Health risk factors assessment using gradual and classic logistics regression analysis

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1 Health risk factors assessment using gradual and classic logistics regression analysis DRAGAN RANDJELOVIC, DRAGAN BOGDANOVIC Academy of Criminalistic and Police Studies Street Cara Dusana 96, 080 Belgrade SERBIA Abstract: - The problem of determining the factor significance is present in analysis of one multifactor or often multivariate experiment. It is necessary to determine individual influence of each factor i.e. criteria on depedent variable/s. Each criteria can be: quantitative, continius or categorical value, when we can determine values in difference or qualitative, when is possible only ranking of criteria weights. First group of problems is frequently solved with application of clasic or disputed gradual technique of regression, which is called logistic in the case if the dependent variables are categorical type. Authors presented an comparsion of two possible logistic methodology and that method clasic logistic regression analysis and method of gradual logistic regression analysis in practical demonstration. Data are used from study enforced on Clinical center Nis, Serbia for solving the problem of determining of the relevant indicators of risk of variceal bleeding and deth income in patients with liver cirrhosis. Key-Words: - Weights of criteria, Multiple regression analysis, Logic regression analysis, Varicael bleeding, Liver cirrhosis Introduction In theory of statistic it is known that experiments are used by scientists and researcher to affirm their hypothesis about differences between two or more groups of variables which are called tests in research, or to determine relation between variables and influence each of them on dependent variable/s which experiments are called valuations in research. In this second group of experiments belongs considering the influence of different factors i.e. criteria on the unit of experimental examination[]. A great number of criteria could make confusion in the process of assessing theirs influence therefore the main task is determination the number of relevant criteria and theirs significance for an event being evaluated and solving of this problem is the subject of this paper. For solving this problem it is available more methodologies which can be divided in two important groups of methodologies[]: methods of standard statistic analysis among which most used is multiple regression analysis of course and logistic regression belongs to this group, methods of operational research among which most known are multiple criteria analysis methods. In practice after one multiple factorial experiment is finished or suitable such problem is defined very often it is necessary to obtain the answer about influence of each factors i.e. criteria on depended variable or variables. Criteria can be quantitative as statistical data or the estimates provided by experts or only qualitative judgments. The presented paper considers the problem of determining the weights of criteria using statistical weighting with classic logistic ISBN:

2 multiple regression method in comparison with disputed gradual logistic multiple regression method which application is interest because it allows necessary decreasing the number of considered criteria. This comparative analysis authors made solving the problem of determining of the relevant indicators of risk of variceal bleeding in patients with liver cirrhosis on Clinic center in Nis, Serbia. Multiple linear regression Method for examining the influence more different independent variables for example x i, x i, x 3i,, x pi on one dependent variable for example y is called multiple regression and can be given in the linear (general polynomial) form[3],[4]: y=a + b x + b x + b 3 x b p x p. where b i i=,,, p are partial coefficients of regression. In the case of fixed values independent variables x when we have and experimental error in each from fully n observation we can present multiple regression in the form y i =β 0 + β x i + β x i + β 3 x 3i + + β k x ki +e i. The calculation of parameters a, b, b, b3,, b p we can made with the method of smallest quadrates with minimization of expression n i= ( y i a b x i b x i b x Practical, algebraic algorithm for solving arising system of equation is rarely in use than known Gaussian method of multiplication all the more so this method are already used in calculation for regression valuation and therefore we consider this method. With differentiation in in relation on a, b, b, b3,, b p and with exchange in notation b 0 =a we obtain next normal equation which must be solved to receive parameters: b 0 (00)+b (0)+ +b p (0p)=(0y) b 0 (0)+b ()+ +b p (p)=(y)... b 0 (p0)+b (p)+ +b p (pp)=(py) where p pi ). n x ji x ki (jk)=(kj)= i= is the sum of products j-th and k-th variables x j and y k, is the sum of products j-th and k-th variables x j and y k, n ( x ji ) (jj)= i= is the sum of squares j-th column variable xi, n x ji yi (jy)= i= is the sum of products j-th column variable xj and variable y. The matrix of independent variables x and vector y is the initial basis for calculation sum of squares and products of variables and can be given like: x y x 0 x º º º x p y x 0 x º º º x p y x 03 x 3 º º º x p3 y 3 x 0n x n º º º x pn y n From this matrix and vector we form sums of square and product of variables x and products of x and y which form system of normal equation: jk=x x jy=x y oo 0 º º º 0p 0y 0 º º º p y 0 º º º p y p0 p º º º pp py With inversion of matrix x x we obtain Gauss multipliers: C jk = C kj = x x C 00 C 0 º º º C 0p C 0 C º º º C p ISBN:

3 C 0 C º º º C p º º º º º º º º º º º º º º º º º º C p0 C p º º º C pp Partial coefficients of regression are: p ( C jk )( jy) b i = i= i.e. the sum of products k-th column of C ij with the column (jy). When the independent variables are mutually orthogonal normal equations are particularly easy to solve therefore in this case all sums of products (jk) vanish (j?k) and the normal equations for b i reduces to: (jj)b j =(jy) and the multiplier in inverse matrix become values C jj /(jj) and C jk =0. Logistic multiple regression model is linear multiple regression model in which is at least one dependent variable categorical[5],[6]. 3 Clinical study Bleeding from esophageal varices is the most common and most serious complication of cirrhosis of the liver and directly lifethreatening. The aim of study was to analyze clinical, biochemical, endoscopic, ultrasound and color Doppler parameters to determine risk indicators of variceal bleeding [7]. The study comprised 96 patients with liver cirrhosis, who were divided into two groups depending on the previous history of bleeding. Patients with variceal bleeding episodes was divided into two subgroups with and without endoscopic sclerotherapy of esophageal varices. Analysis of clinical and biochemical parameters (Child Pugh and MELD scores), endoscopic parameters (size, appearance and location of varices), and ultrasound and Doppler parameters was prformed. Because of the high mortality rate due to variceal bleeding, assessment of risk of bleeding is important for the timely implementation of therapeutic interventions. This study looks at the risks of initial variceal bleeding in patients with liver cirrhosis, and the risks of early and late rebleeding. The results obtained are of great importance for further treatment and prevention of bleeding from esophageal varices, the most common and life-threatening complications of cirrhosis. Proposed clinical and biochemical parameters to assess prognosis and survival rate of patients with cirrhosis, which is important for appropriate therapy, and prioritization on the waiting list for liver transplantation. The study involved 96 subjects, 76 (79.%) men and 0 (0.8%) were women. In the 55 patients without bleeding were 44 (80.0%) males and (0.0%) were women, and in the group of 4 patients with bleeding were 3 (78.0%) males and 9 (.0 %) were women. The average age of all patients was ±.46 years. The youngest patient was 4 and the oldest 80 years. 3. Gradual logistic regression Univariate logistic regression analysis which is given in Table showed that significant predictors of bleeding are: the value of the PC / SD, platelet count, as well as the expression of large esophageal varices, red color signs, gastric varices, and congestive gasthropathy collateral circulation. Any increase in the value of the PC / SD for one unit indicates a reduction in the risk of bleeding by 0.% (from 0. to 0.3%, p <0.05), while any increase in platelet count to x09 / L indicates to decrease the risk of bleeding by 0.8% (from 0. to.5%, p <0.05). Expression of the following factors indicating an increased risk of bleeding: large esophageal varices times (7.368 to times, p <0.00), red color signs times ( to times, p <0.00), gastric varices 4.0 times (.87 times up to 4.35, p <0.05), congestive gastropathy times 0.53 (3.479 to times, p <0.00) and collateral circulation.56 times (.00 to.434 times, p <0.05). Multivariate logistic regression analysis, which is given on the Table, as the most important predictors of bleeding allocated the expression of red color signs and congestive gastropathy. They point to a significantly higher risk for the occurrence of bleeding and red color signs times (9.744 to times, p <0.00), and congestive gastropathy times 6.6 (.3 to 33.6 times, p <0.05). Regression model as independent variables ISBN:

4 contains those two factors and regression constant explains 79.9% of variability in the risk of bleeding (coefficient of determination - R = 0.799). Table. Odds Ratio values for bleeding risk factors (univariate logistic regression) Faktor p OR Female gender 0,86 Age (year) 0,66 Etiology 0,606 Bilirubin (mg/dl) 0,04 Albumin (g/dl) 6 Prothrombin time (sec) 0,393 International normalized ratio (sec) 0,4 Creatinin (mg/dl) 0,90 Ascites 0,63 Neurological disfunction 0,9 Platelet count/spleen diameter ratio 0,00 Body mass index 0,060 Thrombocytes (0 9 /L) 0,03 Large esophageal <0,0 varices 0 Red color signs <0,0 0 Gastric varices 0,06 <0,0 Congestive gastropathy 0, 5 7 0, ,6 40 4,4 7, , , ,99 7 4, 0 0, 53 95% CI for OR Lo we r bo un d Upp er bou nd 0,4 7 3, , 3,66 08, ,6,70 0 3, ,4 5, ,7, ,3 5, ,6 3, , ,3 8, , ,35 3 6, 4, ,4 9, Liver diameter (mm) 0,078 Spleen diameter (mm) 0,390 Portal vein diameter (mm) 0,405 Portal vein wall thickness (mm) 0,859 Lienal + Mesenterical superior vein diameter (mm) 0,600 Collateral circulation 0,049 Flow speed in portal vein (m/sec) 58 Flow speed in lienal vein (m/sec) 0,054 Congestion index in the portal vein (cm/sec) 0,676 Congestion index in the lienal vein (cm/sec) 07 Child Pugh score 0,574 MELD score 0, , ,5, ,8 6, 7,5, ,4 0, ,0 0,0, , 0, ,7 0,0 6, ,8, Table. Odds Ratio values for bleeding risk factors (multivariate logistic regression) Factor p OR Red color signs <0,0 0 6,5 78 Congestive gastropathy 0,037 6,6 Constant <0,0 0 0,00 95% CI for OR Low er boun d 9,7 44, 3 Upper bound 688,3 6 33,6 Univariate logistic regression analysis as significant predictors of death in patients with cirrhosis confirmed: the level of bilirubin, albumin, prothrombin time, INR, creatinine, ascites, neurological disorders, esophageal varices large, red color signs, collateral circulation, bleeding, repeated bleeding early ISBN:

5 rebleeding Child Pugh and MELD scores. Table 3. Odds Ratio values for death income risk factors (univariate logistic regression) Factor p OR Female gender 53 Age (year) 0,88 Etiology 0,55 Bilirubin (mg/dl) 0,00 Albumin (g/dl) 0,03 Prothrombin time (sec) 0,0 International normalized ratio (sec) 0,0 Creatinin (mg/dl) Ascites 0, , 03 0, 53, 30 7,9 6 3,7 57 4,8 45, % CI for OR Lo we r bo un d 0, , , ,5 63,8 4,6 34 Upp er bou nd 4, 6 47, ,7 5, , 78 7,6 67 4, 36 Neurological,9 43, disfunction 8 78 Platelet count/spleen diameter ratio 0, ,5 0,, Body mass index 0, Thrombocytes (0 9 /L) 0, Large esophageal 4,9,4 6, varices 0, ,3,9 44, Red color signs 0, , 4,9 Gastric varices ,6 0,6 4,0 Congestive gastropathy 0, Liver diameter (mm) 0, Spleen diameter (mm) 0, Portal vein diameter (mm) 0,08, Portal vein wall thickness (mm) 0,00,8 9 0,7 9 4,5 37 Lienal + Mesenterical superior vein diameter (mm) 0, , ,0, 74, Collateral circulation 0, Flow speed in portal vein (m/sec) 0,7 37 0,0 0 8,4 35 Flow speed in lienal 0,0 0,0 4, vein (m/sec) 0, Congestion index in the 0,0 0,0 5, portal vein (cm/sec) 0, Congestion index in the lienal vein (cm/sec) 0,479 0,0 00 3,5 5 Bleeding 0,00 5, 0 0 3, 7 00,80 5 Recidive bleeding Early recidive bleeding Late recidive bleeding 0,588 Intervention 0,4 Sclerosation 0,496 Ligation 0,5 Child Pugh score MELD score, , 8 5 0,5 54,4 7,5 5,6 98,8 58, 4 4,6 8 8,8 5 0,0 65 0,7 45 0,4 58 0, ,,40 74,87 3 4,7 0 7,8 47 5,0 7 9, 8, Any increase in the level of albumin g/dl indicates a significant reduction in the risk of occurrence of death for 74.7% (4.9 to 9.5%, p <0.05). Conversely, any increase in bilirubin level, prothrombin time, INR, creatinine, Child Pugh and MELD score for one unit indicate a significantly increased risk of serious and fatal outcome as follows: bilirubin by 0.3% (7.8 to ISBN:

6 34.3 %, p <0.0), prothrombin time 3.0% (4.7 to 44.5%, p <0.05), INR 7.96 times (.563 to times, p <0.05), creatinine times (.84 to times, p <0.00), Child Pugh score.858 times (.345 to.566 times, p <0.00) and MELD score of 4.% (. to 37.%, p <0.00). Expression of ascites, neurological disorders, large esophageal varices, red color signs, collateral circulation, bleeding, recurrent bleeding and early rebleeding showed a statistically significant increase in risk for the occurrence of death compared with patients with cirrhosis in whom these factors are manifested as follows(see Table 3): ascites times (.634 to 4.36 times, p <0.0), neurological disorders times.359 (.98 to times, p <0.00), large esophageal varices times (.48 to 6.54 times, p <0.0), red color road signs (.967 to times, p <0.0), collateral circulation times (.0 to times, p <0.05), bleeding times 5.00 (3.7 to times, p <0.0), repeated bleeding times.939 (4.68 to.40 times, p <0.00), and early rebleeding times (8.85 to times, p <0.00). Multivariate logistic regression analysis as the most important risk factor for mortality in patients with cirrhosis of the liver aside, which is given on Table 4, rebleeding and MELD score values. Table 4. Odds Ratio values for death income risk factors (multivariate logistic regression) Factor p OR 95% CI for OR Lowe r boun d Upper bound Recidive bleeding 0,00 69,60 5,64 938,04 MELD score 0,003 36,03,68 Constant 0,000 Repeated bleeding indicating increased risk for the occurrence of death times (5.64 to times, p <0.0), and any increase in the value of MELD score of indicates an increase in risk by 33.6% (0.3 to 6.8 %, p <0.0). A model that includes as independent variables, and these two factors constant regression explains 67.5% of variability in the occurrence of risk of death in patients with cirrhosis (R = 0.675). 3. Classic logistic regression Classic logistic regression analysis which is given in Table 5 showed that significant predictors of bleeding are: Table 5. Odds Ratio values for bleeding risk factors (classic logistic regression) Variables not in the Equation a Score df Sig. Variables pol() starost etiolog bilirub album protrvr inr kreatin ascites.7.7 neurpor pcsdr uhranj tromb veliki redcols() gastvar() konggas veljetre ISBN:

7 velslez dijvport dzidavp dvldvms kolcirk bpuvp bpuvl konindvp konindvl childps melds a. Residual Chi-Squares are not computed because of redundancies. Table 6. Odds Ratio values for death income risk factors (classic logistic regression) Variables pol() starost.05.7 etiolog bilirub album protrvr inr kreatin ascites neurpor pcsdr uhranj tromb veliki redcols() gastvar() konggas veljetre velslez dijvport dzidavp dvldvms kolcirk bpuvp bpuvl konindvp konindvl krvarenj() recidiv rani kasni interven() skleroz() ligacija() childps melds a. Residual Chi-Squares are not computed because of redundancies. the value of the PC / SD, platelet count, as well as the expression of large esophageal varices, red color signs, gastric varices, and congestive gastropathy collateral circulation. Classic logistic regression analysis which is given on Table 6 as significant predictors of death in patients with cirrhosis confirmed: the level of bilirubin, albumin, prothrombin time, INR, creatinine, ascites, neurological disorders, esophageal varices large, red color signs, collateral circulation, bleeding, repeated bleeding early rebleeding Child Pugh and MELD scores. In this case Hosmer and Lemeshow test shows (see Table 7) that the quality of prediction of this model is not good. Table 7. Hosmer Lemeshow test for death income risk factors(classic logistic regression) Hosmer and Lemeshow Test Step Chi-square df Sig Conclusion In this paper,. In section 3. authors show that determining significant predictors of bleeding and the occurrence of death was performed ISBN:

8 with gradual logistic regression analysis application. Approximate values were calculated relative risk (odds ratio - OR) and their 95% confidence intervals. Estimate the value of statistical significance was performed by calculating the OR Wald (Wald) values. Factors for which the univariate logistic regression analysis showed that were significant predictors of bleeding, recurrence and death were included were in multivariate regression models. Using the method step by step backwards (Backward Wald) from the multivariate model were excluded all those factors for which statistical significance was not confirmed.. In section 3. authors proved that in the case of determination significance of predictors of bleeding application classic and gradual logistic regression give same results but in the case of determination significance of predictor of death authors prove using Hosmer and Lemeshow test that the model of classic logistic regression is not applicable and the using of disputed gradual logistic regression method is obviously. Authors as a subject of the future work to find alternative method for disputed gradual logistic regression model will tray to decrease the number of considered criteria before applying of classic logistic regression model. [6] Kovačić Z.J.Multivarijaciona analiza, Ekonomski fakultet, Beograd 994. [7] Benedeto-Stojanov D. Indikatori rizika varikoznog krvarenja u bolesnika sa cirozom jetre, Medicinski fakultet, Niš, 00. Acknowledgements: This paper was supported by the projects of Ministry of Education and Science Republic of Serbia III44007 and TR3409. References: [] Randjelović D. et. al., Multi criteria analysis application in the analysis of experiment results, Journal WSEAS transaction on mathematics, Issue 7, Volume 6, 007. [] Hadzivuković S., Statistički metodi, Univerzitet u Novom sadu, 99 [3] Kempthorne O, The design and Analysis of Experiments,John Wiley&sons Inc,New York 95. [4] Cohran W. and Cox G., Experimental Designs,John Wiley&sons Inc,New York 957. [5]Kovačić Z.J. Predicting student success by mining enrolment data, Research in Higher Education Journal, Vol. 5 (0) pp. -0, ISSN: ISBN: