Mathematics: The Essentials for All Regulatory Professionals

I consider myself semi-innumerate (lacking many math skills), which may be one of the main reasons I am so enamored with mathematics, including statistics and calculus. The latter was defined by humorist Dave Barry as "the branch of mathematics that is so scary it causes everybody to stop studying mathematics." My home library is filled with books about the history of mathematics, the birth of numbers, innumeracy, the joy of numbers, statistics, biostatistics, the science of measurement, calculus and pre-calculus. I also have DVDs from Great Courses on the History of Mathematics. Despite this vast collection, which has increased my appreciation of mathematics, I remain confounded by the subject. I view it as a lingering challenge despite my advanced age.

I know rigorous mathematical methods have secured science's fidelity to fact and have conferred a timeless reliability to its findings. Understandably, therefore, mathematics underpins everything in medical science, and is a field that all regulatory professionals should embrace.¹ Regression analysis, for example, is the engine that drives the giant randomized controlled studies that increasingly inform every clinical decision.

Pythagoras said, "Mathematics is the way to understand the universe."² Since the days of Galileo and Newton, math has nurtured science. Risk analysis, failure modes effects analysis, design of experiments, pharmacokinetics and other forms of statistical methodology all require math in one form or another. In quality, Six Sigma methods, statistical process control, response surface methodology, sterilization and process validation, geometric dimensioning and tolerancing and sampling plans all are based on mathematical principles. In clinical research, survival analysis and other statistical processes are required to assess safety and effectiveness. For these reasons, I believe all regulatory personnel should know the difference between "frequentist" technique (the null hypothesis approach) and Bayesian statistics. It also is important to become familiar with significance levels, power, Type I and Type II errors and confidence intervals-terms common to clinical and test protocols.

Unfortunately, the US lags far behind other nations in mathematics, and math teachers are in short supply. Studies have shown that US students score significantly lower than students worldwide in math achievement, ranking 25th among 34 countries.³

This all means that regulatory personnel should endeavor to pick up math skills, if they are lacking, through in-house or outside training. In the following paragraphs, I include examples or explanations that helped me and may provide the incentive to learn more about the essentials of mathematics.

Probability

Probability involves weighing the chances or likelihood of something or another taking place.⁴ It often can be full of surprises. The birthday problem is a famously counterintuitive result. If 50 random people are in a room, there is a 97% chance that at least two of them will have the same birthday. In this case, it is easier to compute the probability that all these people have different birthdays. The calculation is as follows:

The product of all the fractions is about 0.03. Thus, the probability that no two people have the same birthday is about 3%. Hence, the probability that at least two people dohave the same birthday is about 97%.⁵ With a similar calculation using 23 people, the chances are even that two of them will have the same birthday.⁶

Of critical importance in probability theory is the notion of independence. Two events are said to be independent when the occurrence of one of them does not make the occurrence of the other more or less probable. If one flips a coin twice, each flip is independent of the other. If one rolls a pair of dice, the top face of one die is independent of the top face of the other. Calculating the probability of two events occurring is easy to do-simply multiply their respective probabilities.

As an example, the probability of obtaining two heads in a row in a coin flip is one-quarter (1/2 x 1/2). The probability of rolling a 2 (1,1) with a pair of dice is 1/36 (1/6 x 1/6), while the probability of rolling a 7 is 6/36 since there are six mutually exclusive ways (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1) the numbers on the faces can add up to 7, and each of these ways has the probability of 1/36, or 1/6 x 1/6.⁷

Risks

Risk management is an approach or process designed to minimize risks. The US Food and Drug Administration (FDA), for example, uses risk-based approaches to prioritize and focus on various activities concerning the oversight of current Good Manufacturing Practice requirements for human drug, biological and veterinary product quality. It is evident that risk management has become the underlying theme or bedrock of FDA regulation in the 21st century.⁸ Unfortunately, many people do not know how to think with numbers and lack the ability to reason about uncertainties and risk.There are a number of explanations for this inability. For one, a person may not know how large a relative risk is.

Risks can presented in a number of ways, and one of the best examples is found in Gerd Gigerenzer's book, Calculated Risks.⁹ British Health authorities were reviewing the benefits of coronary artery bypass surgery versus medical therapy. The actual results of a clinical study were as follows: coronary artery bypass surgery for 1,325 patients resulted in 350 (26.4%) deaths as opposed to non-surgical medical therapy for 1,324 patients, with 404 (30.5%) deaths. The absolute risk reduction from bypass surgery was 4.1% (404-350 = 54; 54/1,325 = 4.1%). In contrast, the relative risk reduction of bypass surgery is 13.4% (4.1/30.5 = 13.4%). Absolute risk reduction is the proportion of patients who die without treatment (placebo) minus those who die with treatment. Relative risk reduction is the absolute risk reduction divided by the proportion of patients who die without treatment.

The survival rates for patients were 73.6% and 69.5%, respectively, for surgery and medical therapy (100 - 26.4 % = 73.6%) (100 - 30.5 = 69.5%). The number of patients who must be treated to save one life is 25 (one of 25 corresponds to 4.1%). In other words, for each 25 patients who receive a bypass, one will have death prevented (within 10 years); the other 24 will have no benefit in mortality reduction from the operation. Finally, the number of event-free patients surviving is virtually the same (73.6% versus 69.5%). All of these are descriptions of the same outcome of an actual randomized clinical trial in which two treatments are compared, despite the fact that 13.4% (relative risk reduction) is more than 4.1% (absolute risk reduction). My point is that different types of calculations of the same data yield different conclusions.

Sensitivity v. Specificity

Two terms that are often confused, especially when the individual is innumerate, are sensitivity and specificity. Sensitivity measures the proportion or percentage of people with a disease, as defined by the gold standard, who are correctly identified in the test. In other words, it measures how sensitive the test is in detecting the disease. Specificity, on the other hand, measures the proportion of people without the disease who are correctly labeled free of the disease by the test. An excellent discussion of the calculation of the measures in found in Richard Reigelman's book, Studying a Study and Testing a Test.¹⁰

Assessing the Effects of an Intervention