发自费城——与一个庞大的听众群体交流医学风险是一件相当艰难的事情，尤其是在官方建议与人们的情绪化描述相抵触的时候。而这也是美国预防医学工作组（USPSTF）在2009年所遇到的问题，当时该机构发布了其乳腺癌医学筛查指引，建议将原有的40岁以上无症状妇女每两年做一次筛查改为所有50岁以上的女性每年拍一张乳房X光片，结果却引发了公众的极大愤怒。

　　其实我们可以在数学和心理学之间的模糊区域中找到解读这一公众回应的关键点。人们对于上述发现的不满在很大程度上源自于一种错误的直觉：如果更早且更频密的筛查可以增加发现某种可致命癌症的几率，那么更多的筛查就是应该的。这么说来倘若更多的筛查可以在40岁年龄段的临床无症状妇女中发现乳腺癌，那何不把它提前到30岁？如果这也可行的话，按照逻辑上的归缪法，干脆从15岁开始每月照一张X光片好了。

　　很显然，如此频密的筛查只会弊大于利。而在此之间寻找一个合适的平衡点也是相当有难度的。其中最不幸的，莫过于我们知道连续多年进行X光照射会产生辐射累积效应，活体组织检查会对人体造成伤害，同时治疗那些缓慢生长且永远不会证明是恶性的肿瘤也将损害患者的健康，但这些因素相对于乳腺癌的危险性该如何权衡则并非易事。

　　美国预防医学工作组最近还发布了一个对前列腺癌特定抗原检测的更明确警告，认为该检测对病人的伤害大于其益处。此外肺癌胸部X光检测与子宫颈癌帕氏抹片检查都受到了类似（但不那么明确）的批评。

　　由于达特茅斯卫生政策研究中心的研究者们宣称乳腺癌筛查的益处往往被过度夸大，而其费用则被刻意低估，因此重新评估癌症筛查的下一步工作也在去年同步展开。事实上，即便是一张可以检测出癌症的乳房X光片（全美每年为此支出达4000万美元）也不一定能拯救一条生命。

　　达特茅斯的研究者们发现，在全美每年检测出的13.8万乳腺癌案例中，乳房X光测试对其中12~13.4万遭受病痛折磨的妇女们并无帮助。那些癌瘤要么太小以至于未能威胁，要么在过一段时间后的临床检测中发现后（或者疯狂扩散而使人无计可施）反而更能得到有效治疗。

　　与之相关则是测量标准的问题。由于病人的存活期是从诊断结论下达后开始计算的，那么更灵敏的筛查肯定能更早得出结论。于是在早期诊断对病人的生存并无实质影响的情况下，存活期似乎却变得更长了。

　　毫无疑问，适用于不同病例的检测和治疗方案各有不同，但对于频繁的筛查来说，另一个要注意的地方就是假阳性的问题。当您在寻找一项相对罕见的东西之时（不管是癌症还是恐怖分子），最好记住某些阳性结论往往会是错误的。要么那个“检测出来的”病理状态根本不存在，要么就并不是致命的。

　　请参见下面这个假想的例子。假设某特定癌症筛查方式的准确率为95%，那么说如果有个人患上了这种癌症，那么该测试呈现阳性的概率为95%。再推一步，假设有人并未患有这种癌症而呈现阳性的概率为1%。最后，我们进一步假设有0.5%的人——也就是1/200——真的会患上这种癌症。那如果你的医生告诉你说测试结果呈阳性，是否就意味着你多半会患有此病？令人吃惊的是，答案是否定的。

　　让我们做一个简单的演算。假设医院共做了10万次检测。其中平均有500人会得癌症。由于患病者检测出阳性的概率是95%，那么平均会475个阳性结果。而对于剩下那9.95万健康人来说，由于出现阳性结果的概率是1%，那么就会有995个假阳性出现，因此这10万人中会有475+995=1470个阳性结果。换句话说，即便你的癌症检测结果呈阳性，你真正患有该病的可能性也只有32%。

　　这个答案显然是违反直觉的，因此也很容易引发抵触心理。其实绝大多数人都是以“50-50”或者“百万分之一”来理解可能性这个词的。但不管可能性几何，事实依然是在筛查某些罕见病症时总会出现高比例的假阳性结果。此外，那些受到错误诊断的患者通常会接受进一步治疗，也往往会带来有害的后果。

　　由于人类倾向于通过联想起相关事例的难易程度来评估一种情况出现的可能性，而这种认知偏好又是普遍存在的——也就是所谓的“可用性启发（availability heuristic）”，通常会令事情变得更加模糊不清。相对于一个陌生人因为筛查测试而受苦的统计概率，人们更容易联想起一个挣扎在癌症死亡线上的熟人。

　　但对此最重要的底线在于目前对癌症筛查的重新评估是有证据支撑的。当这涉及到政策制定的时候，决策必须基于事实和论证，而不是道听途说——不管这些描述究竟有多么活灵活现。翻译：邹痴成

　　约翰•艾伦•鲍罗斯，美国天普大学数学教授，著有《数学盲》和《数学家读报纸》等

[英文版]

Cancer by the Numbers

By John Allen Paulos

PHILADELPHIA – It is difficult to communicate medical risk to a large audience, especially when official recommendations conflict with emotional narratives. That is why, when the United States Preventive Services Task Force (USPSTF) in 2009 presented its guidelines for breast cancer screening, which recommended against routine screenings for asymptomatic women in their 40’s and biennial, rather than annual, mammograms for women over 50, the public responded with confused fury.

The key to understanding this response is to be found in the nebulous zone between mathematics and psychology. People’s discomfort with the findings stemmed largely from faulty intuition: if earlier and more frequent screening increases the likelihood of detecting a possibly fatal cancer, then more screening is always desirable. If more screening can detect breast cancer in asymptomatic women in their 40’s, wouldn’t it also detect cancer in women in their 30’s? And, if so, why not, reductio ad absurdum, begin monthly mammograms at age 15?

The answer, of course, is that such intensive screening would cause more harm than good. But striking the proper balance is challenging. Unfortunately, it is not easy to weigh breast cancer’s dangers against the cumulative effects of radiation from dozens of mammograms over the years, the invasiveness of biopsies, and the debilitating impact of treating slow-growing tumors that would never have proven fatal.

The USPSTF recently issued an even sharper warning about the prostate-specific antigen test for prostate cancer, after concluding that the test’s harms outweigh its benefits. Chest X-rays for lung cancer and Pap tests for cervical cancer have received similar, albeit less definitive, criticism.

The next step in the reevaluation of cancer screening was taken last year, when researchers at the Dartmouth Institute for Health Policy announced that the costs of screening for breast cancer were often minimized, and that the benefits were much exaggerated. Indeed, even a mammogram (almost 40 million are given annually in the US) that detects a cancer does not necessarily save a life.

The Dartmouth researchers found that, of the estimated 138,000 breast cancers detected annually in the US, the test did not help 120,000-134,000 of the afflicted women. The cancers either were growing so slowly that they did not pose a problem, or they would have been treated successfully if discovered clinically later (or they were so aggressive that little could be done).

A related concern is measurement. Since the patient’s duration of survival is calculated from the time of diagnosis, more sensitive screening starts the clock sooner. Survival times can thus appear longer, even if the earlier diagnosis had no real effect on survival.

Naturally, individual cases dictate which tests and treatments are best, but an additional concern about frequent screenings is the problem of false positives. When one is looking for something relatively rare (whether cancer or terrorists), it is wise to remember that a positive result is often false. Either the “detected” pathology is not there, or it is not the sort that will kill you.

Consider the following hypothetical example. Assume that the screening test for a certain cancer is 95% accurate, meaning that if someone has the cancer, the test will be positive 95% of the time. Next, assume that if someone does not have the cancer, the test will be positive only 1% of the time. Finally, assume further that 0.5% of people – one of every 200 – actually have this type of cancer. If your doctor tells you that you have tested positive, does this mean that you are likely to have the cancer? Surprisingly, the answer is no.

A little arithmetic shows why. Suppose that 100,000 screenings are conducted. On average, 500 people will have the cancer. Since 95% of them will test positive, there will be, on average, 475 positive tests. Of the 99,500 people without cancer, 1% will test positive, yielding 995 false positives out of 1,470 positive tests. In other words, even if you tested positive for the cancer, the probability that you actually have it is only about 32%.

That answer is decidedly counterintuitive and hence easy to reject. Most people do not think in terms of probabilities other than “50-50” and “one in a million.” But, whatever the probabilities, the fact remains that there will generally be a high percentage of false positives when screening for rare conditions. Moreover, the patients who receive these faulty diagnoses will usually receive further treatments, which often will have harmful consequences.

The “availability heuristic” – a pervasive cognitive bias caused by people’s tendency to estimate the likelihood of a phenomenon by how easily an example of it comes to mind – routinely clouds the issue. People relate much more readily to a friend dying of cancer than they do to statistics about strangers suffering from the consequences of testing.

But the bottom line is that the ongoing reevaluation of cancer screening is evidence-based. When it comes to policymaking, decisions must be based on facts and argument, not anecdotes and stories, however compelling those narratives may be.

John Allen Paulos is Professor of Mathematics at Temple University and the author of Innumeracy and A Mathematician Reads the Newspaper.