 Symmetric treatment crossover attenuates a risk difference

Assume treatment receipt gives event risk $p_1$, nonreceipt gives risk $p_0$, and a fraction $c$ in each equally sized randomized arm crosses to the opposite treatment. The two assignment risks are $(1-c)p_1+c{p}_0$ and $c{p}_1+(1-c)p_0$. Subtracting yields the displayed attenuation, with ${\Delta }_{\mathrm{receipt}}=p_0-p_1$. It depends on the stated equal crossover fractions and unchanged receipt-specific risks; it is not a universal formula for every form of treatment contamination.