Logistic Regression III: Advanced topics

Conditional Logistic Regression for Matched Data Conditional Logistic Regression for Matched Data

Recall: Matching Matching 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: Agresti example, diabetes 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 DiabetesNo Diabetes MI cases MI controls odds(“favors” case/discordant pair) =

Conditional Logistic Regression

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

Diabetes No diabetes Case (MI)Control Diabetes No diabetes Case (MI)Control Diabetes No diabetes Case (MI)Control Diabetes No diabetes Case (MI)Control

 The conditional likelihood= Each age-gender stratum has the same baseline odds of disease; but these baseline odds may differ across strata

Conditional Logistic Regression

Example: MI and diabetes

Conditional Logistic Regression

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:Prenatal ultrasound examinations and risk of childhood leukemia: case-control study BMJ 2000;320:

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 UltrasoundNo Ultrasound Leukemia cases Myeloid leukemia controls 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 matching

Ordinal 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 Scale 1 = strongly disagree 2 = disagree 3 = neutral 4 = agree 5 = 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 variable 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: 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 groups (2 potential “positive” outcomes) 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 predictors) The intercept-only model, no predictors (two intercepts!): Log odds (amenorrhea)=  amen Log odds (any irregularity)=  amen or oligo

Fitted model: Logit of amenorreha= 8.6% of my sample has amenorrhea Odds = 8.6/91.4=.094 Ln (.094) = 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)= Log odds (any irregularity)= -0.70

Logistic model with predictors: Log odds (amenorrhea)=  amen + β 1 *X 1 + β 2 *X 2 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 (b):

Odds ratio interpretation: Interpretation of the betas: e β = 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: Score on the anorexia subscale of the eating disorder inventory (EDI-A)

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: Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 Intercept <.0001 EDIA <.0001 Log odds (amen)= *EDI-A Log odds (any irregularity)= *EDI-A

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

Compare actual data and fitted model:

Fitted model with EDI-A: Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits EDIA 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: Log odds (outcome)= *EDIA-1 The model predicts that a woman with an EDI-A score of 15 would have:

Predictions: Predicted logit= Predicted probability = 19% Predicted logit=.4281 Predicted probability = 60.5% 50% probability line

Advantages & 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…