User:Psarka/PR1 difference order

This is a proposal to change the difference order in Absolute risk reduction (and subsequently Relative risk reduction). The current order is EER - CER (target - control) and I propose to change it to CER - EER (control - target).

Alignment with the main sources
Three of the four main sources (see User:Psarka/What a mess!) define the Absolute risk reduction as CER - EER.

SMM p. 624:"Any calculations of sampling error are best performed first for the""absolute reduction pC - pT"DoE p. 2:"ABSOLUTE RISK REDUCTION (ARR) The absolute risk of events in the control group""(ARC) minus the absolute risk of events in the exposed or treatment group""(ART): ARR = ARC – ART. The negative of the risk difference."EBtB p. 98:"For example, in a randomized clinical trial A, the incidence in the control/placebo""group is 25%, and that in the active intervention group is 10% [...] evaluated in""trial A (absolute risk reduction = 15%)."The fourth source does not define the Absolute risk reduction, but defines Risk difference instead. It defines it as EER - CER, which is consistent with DoE remark that Absolute risk reduction is the negative of the Risk difference:

EAI p. 59:"The risk difference (RD) is the difference in incidence proportion or risk between""the exposed and the unexposed groups. If the incidence proportion is 0.25 for""the exposed and 0.10 for the unexposed, the RD is 0.15."

Confusion in the NTT
Current definition leads to contradiction in the Number needed to treat, which is everywhere defined as:


 * inverse of the Absolute risk reduction
 * number of people needed to treat in order to cure one

With the current definition, for a useful drug the ARR is negative, so the number of people needed to treat comes out negative as well, which is confusing. This confusion is readily visible in the Number needed to treat page, where most of the text is contradicting the current definition.

With the proposed definition, if the drug is useful, then the ARR is positive, so the number of people needed to treat comes out positive as expected.

Semantics
Statement "the absolute risk reduction of a drug is 20%" reads as if the drug is useful, because it reduces the risk. While with the current definition it would mean that the drug increases the risk by 20%. I think the issue is the presence of the word "reduction" in the definition, which already implies the direction of the change, and we don't expect the direction to be additionally reflected in the numerical value.

Changes
Absolute risk reduction


 * swap subtractions as necessary
 * define risk difference as synonym for absolute risk increase

Number needed to treat


 * remove the remark about the reversal of the signs at the very bottom

ARR RRR worksheet


 * split the formulas for absolute risk reduction vs increase, relative risk reduction vs increase, number needed to treat vs to harm.
 * swap subtractions as necessary