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  • Chapter 16 logistic RegressionAnalysis

  • ContentLogistic regression Conditional logistic regression Application

  • Purpose: Work out the equations for logistic regression which are used to estimate the dependent variable (outcome factor) from the independent variables (risk factors). Logistic regression is a kind of nonlinear regression.Data: 1.The dependent variable is a binary categorical variable that has two values such as "yes" and "no. 2.All of the independent variables, at least, most of which should be categories. Of course, some of them can be numerical variable. The categorical variable should be quantified.

  • Implication: Logistic regression can be used to study the quantitative relations between the happening of some diseases or phenomena and many risk factors. There are some demerits to use test (or u test ): 1. can only study one risk factor. 2. can only educe the qualitative conclusion.

  • Category:1.Between-subjects (non-conditional) logistic regression equation2. Paired (conditional) logistic regression equation

  • 1 logistic regression non-conditional logistic regression

  • I Basic ConceptionThe probability of positive outcome under the function of m independent variables can be marked like this:

  • If: Regression modelProbability: P01logitP Scale:

    While

    is the constant term

    is the coefficient of regression

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    _1076792910.unknown

  • 1

    0.01799

    0.01889

    0.01984

    0.02084

    0.02188

    0.02298

    0.02413

    0.02533

    0.0266

    0.02792

    0.02931

    0.03077

    0.0323

    0.0339

    0.03557

    0.03733

    0.03917

    0.04109

    0.04311

    0.04522

    0.04743

    0.04974

    0.05215

    0.05468

    0.05732

    0.06009

    0.06297

    0.06599

    0.06914

    0.07243

    0.07586

    0.07944

    0.08317

    0.08707

    0.09112

    0.09535

    0.09975

    0.10433

    0.1091

    0.11405

    0.1192

    0.12455

    0.13011

    0.13587

    0.14185

    0.14805

    0.15447

    0.16111

    0.16798

    0.17509

    0.18243

    0.19

    0.19782

    0.20587

    0.21417

    0.2227

    0.23148

    0.24049

    0.24974

    0.25923

    0.26894

    0.27888

    0.28905

    0.29943

    0.31003

    0.32082

    0.33181

    0.34299

    0.35434

    0.36586

    0.37754

    0.38936

    0.40131

    0.41338

    0.42556

    0.43782

    0.45017

    0.46257

    0.47502

    0.4875

    0.5

    0.5125

    0.52498

    0.53743

    0.54983

    0.56218

    0.57444

    0.58662

    0.59869

    0.61064

    0.62246

    0.63414

    0.64566

    0.65701

    0.66819

    0.67918

    0.68997

    0.70057

    0.71095

    0.72112

    0.73106

    0.74077

    0.75026

    0.75951

    0.76852

    0.7773

    0.78583

    0.79413

    0.80218

    0.81

    0.81757

    0.82491

    0.83202

    0.83889

    0.84553

    0.85195

    0.85815

    0.86413

    0.86989

    0.87545

    0.8808

    0.88595

    0.8909

    0.89567

    0.90025

    0.90465

    0.90888

    0.91293

    0.91683

    0.92056

    0.92414

    0.92757

    0.93086

    0.93401

    0.93703

    0.93991

    0.94268

    0.94532

    0.94785

    0.95026

    0.95257

    0.95478

    0.95689

    0.95891

    0.96083

    0.96267

    0.96443

    0.9661

    0.9677

    0.96923

    0.97069

    0.97208

    0.9734

    0.97467

    0.97587

    0.97702

    0.97812

    0.97916

    0.98016

    0.98111

    0.98201

    1

    0.5

    P

    Z

    Sheet1

    -40.01799

    -3.950.01889

    -3.90.01984

    -3.850.02084

    -3.80.02188

    -3.750.02298

    -3.70.02413

    -3.650.02533Z

    -3.60.0266

    -3.550.02792

    -3.50.02931

    -3.450.03077

    -3.40.0323

    -3.350.0339

    -3.30.03557

    -3.250.03733

    -3.20.03917

    -3.150.04109

    -3.10.04311

    -3.050.04522

    -30.04743

    -2.950.04974

    -2.90.05215

    -2.850.05468

    -2.80.05732

    -2.750.06009

    -2.70.06297

    -2.650.06599

    -2.60.06914

    -2.550.07243

    -2.50.07586

    -2.450.07944

    -2.40.08317

    -2.350.08707

    -2.30.09112

    -2.250.09535

    -2.20.09975

    -2.150.10433

    -2.10.1091

    -2.050.11405

    -20.1192

    -1.950.12455

    -1.90.13011

    -1.850.13587

    -1.80.14185

    -1.750.14805

    -1.70.15447

    -1.650.16111

    -1.60.16798

    -1.550.17509

    -1.50.18243

    -1.450.19

    -1.40.19782

    -1.350.20587

    -1.30.21417

    -1.250.2227

    -1.20.23148

    -1.150.24049

    -1.10.24974

    -1.050.25923

    -10.26894

    -0.950.27888

    -0.90.28905

    -0.850.29943

    -0.80.31003

    -0.750.32082

    -0.70.33181

    -0.650.34299

    -0.60.35434

    -0.550.36586

    -0.50.37754

    -0.450.38936

    -0.40.40131

    -0.350.41338

    -0.30.42556

    -0.250.43782

    -0.20.45017

    -0.150.46257

    -0.10.47502

    -0.050.4875

    00.5

    0.050.5125

    0.10.52498

    0.150.53743

    0.20.54983

    0.250.56218

    0.30.57444

    0.350.58662

    0.40.59869

    0.450.61064

    0.50.62246

    0.550.63414

    0.60.64566

    0.650.65701

    0.70.66819

    0.750.67918

    0.80.68997

    0.850.70057

    0.90.71095

    0.950.72112

    10.73106

    1.050.74077

    1.10.75026

    1.150.75951

    1.20.76852

    1.250.7773

    1.30.78583

    1.350.79413

    1.40.80218

    1.450.81

    1.50.81757

    1.550.82491

    1.60.83202

    1.650.83889

    1.70.84553

    1.750.85195

    1.80.85815

    1.850.86413

    1.90.86989

    1.950.87545

    20.8808

    2.050.88595

    2.10.8909

    2.150.89567

    2.20.90025

    2.250.90465

    2.30.90888

    2.350.91293

    2.40.91683

    2.450.92056

    2.50.92414

    2.550.92757

    2.60.93086

    2.650.93401

    2.70.93703

    2.750.93991

    2.80.94268

    2.850.94532

    2.90.94785

    2.950.95026

    30.95257

    3.050.95478

    3.10.95689

    3.150.95891

    3.20.96083

    3.250.96267

    3.30.96443

    3.350.9661

    3.40.9677

    3.450.96923

    3.50.97069

    3.550.97208

    3.60.9734

    3.650.97467

    3.70.97587

    3.750.97702

    3.80.97812

    3.850.97916

    3.90.98016

    3.950.98111

    40.98201

    Sheet1

    0

    0

    0

    0

    0

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    Z

    Sheet2

    Sheet3

  • The meaning of model parameter By constant we mean the natural logarithm of likelihood ratio between happening and non-happening when exposure dose is zero. By regression coefficient we mean the change of logitP when the independent variable changes by one unit.

  • The statistical indicator--odds ratio which is used to measure the function of risk factor in the epidemiology ,the formula of computation is: Odds ratio (OR)

    In the formula ,

    is the incidence of a disease when

    is

    ,and

    is the incidence of a disease when

    is

    .

    is called odds ratio when many variables had been adjusted, it show the function of the risk factors without the influence of the other independent variables.

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    _1208683399.unknown

    _1208683450.unknown

    _1208683486.unknown

    _1208683415.unknown

    _1208683343.unknown

    _1077608162.unknown

  • The relationship with logistic P

    Comparing the conditions of disease when one risk factor has two different exposure levels (

    ,

    ), the natural logarithm of Odds Ratio is:

    _1077608069.unknown

    _1077608080.unknown

  • We often think that

    is an ineffective parameter, because there is no relationship between

    and

    .

    _1208781326.unknown

    _1208781372.unknown

    _1208781297.unknown

  • II the parametric estimation of logistic regression model parametric estimation Theorythe estimation of likelihood

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    _1079443675.unknown

    _1081064387.unknown

    _1077608256.unknown

  • 2.Estimation of OR It can show the OR of two different levels c1c0 of one factor.

    _1077608162.unknown

    _1077608169.unknown

    If the independent variable

    only has two levelsthe exposur