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Lecture 14: Logistic regression

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22 table (contingency table)

10

1

20

29

Divorce

No divorce

30

30

Depression No depression risk

3010

301

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22 table (contingency table)

a

c

b

d

Exposed

Not-exposed

a+b

c+d

Disease Healthy

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Risk (absolute)

• Proportion of individuals initially healthy who contracted the disease during a given periode

• The observation period must be the same for everyone

• Given as a proportion (0-1), or a percentage (0-100%)

• Estimation of the risk

– For exposed

– For non-exposed

baaRe

dccRne

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Relative risk

• Ratio of the risk for the exposed to the risk for the non-exposed (no unit)

• Interpretation– RR>1: exposition increases the risk– RR=1: exposition does not modify the risk– RR<1: exposition decreases the risk

• When RR<1, relative risk reduction (in %)may be presented

RR = Re / Rne

RRR = (1-RR)*100

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Caveat

• Relative risk does not provide any information on the importance of the compared risks

• RR = 3 peut correspondre à:– Re = 45% Rne = 15%

– Re = 3% Rne = 1%

– Re = 0.006% Rne = 0.002%

– …

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Risk difference

• Difference between the risks of the exposed and non-exposed individuals

• Given as a proportion or a percentage

• Interpretation– DR>0: exposition increases the risk– DR=0: exposition does not modify the risk– DR<0: exposition decreases the risk

DR = Re – Rne

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Test and estimation

• Null hypothesis: no effect of exposition:– RR = 1

– DR = 0

• It is also possible to compute a standard-error for each of these statistics, and to obtain a confidence interval

H0: Re = Rne

2 test

BMJ 2003;327:1254-7

RR=1.53

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Prevalence

• Proportion of individuals having a disease (or presenting a characteristic) at a given moment

• Moment is defined by• Date (eg: 19 october 2011)

• Event (eg: at birth)

Prevalence of smoking in an freshman class

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Incidence rate

• Risk of contracting a disease during one time unit

• Numerator: new cases

• Denominator: sum of the time-person at risk– Person-years

– Person-days

– …

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Comparison with prospective study: case control study…

– Allows to study very rare conditions (e.g., autism, suicide)– Can be made more quickly– Requires less observations (cost )– Allows to test several risk factors for the outcome

• But…– Does not allow to compute a risk, or an incidence rate

(question: why)– Risk of bias in the measure of risk factor– Difficult to choose appropriate control– Can only study one condition at a time

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Case control study

a

c

b

d

Case Control

We take everyone

We take only a sample

Exposed

Non-exposed

The sums: a+b and c+d have no sense!

Impossible to compute risk and relative risk

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Solution: odds ratio

• Transform a proportion in odds

• Transform odds in proportion:

ppodds

1

1oddsoddsp

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Cas control study

a

c

b

d

Exposed

Non-exposed

Case Control

a/c b/d exposition odds

cbda

dbca

dbddbb

caccaa

ratioodds

//

)/()/(

)/()/(

denominator

numerator

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Prospective study

a

c

b

d

Exposed

Non-exposed

Disease Healthy

a/b

c/d

Disease odds

a/c b/dExposition odds

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Signification a-b-c-d

Case control Prospective study

a Case exposed to the risk factor

People exposed who develop the disease

b Control exposed People exposed remaining healthy

c Cases non-exposed

People non exposed who develop the disease

d Control non exposed

People non exposed remaining healthy

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Property of odds ratio

• Exposition odds ratio = disease odds ratio

..exp //

//

malORdcba

cbda

dbcaOR

Exposition odds Disease oddsodds ratio

Odds ratio are the same, computed from a prospective study or a case control study

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Odds ratio and relative risk

• When the condition is rare, (a<<b et c<<d), OR is approximately equal to RR

RRdccbaa

dcbaOR

)/()/(

//

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• 573 patients with facial clefts

• 763 controls

• Exposition:– Taking more than 400 mg of folic acid

supplements

BMJ 2007; 334:464-470

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Results

4.161882145491

ratioodds

Cleft No cleft

Folic acid < 400 mg

491 618

Folic acid ≥ 400 mg

82 145

Total 573 763

Odds 491 / 82 618 / 145

Odds ratio for folic acid < 400 mg

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Why not compute the RR?

• The prevalence is not correct since the goal of the case-control design is to have as many cases as controls.

• Thus, a/(a+b) does not make sense.

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Interpretation of odds ratio

• Similar to relative risk:– OR>1: IV is associated with DV (e.g., exposition is

associated with disease)– OR=1: IV is not associated with DV– OR<1: IV is negatively associated with DV

• Folic acid and facial cleft:– Not taking folic acid supplements (>400 mg) increases

the risk of facial cleft (by 40%)

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Continuous IV

1. Compare means of cases and controls (t test)

2. Divide the IV in several categories and compute an odds ratio for each category

3. Model:

2740.00 50.00 60.00 70.00 80.00 90.00 100.00

poids

-0.50

0.00

0.50

1.00

1.50

sex

R Sq Linear = 0.441

Model: linear regression

male

female

Proportion of males >100%

Distribution of the residuals [y - (a+bx)] is not normal

Proportion of females <0%

This is not the right method!

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Logistic regression

• The DV must be transformed– Y: probability that sex=1 (female), instead of 0 (male)

• “logit”

odds!

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Interpretation of « b »

• In linear regression:– b: mean change of Y expected for an increase of

one unit of X

• By analogy, logistic regression:– b: mean change of logit(Y=1) expected for an

increase of one unit of X

eb = odds ratio of Y for one unit of X

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Odds ratio and logistic regression coefficients

• Dependent variable: Y = 1 (case) or = 0 (control)

• Independent variable: X = 1 (exposed) or = 0 (non-exposed)

• Model: logit(y) = a + bx

• Equation among exposed: logit(yexp) = a + b*1 = a + b

• Equation among non-exposed: logit(ynon-exp) = a + b*0 = a

• Equation for b = (a + b) – a = logit(yexp) – logit(ynon-exp)

)ORlog()0y(Pr)1y(Pr

)0y(Pr)1y(Pr

log)0y(Pr)1y(Pr

log)0y(Pr)1y(Pr

logbexpnon

expnon

exp

exp

expnon

expnon

exp

exp

OReb

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Example weight - sex

• Weight in 4 categories: – odds ratio = 8.9– Odds of being a man is multiplied by 8.9 for each

increase to a higher category of weight

• Weight in kilos: – odds ratio = 1.2– Odds of being a man is multiplied by 1.2 for each

additional kilo de poids

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Multiple IVs

• Example:

• Results:– Odds ratio for one additional kg: 1.15

(p<0.001)– Odds ratio for one additional cm: 1.20

(p<0.001)

adjusted for height

Adjusted for weight

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Conclusions

• Case-control study: design to examine associations between risk factors and disease– Mostly for rare disease– Efficient and cost effective

• Odds ratio: measure of association – often similar to relative risk

• Logistic regression: modeling method for binary dependent variables

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Multilevel logistic regression

This is a random intercept multilevel logistic regression

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Correct judgment of normality

• Statistical normality test (KS) was correct for 71.4% of the distributions.

• Only 57.1% for AD and JB.• Levene test correct in both cases (because data

were normally distributed)• You were correct for 71.4% of the distributions,

but your errors were not the same as the statistical tests.– All of you correctly found the bimodal distribution– You were not influenced by sample size