Hazard Ratio Meta-Analysis Guide: HR Extraction, Calculation & Forest Plot

What Is Hazard Ratio Meta-Analysis?

In medical research, many clinical outcomes are time-to-event data: the time from treatment initiation to disease progression, death, or another endpoint. These data are analyzed using survival analysis methods.

When multiple studies report the effect of the same intervention on a survival outcome, meta-analysis pools them to obtain a more precise overall estimate. The core effect measure for survival data meta-analysis is the Hazard Ratio (HR).

What Is a Hazard Ratio?

HR is the ratio of instantaneous event rates (hazards) between two groups. It comes from the Cox proportional hazards regression model and accounts for follow-up time and censoring. HR = 0.70 means that at any given time point, the intervention group's event rate is 70% of the control group's — a 30% risk reduction.

When Do You Need HR Meta-Analysis?

• Oncology — Overall survival (OS), progression-free survival (PFS), disease-free survival (DFS). Nearly all oncology RCTs report HR.
• Cardiovascular disease — Time to major adverse cardiovascular events (MACE), composite endpoints of heart failure hospitalization or death.
• Infectious disease — Time to virological cure, time to infection recurrence.
• Organ transplantation — Graft survival time, time to rejection.
• Chronic disease — Diabetic nephropathy progression time, COPD exacerbation interval.

HR vs. RR: Don't Confuse Them

How to Extract HR Data from Papers

Extracting HR data is the most critical and error-prone step in survival meta-analysis. Depending on what the paper reports, there are three extraction scenarios.

Scenario 1: Paper Reports HR + 95% CI (Most Common, Easiest)

This is the ideal case. Most high-quality RCTs and cohort studies directly report the HR and 95% CI from Cox regression in the text or tables.

Example: Paper reports "HR = 0.72, 95% CI: 0.58–0.90." Record these three numbers:

• HR = 0.72
• CI Lower = 0.58
• CI Upper = 0.90

MetaReview handles all subsequent log transformation and SE calculations automatically.

Scenario 2: Paper Reports HR + p-value (No CI)

Some older papers or conference abstracts report only the HR and p-value without a 95% CI. You can back-calculate the SE from the p-value:

1. Convert the p-value to a z-score (e.g., p = 0.05 → z = 1.96, p = 0.01 → z = 2.576)
2. Calculate SE: SE[log(HR)] = |log(HR)| / z
3. Reconstruct 95% CI: CI = exp(log(HR) ± 1.96 × SE)

Example: Paper reports HR = 0.65, p = 0.003.

1. p = 0.003 (two-sided) → z = 2.968
2. log(0.65) = −0.431
3. SE = |−0.431| / 2.968 = 0.145
4. 95% CI = exp(−0.431 ± 1.96 × 0.145) = (0.489, 0.863)

Scenario 3: Paper Only Shows Kaplan-Meier Curves (No HR Reported)

This is the most challenging scenario. Some older publications or studies with negative results only show KM curves without reporting HR.

Use the Tierney et al. (2007) method to estimate HR from KM curves:

1. Digitize the KM curve using a tool like WebPlotDigitizer to extract survival probability coordinates at each time point
2. Combine with the number-at-risk table (usually displayed below the KM plot)
3. Reconstruct event and censoring counts for each time interval
4. Estimate the HR from the reconstructed approximate individual patient data (IPD)

Reference: Tierney JF, Stewart LA, Ghersi D, Burdett S, Sydes MR. Practical methods for incorporating summary time-to-event data into meta-analysis. Trials. 2007;8:16.

Data Extraction Template

Statistical Principles of HR Meta-Analysis

Understanding the statistical foundation helps you correctly perform the analysis and identify errors. The key point: HRs are pooled on the log scale, not the original scale.

Why Log Transformation?

HR values range from (0, +∞) and have a right-skewed distribution. For example, HR = 0.5 (risk halved) and HR = 2.0 (risk doubled) represent effects of equal magnitude but opposite direction, yet they are unequally distant from the null value of 1 (distances: 0.5 vs. 1.0).

After log transformation:

• log(0.5) = −0.693, log(2.0) = 0.693 — equidistant from zero
• log(1.0) = 0 — null value becomes 0, convenient for statistical testing
• log(HR) is approximately normally distributed — satisfying the assumptions required for meta-analysis

Calculating the Standard Error

When you know the HR and 95% CI:

Where 3.92 = 2 × 1.96, because 95% CI = log(HR) ± 1.96 × SE, so CI width = 2 × 1.96 × SE.

Example: HR = 0.72, 95% CI (0.58, 0.90):

1. log(HR) = log(0.72) = −0.329
2. log(0.90) = −0.105
3. log(0.58) = −0.545
4. SE = (−0.105 − (−0.545)) / 3.92 = 0.440 / 3.92 = 0.112

Inverse-Variance Weighted Pooling

Meta-analysis uses the inverse-variance method to pool log(HR) values across studies:

Studies with smaller SE (typically larger sample size, more events) receive greater weight. After pooling, the combined log(HR) and its CI are back-transformed via exp():

Fixed-Effect vs. Random-Effects Model

• Fixed-effect model: Assumes all studies estimate the same true HR. Appropriate when studies are highly homogeneous.
• Random-effects model: Assumes the true HR varies across studies and pools the average HR. Appropriate when heterogeneity exists — the more common choice.

Decision guide: Use random-effects when I² > 50% or Q-test p < 0.10.

Forest Plot Interpretation for HR

The forest plot is the most important visual output of any meta-analysis. HR forest plots share the same structure as OR/RR plots but have specific interpretation details.

Basic Elements

Direction Interpretation

HR forest plots typically use a log-scale x-axis. Key judgments:

• HR < 1 (left of null line): Lower event rate in intervention group = favours treatment
• HR > 1 (right of null line): Higher event rate in intervention group = favours control
• CI crosses null line 1: The study result is not statistically significant (p > 0.05)
• Diamond entirely on one side: The pooled result is statistically significant

Heterogeneity Indicators

• I²: Percentage of total variation due to heterogeneity. 0% = no heterogeneity, >50% = moderate-to-high, >75% = high.
• Q-test p-value: p < 0.10 suggests significant heterogeneity (threshold is 0.10, not 0.05, because Q-test has low statistical power).
• τ²: Estimated between-study variance (random-effects model).

Step-by-Step: HR Meta-Analysis in MetaReview

Step 1: Select Effect Measure

Open MetaReview and select "Hazard Ratio" from the effect measure dropdown at the top. The data entry area automatically switches to HR-specific input fields.

Step 2: Enter Study Information

For each row, fill in:

• Study: Study name, typically "First Author Year" (e.g., "Wang 2023")
• Year: Publication year, used for sorting and subgroup analysis

Step 3: Enter HR and Confidence Interval

• HR: Hazard Ratio point estimate (e.g., 0.72)
• CI Lower: 95% CI lower bound (e.g., 0.58)
• CI Upper: 95% CI upper bound (e.g., 0.90)

MetaReview automatically performs log(HR) transformation and SE calculation in the background.

Step 4: Run the Analysis

After entering all study data, click "Run Meta-Analysis". Results are computed within seconds.

Step 5: Review Results

1. Forest Plot — Displays each study's HR, 95% CI, weight, and pooled result. X-axis uses log scale with the null line at HR = 1.
2. Funnel Plot — Assesses publication bias. A symmetric funnel shape suggests no obvious bias; asymmetry suggests possible small-study effects.
3. Sensitivity Analysis (Leave-one-out) — Recalculates the pooled HR after removing each study one at a time. If removing a study substantially changes the result, that study has high influence.
4. Heterogeneity Statistics — I², Q-test p-value, and τ².

Keyboard Shortcuts for Faster Data Entry

• Tab: Jump between input fields
• Enter: Add a new study row or run the analysis
• Ctrl+Z: Undo the last change

Common Pitfalls and How to Avoid Them

Pitfall 1: Mixing HR with OR/RR

This is the most serious methodological error. HR comes from survival analysis (Cox regression) and accounts for time and censoring. OR comes from logistic regression, RR from frequency tables — they only consider cumulative events at a fixed time point. Their mathematical foundations are entirely different.

Solution: Specify in your inclusion criteria that "studies must report HR or provide data from which HR can be calculated." Analyze studies reporting only OR separately.

Pitfall 2: Mixing Adjusted and Unadjusted HRs

The same paper may report both unadjusted (univariate) and adjusted (multivariable) HRs. They can differ substantially.

Solution: Pre-specify in your protocol which type of HR to extract. Prefer adjusted HRs because they control for confounders. Maintain consistency across all included studies. If mixing is unavoidable, conduct a sensitivity analysis.

Pitfall 3: Ignoring the Proportional Hazards Assumption

The Cox model assumes the HR remains constant throughout follow-up (proportional hazards assumption). If KM curves cross, this assumption is violated and a single HR cannot accurately describe the treatment effect.

How to check:

• Look for crossing KM curves in the original papers
• Check if the paper reports Schoenfeld residual tests
• If curves separate early but cross later, the treatment effect changes over time

Solution: Discuss the PH limitation in your Discussion section. If multiple studies show crossing curves, consider restricted mean survival time (RMST) as an alternative effect measure.

Pitfall 4: Incorrect Subgroup HR Extraction

Papers often report subgroup HRs (by tumor type, age group, PD-L1 expression, etc.) in forest plot format.

• If your meta-analysis targets the overall effect, extract the overall HR, not a subgroup HR
• If your meta-analysis specifically targets a subgroup (e.g., "immunotherapy effect in PD-L1 high expressors"), extract the corresponding subgroup HR
• Never treat multiple subgroup HRs from the same study as independent studies — this causes double-counting and statistical bias

Pitfall 5: Ignoring Follow-Up Duration Differences

Different studies may have vastly different median follow-up times (e.g., 12 months vs. 60 months). Short follow-up may underestimate long-term effects, while long follow-up may be affected by treatment crossover.

Solution: Record median follow-up time during data extraction and explore it as a potential source of heterogeneity via meta-regression.

Pitfall 6: Publication Bias

Studies with positive results (HR significantly < 1) are more likely to be published, while negative results may remain in file drawers.

Solution:

• Search trial registries (ClinicalTrials.gov) for completed but unpublished studies
• Examine funnel plot symmetry
• Use Egger's test or Begg's test to quantitatively assess publication bias
• MetaReview automatically generates funnel plots and supports both Egger's and Begg's tests