffcent s assumed as =0.5. The AR (1) structure s ρ ρ ρ ρ 3 ρ 1 ρ ρ ρ σ ρ ρ 1 ρ ρ 3 ρ ρ ρ 1 ρ 4 3 ρ ρ ρ ρ 1 In general, generatng each dataset s based on the followng assumpton: 1. The measurement at the frst tme pont ( t = 1) s fully observed,. The mssngness data mechansm are MCAR, MAR and MNAR, 3. The mssngness pattern s monotone, and 4. The number of replcatons s fxed at The comparson between methods depends on two measures; the Relatve Bas (RB) and Mean Square Error (MSE). The GLS method s used for estmatng the unknown parameters n the lnear regresson model. The parameter estmates have been obtaned for the selected mputaton methods: the CCA, the Mean, the LOCF, the HOT, the RM, the KNN, the EM, and MI methods. For the MCAR stuaton, the data are smulated wth dropout rates of 0%, 5%, 50%, 75%, and 87.5% at tme ponts 0, 1,, 3, 4 respectvely. If subject s mssng at a gven tme pont, then t s consdered mssng at all latter tme ponts. These rates ndcate the percentages of the orgnal sample that are mssng at each tme pont. For the MAR settng, f the value of the dependent varable s greater than the thrd quartle of the observatons, then the subject s dropped out at the next tme pont. For MNAR settng, after the frst tme pont, f the value of the dependent varable s greater than the thrd quartle of the observatons, then the subject s dropout at that tme pont and all subsequent tme ponts. 4.. Smulaton Results The smulaton results are presented n Table 1 to Table 9. The MSE results are not presented for the sake of parsmony, but ther qualtatve conclusons are dscussed. It s noted, for all methods, that the relatve bas has a negatve relaton wth the sample sze. The behavor of the dfferent methods s dscussed below. Table 1. The RB% and the MSE of the estmates under MCAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n= σ ρ

6 Internatonal Journal of Statstcal Dstrbutons and Applcatons 017; 3(4): Table. The RB% and the MSE of the estmates under MCAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n=50. Method σ ρ Table 3. The RB% and the MSE of the estmates under MCAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n=100. Methods σ ρ The Tables 1-3 show that for all sample szes both the CCA and the RM methods subdue the other methods n performance for MCAR settng. So, they get the best estmates and the smallest RB and MSE. It s noted that as the sample sze ncreases, the value of both the RB and the MSE decrease for most mputaton methods. Both the CCA and the Regress methods predct the mssng values very well. They can be used for small samples wth small rate of mssng values. Moreover they can be appled to large samples wth small percentage of mssngness. The MI provdes effcent estmates especally for large samples rrespectve of underestmatng of the varance. The HOT method greatly responds to the ncrease n the sample sze. The Mean, the LOCF, and the KNN methods gve reasonable results except n the coeffcent of the tme covarate ( ). The correlaton between each tme nterval and the adjacent one affects the performance of these methods. Concernng the EM, t s obvous that t could not predct the mssng values. Table 4. The RB% and the MSE of the estmates under MAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n=10. Method σ ρ Table 5. The RB% and the MSE of the estmates under MAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n=50. Method σ ρ Table 6. The RB% and the MSE of the estmates under MAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n= σ ρ

7 78 Ahmed Mahmoud Gad1 and Rana Hassan Mohamed Abdelkhalek: Imputaton Methods for Longtudnal Data: A Comparatve Study Accordng to Tables 4 6 the Mean method s superor to other methods for all sample szes. It s recommended to use the Mean mputaton for small sample szes under the MAR settng. The CCA method ndcates also good performance but t does not provde a good estmate for. 1 It s preferable to use the CCA wth MCAR settng rather than the MAR mechansm. All other methods except the Mean and the EM methods sustan from large RB for ˆ 1. The RM, the KNN, and the EM have a bad estmate for the varance. The LOCF, the HOT, the EM, and the MI methods underestmate the value of ρ. The EM method performs well n the MAR mechansm rather than the MCAR mechansm. Table 7. The RB% and the MSE of the estmates under MNAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n= σ ρ Table 8. The RB% and the MSE of the estmates under MNAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n= σ ρ Table 9. The RB% and the MSE of the estmates under MNAR mssngness at 0%, 5%, 50%, 75%, and 87.5% mssngness rates wth n= σ ρ Tables 7-9 show the results of the MNAR mechansm, n whch the estmates are dfferent from the underlyng values. It s obvous that the results are mpacted by the choce of the mssngness mechansm because the methods that performed well n the MCAR and the MAR settng worsen n the MNAR mechansm. 5. Applcaton (Breast Cancer Data) Breast cancer data concerns wth qualty of lfe among breast cancer patents n a clncal tral taken by the Internatonal Breast Cancer Study Group (IBCSG). In the IBCSG tral VI (Hurny et al. 199) premenopausal women wth breast cancer are followed for tradtonal outcomes such as relapse, death and also focused on qualty of lfe. The Patents were chosen at random to represent four groups under four dfferent chemotherapy regmes denoted by A, B, C and D. It s ntended to compare the qualty of lfe among the four dfferent treatments. The patents were asked to complete qualty of lfe questonnares at baselne (before startng treatment) and at three months ntervals for 15 months. Hence, each questonnare should be flled out sx tmes. It s planned that the sx tme ponts cover the tme durng the admnstraton of chemotherapy across all the four treatments. One of the relevant determnants of qualty of lfe was the Perceved Adjustment to Chronc Illness Scale (PACIS). Ths s a onetem scale comprsng a global patent ratng of the amount of effort costs to cope wth llness. The PACIS measured the response of the patents n dfferent groups. The total number of patents who start the study s 446 patents. The patents wth mssng response at the frst assessment (64 cases) are excluded from the analyss, leavng 38 patents. The patents dd not complete the study untl the 15 months of the study for many reasons. Some patents refused to complete the assessment, other patent not appear to fll the questonnare f her mood s poor. Wthdrawal from the study occurred by many patents who had already ded wthn the study perod. Thus, the structure of ths tral results n dropout pattern of mssng data. The amount of mssng data ncreased over tme, wth 9%, 36%, 47%, 54% and 6% for the consecutve vsts startng from the second tme pont. The

8 Internatonal Journal of Statstcal Dstrbutons and Applcatons 017; 3(4): percentages of patents wth 0, 1,, 3, 4, 5 mssng responses were, respectvely, 3%, 18%, 13%, 13%, 14% and 19%. The PACIS measured on a contnuous scale from 0 to 100 where, a larger score ndcates that greater amount of effort are requred for the patent to cope wth her llness. Followng Hürny et al. (199) and Gad and Ahmed (006) a square-root transformaton s used to normalze the data. The averages of the assessments usng all avalable transformed data are 6.1, 5.7, 5.6, 5.1, 4.7, 5.1, respectvely, and the standard devatons are.50,.46,.49,.51,.51, and.51. A prelmnary verson of these data, the responses for the frst 9 months of the study, was analyzed by Hürny et al. (199). Only patents wth complete responses are ncluded n the analyss (complete cases analyss). Ths analyss showed that the treatment dfferences are not statstcally sgnfcant. Dfferent versons of these data have been analyzed as Troxel et al. (1998) and Ibrahm et al. (001) Gad and Ahmed (006) adopted the mean model for the responses suggestng the AR (1) covarance structure and the unstructured covarance matrx to analyze ths data. ( = ψ ) = ψ 0 + ψ ψ logt r 1 Y Y, for = 1, 38 and j = 1,,6. In ths artcle, we depend on the mean model used n Gad and Ahmed (006). The response for the frst 15 months of the study s determned by PACIS response varable whch are of man nterest. The mean model allows each treatment to have ts own effect, that s µ = µ + α + α + α, = 1,,6, j 0j 1x1 x 3x3 j where µ 0j s a constant shft at each assessment tme and ( x, x, x ) 1 3 ( ) ( ) ( ) ( ) 1,0,0 treatment( A) 0,1,0 treatment( B) = 0,0,1 treatment( C) 0,0,0 treatment( D) The frst order autoregressve model s adopted for the covarance structure. In ths model the (, j) th element of the j covarance matrx s σ = σ ρ for, j = 1,,6. Based on the prevous model, a comparson s conducted among dfferent mputaton methods to compensate for mssng values n the Breast Cancer Data. The standard error s calculated for each mputaton method to evaluate the estmator performance. Table 10 dsplays the estmated parameters usng the Generalzed Least Square (GLS) method for the eght methods to the Breast Cancer Data. In addton to standard error are calculated to each mputaton method for the sake of comparson among them. Table 10. The parameter estmates of dfferent mputaton methods wth the values of ther standard errors. Method Par. Est. SE Est. SE Est. SE Est. SE Est. SE Est. SE Est. SE Est. SE The results show that the LOCF method approxmately gves the largest standard errors. So, t has the lowest effcency. In the LOCF method the mssng value s replaced wth the prevous observaton. Ths mples that the mputed value dd not predct the mssng value well. Unless the values for each tme pont are close to each other, the LOCF may not be an effcent mputaton. The CCA also shows bad performance. In ths method there s much loss of sample sze because the subject that has any mssng value s removed from the data. Hence, the CCA s recommended for large samples but wth small mssng values. These data have large number of subjects and hgh percentage of mssng. Moreover, ths experment confrms that the mean mputaton s a sutable mputaton method when the number of subjects s small and less mssng values. The HOT, the Regress, and the EM estmates are approxmately close to the mean mputaton. They have also large standard errors. It s noted that the KNN method s more effcent than the HOT, the Regress, and the EM methods for ths experment. The MI method s the most effcent method throughout ths experment. It has the least standard errors value. 6. Concluson The CCA method should be consdered as the frst choce of mputaton even n MCAR. It has the least relatve bas compared to the other methods. The performance of CCA was trembled n the MAR and the MNAR settng. The CCA method gves based estmates but have small MSE. The Regress method performs well especally under the MCAR but the sample varance and covarance are underestmated whch leads to small standard error and P-value. The Mean mputaton method s not a good choce for the dropout pattern under the MCAR assumpton. It performs slghtly well for the MAR and the MNAR assumptons and produced less MSE compared to other methods. The LOCF estmates the parameter very well and gves small MSE except under MNAR assumpton. However t shows large bas n some parameters. The HOT method sustans from large bas especally wth MNAR mssngness. It s performance gets better for large samples under the MCAR and the MNAR. However, the HOT has small MSE under the three mssngness mechansms. The KNN gves reasonable results

9 80 Ahmed Mahmoud Gad1 and Rana Hassan Mohamed Abdelkhalek: Imputaton Methods for Longtudnal Data: A Comparatve Study for the MCAR and the MAR mechansms. It gets better results as the sample sze ncrease, n other word t should be appled for large sample szes rather than small sample szes. The EM algorthm provdes a poor predcton to mssng values under the three mssng data mechansms especally the MCAR. However, t gves small MSE compared to the other methods. The MI method estmates are relatvely based, but under the MCAR mechansm t has the least bas. The MI method provdes small MSE. Acknowledgements The authors would lke to thank the Edtor and anonymous referees for ther helpful comments on the manuscrpt. References [1] Allson, P. D. (00) Mssng data, quanttatve applcatons n the socal scences, SAGE Unversty Papers. [] Blankers, M., Koeter, M. W. J., and Schppers, G. M. 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