statistik regresi logistik

8/7/2019 statistik regresi logistik

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Introduction to

Logistic Regression

Rachid Salmi, Jean-Claude Desenclos, Alain Moren, Thomas Grein


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Content

Simple and multiple linear regression

Simple logistic regression

The logistic function

Estimation of parameters

Interpretation of coefficients

Multiple logistic regression

Interpretation of coefficients

Coding of variables

Examples in Epiinfo 2002


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Simple linear regression

Age SBP Age SBP Age SBP

22 131 41 139 52 128

23 128 41 171 54 105

24 116 46 137 56 145

27 106 47 111 57 141

28 114 48 115 58 15329 123 49 133 59 157

30 117 49 128 63 155

32 122 50 183 67 176

33 99 51 130 71 172

35 121 51 133 77 178

40 147 51 144 81 217

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


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80

100

120

140

160

180

200

220

20 30 40 50 60 70 80 90

SBP (mm Hg)

Age (years)

adapted from Colton T. Statistics in Medicine. Boston: Little Brown, 1974


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Simple linear regression

y

x

xy 11+=Slope


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Multiple linear regression

Relation between a continuous variable and a set ofi continuous variables

Partial regression coefficients i Amount by which y changes on average when xi changes by one

unit and all the other xis remain constant

Measures association between xi and y adjusted for all other xi

Example SBP versus age, weight, height, etc

x...xxy ii2211 ++++=


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Multiple linear regression

Predicted Predictor variables

Response variable Explanatory variables

Outcome variable Covariables

Dependent Independent variables

x...xxy ii2211 ++++=


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General linear models

Family of regression models

Outcome variable determines choice of model

Uses

Control of confounding

Model building, risk prediction

Outcome Model

Continuous Linear regression

Counts Poisson regression

Survival Cox model

Binomial Logistic regression


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Logistic regression

Models relationship between set of variables xi dichotomous (yes/no)

categorical (social class, ... )

continuous (age, ...)

and

dichotomous (binary) variable Y

Dichotomous outcome most common situation inbiology and epidemiology


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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


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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

0

20

40

60

80

100

0 1 2 3 4 5 6 7

Diseased %

Age group


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Logistic function (1)

0.0

0.2

0.4

0.6

0.8

1.0Probability ofdisease

x


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Logistic transformation

logit ofP(y|x)

{


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Advantages of Logit

Properties of a linear regression model

Logit between - and + Probability (P) constrained between 0 and 1

Directly related to notion of odds of disease

xP-1

Pln +=

e

P-1

P x+=


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Interpretation of coefficient

eP-1

P x+=


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Example

Risk of developing coronary heart disease (CD)by age (


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Logistic Regression Model

Age2.0940.841-AgeP-1

Pln 1 +=+=


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Fitting equation to the data

Linear regression: Least squares

Logistic regression: Maximum likelihood

Likelihood function

Estimates parameters and with property that likelihood(probability) of observed data is higher than for any other values

Practically easier to work with log-likelihood

[ ] [ ] [ ]{ }=

+==n

i

iiii xyxylL

1

)(1ln)1()(ln)(ln)(


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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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Effect modification

2132211 xxxx

P-1

Pln +++=


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Statistical testing

Question Does model including given independent variable

provide more information about dependent variable thanmodel without this variable?

Three tests

Likelihood ratio statistic (LRS)

Wald test

Score test


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Likelihood ratio statistic

Compares two nested modelsLog(odds) = + 1x1 + 2x2 + 3x3 + 4x4 (model 1)Log(odds) = + 1x1 + 2x2 (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 parametersin model


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Example

0.2664)(SE0.2614)(SESmk0.7005Exc1.00470.7102

SmkExcP-1

Pln 21

++=

++=

P Probability for cardiac arrest

Exc 1= lack of exercise, 0 = exerciseSmk 1= smokers, 0= non-smokers

adapted from Kerr, Handbook of Public Health Methods, McGraw-Hill, 1998


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Interaction between smoking and exercise?

Product term 3 = -0.4604 (SE 0.5332)Wald test = 0.75 (1df)

-2log(L) = 342.092 with interaction term

= 342 .836 without interacti on term

LR statistic = 0.74 (1df), p = 0.39No evidence of any interaction

ExcSmkSmkExcP-1

Pln 321 +++=


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Coding of variables (1)

Dichotomous variables: yes = 1, no = 0 Continuous variables

Increase in OR for a one unit change in exposurevariable

Logistic model is multiplicative OR increases exponentially with x

If OR = 2 for a one unit change in exposure and x increasesfrom 2 to 5: OR = 2 x 2 x 2 = 23 = 8

Verify that OR increases exponentially with x.

When in doubt, treat as qualitative variable


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Continuous variable?

Relationship between SBP>160 mmHg and body weight

Introduce BW as continuous variable?

Code weight as single variable, eg. 3 equal classes:40-60 kg = 0, 60-80 kg = 1, 80-100 kg = 2

Compatible with assumption of multiplicative model

If not compatible, use indicator variables


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Indicator variables: Type oftobacco

Neutralises artificial hierarchy between classes in thevariable "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 inreference to non-smoking


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i k f d h f b i l


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Risk of death from bacterialmeningitis according to treatment

161 observations Death (yes, no)

Treatment

1=Chloramphenicol, 2=Ampicillin

Delay before treatment (onset, in days)

Convulsions (1,0)

Level of consciousness (1-3)

Severity of dehydration (1-3)

Age in years

Pathogen

1 Others, 2 HiB, 3 Streptococcus pneumoniae


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Reference

Hosmer DW, Lemeshow S. Applied logisticregression. Wiley & Sons, New York, 1989

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