I still don't know what the answer is, but I've always wondered how I would define "correlation to top 25" using the latter approach. Actually, I knew how I would do it, but was too ~~lazy~~ busy to implement it until I was prompted to by compare the human top 25s to the computers.

Presuming that Dr. Massey's "Top 25 Cor to Con" is a correlation of the teams in a rating's top 25 to the "consensus" ranking of all teams by all ratings

The "average" or "consensus" ranking for each team is determined using a least squares fit based on paired comparisons between teams for each of the listed ranking systems. If a team is ranked by all systems, the consensus is equal to the arithmetic average ranking. When a team is not ranked by a particular system, its consensus will be lowered accordingly.I choose to calculate the "consensus of top 25s" by only considering ranks 1-25 for each of the computers, as reported in Computer Top 25. The notion of correlation is made more complicated by the fact that different ratings' top 25s can (invariably do) include different teams.

For rank-based correlations I use the *distance* metric to characterize the correlation between two ratings. This is the number of team-pair swaps it would take to re-order one list to match the other. That doesn't work for lists that don't have the same elements, as is usually the case with the computer rankings' top 25s. No number of swaps of teams in one list can add a team to it!

So we add *percentage of total team-pairs that are "concordant"* as a measure of how close two rankings are. Instead of -1 ≤ x ≤ 1 we have 0 ≤ x ≤ 1.

- Kendall's
tau:

τ = #Concordant pairs − #Discordant pairs #pairs - Goodman and Kruskal's
gamma:

γ = #Concordant pairs − #Discordant pairs #Concordant pairs + #Discordant pairs - Percentage of Pairs that are Concordant

%Concordant = #Concordant pairs #Total pairs

I calculate the "top 25 consensus" by just summing 26-*team's rank by rating* over all ratings that rank the team. This is **not** a good method for counting votes when all ranks 1-130 are considered, but it has the same properties as the consensus described by Dr. Massey when applied to truncated rankings. Using the ranings from Massey Ratings `College Football Ranking Composite` as of Sun Aug 27 05:28:22 we have 64 teams in at least one of 40 ratings' top 25.

## Top 25 Correlation to Top 25 Consensus

#Common#TeamsConcDisc#Pairsγτ%ConcBorda 23 27 334 14 351 0.9195 0.9117 0.9516 Mix 23 27 331 17 351 0.9023 0.8946 0.9430 DES 23 27 322 26 351 0.8506 0.8433 0.9174 PIR 23 27 313 35 351 0.7989 0.7920 0.8917 KPK 22 28 324 47 378 0.7466 0.7328 0.8571 PGH 21 29 346 47 406 0.7608 0.7365 0.8522 DII 21 29 346 47 406 0.7608 0.7365 0.8522 BIL 22 28 322 49 378 0.7358 0.7222 0.8519 PIG 21 29 345 48 406 0.7557 0.7315 0.8498 DOK 22 28 319 52 378 0.7197 0.7063 0.8439 KAM 22 28 318 53 378 0.7143 0.7011 0.8413 YAG 21 29 335 59 406 0.7005 0.6798 0.8251 KEL 21 29 334 59 406 0.6997 0.6773 0.8227 HOW 21 29 334 60 406 0.6954 0.6749 0.8227 SAG 21 29 331 62 406 0.6845 0.6626 0.8153 MAS 20 30 353 61 435 0.7053 0.6713 0.8115 BRN 21 29 329 64 406 0.6743 0.6527 0.8103 ARG 21 29 329 64 406 0.6743 0.6527 0.8103 TPR 21 29 328 66 406 0.6650 0.6453 0.8079 MOR 21 29 325 68 406 0.6539 0.6330 0.8005 FEI 21 29 314 79 406 0.5980 0.5788 0.7734 DWI 20 30 333 81 435 0.6087 0.5793 0.7655 FPI 19 31 354 81 465 0.6276 0.5871 0.7613 HAT 19 31 352 82 465 0.6221 0.5806 0.7570 MAR 20 30 329 85 435 0.5894 0.5609 0.7563 #Common#TeamsConcDisc#Pairsγτ%ConcDCI 20 30 329 85 435 0.5894 0.5609 0.7563 RTP 19 31 346 88 465 0.5945 0.5548 0.7441 MGS 20 30 323 92 435 0.5566 0.5310 0.7425 DEZ 22 28 280 91 378 0.5094 0.5000 0.7407 BWE 18 32 364 90 496 0.6035 0.5524 0.7339 RUD 19 31 341 94 465 0.5678 0.5312 0.7333 CTW 19 31 337 97 465 0.5530 0.5161 0.7247 BDF 20 30 309 105 435 0.4928 0.4690 0.7103 RWP 20 30 305 109 435 0.4734 0.4506 0.7011 CGV 19 31 321 114 465 0.4759 0.4452 0.6903 PFZ 18 32 328 126 496 0.4449 0.4073 0.6613 NGS 18 32 306 148 496 0.3480 0.3185 0.6169 LSD 17 33 316 155 528 0.3418 0.3049 0.5985 ENG 17 33 307 164 528 0.3036 0.2708 0.5814 LOG 17 33 303 168 528 0.2866 0.2557 0.5739 NUT 17 33 279 192 528 0.1847 0.1648 0.5284 PPP 13 37 318 215 666 0.1932 0.1547 0.4775

I've added a Top 25 Consensus page with three reports.

**Top 25 Rank Correlations to Consensus**- Lists the actual top-25 ranks for each rating, the consensus top 25, and what the top-25 list would be based upon average (
**Borda**) and median (**Mix**) rankings using all team ranks (not just the top 25) from all ratings. The latter two are**not**used to determine the consensus.For each

*team*in any ratings' top-25: **Points**is the usual sum of 25 points for each #1 ranking, 24 for each #2, and so on down to zero points for ranks worse than #25.**∑Dist**is the sum of the number of discordant pairs over all ratings. Lower numbers indicate stronger agreement among the computers about the team's rank.**#Votes**is the number of ratings that include the team in their top 25.**Team**links to the list of ratings that rank the team at each rank that any does.For each

*Rating**∑Dist*is the sum of the number of discordant pairs when the rating is compared to every other rating. The lower the number the more representative the ranking is of all rankings.*Dist from consensus*is the number of discordant pairs when the rating's top 25 is compared to the consensus top 25.*Common w/ Cons*is the number of teams in both the rating's top 25 and the consensus top 25 (the intersection of teams ranked by each.) To find the number of teams ranked by either (the union) just subtract this number from 50.is (10000×) the ratio of concordant pairs to total pairs.*%Concordant w/ Cons***Computer Ratings Top 25 % Concordant**- For any two ratings displays the percentage of team-pairs that are concordant, to four decimal places with the leading zero and decimal point removed. The column-rating that matches the best agreement with the row-rating is displayed as blue and those which least agree in red. Higher numbers indicate better agreement between the ratings.
**Computer Ratings Top 25 # Common Teams**- The order of the intersection of teams in row-rating's top 25 and column-rating's top 25 is displayed in row, column. If this value is
*c*the number of pairs used to derive the previous report is`(50 −`*c*) × (50 −*c*− 1) ⁄ 2

= (*c*^{2}− 99×*c*+ 2450 ) ⁄ 2

The "top 25" is not as significant as its popularity might suggest. It approximately selects the top quintile of the 1A field, but the main value in voting for a top 25 is to rank some smaller number of teams.

© Copyright 2017, Paul Kislanko

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