MONSTERPET and PETPLUMP are two different brands of pet food. Each is guaranteed to make your animal grow larger. The following study involves 11 cats and 11 dogs. 4 cats eat Monsterpet. 3 of them double in size; 1 remains puny. 7 cats eat Petplump. 5 of them double in size; 2 remain puny. 7 dogs eat Monsterpet. 2 of them double in size; 5 remain puny. 4 dogs eat Petplump. 1 of them doubles in size; 3 remain puny.
MONSTERPETPETPLUMP Success rate =3/4=75% Success rate =5/7=71% Success rate =1/4=25% Success rate =2/7=28%
MONSTERPETPETPLUMP Total success rate =5/11= 45% Total success rate =6/11= 54%
Overall, PETLUMP outperforms MONSTERPET 54% grow with Petplump while 45% grow with Monsterpet Perhaps we can unravel this paradox. Notice that with both brands of food, the cats respond much more than the dogs: Seventy some percent of cats grow with each product. Twenty some percent of dogs grow with each product. But a higher percentage of cats grow with Monsterpet: 75% compared to 71% And a higher percentage of dogs grow with Monsterpet: 28% compared to 25%
MONSTERPETPETPLUMP Because there are more cats on the Petplump side, the overall percentages for Petplump are higher
This puny creature would not deter a burglar. But feed her Monsterpet and see results MEOW
We are going to rework this example, turning the animals into people participating in a drug trial. Monsterpet becomes drug A Petplump becomes drug B The animals who responded favorably to the petfoods by growing large will become people who were cured by the drug. The small animals who did not respond become people who were not cured.
DRUG ADRUG B Cure rate = 5/11 = 45%Cure rate = 6/11 = 54%
Drug B appears to be more effective. Now, suppose the people who used to be cats share a genetic trait that predisposes them to respond more favorably to treatment. Suppose this trait is invisible. Drug A really works better for both groups of people - those with the trait and those without the trait. But, because more of the people with the trait are assigned to the sample being treated with B, it makes B look better.
DRUG ADRUG B 3/4=75% cure rate 5/7=71% cure rate 2/7=28% cure rate 1/4=25% cure rate