Sample Size for Odds Ratios in Case-Control Studies

Introduction

Case-control studies are commonly used to investigate whether an exposure is associated with a disease or other binary outcome. Rather than following a population forward in time, investigators begin by identifying individuals who have the outcome (cases) and individuals who do not have the outcome (controls), and then compare their histories of exposure.

The principal measure of association is typically the odds ratio (OR). For a simple unmatched case-control study, the odds ratio compares the odds of exposure among cases with the odds of exposure among controls.

Sample size planning therefore asks a different question from a conventional two-arm clinical trial. The investigator wants enough cases and controls to detect a prespecified difference in exposure prevalence, expressed through an anticipated odds ratio, with adequate statistical power.

The 2 × 2 Case-Control Table

Suppose investigators are studying whether an exposure is associated with a binary disease outcome. The data can be represented as:

The odds of exposure among cases are:

The odds of exposure among controls are:

The odds ratio is therefore:

An odds ratio of 1 indicates no association. An odds ratio greater than 1 indicates higher odds of exposure among cases, whereas an odds ratio below 1 indicates lower odds of exposure among cases.

Why the Odds Ratio Is Natural for Case-Control Studies

In a case-control study, the investigator generally fixes the number of cases and controls through the sampling design. The proportion of participants who are cases is therefore determined by the study design rather than by the disease prevalence in the source population.

This means that quantities such as disease risk or disease incidence generally cannot be estimated directly from an ordinary case-control sample. The odds ratio, however, remains a natural measure of association.

Under appropriate sampling conditions, the case-control odds ratio estimates the corresponding odds ratio in the underlying source population.

The Three Core Inputs

A basic unmatched case-control sample size calculation requires:

The exposure prevalence among cases does not usually need to be specified independently. It can be derived from the anticipated odds ratio and the exposure prevalence among controls.

From Control Exposure Prevalence to Case Exposure Prevalence

Let:

• \(p_0\) = exposure prevalence among controls
• \(p_1\) = exposure prevalence among cases
• \(OR\) = anticipated odds ratio

The odds of exposure among controls are:

If the anticipated odds ratio is \(OR\), the expected odds of exposure among cases are:

Solving for \(p_1\) gives:

This relationship is fundamental to case-control sample size planning. Once the control exposure prevalence and anticipated odds ratio are specified, the expected exposure prevalence among cases follows directly.

A Complete Worked Example

Suppose investigators want to determine whether a particular occupational exposure is associated with a disease. They plan an unmatched case-control study with:

The objective is to determine how many cases and controls are required to have 80% power to detect an odds ratio of 2.0.

Step 1: Calculate the Expected Exposure Prevalence Among Cases

The expected exposure prevalence among controls is:

The anticipated odds ratio is:

Therefore:

Thus, the study is implicitly assuming an exposure prevalence of approximately 33.3% among cases.

The anticipated odds ratio of 2.0 therefore corresponds to a difference in exposure prevalence from 20% among controls to approximately 33.3% among cases.

Step 2: Understand the Statistical Test

For an unmatched case-control study, the primary hypothesis can be written as:

versus:

The null hypothesis represents no association between exposure and disease. The alternative hypothesis represents an association.

For planning purposes, the anticipated effect is:

The study should therefore have sufficient information to distinguish an odds ratio of 2.0 from the null value of 1.0.

Step 3: Approximate Sample Size Using the Log Odds Ratio

The logarithm of the odds ratio is especially useful because logistic regression and large-sample inference are naturally expressed on the log-odds scale.

For an odds ratio of 2:

For a simple two-group comparison of exposure proportions, the variance of the estimated log odds ratio can be approximated by:

where:

• \(n_1\) = number of cases
• \(n_0\) = number of controls
• \(p_1\) = exposure prevalence among cases
• \(p_0\) = exposure prevalence among controls

For a 1:1 design:

and the variance becomes:

The corresponding approximate sample size per group is:

Equivalently:

This is an approximate large-sample calculation based on the variance of the log odds ratio.

Step 4: Insert the Worked Example Values

For a two-sided alpha of 0.05:

For 80% power:

We already calculated:

and:

The resulting approximate sample size per group is about:

Thus, approximately:

or approximately:

participants in total.

Why the Exposure Prevalence Matters

The exposure prevalence among controls has a major influence on sample size. An exposure that is extremely rare provides relatively little information about the association unless the study is substantially enlarged.

For example, an anticipated odds ratio of 2.0 could correspond to very different absolute exposure differences depending on the baseline exposure prevalence.

When exposure is uncommon, even a meaningful odds ratio can correspond to a relatively small absolute difference in exposure prevalence. That makes the association harder to detect statistically.

Effect Size: Why Smaller Odds Ratios Require Larger Studies

The anticipated odds ratio is the primary effect-size assumption. The sample size is strongly influenced by:

Because the log odds ratio appears in the denominator, effects closer to the null value of 1 require substantially larger sample sizes.

For example, with the same exposure prevalence and power:

This illustrates why epidemiologic studies designed to detect modest associations often require hundreds or thousands of cases.

Power and Type I Error

Statistical power is the probability of rejecting the null hypothesis when the anticipated alternative effect is truly present.

The conventional relationship is:

where \(\beta\) is the probability of a Type II error.

Common planning values include 80%, 90%, and 95% power. Higher power requires more participants.

Similarly, a more stringent significance level increases sample size because the study must provide stronger evidence against the null hypothesis.

Case-Control Ratios

A case-control study does not necessarily require a 1:1 ratio. Investigators can recruit multiple controls for each case.

Let:

where \(r\) is the number of controls per case. Then:

Increasing the number of controls per case generally increases statistical efficiency, but the improvement diminishes as the ratio becomes large.

The exact efficiency depends on the exposure distribution and the analysis. The important practical point is that adding controls is most valuable when cases are difficult or expensive to recruit.

Why More Than Four Controls Per Case Often Has Limited Benefit

For a fixed number of cases, the information contributed by controls increases as more controls are added, but the marginal gain becomes progressively smaller.

Conceptually, the variance of an odds ratio estimator can be written in terms of both case and control information:

Once the control component becomes relatively small, adding still more controls has limited impact on the total variance.

Matched Case-Control Studies

Many case-control studies use matching. A case might be matched to one or more controls based on factors such as age, sex, geographic region, calendar time, or another important design variable.

Matching changes the statistical structure of the study. The analysis should account for the matched sets rather than treating all participants as independent observations.

For example, a 1:1 matched case-control study may consist of pairs:

For a matched analysis, the information comes especially from discordant matched sets. Pairs in which both members have the same exposure do not contribute the same way to estimation of the matched odds ratio.

Conditional Logistic Regression

Matched case-control studies are frequently analyzed using conditional logistic regression. The model conditions on the matched sets and estimates exposure effects while respecting the matching structure.

For example:

library(survival)

fit <- clogit(
  case ~ exposure + age + sex + strata(match_id),
  data = dat
)

summary(fit)

The coefficient for exposure is interpreted on the log odds scale:

and exponentiating gives:

Sample size planning should therefore be aligned with the conditional logistic regression analysis that will actually be performed.

Unmatched Logistic Regression

For an unmatched case-control study, logistic regression is commonly used to estimate adjusted odds ratios.

A simple logistic regression model is:

For a binary exposure:

When covariates are added:

The sample size calculation should ideally reflect the planned adjusted analysis, especially when the study is intended to estimate an adjusted odds ratio rather than a crude association.

Confounding and Sample Size

Confounding is one of the major reasons epidemiologic studies require multivariable analysis. Suppose the primary exposure is associated with age, smoking, and occupational history, all of which may also be associated with disease.

A multivariable logistic regression model may therefore be required.

The sample size should provide sufficient information not only for the primary exposure but also for the planned adjustment variables.

Rare Exposures

Rare exposures create a particularly difficult sample size problem. Suppose only 2% of controls are expected to be exposed. Even an odds ratio of 2.0 would produce only approximately 3.9% exposure among cases:

Thus, in a study with 200 cases, only about eight cases would be expected to be exposed. The resulting data are sparse.

Sparse exposure categories can produce unstable odds ratio estimates and very wide confidence intervals.

Rare Outcomes and the Case-Control Advantage

One of the greatest strengths of the case-control design is efficiency for rare diseases. If the outcome is uncommon in the source population, a prospective cohort study may require tens of thousands of participants to accumulate enough cases.

A case-control study can instead identify the required cases directly and sample controls from the same source population.

For example, suppose a disease has an incidence of only 0.5%. A cohort study may need a very large population to observe several hundred cases. A case-control study can deliberately recruit 300 cases and an appropriate number of controls.

Odds Ratios and Rare Disease

The odds ratio is sometimes interpreted as though it were a risk ratio. That interpretation requires caution.

When the outcome is rare, the odds ratio can approximate the risk ratio under appropriate conditions:

But the approximation becomes poor when the outcome is common. An odds ratio of 2 does not necessarily mean that disease risk is doubled.

The study report should therefore describe the estimated quantity accurately as an odds ratio unless a justified rare-outcome approximation is being used.

Two-Sided vs. One-Sided Testing

Most confirmatory epidemiologic studies use a two-sided significance test:

A one-sided test might instead be:

A one-sided design can require fewer participants for the same nominal alpha and power, but it should only be used when a directionally restricted hypothesis is scientifically justified and prespecified.

Confidence Intervals and Sample Size

A power-based sample size calculation can be viewed as a way of controlling the precision of inference around an anticipated effect.

The estimated odds ratio is commonly reported with a confidence interval:

This is why the logarithm of the odds ratio is central to inference. The confidence interval is symmetric on the log scale but asymmetric after exponentiation back to the odds-ratio scale.

A sufficiently large study produces a smaller standard error and therefore a narrower confidence interval.

Sample Size for a Prespecified Confidence Interval

Not every case-control study is primarily designed as a hypothesis test. An investigator may instead want to estimate the odds ratio with a specified precision.

For example, the study might seek a 95% confidence interval around the odds ratio that is reasonably narrow.

On the log scale, suppose the desired half-width is:

Then an approximate information requirement is:

The required case and control counts can then be derived from the anticipated exposure prevalences.

This is conceptually different from specifying power to reject \(OR=1\).

Power-Based vs. Precision-Based Case-Control Designs

Accounting for Missing Exposure Information

Suppose a study requires 380 evaluable subjects but expects 5% of participants to have missing or unusable exposure information. The enrollment target can be inflated as:

where \(r\) is the expected proportion of non-evaluable participants.

For a 5% non-evaluable rate:

Thus:

If cases and controls are recruited separately, the inflation should be applied appropriately to each group rather than simply adding arbitrary subjects to one group.

Accounting for Unequal Losses by Group

Suppose 190 evaluable cases and 190 evaluable controls are required, but the investigator expects:

The required enrollment becomes approximately:

Therefore, the investigator should plan for approximately:

This approach is preferable to applying a single overall inflation factor when the expected loss rates differ materially between groups.

Continuity Corrections and Exact Methods

Large-sample formulas are useful for understanding the basic relationships, but they can be unreliable when exposure is rare, the sample size is small, or the expected cell counts are sparse.

Potential approaches include:

• Exact conditional methods
• Score-based methods
• Continuity-corrected approximations
• Simulation-based power calculations
• Direct logistic-regression simulation

Simulation is especially useful when the planned analysis includes multiple covariates, interactions, matching, clustering, or complex sampling.

Sample Size for Multiple Predictors

A common misconception is that an odds-ratio sample size calculation for one exposure automatically guarantees adequate power for a multivariable logistic regression model. It does not.

Suppose the primary exposure is accompanied by:

• Age
• Sex
• Smoking status
• Body mass index
• Occupational category
• Family history
• Several interaction terms

The required sample size may be substantially larger than that required for a single exposure comparison.

The difficulty becomes particularly severe when predictors are rare or strongly correlated.

Events Per Variable: A Useful but Limited Concept

Traditional logistic regression guidance has sometimes relied on the concept of events per variable (EPV), such as requiring approximately 10 outcome events per candidate predictor.

This is a useful heuristic but should not be treated as a universal mathematical rule. Modern work emphasizes the importance of:

• Outcome prevalence
• Predictor strength
• Model complexity
• Collinearity
• Separation
• Expected model shrinkage
• Target prediction error

In a case-control study, the number of cases is therefore particularly important when the final model contains many predictors.

Separation in Logistic Regression

If an exposure is extremely rare and occurs almost exclusively among cases, logistic regression can experience complete or quasi-complete separation.

For example, suppose only three subjects in the entire study have a particular exposure and all three are cases. The maximum-likelihood estimate of the odds ratio may become extremely large or effectively infinite.

This is not solved simply by interpreting the resulting odds ratio. The model may require penalized or exact methods.

Case-Control Matching and Efficiency

Matching can improve efficiency when the matching factors are strong predictors of disease. However, matching does not automatically improve efficiency in every setting.

More importantly, matching changes the analysis. If a study matches on a factor, that matching structure should be incorporated into the primary statistical model.

For 1:1 matched studies, the discordant pairs are especially important. If both the case and control are exposed or both are unexposed, the pair does not provide the same information for estimating the exposure effect.

Overmatching

Matching on variables that are strongly related to the exposure but not necessary for controlling confounding can reduce efficiency or make recruitment difficult.

For example, matching too aggressively on highly specific characteristics can make it difficult to find appropriate controls.

The matching strategy should therefore be based on the causal structure of the study rather than on the desire to make cases and controls appear as similar as possible.

Exposure Misclassification

Sample size calculations generally assume that exposure status is measured correctly. In practice, exposure measurement can be imperfect.

If exposure misclassification is substantial, the observed association can be attenuated or otherwise distorted.

A larger sample does not necessarily correct systematic measurement error.

Selection of Controls

Control selection is one of the most important components of case-control design. Controls should arise from the same source population that produced the cases.

If controls are selected from a population with a different exposure distribution than the source population, the estimated odds ratio can be biased.

This is why sample size calculations based on an assumed control exposure prevalence are only as good as the epidemiologic assumptions underlying that prevalence.

Population-Based vs. Hospital-Based Controls

Controls can be selected using different strategies. For example:

The choice can affect the expected exposure prevalence among controls and therefore directly affect the sample size calculation.

Scenario Analysis

Because exposure prevalence and the anticipated odds ratio are uncertain, a robust protocol should often evaluate multiple plausible scenarios.

The conservative scenario is often more useful for ensuring that the study remains adequately powered if the true association is weaker than anticipated.

Sample Size Planning in R

The following R code implements the basic large-sample calculation using the anticipated exposure prevalence among controls and the odds ratio.

alpha <- 0.05
power <- 0.80

p0 <- 0.20
OR <- 2.0

z.alpha <- qnorm(1 - alpha / 2)
z.beta  <- qnorm(power)

p1 <- (OR * p0) /
      (1 - p0 + OR * p0)

n.case <- ceiling(
  (z.alpha + z.beta)^2 *
  (
    1 / (p1 * (1 - p1)) +
    1 / (p0 * (1 - p0))
  ) /
  log(OR)^2
)

n.control <- n.case

n.case
n.control

The calculation gives approximately:

n.case
# approximately 190

n.control
# approximately 190

The total sample size is therefore approximately:

N.total <- n.case + n.control

N.total
# approximately 380

Changing the Control-to-Case Ratio in R

For a simple planning calculation, investigators can evaluate several control ratios. The following code calculates the approximate information available for a fixed number of cases and different numbers of controls:

alpha <- 0.05
power <- 0.80

p0 <- 0.20
OR <- 2.0

z.alpha <- qnorm(1 - alpha / 2)
z.beta  <- qnorm(power)

p1 <- (OR * p0) /
      (1 - p0 + OR * p0)

ratios <- c(1, 2, 3, 4)

results <- data.frame(
  controls_per_case = ratios
)

results

For final protocol planning, a validated package or dedicated statistical software should be used for the selected case-control ratio and exact analysis method.

Simulation-Based Power Calculation

Simulation becomes particularly valuable when the planned analysis is more complex than a simple 2 × 2 comparison.

A simulation framework can:

1. Generate exposure status according to the assumed control prevalence.
2. Generate disease status according to the anticipated odds ratio.
3. Apply matching or sampling restrictions.
4. Add confounders and interactions.
5. Fit the planned logistic regression model.
6. Record whether the primary hypothesis is statistically significant.
7. Repeat the process thousands of times.

Estimated power is then:

Simulation is often the most defensible approach when the final analysis includes complex covariate adjustment or a matched design.

Sample Size vs. Detectable Odds Ratio

The relationship can also be reversed. Instead of asking:

"How many cases and controls are needed to detect OR = 2.0?"

the investigator can ask:

"What is the smallest odds ratio this study can reliably detect?"

This is particularly useful when the number of cases is constrained by the availability of eligible patients.

For example, a registry may provide only 150 cases. The study team can calculate the minimum detectable odds ratio given:

• 150 cases
• A fixed number of controls
• Expected exposure prevalence
• Alpha
• Target power

This provides a more realistic assessment of what the study can accomplish.

Power Curves

A useful sensitivity analysis is to evaluate statistical power across a range of plausible odds ratios.

The resulting power curve can help investigators understand how sensitive the study is to uncertainty in the true effect size.

Common Sample Size Mistakes

1. Using disease prevalence instead of exposure prevalence. For a standard case-control OR calculation, the relevant baseline proportion is usually exposure prevalence among controls.
2. Assuming the odds ratio is the same as the risk ratio. An OR of 2 does not automatically mean that disease risk is doubled.
3. Ignoring the control exposure prevalence. The same OR can correspond to very different exposure differences depending on the baseline exposure frequency.
4. Using an unmatched formula for a matched study. Matched designs require methods that account for the matched sets.
5. Assuming 1:1 is always optimal. Additional controls can improve efficiency when cases are difficult to recruit.
6. Adding unlimited controls. The efficiency gain from additional controls diminishes substantially.
7. Ignoring rare exposures. Sparse exposure categories can produce unstable odds ratio estimates.
8. Planning only for the crude analysis. A multivariable logistic regression model may require substantially more information.
9. Ignoring missing exposure data. Non-evaluable participants reduce the effective sample size.
10. Choosing an implausibly large odds ratio. Designing a study around an exaggerated effect can result in an underpowered study if the true effect is more modest.
11. Assuming a large N eliminates bias. Selection bias, confounding, and exposure misclassification are not fixed simply by recruiting more subjects.

A Practical Case-Control Sample Size Workflow

A useful workflow is:

1. Define the disease outcome and source population.
2. Define the primary exposure.
3. Specify whether the study is matched or unmatched.
4. Estimate the exposure prevalence among controls.
5. Specify the smallest scientifically meaningful odds ratio.
6. Specify the alpha level.
7. Specify the desired statistical power.
8. Specify the control-to-case ratio.
9. Derive the expected exposure prevalence among cases.
10. Calculate the required number of cases and controls.
11. Consider additional information required for covariate adjustment.
12. Inflate for missing or unusable exposure data.
13. Evaluate sensitivity of the result to plausible assumptions.
14. Use simulation when matching or multivariable modeling makes simple formulas inadequate.

What a Protocol Should State

A case-control study protocol should make the sample size assumptions transparent. At minimum, the statistical section should identify:

• Study design: matched or unmatched
• Primary exposure
• Expected exposure prevalence among controls
• Anticipated odds ratio
• Alpha level
• Target statistical power
• Case-control ratio
• Expected exposure prevalence among cases
• Required number of cases
• Required number of controls
• Expected non-evaluable rate
• Total planned enrollment
• Primary statistical model
• Covariates planned for adjustment

This allows reviewers to reproduce the calculation and assess whether the assumptions are clinically and epidemiologically credible.

Worked Example: What Happens if the OR Is Only 1.5?

Suppose the study team initially planned around an odds ratio of 2.0, but the scientifically meaningful effect is actually closer to 1.5. The control exposure prevalence remains 20%.

The expected case exposure prevalence becomes:

The exposure prevalence among cases is now approximately 27.3%, compared with 20% among controls. The absolute difference is much smaller than in the OR = 2.0 scenario.

Because:

rather than:

the study requires substantially more participants.

What If the Exposure Is Very Common?

The same principle applies at the other extreme. Suppose 80% of controls are exposed and the anticipated odds ratio is 2.0. Then:

The expected exposure prevalence among cases is approximately 88.9%.

Although the odds ratio is large, the two exposure proportions are already close to the upper boundary of 1. This affects the amount of information available for estimating the odds ratio.

The relationship between exposure prevalence and information is therefore nonlinear. Sample size planning should not rely solely on the magnitude of the OR.

Case-Control Studies With Multiple Exposure Categories

Exposure may not be binary. For example, investigators might classify exposure as:

The sample size must then provide enough observations in each exposure category to estimate the corresponding odds ratios.

If the high-exposure category is rare, it may become the limiting component of the design.

For categorical predictors with several levels, a simulation-based approach can be particularly useful because the joint distribution of exposure categories and covariates becomes important.

Trend Tests and Continuous Exposures

Some epidemiologic questions concern dose-response relationships rather than a simple binary exposure. For example, the exposure may be measured as cumulative dose or years of exposure.

The logistic model might be:

where \(X\) is continuous. The corresponding odds ratio for a one-unit increase in exposure is:

Sample size planning for continuous exposures requires assumptions about the distribution and variance of the exposure, not merely an exposure prevalence.

In these settings, regression-based or simulation-based calculations are often more appropriate than a simple 2 × 2 formula.

Interaction Effects

Investigators sometimes want to determine whether an exposure effect differs between subgroups. For example:

Here, \(\beta_3\) represents the interaction between exposure \(X\) and subgroup \(Z\).

Interaction tests generally require considerably more information than tests of the main exposure effect.

Study Feasibility and Recruitment

Statistical sample size is only one component of feasibility. A study requiring 1,000 cases may be statistically attractive but operationally impossible if only 300 eligible cases are available during the recruitment period.

Before finalizing the sample size, investigators should consider:

• Number of eligible cases available
• Expected recruitment rate
• Availability of appropriate controls
• Expected exposure prevalence
• Expected missing data
• Recruitment duration
• Cost per participant
• Availability of exposure records

If the maximum feasible sample is known, a minimum detectable odds ratio analysis can be more informative than simply selecting an arbitrary target N.

Sample Size and Statistical Significance

A statistically significant result is not necessarily a clinically important result. With a very large case-control study, a small odds ratio such as 1.10 may be estimated with high precision and produce a very small p-value.

Conversely, an odds ratio of 2.0 may be scientifically important but fail to reach statistical significance in a small study.

Sample size planning should therefore distinguish:

• Statistical significance
• Magnitude of association
• Precision of estimation
• Scientific importance

Interpreting the Final Odds Ratio

Suppose the completed study produces:

with a 95% confidence interval:

The estimated odds of exposure are approximately 2.1 times as high among cases as among controls, under the study's sampling and modeling assumptions.

Because the confidence interval excludes 1, the result provides evidence against the null hypothesis of no association at the 5% significance level.

However, the odds ratio should not automatically be interpreted as a causal effect. Causal interpretation requires appropriate consideration of confounding, selection, measurement, temporality, and the underlying study design.

Common Interpretation Mistakes

1. "OR = 2 means twice the risk." Not necessarily. An odds ratio is not generally equal to a risk ratio.
2. "A significant OR proves causation." Association does not establish causality.
3. "More controls always solve the problem." Additional controls have diminishing statistical returns.
4. "A large total sample guarantees adequate power." Rare exposures or sparse strata can still produce unstable estimates.
5. "The control exposure prevalence is the disease prevalence." These are entirely different quantities.
6. "The same formula works for matched and unmatched designs." Matching changes the information structure and analysis.

Summary

Sample size planning for odds ratios in case-control studies begins with the exposure distribution rather than disease prevalence. The investigator typically specifies the expected exposure prevalence among controls, the odds ratio that the study should be able to detect, the desired power, the significance level, and the case-control ratio.

The expected exposure prevalence among cases can be derived from:

For an unmatched 1:1 case-control study, a large-sample approximation based on the variance of the log odds ratio is:

where \(n\) is the approximate number of cases and, for a 1:1 design, also the number of controls.

The sample size increases when:

• The anticipated odds ratio is closer to 1.
• The exposure is rare.
• Higher statistical power is required.
• A smaller Type I error rate is specified.
• Additional covariate adjustment is required.
• Exposure categories become sparse.
• Matching or complex sampling reduces effective information.

Adding controls can improve efficiency when cases are difficult to recruit, but the benefit diminishes as the number of controls per case increases.

References

Dupont, W.D. (1988). Power calculations for matched case-control studies. Biometrics, 44(4), 1157–1168.
Fleiss, J.L., Levin, B. & Paik, M.C. (2003). Statistical Methods for Rates and Proportions, 3rd ed. Wiley.
Hsieh, F.Y., Bloch, D.A. & Larsen, M.D. (1998). A simple method of sample size calculation for linear and logistic regression. Statistics in Medicine, 17(14), 1623–1634.
Kelsey, J.L., Whittemore, A.S., Evans, A.S. & Thompson, W.D. (1996). Methods in Observational Epidemiology, 2nd ed. Lippincott Williams & Wilkins.
Schlesselman, J.J. (1982). Case-Control Studies: Design, Conduct, Analysis. Oxford University Press.
Woodward, M. (2014). Epidemiology: Study Design and Data Analysis, 3rd ed. CRC Press.