THE APPLICATION OF LINEAR MIXED-EFFECTS MODEL TO THE EFFECT OF MICROCURRENT ON DECUBITUS WOUNDS. A STUDY IN LIMBURG PROVINCE OF BELGIUM

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1 The 3 rd Internatonal Conference on Mathematcs and Statstcs (ICoMS-3) Insttut Pertanan Bogor, Indonesa, 5-6 August 008 THE APPLICATIO OF LIEAR MIXED-EFFECTS MODEL TO THE EFFECT OF MICROCURRET O DECUBITUS WOUDS. A STUDY I LIMBURG PROVICE OF BELGIUM Seta Pramana,, Meke urmalasar 3, Center for Statstcs Hasselt Unversty, Depenbeek, Belgum Sekolah Tngg Ilmu Statstk (STIS) Jakarta Indonesa 3 Sekolah Tngg Ilmu Ekonom Indonesa (STEI) Jl. Kayujat Raya, Jakarta-Indonesa e-mal : seta.pramana@uhasselt.be, meke_hafdz@yahoo.co.d Abstract. A decubtus wound, s areas of njured skn and tssue. One of promsng strateges for healng of decubtus wound s electrcal stmulaton. The effectveness of two electrcal stmulatons (Mcrocurrent) n decubtus wounds closure was compared. General Lnear Mxed-effects Model was used to deal wth longtudnal data due to the wounds area were measured repeatedly over tme. To get the most sutable model, a number of models wth several possble mean effects, varance and seral correlaton structures were ftted and compared. The wound closure evoluton n both treatments s smlar and decreases durng the treatment. A wound closure depends on how long t has been treated, but a wound surface area after treatment depends also on the ntal area. Therefore, a large wound need more tme to heal than a small wound. Keywords: Mcrocurrent, decubtus ulcers, Lnear Mxed Effects Models.. Introducton Pressure sores also called bed sores, pressure ulcers and decubtus ulcers, are areas of njured skn and tssue. They are usually caused by sttng or lyng too long n one poston. Ths puts pressure on certan areas of the body. The pressure can reduce the blood supply to the skn and the tssues under the skn. When a change n poston does not occur often enough and the blood supply gets too low, a sore may form. Once pressure sores develop, they can take months to heal, f they heal at all. Varous strateges have been used to heal pressure sores. However, some therapes are often unsuccessful. A promsng strategy for healng of pressure sores and leg ulcers s electrcal stmulaton (Mcrocurrent). The concept s the tssue wll grow n an electrc feld. However ths concept s not new snce t has been known for over 00 years that a wound generates an electrc current. It was felt that weak electrc currents were generated by a wound and were a trgger n the body to promote cell growth. Electrcal stmulaton ncreases blood flow to the tssue and may ncrease whte cell dapedess to the area as well. The am of ths study s to compare the effectveness of two mcrocurrent methods; Twn Peak Hgh Voltage (TPHV) and Bphasc n decubtus wound closure.. Data Descrpton The data comes from a study of the effectveness of mcrocurrent whch s conducted n several hosptals n Lmburg, provnce, Belgum. In the study, the patents wth therapy resstant leg ulcers or decubtus wounds durng 3 months or longer are selected. The patents were treated daly 5 days a week for 60 mnutes. The photographc evaluatons of wound area were taken every week usng a dgtal camera. The

2 wound surface (mm ) was calculated usng Surface Measurement software. There were 7 wounds treated by TPHV current and 0 wounds by Bphasc exponental current. 3. Methods In ths study we are dealng wth a longtudnal data snce for each subject the wound area was measured at dfferent tme ponts durng the treatments. The evoluton of wound area over tme s of prmary nterest. The measurements on the same subject are not ndependent but clustered wthn subjects. In our dataset, we have therefore as many clusters as there are subjects. The ncompleteness (mssngness and dropouts) also appears n ths study whch leads us to choose a statstcal method that can handle these problems properly. For these reasons, Lnear Mxed Models s preferable. Lnear Mxed Models The lnear mxed-effects model was used to study the relatonshp between ln(wound area) and tme (week). The general lnear mxed model s gven by: Y X Z b ( ) () () Where b ~ (0, D), ~ (0, ), ( ) I n ~ (0, H ) ( ) b, b,..., b n, (),..., (), (),..., () are ndependent. X and Z are the desgn matrces for the fxed effects and the random effects for the -th subject. The represents the fxed effects that descrbe the average trend n the populaton. The b contans all subjectspecfc parameters whch descrbes how a subject devates from the average trend. These subject-specfc parameters are assumed to be normally dstrbuted wth mean zero and covarance matrx D. The s the measurement error for the -th subject. The () () s the seral correlaton component whch represents the belef that part of a subject s observed profle s a response to a tme varyng stochastc processes operatng wthn that ndvdual. The above formulaton of the lnear mxed model called the herarchcal formulaton of the lnear mxed model. The correspondng margnal normal dstrbuton wth mean X and covarance matrx Z DZ I n H s called the margnal formulaton of the lnear mxed model,.e. Y ~ ( X, Z DZ I H ). D s varance component of random effects, the H n I n s the varance covarance matrx of the error term and s the varance covarance matrx of the seral correlaton component. Checkng for Seral Correlaton Model () assumes that the error term ( ) can be decomposed as ( ) () n whch () s a component of measurement error and () s a component of seral correlaton. The model s then completed by specfyng a seral correlaton functon g(.). To get the approprate seral correlaton functon, lnear mxed models wth the same mean and random effects structure but wth dfferent seral correlaton was ftted. Then the model wth the maxmum log lkelhood wll be chosen as the approprate model.. Inferences of Varance Components In a lnear mxed model (), random effects represent the varablty n ndvdual ntercepts and slopes, whch s not explaned by the covarates ncluded n the model. Under the herarchcal nterpretaton of the model, t may therefore be of scentfc nterest to test for the need of random effects n the model. Snce n ths study the herarchcal nterpretaton s the man focus, the Wald test and Lkelhood Rato test statstcs constructed wth the maxmum lkelhood are not vald for nference n the varance components due to boundary problem. In ths case the lkelhood rato statstc constructed usng restrcted maxmum lkelhood approach provdes vald nferences.

3 Generally we are to check for the sgnfcance of k random effects and the hypotheses to be tested are as follows: D 0 D D H 0 : D versus H : D. 0 0 D D Where D s a postve defnte matrx and D n the alternatve hypothess s a general ((q+k)(q+k)) postve semdefnte matrx. The asymptotc null dstrbuton of -log-lkelhood rato ( ln ) s a mxture of random varables as well as other random varables. For the case of q versus q+ random effects ln s a mxture of q and q wth equal weght 0.5. Hence the probablty value s calculated as follow: p P( q: q ln ) P q ln P( q ln Inferences of Fxed Effects After we obtan the approprate varance component then the lkelhood rato test s used to asses the sgnfcance of the fxed effects. In ths step the result of the lkelhood rato test are vald under maxmum lkelhood estmaton and not under restrcted maxmum lkelhood estmaton. Hgher terms are removed f the correspondng p-value of the Wald test s large, and t s performed n a herarchcal way usng the maxmum lkelhood. The fxed effect whch s found not sgnfcant at 5% level of sgnfcance s removed one by one from the model. 4. Results The ndvdual profle graph presented n Fgure shows that almost all of the wound area decreased over tme except for some wounds that ncreased. For those wounds wth large sze at the begnnng reman large untl the end of the study. Moreover, there s much varablty between wounds, but there s a small varablty wthn wounds. Lookng at the ndvdual profle n dfferent treatment n Fgure t can be seen that the varablty between wounds treated wth TPHV s larger than n bphasc. It s due to n the orgnal data there are three wounds that more than 000 cm n TPHV area, on the other hand almost all of the wound area n bphasc s less than 000 cm, except only for one wound whch s the largest wound. TPHV Bphasc Fgure. Indvdual profles of ln(wound area) by treatment In order to observe the average evoluton of n dfferent current, the average ln(wound area) at each tme ponts s drawn n Fgure. We can see that n the frst four weeks, the average of ln(wound area) treated wth Bphasc s larger than TPHV. It can be also observed that on average, the ln (wound area) treated wth TPHV get smaller after the frst week treatment up to the thrd week, before gettng larger after the fourth week to the sxth week. Ths ncrease s because after the 5 th week only 3 large wounds left whch lead to the hgher average of evoluton. 3

4 Fgure. Average evoluton of ln(wound area) by treatment For bphasc, on average the ln(wound area) decreases steadly from the begnnng to the 7 th week of the treatment then slghtly ncreases n the 8 th week before t decreases extensvely after the 9 th week. The ncreasng of ln(wound area) for th to 5 th week due to small number of observaton. Fgure shows that the ln(wound area) follow a quadratc trend n tme. We also observe that there s an nteracton between treatment tme tme. Lnear Mxed Models The nformaton from exploratory data analyss part suggests the followng model: b b tme b tme f TPHV b b j ( 3 3 ) () j () j Y j 4 5 tmej ( 6 b3 ) tme () j () j f Bphasc Let Y j be the ln(wound area) of the th wound n the j th week and β, β, β 3, β 4, β 5 β 6,are fxed effects. β and β 4 represent the average ntercepts, β, β 3, β 5 and β 6 represent the average slopes, and b, b and b 3 are random effects. Respectvely, these random effects correspond to the devaton of the ntercept and slope of ndvduals from the average ntercept and slope. The measurement error and seral correlaton are denoted by () j and () j, respectvely. The nferences on the parameter are conducted to reduce the number of parameters n the fnal model. Inference on Seral Correlaton Structure The correlaton between two measurements on the same subject can be affected by the dfference n tme lag that the closer each other the more they are correlated, wth decreasng correlaton f tme lag ncrease. To get the approprate seral correlaton functon, we ft a lnear mxed model wth the same mean and random effects structure but wth dfferent seral correlaton, whch are; whether t can be plausble to neglect the seral correlaton, whether there s only Gauss seral correlaton, and whether to consder Gauss decay wth measurement error. Table. REML Log-lkelhood, AIC, BIC for three models wth the same mean structure but dfferent resdual covarance structure Resdual Covarance Structure REML Log-lkelhood AIC BIC Measurement error Gaussan Measurement error + Gaussan From Table we can see that the restrcted maxmum lkelhood for the model whch only consders Gaussan decay s larger than the model wth only measurement error. However when they are combned, the model has the hghest restrcted maxmum lkelhood. We also can observe that the resdual 4

5 covarance structure s explaned more by the presence of seral correlaton than measurement error. Thus the model wth measurement error and Gaussan seral correlaton s used. Inference for Varance Components In order to test for the need of random effects, four models wth the same fxed effect but dfferent random effects were ftted and then compared. The results obtaned from ths analyss are shown n Table and 3. Table. Several random effects model wth the assocated -log-lkelhood (Restrcted Maxmum Lkelhood) Random effects -ln REML (ˆ) L AIC BIC Model : Intercepts, week, week Model : Intercepts, week Model 3: Intercepts Model 4: Table 3. Lkelhood Rato Statstc wth correct asymptotc null dstrbuton for comparng random effects model (REML) Asymptotc null Hypothess -ln REML (ˆ) p-value dstrbuton Model vs Model 3.7 : Model 3 vs Model 8 : < Model 4 vs Model : Accordng to AIC and BIC n Table, model s the most approprate model. Table 3 shows that the ncluson of random ntercepts and slopes of week s needed, however addng random effects of week have no sgnfcant effect n the model. Thus we conclude to use model whch consst random ntercepts and slopes of week. Inference for Fxed Effects After we obtan the approprate varance component then we ftted several models wth dfferent fxed effects and then used the lkelhood rato test to asses the sgnfcance of the fxed effects. From Table 4 the treat*week fxed effect can be removed from the model. Thus the fnal model base on ths result can be wrtten as: b b tme f TPHV Ln b b j () j () j ( Y ) j 4 5 tmej () j () j f Bphasc Where the random effects are assumed to be stochastc and are obtaned through Emprcal Bayes estmaton. The result of the estmates for each parameter n the fnal model s shown n Table 5. Table 4. The value of Lkelhood rato statstc, AIC and BIC for model wth dfferent fxed effects Model -ln -ln L ML (ˆ) ML (ˆ) AIC BIC (p-value) Treat Treat*week Treat*week Treat Treat*week (0.7408) Accordng to the fnal selected model, the ln(wound area) depends on the ln(ntercept/ntal wound sze) and the lnear tme effect (week). The average evoluton (slopes) of wound treated wth TPHV and Bphasc do not show any sgnfcant dfference (p-value = 0.878). 5

6 Table 5. Result from fttng the fnal model to the dataset Effect Parameter Estmates (s.e.) Intercepts TPHV (0.459) Bphasc (0.545) Tme effects TPHV (0.0875) Bphasc (0.) Covarance of b : var (b ) d.667 (0.8733) var (b ) d (0.079) cov (b,b ) d d (0.066) Measurement error varance var ( () j ) 0.076(0.07) Gaussan seral correlaton var ( () j ) (0.0833).338(0.008) Rate of exponental decrease Observaton 9 - REML log-lkelhood 4.9 Akake's Informaton Crtera (AIC) 36.9 Sawa's Bayesan Informaton Crteron (BIC) Conclusons From a populaton averaged profle, we observed that ln(wound area), especally Bphasc, follow a quadratc trend n tme, however after checkng usng Wald test, t does not appear sgnfcant n level of confdence 5%. It mght be due to dropouts at the end of the study. The sources of the varablty n ths study are subject-specfc effects (random ntercepts and random slopes), Gaussan seral correlaton and measurement error. Snce there was no sgnfcant dfference between the ntercepts of the two treatments t can be mpled that on average the ntal wound area of the treatments are the same. We also found no sgnfcant dfference of the effect to the wound closure of two mcrocurrent methods. The wound closure evoluton n both treatments s smlar and t decreases durng treatment. A wound closure depends on how long t has been treated, but a wound surface area after treatment depends also on the ntal area. Therefore, wound closure n large wound need more tme to get smaller than small one. The measurement error and seral correlaton are the varablty components that explan a varaton of wound measurement n each observaton. 6

7 References: Braddock M, Campbell CJ, Zuder D. (999) Current therapes for wound healng: electrcal stmulaton, bologcal therapeutcs and the potental for gene therapy. Internatonal Journal Dermatol ; 38: Brown, H., Prescott, R. (999) Appled Lnear Models n Medcne. West Sussex: John Wley & Son s ltd. Molenberghs, G. and Verbeke, G. (000) Lnear Mxed Models for Longtudnal Data. Sprnger Seres n Statstcs. ew York: Sprnger-Verlag. Molenberghs, G. and Verbeke, G. (005) Models for Dscrete Longtudnal Data. Sprnger Seres n Statstcs. ew York: Sprnger-Verlag 7