# NNT/NNH Calculator & Log-rank to Hazard Ratio

## Overview

Systematic reviews frequently encounter trials that report incomplete survival data – a log-rank p-value but no Hazard Ratio, or probabilities without a directly computed NNT. ParCC bridges these gaps with two tools in the HR Converter module.

## Tutorial A: Extracting a Hazard Ratio from a Log-rank Test

### The Scenario – Adjuvant Chemotherapy in Colon Cancer

An older trial (published 2005) reports:

- Log-rank chi-squared = 6.8
- Total events (deaths) across both arms = 142
- The treatment arm had better outcomes

The paper does not report a Hazard Ratio, which you need for your meta-analysis.

### The Peto Approximation

When only summary log-rank statistics are available, the Peto method estimates:

\[\ln(HR) = \pm \frac{\sqrt{\chi^2}}{\sqrt{E/4}}\]

with a 95% confidence interval:

\[\ln(HR) \pm \frac{1.96}{\sqrt{E/4}}\]

where \(E\) is the total number of events.

### In ParCC

1. Navigate to Convert > HR -> Probability & NNT > Log-rank -> HR tab.
2. Select input type: Chi-squared statistic.
3. Enter chi-squared = 6.8, Total events = 142.
4. Select direction: Treatment is better (HR < 1).
5. Result: HR = 0.68 (95% CI: 0.51 - 0.91).

### Alternative: From a p-value

If the paper reports only “log-rank p = 0.009”:

1. Select input type: p-value.
2. Enter p = 0.009, Total events = 142.
3. ParCC converts the p-value to a z-statistic via \(z = \Phi^{-1}(1 - p/2)\), then applies the same Peto formula.

## Tutorial B: Computing NNT for a Formulary Decision

### The Scenario – Hospital P&T Committee

A Pharmacy & Therapeutics committee asks: “How many patients must we treat with Drug X to prevent one additional death?” The trial reports:

- 12-month mortality: Control = 18%, Intervention = 12%

### The Formula

\[NNT = \left\lceil \frac{1}{ARR} \right\rceil = \left\lceil \frac{1}{p_{control} - p_{intervention}} \right\rceil\]

### In ParCC

1. Navigate to Convert > HR -> Probability & NNT > NNT/NNH tab.
2. Select input mode: Two Probabilities.
3. Enter Control = 0.18, Intervention = 0.12.
4. Result: ARR = 6.0%, NNT = 17.

Interpretation: For every 17 patients treated with Drug X for 12 months, one additional death is prevented.

### Other Input Modes

ParCC supports four ways to compute NNT:

Input Mode | You provide | ParCC calculates
--- | --- | ---
Direct ARR | Absolute risk reduction | NNT = ceil(1/ARR)
Two Probabilities | Control & intervention probabilities | ARR, then NNT
RR + Baseline | Relative Risk + control probability | ARR = p0 x (1 - RR), then NNT
OR + Baseline | Odds Ratio + control probability | Converts to probabilities via Zhang & Yu, then NNT

### NNT vs NNH

When the intervention increases risk (ARR < 0), the result is reported as NNH (Number Needed to Harm) with an orange warning. This happens when testing safety endpoints rather than efficacy.

## When to Use These Tools

- Log-rank -> HR: Systematic reviews where older trials lack HR estimates; indirect treatment comparisons needing HR inputs from published statistics.
- NNT Calculator: Communicating treatment effects to clinicians and formulary committees; sensitivity analyses varying NNT across plausible baseline risk ranges.

## References

1. 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.
2. Parmar MKB, Torri V, Stewart L. Extracting summary statistics to perform meta-analyses of the published literature for survival endpoints. Statistics in Medicine. 1998;17(24):2815-2834.
3. Altman DG, Andersen PK. Calculating the number needed to treat for trials where the outcome is time to an event. BMJ. 1999;319(7223):1492-1495.