Two analysts, two quarters, a reversal

Question

Two analysts make directional calls. You tally correct calls over two quarters:

	Q1	Q2	Full year
Analyst P	104/411 (25.3%)	45/140 (32.1%)	149/551 (27.0%)
Analyst Q	12/48 (25.0%)	183/582 (31.4%)	195/630 (31.0%)

Analyst P has the higher hit rate in Q1 (25.3% vs 25.0%) and in Q2 (32.1% vs 31.4%), yet Analyst Q wins the full year (31.0% vs 27.0%).

How can P lead every quarter yet lose overall? Who is the better forecaster?

Accepted Answer

What's happening. Classic Simpson's paradox (the same structure as the famous baseball batting-average reversal). Two facts combine: 1. Q2 was easier for both analysts (roughly 31 to 32\% hit rates versus 25\% in Q1). 2. The analysts concentrated their calls in opposite quarters. Analyst P made most of his calls in the hard Q1 (411 of 551), while Analyst Q made most of hers in the easy Q2 (582 of 630). So Q's overall average is pulled up toward the easy-quarter rate, while P's is dragged down toward the hard-quarter rate: 	ext{P: } 	frac{411}{551}(0.253) + 	frac{140}{551}(0.321) \approx 0.…

Two analysts, two quarters, a reversal

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