If the screening result is positive, does it mean that I am sick?

Author: Luan Shenghua (Institute of Psychology, Chinese Academy of Sciences)

The article comes from the Science Academy official account (ID: kexuedayuan)

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Nowadays, "disease screening" is being gradually promoted in my country and accepted by more and more people. The purpose of screening is to detect problems as early as possible, treat them as early as possible, and reduce the adverse consequences of the disease. However, do you know how to view the results of the screening?

For example, breast cancer has become the most common cancer in women worldwide and in my country. In 2015 alone, there were nearly 300,000 new female breast cancer patients and nearly 70,000 women who died of breast cancer in China (Chen et al., 2016). Many medical experts believe that if breast cancer can be screened in time, patients can receive earlier treatment, thereby reducing mortality. Currently, a popular breast cancer screening method is mammography.

So, when a person's breast mammography X-ray screening result is positive, does it mean that she has breast cancer?

(Photo source: China Centers for Disease Control and Prevention website)

“True Positive” vs. “False Positive”

Like other screening methods, mammography results are not 100% accurate. Specifically, if a woman has breast cancer, mammography will show that she has breast cancer about 90% of the time; if she does not have breast cancer, mammography will show that she has breast cancer about 9% of the time. The former is the so-called "true positive" rate, and the latter is the "false positive" rate. These two probabilities are the basis for answering a very important question for both patients and doctors, which is: "After knowing the positive test result, what is the probability that the person being tested actually has the disease?"

To answer this question correctly, we also need to know the incidence of the disease. In my country, the incidence of breast cancer in urban women over 45 years old is about 0.1%, or one in a thousand (Zuo et al., 2017). Based on this information, we can apply Bayes' theorem to deduce the answer to the question:

In the case of breast cancer, P(prevalence) is the incidence rate of 0.001, and P(no prevalence) is 0.999; P(positive|prevalence) is the true positive rate of mammography, which is 0.90, and P(positive|no prevalence) is the false positive rate of the test, which is 0.09. Substituting these values ​​into the formula, the answer is:

This 0.01, or 1%, result may have two impacts on people's thinking.

Why only 1%?

The overall accuracy of molybdenum target X-ray imaging is over 90%. Why does it give a positive result, showing that the person being tested has breast cancer, when there is only a 1% chance that this person actually has breast cancer?

The main reason for this result is that the incidence of breast cancer is only 0.1%. Finding the one person with the disease in 1,000 people without any other clues is like finding a needle in a haystack. It is very difficult. Screening tests are a tool that can help us narrow the search. But if this tool is still far from 100% accurate, even if it gives a positive result, the possibility of finding the "needle" will be relatively low.

There are two ways to improve the certainty of diagnosis: first, after getting a positive result, retest; you can then use mammography or another technical means, such as biopsy, which is more accurate but more dangerous and more harmful to the human body; second, only do mammography when there are suspected symptoms of breast cancer. The incidence of breast cancer is much higher in women with suspected symptoms than in ordinary women. Continuing the above metaphor, for these people, diagnosing breast cancer is not like finding a needle in a haystack, but more like finding a needle in a small lake, which is much easier.

This is too hard to calculate!

The second impact is the difficulty of calculation. Although Bayes' theorem is widely used in statistics and some related disciplines, ordinary people, including those with higher education, know very little about it. Moreover, after knowing the numbers of incidence, true positive rate, and false positive rate, they do not know whether they should use Bayes' theorem to integrate these numbers and deduce the true probability of illness when the result is positive.

Furthermore, even if people know that Bayes’ theorem should be used, the calculation process is cumbersome without the help of pen, paper or calculator, and it is easy to make mistakes, leading to wrong conclusions. Fortunately, psychologists have recognized this problem in the 1990s and have given a simple and feasible solution (Gigerenzer & Hoffrage, 1995; McDowell & Jacobs, 2017).

Bayesian

The calculation can be so simple

For the breast cancer screening problem, we previously used the "probability" statement method (ie, 90%, 9%, 0.1%). This method is more common in real life and is also the input required for Bayesian calculations, but it brings a lot of difficulties to our cognition and is not easy to apply. For the same problem, we can also express it in the "frequency" way. For example, we can describe the breast cancer screening problem like this:

10 out of every 10,000 urban women aged 45 and over suffer from breast cancer

9 out of 10 women with breast cancer will screen positive

Of the 9,990 women without breast cancer, 899 would have a positive screening result.

So, when an urban woman over 45 years old screens positive, what is the probability that she actually has breast cancer?

The numbers in these statements and the relationship between them can be represented by the following graph. In this graph, we see that there are a total of (9+899) = 908 women who tested positive, but among these women, only 9 actually had breast cancer. Therefore, the answer to the question is 9/908 = 0.01. Isn't that much simpler?

How to calculate the probability of a person having a positive screening result but actually having breast cancer using a frequency-based method (Image source: drawn by the author)

If you want to deepen your understanding and memory, here is a similar question, stated in probability terms. You can try to convert it into frequency terms to get the answer (the answer is at the end of this article).

Computational Challenges

Research shows that using frequency can significantly improve people's success rate in solving similar problems (at least 20%). If there are more vivid visual aids or people receive less than 2 hours of training, the success rate will be higher and the effect will be more lasting (McDowell & Jacobs, 2017; Sedlmeier & Gigerenzer, 2001). A study of elementary school students (Zhu & Gigerenzer, 2006) showed that the frequency method allows sixth-grade children to answer 60% of the questions correctly on average, while this proportion is 0 under the probability method!

Frequency is effective because it makes the problem easier to understand and calculate, and the deeper reason is that it is the most commonly used numerical representation of risk and uncertainty in the long history of human evolution. Whether it is natural or social phenomena, our ancestors observed and recorded them using frequency and frequency (for example, in the past 100 sunrises and sunsets, wolves appeared on the east mountain 5 times, 3 of which were the same wolf pack; in 10 transactions with the tribe across the river, we lost 3 times but gained 4 times, etc.).

Long-term accumulation makes us more skilled and handy in processing such information. "Probability" is a concept that only appeared after the Enlightenment in the 18th century. Its application has greatly promoted the progress of all aspects of human society, but it is not a natural way for ordinary people to understand numbers. It needs formal education to be gradually accepted and mastered.

Conclusion

A healthy life is inseparable from medical treatment. But like many fields, medical treatment is full of risks and uncertainties. This article discusses one of the uncertainties that is closely related to everyone, that is: when we get a positive test result, what is the probability that we are actually sick. Because almost no test is 100% accurate, this probability is not 100% in most cases.

Once we have or estimate the relevant information (including the prevalence of the disease and the true-positive and false-positive rates of the test), we recommend using frequencies to extrapolate the answers. This applies to the general public, but especially to the physicians who interpret the test results.

(Photo source: veer photo gallery)

Finally, the answer to the previous question about a virus is: 16.7%. Did you get it right?

References:

[1]Chen, W., Zheng, R., Baade, PD, Zhang, S., Zeng, H., Bray, F., ... & He, J. (2016). Cancer statistics in China, 2015. CA: A cancer journal for clinicians, 66, 115-132.

[2]Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102, 684-704.

[3]McDowell, M., & Jacobs, P. (2017). Meta-analysis of the effect of natural frequencies on Bayesian reasoning. Psychological bulletin, 143, 1273-1312.

[4]Sedlmeier, P., & Gigerenzer, G. (2001). Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology: General, 130, 380-400.

[5]Zhu, LQ, & Gigerenzer, G. (2006). Children can solve Bayesian problems: The role of representation in mental computation. Cognition, 98, 287-308.

[6]Zuo, TT, Zheng, RS, Zeng, HM, Zhang, SW, & Chen, WQ (2017). Female breast cancer incidence and mortality in China, 2013. Thoracic Cancer, 8, 214-218.