Rabbi Akiva and Regression toward the Mean
In yesterday’s daf, Avodah Zarah 55a, we saw Rabbi Akiva or Rabbi Yochanan express a proto-version of “regression towards the mean”. It may require some careful reading — or you may feel a kvetch, but I believe that it is there.
First, let us read the gemara:
אֲמַר לֵיהּ זוּנִין לְרַבִּי עֲקִיבָא: לִבִּי וְלִבָּךְ יָדַע דַּעֲבוֹדָה זָרָה לֵית בַּהּ מְשָׁשָׁא, וְהָא קָחָזֵינַן גַּבְרֵי דְּאָזְלִי כִּי מְתַבְּרִי וְאָתוּ כִּי מְצַמְּדִי! מַאי טַעְמָא?
Zunin said to Rabbi Akiva: Both my heart and your heart know that there is no substance to idol worship. Nevertheless, don’t we see people who go with broken limbs to worship idols and come back when they are whole? What is the reason for this?
אָמַר לוֹ: אֶמְשׁוֹל לְךָ מָשָׁל, לְמָה הַדָּבָר דּוֹמֶה? לְאָדָם נֶאֱמָן שֶׁהָיָה בָּעִיר, וְכׇל בְּנֵי עִירוֹ הָיוּ מַפְקִידִין אֶצְלוֹ שֶׁלֹּא בְּעֵדִים, וּבָא אָדָם אֶחָד וְהִפְקִיד לוֹ בְּעֵדִים. פַּעַם אֶחָד שָׁכַח וְהִפְקִיד אֶצְלוֹ שֶׁלֹּא בְּעֵדִים, אָמְרָה לוֹ אִשְׁתּוֹ: בּוֹא וְנִכְפְּרֶנּוּ, אָמַר לָהּ: וְכִי מִפְּנֵי שֶׁשּׁוֹטֶה זֶה עָשָׂה שֶׁלֹּא כַּהוֹגֶן אָנוּ נְאַבֵּד אֶת אֱמוּנָתֵינוּ?
Rabbi Akiva said to Zunin: I will relate a parable to you. To what is this matter comparable? It can be compared to a trusted person who was in a certain city, and all the residents of his city would deposit items and money with him, even not in the presence of witnesses. And there was one man who did not trust him, who came and specifically deposited money with him in the presence of witnesses. On one occasion, that person forgot and deposited money with him not in the presence of witnesses. The trusted man’s wife said to him: Come, let us deny that he deposited the money with us, as there are no witnesses. The man said to her: Should we lose our credibility and act deceitfully just because this fool acted improperly and did not require the presence of witnesses?
אַף כָּךְ יִסּוּרִין, בְּשָׁעָה שֶׁמְּשַׁגְּרִין אוֹתָן עַל הָאָדָם, מַשְׁבִּיעִין אוֹתָן: שֶׁלֹּא תֵּלְכוּ אֶלָּא בְּיוֹם פְּלוֹנִי, וְלֹא תֵּצְאוּ אֶלָּא בְּיוֹם פְּלוֹנִי וּבְשָׁעָה פְּלוֹנִית, וְעַל יְדֵי פְּלוֹנִי וְעַל יְדֵי סַם פְּלוֹנִי. כֵּיוָן שֶׁהִגִּיעַ זְמַנָּן לָצֵאת, הָלַךְ זֶה לְבֵית עֲבוֹדָה זָרָה, אָמְרוּ יִסּוּרִין: דִּין הוּא שֶׁלֹּא נֵצֵא, וְחוֹזְרִין וְאוֹמְרִים: וְכִי מִפְּנֵי שֶׁשּׁוֹטֶה זֶה עוֹשֶׂה שֶׁלֹּא כַּהוֹגֶן אָנוּ נְאַבֵּד שְׁבוּעָתֵנוּ?
So too, with regard to different forms of suffering, at the time when they are sent to afflict the person, an oath is administered to them as follows: Take an oath that you shall not go and afflict the person except on such and such a day. And you shall not leave him except on such and such a day, at such and such an hour, by means of so-and-so, a specific doctor, and by means of such and such a medicine. When the time came for the suffering to leave him, this sick man went to a temple of idol worship. The forms of suffering said: By right we should not leave him. But then they say: Should we lose the fulfillment of our oath just because this fool is acting improperly?
וְהַיְינוּ דְּאָמַר רַבִּי יוֹחָנָן: מַאי דִּכְתִיב ״וׇחֳלָיִם רָעִים וְנֶאֱמָנִים״? רָעִים בִּשְׁלִיחוּתָן, וְנֶאֱמָנִים בִּשְׁבוּעָתָן.
And this is consistent with that which Rabbi Yoḥanan says: What is the meaning of that which is written: “Then the Lord will make your plagues wonderful, and the plagues of your seed, even great plagues and faithful, and evil and faithful sicknesses” (Deuteronomy 28:59)? The term “evil” does not mean that the agent of sickness is in itself evil. Rather, this is referring to its mission, which is to cause harm. The term “and faithful sicknesses” indicates that the illnesses are faithful to their oath and adhere to the times designated for the affliction.
To summarize, Zunin wanted to know how idolatry-related healing could be effective. Doesn’t this show that there is substance? Rabbi Akiva (a Tanna) answered that there is a “faithfulness” to illnesses such that they will disappear at a set time. Just because someone went to the idolatry-related healing will not then cause the illness to suddenly NOT disappear at the set time. And Rabbi Yochanan (an early Amora) supported the idea with a verse, in which ra’a / harmful forces of illness are ne’emanim / faithful in leaving.
In other words, these particular illnesses would have left anyway according to the derech hateva, natural order, and that won’t change just because someone took a foolish idolatrous step.
How is this “regression towards the mean”? That is a modern principle of statistics and medicine. I’ll quote Wikipedia, but also see this article about the impact on medical decision making.
Consider a class of students taking a 100-item true/false test on a subject. Suppose that all students choose randomly on all questions. Then, each student's score would be a realization of one of a set of independent and identically distributed random variables, with an expected mean of 50. Naturally, some students will score substantially above 50 and some substantially below 50 just by chance. If one selects only the top scoring 10% of the students and gives them a second test on which they again choose randomly on all items, the mean score would again be expected to be close to 50. Thus the mean of these students would "regress" all the way back to the mean of all students who took the original test. No matter what a student scores on the original test, the best prediction of their score on the second test is 50.
To take an example of illness, people can be sick and be healthy somewhat randomly, and can switch from one to the other without specific medical intervention. Say you wanted to find out if a particular cure was effective, say Tylenol. But, you specifically selected a non-random sample, namely people who were currently sick. Even without medical intervention, if you would measure them a week later, they might be cured. You should not necessarily attribute that to the Tylenol, because regression towards the means is a confounder.
To take an example from the medical article:
Pilot instructors noted that when a trainee pilot was praised for a good landing they invariably made a subsequent poor landing. This was misinterpreted as praise lulling pilots into complacency when the real explanation was regression towards the mean.
Because people interpret this gemara more literally, some commentators grapple with the idea of how prayer to Hashem could be effective. After all, these diseases have a specific destined time when they are to disappear, and they are sworn to disappear (specifically) at that time. No cure should be effective before then!
I think that that is not a correct read of the gemara. The point was to discount the idolatrous healing of definitive proof of its efficacy, either of the cure or the idol. But that the disease will leave of its own accord at a set time does not mean that a cure beforehand could not work. Or, that outside the realm of derech hateva, a prayer to Hashem could not work.