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# Biostatistics - Log Rank versus Cox Regression

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## If you don't understand, you shouldn't invest

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I've written in some detail how biotech can be a very difficult sector to profit in. The key is to pick that one company of every ten that succeeds. Picking that one company requires an understanding of rather arcane science. Increasingly often, it requires an understanding of the black art of biostatistics. Helping people understand the science requires a lifetime of work. Understanding biostats may be a less time consuming process so this is the first in what will be an irregular series of articles on the subject.

I will note up front that those of you who are familiar with statistics (particularly engineers, in my experience) will struggle with this as much as those who have little background in this art. Biostats are different from "normal" statistics and rationalizing those differences will be a particular challenge to those with some prior knowledge.

In an oncology trial, the endpoint that usually requires the least interpretation is survival. Is the patient dead or alive? Pretty easy to determine with some degree of accuracy.

Survival is depicted via a Kaplan-Meier (KM) curve that (roughly) plots the percentage of people still alive in a trial on the vertical y-axis (as a percentage) versus time from randomization on the horizontal x-axis. How to calculate a KM curve is a discussion for another time - next time, probably.

Statistical significance in a survival trial is calculated using something called a log-rank test, which takes into consideration the area between the curves on a KM plot. The wider the area, the more likely the trial reaches statistical significance. Statistical significance is expressed as a "p-value." As I've noted before, scientists and regulators have arbitrarily decided a p-value of p=0.05 or lower is statistically significant. A p-value of p=0.05 means there is a 95% chance the results observed in a trial are not due to random chance. A p-value of p=0.001 means there is a 99.9% chance the results are not due to random chance.

The goal of any randomized clinical trial is to ensure that patients enrolling in the arms of the trial are as identical as possible before they receive the drug under investigation. In this way, observers can truly test whether any results seen are due to the drug. If there are imbalances - like patients in the control arm being healthier than patients in the arm that receives the drug - then this adversely affects the ability of the log-rank analysis to accurately portray the results of the trial.

Enter the Cox regression analysis.

The Cox regression analysis is almost always specified as a secondary analysis in a randomized trial. This calculation goes beyond the simple log-rank analysis by correcting for any imbalances between the arms in a trial. Cox regression analyses are always performed by FDA reviewers in their effort to determine whether the results claimed by the company making the application are due to the drug or due to an imbalance between the patients.

The biostatistician collects all the pre-treatment demographic information available (age, gender, disease state, initial health, number of tumors, prior therapies, etc.) and runs each metric through a calculation to compare whether it benefited the study arm or the control arm. Those demographics that show statistical significance or a strong trend are then chosen to plug into a Cox regression analysis.

The long rank analysis answers the question of whether the two arms of a trial were different enough to be statistically significant. It place no conditions or assumptions on this analysis and ignores the fact that even the most well run clinical trials have imbalances between arms.

The Cox analysis answers a slightly different question: If we mathematically control for all imbalances between the arms, were the two arms of the trial different enough to be statistically significant. It is a subtle, but important, difference.

As an investor in a company, you always want to see a Cox analysis produce a p-value equal to or lower than that of the log rank analysis of the survival data. This means that healthier patients were enrolled in the control arm. Logically, it means the drug under investigation had to work extra hard to produce a survival benefit because it was given to sicker patients.

When the Cox analysis produces a higher p-value - particularly a p-value greater than p=0.05 - then you have problems. This begins to indicate the reason the drug was able to show a survival benefit was it was given to healthier people. This raises doubts about the drug's efficacy and doubts are rarely positive for share prices.

So why use log rank at all? Custom. Log rank is also considered by some to be the more pure measure of statistical significance since the Cox analysis depends on a qualitative judgment whether a particular demographic difference is likely to increase or decrease survival.

How the FDA views the two is largely a product of how the statistical analysis plan filed before the trial began describes their use. If the log-rank analysis is specified as the primary analysis, the FDA finds that more persuasive. Oddly, the FDA has little problem rejecting drugs with negative Cox analyses despite a positive log-rank analysis, but more issues with accepting a positive Cox analysis in the face of a negative log-rank analysis. Go figure.
No positions in stocks mentioned.

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