Logistic Regression III: Advanced topics Conditional Logistic Regression for Matched Data...

Logistic Regression III: Advanced Logistic Regression III: Advanced topics topics

Conditional Logistic Regression for Conditional Logistic Regression for Matched DataMatched Data

Recall: MatchingRecall: MatchingMatching can control for extraneous

sources of variability and increase the power of a statistical test.

Match M controls to each case based on potential confounders, such as age and gender.

If the data are matched, you must account for the matching in the statistical analysis!!

Recall: Recall: Agresti Agresti example, example, diabetes and MIdiabetes and MI

Match each MI case to an MI control based on age and gender.

Ask about history of diabetes to find out if diabetes increases your risk for MI.

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

odds(“favors” case/discordant pair) =

16

37

c

bOR

Conditional Logistic RegressionConditional Logistic Regression

The Conditional Likelihood: The Conditional Likelihood: each each discordant discordant stratumstratum (rather than individual) (rather than individual) gets 1 term in the likelihoodgets 1 term in the likelihood

xDPDPDPDP

DPDP

i

)exposurescontrol/(*)exposurescase/(~)exposurescontrol/(~*)exposurescase/(

)exposurescontrol/(~*)exposurescase/(strata all

1

Note: the marginal probability of disease may differ in each age-gender stratum, but we assume that the (multiplicative) increase in disease risk due to exposure is constant across strata.

For each stratum, we add to the likelihood: the CONDITIONAL probability that the case got disease and the control did not, given that we have a case-control pair.

The numerator is the probability (as a function of exposures) that the case gets disease and the control does not.

The denominator is the probability that the case gets disease and the control does not OR that the control (with all her exposures) gets disease and the case doesn’t (with all her exposure).

Recall probability terms:Recall probability terms:

e

eEDP

1)/(

e

eEDP

1)~/(

eEDP

1

1)/(~

eEDP

1

1)~/(~

α)0(α))~/(1

)~/(ln(

)1(α))/(1

)/(ln(

EDP

EDP

EDP

EDP

Diabetes

No diabetes

Case (MI) Control

1 1

0 0

Diabetes

No diabetes

Case (MI) Control

1 0

0 1

Diabetes

No diabetes

Case (MI) Control

0 1

1 0

Diabetes

No diabetes

Case (MI) Control

0 0

1 1

iiii

i

ii

i

ee

e

ee

eee

e

L

1

1*

11

1*

1

1

1*

1),(

ii

i

ii

i

ii

i

ee

e

ee

eee

e

L

1

1*

11

1*

1

1

1*

1),(

1

1*

11

11

*1

11

*1),(

i

i

iii

i

ii

i

ee

eeee

eee

L

1

1*

1

1

1

1*

1

1

1*

1),(

i

i

iii

i

ii

i

e

e

eee

eee

e

L

The conditional likelihood=The conditional likelihood=

case favor thethat strata discordant

1i

control favor the that strata discordant

1

1*

1

1

1

1*

1

1

1*

1

1

1*

11*

1

11

*1

1

n

m

j

i

i

iii

i

ii

i

jj

j

j

j

j

j

j

j

e

e

eee

eee

e

x

ee

e

e

e

e

e

e

e

Each age-gender stratum has the same baseline odds of disease; but these

baseline odds may differ across strata


case favor thethat

strata discordant

1

control favor thethat strata discordant

1j

n

i

m

ii

i

jj

j

ee

ex

ee

e

nmn

i

m

j e

e

ee

ex

e

e i

)1

()1

1(

1

1

1

parameter) nuisance of rid (gets !cancel! s' The***

11

Example: MI and diabetesExample: MI and diabetes

3716 )1

()1

1()L(

e

e

e


16

37

1637

53)137(

01

53-37

dlog(L)

)1log(*5337)log(

e

e

ee

e

e

d

eL

In SAS…In SAS…

proc logistic data = YourData;model MI (event = "Yes") = diabetes;strata PairID;run;

Could there be an association between exposure to ultrasound in utero and an increased risk of childhood malignancies?

Previous studies have found no association, but they have had poor statistical power to detect an association.

Swedish researchers performed a nationwide population based case-control study using prospectively assembled data on prenatal exposure to ultrasound.

Example:Example:Prenatal ultrasound examinations and risk Prenatal ultrasound examinations and risk of childhood leukemia: case-control study of childhood leukemia: case-control study

BMJBMJ 2000;320:282-283 2000;320:282-283

Example:Example:Prenatal ultrasound examinations and risk Prenatal ultrasound examinations and risk of childhood leukemia: case-control study of childhood leukemia: case-control study

BMJBMJ 2000;320:282-283 2000;320:282-283

535 cases: all children born and diagnosed as having myeloid leukemia between 1973 and 1989 in Swedish registers of birth, cancer, and causes of death.

535 matched controls: 1 control was randomly selected for each case from the Swedish Birth Registry, matched by sex and year and month of birth.

Ultrasound

No ultrasound

215 320

Ultrasound No Ultrasound

200

335

535

Leukemia cases

Myeloid leukemia controls

235100

115 85

85.100

85

c

bOR

But this type of analysis is limited to single dichotomous exposure…

Used conditional logistic regression to look at dose-response with number of ultrasounds:

Results: Reference OR = 1.0; no ultrasounds OR =.91 for 1-2 ultrasounds OR=.64 for >=3 ultrasounds

Conclusion: no evidence of a positive association between prenatal ultrasound and childhood leukemia; even evidence of inverse association (which could be explained by reasons for frequent ultrasound)

Each term in the likelihood represents a stratum of 1+M individuals

More complicated likelihood expression! Just as easy to implement in SAS as we’ll

see Wednesday…

Extension: 1:M matchingExtension: 1:M matching

Ordinal Logistic Regression Ordinal Logistic Regression

Ordinal Logistic RegressionOrdinal Logistic Regression

What if your outcome variable has more than two levels?

For ordinal outcomes, use ordinal logistic regression:

*Relies on the cumulative logit*Models the predicted probability of multiple

outcomes*Also known as the “proportional odds model”

Ordinal Variable Example: Likert Ordinal Variable Example: Likert ScaleScale

1 = strongly disagree2 = disagree3 = neutral 4 = agree5 = strongly agree

Cumulative outcomes:

*strongly agree vs. the rest

*agree or strongly agree vs. neutral or negative

*agree or neutral vs. negative

*the rest vs. strongly negative

Ordinal logistic regression gives you a way to model these cumulative outcomes all at once!

Ordinal Variable Example: Continuous Ordinal Variable Example: Continuous variable measured crudelyvariable measured crudely

1 = breastfed >=6 months

2 = breastfed 4-5 months

3 = breastfed 2-3 months

4 = breastfed <2 months

The outcome variable, breastfeeding, was only measured at limited time

points. So, may not be best modeled as continuous variable in linear regression. Use ordinal logistic!

More inclusive

definition of a “positive”

outcome

Another example, 3 levels:Another example, 3 levels:

1 = eumenorrhea (normal menses) (66.6%)

2 = oligomenorrhea (mild irregularity) (24.6%)

3 = amenorrhea (severe irregularity) (8.6%)

From my data on runners:

Most “severe” outcome

Cumulative logit, 3 groupsCumulative logit, 3 groups(2 potential “positive” outcomes)(2 potential “positive” outcomes)

normal

rheaoligomenoror amenorrhea

normalor reaoligomenor

amenorrhea

logty irregulariany for logit cumulative

log amenorrheafor logit cumulative

p

p

p

p

In words:

The log odds of having amenorrhea (versus everything else).

And the log odds of having any irregularity (versus normal).

Corresponding logistic model (no Corresponding logistic model (no predictors)predictors)

The intercept-only model, no predictors (two intercepts!):

Log odds (amenorrhea)= amen

Log odds (any irregularity)= amen or oligo

Fitted model:Fitted model:Logit of amenorreha=

8.6% of my sample has amenorrhea

Odds = 8.6/91.4=.094

Ln (.094) = -2.3623

Logit of any irregularity=

33.3% has any irregularity (24.6% + 8.6%)

Odds=(1/3)/(2/3) = 1/2

Ln(1/2) = -.70

Fitted models are: Log odds (amenorrhea)= -2.36 Log odds (any irregularity)= -0.70

Logistic model with predictors:Logistic model with predictors:

Log odds (amenorrhea)= amen + β1*X1 + β2*X2

Log odds (any irregularity)= amen or oligo + β1*X1 + β2*X2

Note, different intercepts but shared betas (shared slopes)!

Odds ratio interpretation (a):Odds ratio interpretation (a):

unexposed for the amenorrhea of odds

exposed for the amenorrhea of oddsOR

)1()0(

)1()1(

exp

exp

confounderosureamen

confounderosureamen

e

e

)1()1(

exp

exp

1osure

osure

ee

)1()0(

)1()1(

exp

exp

confounderosureamen

confounderosureamen

e

e

Odds ratio interpretation (b):Odds ratio interpretation (b):

unexposed for thety irregulari menstrualany of odds

exposed for thety irregulari menstrualany of oddsOR

)1()0(

)1()1(

exp

exp

confounderosureoamenorolig


e

e

)1()1(

exp

exp

1osure

osure

ee

)1()0(

)1()1(

exp

exp



e

e

Odds ratio interpretation:Odds ratio interpretation:

Interpretation of the betas:

eβ = adjusted odds ratio

For every 1-unit increase in X, it’s the increase in the odds of any menstrual irregularity compared with none and it’s also the increase in the odds of amenorrhea compared with the other two categories (adjusted for any other predictors in the model).

Note: proportional odds assumption! The odds ratios are the same across different levels of the outcome.

Example predictor, EDI-A:Example predictor, EDI-A:

Score on the anorexia subscale of the eating disorder inventory (EDI-A)

Cumulative logit plot (4 bins)Cumulative logit plot (4 bins)

The intercept for any irregularity (the log odds of any irregularity where EDI-A=0)

The intercept for amenorrhea (the

log odds of amenorrhea where

EDI-A=0)

These lines should be linear and parallel (equal slopes, one beta!)

The slopes represent the increase in the log odds of either outcome for every 1-unit increase in EDI-A score.

Fitted model with EDI-A:Fitted model with EDI-A:

Analysis of Maximum Likelihood Estimates

Standard WaldParameter DF Estimate Error Chi-Square Pr > ChiSq

Intercept 1 1 -3.2630 0.3823 72.8648 <.0001Intercept 2 1 -1.3888 0.2478 31.4220 <.0001EDIA 1 0.1211 0.0265 20.9065 <.0001

Log odds (amen)= -3.2630 + 0.1211*EDI-A

Log odds (any irregularity)= -1.3888 + 0.1211*EDI-A

Fitted Model: Predicted logit at Fitted Model: Predicted logit at every level of EDI-Aevery level of EDI-A

Compare actual data and fitted Compare actual data and fitted model:model:

Fitted model with EDI-A:Fitted model with EDI-A:

Odds Ratio Estimates

Point 95% WaldEffect Estimate Confidence Limits

EDIA 1.129 1.072 1.189

For every 1-unit increase in EDI-A score, there’s a 13% increase in the odds of being amenorrheic versus the other two categories and a 13% increase in the odds of being amenorrheic or oligomenorrheic versus normal.

Predictions:Predictions:

Log odds (outcome)= -3.2630 + -1.3888 + 0.1211*EDIA-1

The model predicts that a woman with an EDI-A score of 15 would have:

%5.6011

1)tyirregulariP(any

%1911

1)P(amen

4281.

4281.

)15(1211.3888.1

)15(1211.3888.1

4461.1

4461.1

)15(1211.2630.3

)15(1211.2630.3

e

e

e

e

e

e

e

e

Predictions:Predictions:

Predicted logit=-1.446

Predicted probability = 19%

Predicted logit=.4281

Predicted probability = 60.5%50%

probability line

Advantages & disadvantagesAdvantages & disadvantages

Ordinal logistic is better than running separate logistic models for different outcomes (e.g., one model for amenorrhea, one model for any irregularity) because of the improvement in statistical power!

Ordinal logistic prevents you from having to arbitrarily turn an ordinal variable into a binary variable!

But does require that you meet the proportional odds assumption…

Logistic Regression III: Advanced topics Conditional Logistic Regression for Matched Data...

Documents

Transcript of Logistic Regression III: Advanced topics Conditional Logistic Regression for Matched Data...