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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

0

0

0

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1

0.5

P

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.

_1077608169.unknown

_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

_1077608381.unknown

_1079443675.unknown

_1081064387.unknown

_1077608256.unknown

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

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_1077608169.unknown

If the independent variable

only has two levelsthe exposur