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Introduction to Logistic Regression Rachid Salmi, Jean-Claude Desenclos, Thomas Grein, Alain Moren

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Oral contraceptives (OC) and myocardial infarction (MI) Case-control study, unstratified data OC MIControlsOR Yes No Ref. Total

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Oral contraceptives (OC) and myocardial infarction (MI) Case-control study, unstratified data Smoking MIControlsOR Yes No Ref. Total

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Odds ratio for OC adjusted for smoking = 4.5

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Number of cases One case Days Cases of gastroenteritis among residents of a nursing home, by date of onset, Pennsylvania, October 1986

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ProteinTotalCasesAR%RR suppl. YES NO Total Cases of gastroenteritis among residents of a nursing home according to protein supplement consumption, Pa, 1986

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Sex-specific attack rates of gastroenteritis among residents of a nursing home, Pa, 1986 SexTotalCases AR(%)RR & 95% CI Male Reference Female ( ) Total

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Attack rates of gastroenteritis among residents of a nursing home, by place of meal, Pa, 1986 MealTotal CasesAR(%)RR & 95% CI Dining room Reference Bedroom ( ) Total

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Age – specific attack rates of gastroenteritis among residents of a nursing home, Pa, 1986 Age groupTotalCasesAR(%) Total

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Attack rates of gastroenteritis among residents of a nursing home, by floor of residence, Pa, 1986 FloorTotalCasesAR (%) One Two Three Four Total

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Multivariate analysis Multiple models –Linear regression –Logistic regression –Cox model –Poisson regression –Loglinear model –Discriminant analysis – Choice of the tool according to the objectives, the study, and the variables

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Simple linear regression Table 1 Age and systolic blood pressure (SBP) among 33 adult women

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SBP (mm Hg) Age (years) adapted from Colton T. Statistics in Medicine. Boston: Little Brown, 1974

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Simple linear regression Relation between 2 continuous variables (SBP and age) Regression coefficient 1 –Measures association between y and x –Amount by which y changes on average when x changes by one unit –Least squares method y x Slope

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Multiple linear regression Relation between a continuous variable and a set of i continuous variables Partial regression coefficients i –Amount by which y changes on average when x i changes by one unit and all the other x i s remain constant –Measures association between x i and y adjusted for all other x i Example –SBP versus age, weight, height, etc

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Multiple linear regression Predicted Predictor variables Response variable Explanatory variables Outcome variable Covariables Dependent Independent variables

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Logistic regression (1) Table 2 Age and signs of coronary heart disease (CD)

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How can we analyse these data? Compare mean age of diseased and non-diseased –Non-diseased: 38.6 years –Diseased: 58.7 years (p<0.0001) Linear regression?

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Dot-plot: Data from Table 2

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Logistic regression (2) Table 3 Prevalence (%) of signs of CD according to age group

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Dot-plot: Data from Table 3 Diseased % Age group

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Logistic function (1) Probability of disease x

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Transformation logit of P(y|x) { = log odds of disease in unexposed = log odds ratio associated with being exposed e = odds ratio

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Fitting equation to the data Linear regression: Least squares Logistic regression: Maximum likelihood Likelihood function –Estimates parameters and –Practically easier to work with log-likelihood

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Maximum likelihood Iterative computing –Choice of an arbitrary value for the coefficients (usually 0) –Computing of log-likelihood –Variation of coefficients values –Reiteration until maximisation (plateau) Results –Maximum Likelihood Estimates (MLE) for and –Estimates of P(y) for a given value of x

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Multiple logistic regression More than one independent variable –Dichotomous, ordinal, nominal, continuous … Interpretation of i –Increase in log-odds for a one unit increase in x i with all the other x i s constant –Measures association between x i and log-odds adjusted for all other x i

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Statistical testing Question –Does model including given independent variable provide more information about dependent variable than model without this variable? Three tests –Likelihood ratio statistic (LRS) –Wald test –Score test

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Likelihood ratio statistic Compares two nested models Log(odds) = + 1 x x x 3 (model 1) Log(odds) = + 1 x x 2 (model 2) LR statistic -2 log (likelihood model 2 / likelihood model 1) = -2 log (likelihood model 2) minus -2log (likelihood model 1) LR statistic is a 2 with DF = number of extra parameters in model

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Coding of variables (2) Nominal variables or ordinal with unequal classes: –Tobacco smoked: no=0, grey=1, brown=2, blond=3 –Model assumes that OR for blond tobacco = OR for grey tobacco 3 –Use indicator variables (dummy variables)

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Indicator variables: Type of tobacco Neutralises artificial hierarchy between classes in the variable "type of tobacco" No assumptions made 3 variables (3 df) in model using same reference OR for each type of tobacco adjusted for the others in reference to non-smoking

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Reference Hosmer DW, Lemeshow S. Applied logistic regression. Wiley & Sons, New York, 1989

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Logistic regression Synthesis

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Salmonella enteritidis Protein supplement S. Enteritidis gastroenteritis Sex Floor Age Place of meal Blended diet

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Unconditional Logistic Regression Term Odds Ratio 95% C.I.Coef.S. E. Z- Statistic P- Value AGG (2/1)1,67950,263410,70820,51850,94520,54860,5833 AGG (3/1)1,75700,32499,50220,56360,86120,65450,5128 Blended (Yes/No)1,03450,32773,26600,03390,58660,05780,9539 Floor (2/1)1,61260,26759,72200,47780,91660,52130,6022 Floor (3/1)0,72910,09915,3668-0,31591,0185-0,31020,7564 Floor (4/1)1,11370,15737,88700,10760,99880,10780,9142 Meal1,59420,49535,13170,46640,59650,78190,4343 Protein (Yes/No)9,09183,021927,35332,20740,56203,92780,0001 Sex1,30240,22787,44680,26420,88960,29700,7665 CONSTANT***-3,00802,0559-1,46310,1434

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Unconditional Logistic Regression TermOdds Ratio 95% C.I.CoefficientS. E.Z-StatisticP-Value Age1,02340,96601,08420,02310,02940,78480,4326 Blended (Yes/No)1,01840,32203,22070,01830,58740,03110,9752 Floor (2/1)1,64400,27459,84680,49710,91330,54430,5862 Floor (3/1)0,71320,09725,2321-0,33791,0167-0,33240,7396 Floor (4/1)1,07080,15227,53220,06840,99530,06870,9452 Meal1,65610,52365,23790,50450,58750,85870,3905 Protein (Yes/No)8,76782,952126,04032,17110,55543,90910,0001 Sex1,19570,21356,69810,17870,87910,20330,8389 CONSTANT***-4,28962,8908-1,48390,1378

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Logistic Regression Model Summary Statistics ValueDFp-value Deviance107, Likelihood ratio test34,80688< Parameter Estimates 95% C.I. TermsCoefficientStd.Errorp-valueORLowerUpper %GM-1,88571,04200,07030,15170,01971,1695 SEX ='2'0,21390,88120,80821,23850,22026,9662 FLOOR ='2'0,49870,90830,58291,64660,27769,7659 ²FLOOR ='3'-0,32351,01500,75000,72360,09905,2909 FLOOR ='4'0,10880,98390,91191,11500,16217,6698 MEAL ='2'0,53080,56130,34431,70020,56595,1081 Protein ='1'2,18090,5303< ,85413,131625,034 TWOAGG ='2'0,19040,51620,71221,20980,43993,3272 Termwise Wald Test TermWald Stat.DFp-value FLOOR1,081230,7816

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Poisson Regression Model Summary Statistics ValueDFp-value Deviance60, Likelihood ratio test67,73788< Parameter Estimates 95% C.I. TermsCoefficientStd.Errorp-valueRRLowerUpper %GM-1,82130,84460,03100,16180,03090,8471 SEX ='2'0,12950,71060,85541,13830,28274,5828 FLOOR ='2'0,25030,68670,71541,28440,33444,9343 FLOOR ='3'-0,14220,80320,85950,86740,17974,1877 FLOOR ='4'0,13680,72630,85061,14660,27614,7608 MEAL ='2'0,23730,38540,53811,26780,59562,6987 Protein ='1'1,06580,34130,00182,90321,48715,6679 TWOAGG ='2'0,06450,36820,86111,06660,51822,1951 Termwise Wald Test TermWald Stat.DFp-value FLOOR0,417830,9365

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Cox Proportional Hazards TermHazard Ratio95%C.I.CoefficientS. E.Z-StatisticP-Value _AGG (2/1)1,06660,51832,1950,06450,36820,1750,8611 Floor(2/1)1,28440,33444,93420,25030,68670,36460,7154 Floor(3/1)0,86740,17974,1876-0,14220,8032-0,1770,8595 Floor(4/1)1,14660,27614,76070,13680,72630,18830,8506 Meal (2/1)1,26780,59572,69860,23730,38540,61570,5381 Protein(Yes/No)2,90321,48715,66781,06580,34133,12250,0018 Sex (2/1)1,13830,28274,58270,12950,71060,18220,8554 Convergence:Converged Iterations:5 -2 * Log-Likelihood:346,0200 TestStatisticD.F.P-Value Score17,172770,0163 Likelihood Ratio15,488970,0302

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