Conditional Logistic Regression for Matched Case-Control Studies

Setting Up the Analysis: The Data Structure

Before we can fit a model, we need to ensure our data is structured correctly. For CLR, especially in a matched case-control context, the data must be in a "long" format. This means each row is an individual, not a pair.

Specifically, you need at least three key columns:

1. A subject identifier: A unique ID for each person.
2. A stratum or group identifier: A column that links the matched subjects together. For a 1-to-1 matched study, two subjects (one case, one control) will share the same stratum ID.
3. An outcome variable: A binary indicator (0/1) identifying who is a case and who is a control.
4. Predictor variables: The exposures or covariates you want to investigate.

Let's look at a reference document that illustrates this structure.

As you can see in that example, each person has their own row, but they are linked to their matched counterparts via the Match variable. This structure is essential for the software to know which individuals to compare within each stratum.

Fitting and Interpreting a CLR Model: An R Example

Now, let's walk through a complete example of fitting and interpreting a CLR model. We will use a matched case-control dataset to assess the risk factors for admission to a newborn intensive care unit (NICU).

The following reading from an R-based public health textbook provides a perfect, step-by-step walkthrough.

Let's break down the key steps and results from that reading.

1. The Model Formula

The core of the analysis is the clogit() function from R's survival package. The formula used was:
clogit(NICU ~ RF_PPTERM + strata(MEDUC, MAGER), data = natality_CC)

• NICU: This is the outcome variable (1 for case, 0 for control).
• RF_PPTERM: This is the predictor of interest (previous preterm birth).
• strata(MEDUC, MAGER): This is the crucial part for CLR. It tells the function that the matching was performed on maternal education (MEDUC) and age (MAGER). The model will condition on these strata.

Notice that the matching variables are specified in strata() and not as regular predictors. This is because, as we learned last lesson, their effects cannot be estimated by CLR.

2. Interpreting the Output

The R output provides a table of coefficients. Let's focus on the key values for the RF_PPTERM predictor.

| | coef | exp(coef) | se(coef) | z | Pr(>|z|) |
| :--- | :---: | :---: | :---: | :---: | :---: |
| RF_PPTERMYes | 1.123 | 3.074 | 0.316 | 3.554 | 0.0004 |

This table contains everything we need to interpret the effect of having a previous preterm birth.

• coef (The Log-Odds Ratio): The value 1.123 is the estimated regression coefficient, $\hat{\beta }$. It represents the change in the log-odds of being a case (NICU admission) for a mother with a previous preterm birth compared to a mother without one within the same matched stratum.

• exp(coef) (The Odds Ratio): This is the most common and intuitive way to interpret the result. By exponentiating the coefficient ($e^{1.123}=3.074$), we get the odds ratio (OR).

The interpretation is as follows:

Infants born to mothers with a history of preterm birth have 3.07 times the odds of being admitted to the NICU compared to infants whose mothers did not have a previous preterm birth, after accounting for the matching on maternal age and education.

This odds ratio is a relative measure. It compares the odds of the outcome for two individuals within the same stratum, one exposed and one unexposed.

• Pr(>|z|) (The p-value): The p-value of 0.0004 is very small, indicating that this association is statistically significant. We can reject the null hypothesis that there is no association between previous preterm birth and NICU admission.

The NCSS document you reviewed earlier provides another clear breakdown of how to interpret each of these components from a typical software output.

Reinforcing Interpretation with a Video Example

To further solidify this, let's watch a segment of the video from our previous lesson. It discusses a very similar matched case-control study and interprets the coefficients in the same way.

As the video explains, the exponentiated coefficient for a predictor can be interpreted as the odds ratio, which is a highly practical and standard way to report findings from a conditional logistic regression analysis.

Conclusion

In this lesson, we successfully bridged the gap from theory to practice. You learned how to set up, fit, and—most importantly—interpret a conditional logistic regression model for a matched case-control study.

Key Takeaways:

• CLR requires data in a "long" format with a specific variable identifying the matched strata.
• In R, the clogit() function is used, with the matching variables specified inside a strata() term.
• The primary output is the regression coefficient ($\beta$), which represents the log-odds ratio.
• The most practical interpretation comes from exponentiating the coefficient ($e^{\beta }$) to get the odds ratio (OR).
• The OR from a CLR model quantifies the relative odds of being a case for an exposed vs. unexposed individual within the same matched set.
• As expected from the theory, the model does not produce estimates for the matching variables themselves.

Preview of the Next Module:
This lesson concludes our deep dive into methods for paired and matched categorical data. We are now ready to move to the final major topic of the course: Poisson Regression and Log-Linear Models. In the next module, we will shift our focus from binary outcomes (case/control) to analyzing count data, opening up a new set of analytical tools and research questions.