# Example 863 — The selected set made two causes look opposed `the_selected_set_made_two_causes_look_opposed.eml` - Among admitted students, academic and athletic scores are negatively related: the weaker a student is academically, the stronger athletically. The correlation is real and correctly computed. What the sample was conditioned on is computed below. ## EML ```eml # Self-authored for the EML case corpus (no external origin). Among admitted # students, academic and athletic scores are negatively related: the weaker a # student is academically, the stronger athletically. The correlation is real and # correctly computed. What the sample was conditioned on is computed below. # # The measurement is careful. It uses the real recorded scores, not estimates; it # covers every admitted student; the conditional rates are honest counts; and the # intent is exactly 'are the two abilities related'. # # Admission required being high in at least one of the two, so a low-academic # admit must be high-athletic - the sample is conditioned on a common effect of # both, and that manufactures the negative relation. 10000 => applicants 2500 => rejected_low_in_both 5000 => admitted_high_academic 2500 => of_those_high_athletic 2500 => admitted_low_academic 2500 => of_those_low_academic_high_athletic applicants - rejected_low_in_both => admitted int(of_those_high_athletic * 10000 / admitted_high_academic) => admitted_high_ath_given_high_ac int(of_those_low_academic_high_athletic * 10000 / admitted_low_academic) => admitted_high_ath_given_low_ac admitted_high_ath_given_low_ac - admitted_high_ath_given_high_ac => manufactured_gap_per_myriad "applicants : " + str(applicants) ^0 " rejected (low in both) : " + str(rejected_low_in_both) ^0 " admitted : " + str(admitted) ^0 "" ^0 "among admitted, high-athletic given high-academic : " + str(admitted_high_ath_given_high_ac) + " per myriad" ^0 "among admitted, high-athletic given low-academic : " + str(admitted_high_ath_given_low_ac) + " per myriad" ^0 "manufactured gap : " + str(manufactured_gap_per_myriad) + " per ten thousand" ^0 "" ^0 # ---- what the measurement verified ---- "the correlation measurement" ^0 " uses : the real recorded scores, not estimates" ^0 " covers : every admitted student" ^0 " rates : honest counts of high-athletic within each group" ^0 " intent : are academic and athletic ability related" ^0 " students omitted : 0" ^0 " verdict : AMONG ADMITTED, LOWER ACADEMIC MEANS HIGHER ATHLETIC" ^0 "" ^0 " computing the conditional rates from real counts over" ^0 " every admitted student is the part done right here, and" ^0 " it is why the negative relation in that sample is genuine" ^0 "" ^0 # ---- what the sample was conditioned on ---- "the set the rates were computed over" ^0 " who is in it : admitted students only" ^0 " the admission rule : high in academics OR athletics" ^0 " what that rule is : a common effect of both abilities" ^0 " a low-academic student who was admitted : must have been" ^0 " high-athletic, or would have been rejected" ^0 " so within the admitted : low academic forces high" ^0 " athletic, a relation the rule created, not the abilities" ^0 "" ^0 # ---- what the caller concluded ---- "the conclusion drawn" ^0 " claim : athletic training crowds out academics" ^0 " gap among admitted : " + str(manufactured_gap_per_myriad) + " per myriad" ^0 " gap in the full applicant pool : 0, the two are" ^0 " independent" ^0 " are the admitted rates wrong : no; they are exact" ^0 " do the abilities oppose each other : no; conditioning on" ^0 " admission, a collider, made independent traits look" ^0 " opposed" ^0 "" ^0 # ---- null control ---- # The same scores, with the rates computed over the full applicant pool (not # conditioned on admission), or within a single ability's admits. 5000 => nc_high_ath_given_high_ac_in_full_pool 5000 => nc_high_ath_given_low_ac_in_full_pool 0 => nc_gap_in_the_full_pool "null control - measure the full applicant pool, not the admitted" ^0 " high-athletic given high-academic, full pool : " + str(nc_high_ath_given_high_ac_in_full_pool) + " per myriad" ^0 " high-athletic given low-academic, full pool : " + str(nc_high_ath_given_low_ac_in_full_pool) + " per myriad" ^0 " gap in the full pool : " + str(nc_gap_in_the_full_pool) ^0 " no student and no score changed; the rates stopped being" ^0 " taken over the common effect and started being taken over" ^0 " everyone" ^0 "" ^0 # ---- the rule ---- "what a correlation over a selected sample guarantees" ^0 " the relation within the sample is correctly computed :" ^0 " exactly, real scores, every admit, honest rates" ^0 " the two abilities are related in the world : not" ^0 " addressed; admission requires high in one or the other," ^0 " so conditioning on it forces a low-academic admit to be" ^0 " high-athletic - a " + str(manufactured_gap_per_myriad) + "-per-myriad gap that is 0 in the pool" ^0 "" ^0 "selection on a common effect of two independent causes correlates them among the" ^0 "selected; requiring at least one to be high means whoever is low on one is high" ^0 "on the other, so the relation is a fact about the gate, not about the causes" ^0 "" ^0 "It computes honest conditional rates over every admitted student - the negative" ^0 "relation in that sample is real. But admission requires being high in one ability" ^0 "or the other, a collider, so a low-academic admit must be high-athletic; over the" ^0 "full pool the gap is " + str(nc_gap_in_the_full_pool) + ", and the whole " + str(manufactured_gap_per_myriad) + " per myriad is the selection." ^0 ``` ## Python (deterministic transpilation) ```python applicants = 10000 rejected_low_in_both = 2500 admitted_high_academic = 5000 of_those_high_athletic = 2500 admitted_low_academic = 2500 of_those_low_academic_high_athletic = 2500 admitted = applicants - rejected_low_in_both admitted_high_ath_given_high_ac = int(of_those_high_athletic * 10000 / admitted_high_academic) admitted_high_ath_given_low_ac = int(of_those_low_academic_high_athletic * 10000 / admitted_low_academic) manufactured_gap_per_myriad = admitted_high_ath_given_low_ac - admitted_high_ath_given_high_ac print("applicants : " + str(applicants)) print(" rejected (low in both) : " + str(rejected_low_in_both)) print(" admitted : " + str(admitted)) print("") print("among admitted, high-athletic given high-academic : " + str(admitted_high_ath_given_high_ac) + " per myriad") print("among admitted, high-athletic given low-academic : " + str(admitted_high_ath_given_low_ac) + " per myriad") print("manufactured gap : " + str(manufactured_gap_per_myriad) + " per ten thousand") print("") print("the correlation measurement") print(" uses : the real recorded scores, not estimates") print(" covers : every admitted student") print(" rates : honest counts of high-athletic within each group") print(" intent : are academic and athletic ability related") print(" students omitted : 0") print(" verdict : AMONG ADMITTED, LOWER ACADEMIC MEANS HIGHER ATHLETIC") print("") print(" computing the conditional rates from real counts over") print(" every admitted student is the part done right here, and") print(" it is why the negative relation in that sample is genuine") print("") print("the set the rates were computed over") print(" who is in it : admitted students only") print(" the admission rule : high in academics OR athletics") print(" what that rule is : a common effect of both abilities") print(" a low-academic student who was admitted : must have been") print(" high-athletic, or would have been rejected") print(" so within the admitted : low academic forces high") print(" athletic, a relation the rule created, not the abilities") print("") print("the conclusion drawn") print(" claim : athletic training crowds out academics") print(" gap among admitted : " + str(manufactured_gap_per_myriad) + " per myriad") print(" gap in the full applicant pool : 0, the two are") print(" independent") print(" are the admitted rates wrong : no; they are exact") print(" do the abilities oppose each other : no; conditioning on") print(" admission, a collider, made independent traits look") print(" opposed") print("") nc_high_ath_given_high_ac_in_full_pool = 5000 nc_high_ath_given_low_ac_in_full_pool = 5000 nc_gap_in_the_full_pool = 0 print("null control - measure the full applicant pool, not the admitted") print(" high-athletic given high-academic, full pool : " + str(nc_high_ath_given_high_ac_in_full_pool) + " per myriad") print(" high-athletic given low-academic, full pool : " + str(nc_high_ath_given_low_ac_in_full_pool) + " per myriad") print(" gap in the full pool : " + str(nc_gap_in_the_full_pool)) print(" no student and no score changed; the rates stopped being") print(" taken over the common effect and started being taken over") print(" everyone") print("") print("what a correlation over a selected sample guarantees") print(" the relation within the sample is correctly computed :") print(" exactly, real scores, every admit, honest rates") print(" the two abilities are related in the world : not") print(" addressed; admission requires high in one or the other,") print(" so conditioning on it forces a low-academic admit to be") print(" high-athletic - a " + str(manufactured_gap_per_myriad) + "-per-myriad gap that is 0 in the pool") print("") print("selection on a common effect of two independent causes correlates them among the") print("selected; requiring at least one to be high means whoever is low on one is high") print("on the other, so the relation is a fact about the gate, not about the causes") print("") print("It computes honest conditional rates over every admitted student - the negative") print("relation in that sample is real. But admission requires being high in one ability") print("or the other, a collider, so a low-academic admit must be high-athletic; over the") print("full pool the gap is " + str(nc_gap_in_the_full_pool) + ", and the whole " + str(manufactured_gap_per_myriad) + " per myriad is the selection.") ``` ## stdout (executed) ```text applicants : 10000 rejected (low in both) : 2500 admitted : 7500 among admitted, high-athletic given high-academic : 5000 per myriad among admitted, high-athletic given low-academic : 10000 per myriad manufactured gap : 5000 per ten thousand the correlation measurement uses : the real recorded scores, not estimates covers : every admitted student rates : honest counts of high-athletic within each group intent : are academic and athletic ability related students omitted : 0 verdict : AMONG ADMITTED, LOWER ACADEMIC MEANS HIGHER ATHLETIC computing the conditional rates from real counts over every admitted student is the part done right here, and it is why the negative relation in that sample is genuine the set the rates were computed over who is in it : admitted students only the admission rule : high in academics OR athletics what that rule is : a common effect of both abilities a low-academic student who was admitted : must have been high-athletic, or would have been rejected so within the admitted : low academic forces high athletic, a relation the rule created, not the abilities the conclusion drawn claim : athletic training crowds out academics gap among admitted : 5000 per myriad gap in the full applicant pool : 0, the two are independent are the admitted rates wrong : no; they are exact do the abilities oppose each other : no; conditioning on admission, a collider, made independent traits look opposed null control - measure the full applicant pool, not the admitted high-athletic given high-academic, full pool : 5000 per myriad high-athletic given low-academic, full pool : 5000 per myriad gap in the full pool : 0 no student and no score changed; the rates stopped being taken over the common effect and started being taken over everyone what a correlation over a selected sample guarantees the relation within the sample is correctly computed : exactly, real scores, every admit, honest rates the two abilities are related in the world : not addressed; admission requires high in one or the other, so conditioning on it forces a low-academic admit to be high-athletic - a 5000-per-myriad gap that is 0 in the pool selection on a common effect of two independent causes correlates them among the selected; requiring at least one to be high means whoever is low on one is high on the other, so the relation is a fact about the gate, not about the causes It computes honest conditional rates over every admitted student - the negative relation in that sample is real. But admission requires being high in one ability or the other, a collider, so a low-academic admit must be high-athletic; over the full pool the gap is 0, and the whole 5000 per myriad is the selection. ``` ## Round-trip `ok: true` — round-trip fixpoint reached (python1 == python2) ## Trace event types eml:run:start · eml:assign · eml:output · eml:run:done