Introduction to Logistic Regression

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

Objectives

• When do we need to use logistic regression

• Principles of logistic regression

• Uses of logistic regression

• What to keep in mind

Chlamorea

• Sexually transmitted infection –Virus recently identified

–Leads to general rash, blush, pimples and feeling of shame

–Increasing prevalence with age

–Risk factors unknown so far

Case control study

• Population of Berlin

• 150 cases, 150 controls

• Hypothesis: Consistent use of condoms protects against chlamorea

• Questionnaire with questions on demographic characteristics, sexual behaviour

• OR, t-test

Results bivariate analysis

Cases n=150

Controls n=150

Odds ratio

Used condoms at last sex 40 90 0.17

Did not use condoms 110 60 Ref

Results bivariate analysis

Cases n=150

Controls n=150

Odds ratio

Single 125 50 4.7

Currently in a relationship 25 100 Ref

Results bivariate analysis

Cases n=150

Controls n=150

T-test

nr partners during last year 4 2 p=0.001

Mean age in years 39 26 p=0.001

Confounding?

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Chlamorea and condom use

Single statusAgegroup

Number of partnersStratification

Let’s go one step back

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 58 153 29 123 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

80

100

120

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160

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20 30 40 50 60 70 80 90

SBP (mm Hg)

Age (years)

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

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

11xβαy Slope

α

What if we have more than one independent variable?

Multiple risk factors

• Objective:To attribute to each risk factors the respective effect (RR) it has on the occurrence of disease.

Types of multivariable analysis• Multiple models

–Linear regression–Logistic regression–Cox model–Poisson regression–Loglinear model–Discriminant analysis…

• Choice of the tool according objectives, study design and variables

Multiple linear regression

• Relation between a continuous variable and a set of i variables

• Partial regression coefficients i

–Amount by which y changes when xi changes by one unit and all the other xi remain constant

–Measures association between xi and y adjusted for all other xi

• Example–Number of partners in relation to age & income

xβ ... xβ xβαy ii2211

Multiple linear regression

Predicted Predictor variables

Response variable Explanatory variablesOutcome variable CovariablesDependent Independent variables

xβ ... xβ xβα y ii2211

y (number of partners) = α + β1 age + β2 income + β3 gender

What if our outcome variable is dichotomous?

Logistic regression (1)Table 2 Age and chlamorea

How can we analyse these data?

• Compare mean age of diseased and non-diseased

–Non-diseased: 26 years

–Diseased: 39 years (p=0.0001)

• Linear regression?

Dot-plot: Data from Table 2P

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ore

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

Dot-plot: Data from Table 3

0

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0 2 4 6 8

Diseased %

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Logistic function (1)

0.0

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1.0Probability of disease

x

Logistic function

• Logistic regression models the logit of the outcome=natural logarithm of the odds of the outcome

Probability of the outcome (p)

Probability of not having the outcome (1-p)ln

ii2211 xβ ... xβ xβαP-1

P ln

Logistic function

= log odds of disease in unexposed

= log odds ratio associated with being exposed

e = odds ratio

ii2211 xβ ... xβ xβαP-1

P ln

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 xis constant–Measures association between xi and log-odds

adjusted for all other xi

ii2211 xβ ... xβ xβαP-1

P ln

Uses of multivariable analysis

• Etiologic models–Identify risk factors adjusted for

confounders–Adjust for differences in baseline

characteristics

• Predictive models –Determine diagnosis–Determine prognosis

Fitting equation to the data

• Linear regression: –Least squares

• Logistic regression: –Maximum likelihood

Elaborating eβ

• eβ = OR What if the independent variable

is continuous?

what’s the effect of a change in x by more than one unit?

The Q fever example

• Distance to farm as independent continuous variable counted in meters–β in logistic regression was -0.00050013 and

statistically significant

• OR for each 1 meter distance is 0.9995 –Too small to use

• What’s the OR for every 1000 meters?

–e1000*β = e-1000*0.00050013 = 0.6064

Continuous variables

• Increase in OR for a one unit change in exposure variable

• Logistic model is multiplicative OR increases exponentially with x–If OR = 2 for a one unit change in exposure

and x increases from 2 to 5: OR = 2 x 2 x 2 = 23 = 8

• Verify if OR increases exponentially with x –When in doubt, treat as qualitative variable

Coding of variables (2)

• Nominal variables or ordinal with unequal classes:–Preferred hair colour of partners:

» No hair=0, grey=1, brown=2, blond=3

–Model assumes that OR for blond partners = OR for grey-haired partners3

–Use indicator variables (dummy variables)

Indicator variables: Hair colour

• Neutralises artificial hierarchy between classes in variable “hair colour of partners"

• No assumptions made• 3 variables in model using same reference • OR for each type of hair adjusted for the

others in reference to “no hair”

Classes

• Relationship between number of partners during last year and chlamorea

– Code number of partners: 0-1 = 1, 2-3 = 2, 4-5 = 3

• Compatible with assumption of multiplicative model – If not compatible, use indicator variables

Code nr partners

Cases Controls OR

1 20 40 1.0

2 22 30 1.5

3 12 11 2.2

1.52 2.2

Risk factors for Chlamorea

No condom use

Chlamorea

SexHair colourAgegroupSingleVisiting barsNumber of partners

Unconditional Logistic RegressionTerm Odds

Ratio 95% C.I. Coef. S. E. Z-Statistic

P-Value

# partners 1,2664 0,2634 10,7082 0,2362 0,9452 0,5486 0,5833

Single (Yes/No) 1,0345 0,3277 3,2660 0,0339 0,5866 0,0578 0,9539

Hair colour (1/0) 1,6126 0,2675 9,7220 0,4778 0,9166 0,5213 0,6022

Hair colour (2/0) 0,7291 0,0991 5,3668 -0,3159 1,0185 -0,3102 0,7564

Hair colour (3/0) 1,1137 0,1573 7,8870 0,1076 0,9988 0,1078 0,9142

Visiting bars 1,5942 0,4953 5,1317 0,4664 0,5965 0,7819 0,4343

Used no Condoms 9,0918 3,0219 27,3533 2,2074 0,5620 3,9278 0,0001

Sex (f/m) 1,3024 0,2278 7,4468 0,2642 0,8896 0,2970 0,7665

CONSTANT * * * -3,0080 2,0559 -1,4631 0,1434

Last but not least

Why do we need multivariable analysis?

• Our real world is multivariable

• Multivariable analysis is a tool to determine the relative contribution of all factors

Sequence of analysis

• Descriptive analysis–Know your dataset

• Bivariate analysis–Identify associations

• Stratified analysis–Confounding and effect modifiers

• Multivariable analysis–Control for confounding

What can go wrong

• Small sample size and too few cases

• Wrong coding

• Skewed distribution of independent variables–Empty “subgroups”

• Collinearity–Independent variables express the same

Do not forget

• Rubbish in - rubbish out

• Check for confounders first

• Number of subjects >> variables in the model

• Keep the model simple–Statisticians can help with the model but

you need to understand the interpretation

• You will need several attempts to find the “best” model

• If in doubt…

Really call a statistician !!!!

References

• Norman GR, Steiner DL. Biostatistics. The Bare Essentials. BC Decker, London, 2000

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

• Schwartz MH. Multivariable analysis. Cambridge University Press, 2006