Composite Computer Rankings and Correlations

3 April 2019

My initial analysis of the computer ratings included nine rankings. With the input compiled by Dr. Massey on April 3^nd we now have 12 input rankings. This directly affects the Bucklin Majority composite: instead of the 5^th-best rank for each team it selects the 7^th-best.

∑Dist ISR KLK MAS DII FMG MGS RT SAG DOK MOR NOL RPI

26766 BMaj 1014 1028 1184 1311 1415 1966 2181 2356 2965 3312 3969 4065

26823 Cond 1011 987 1135 1255 1460 2055 2160 2359 2980 3328 3973 4120

27525 Borda 1195 1215 1345 1393 1551 2234 2028 2429 2966 3223 3870 4076

27851 Mcomp 1245 1285 1364 1440 1600 2280 2035 2462 3004 3240 3816 4080

Having ordered the composite rankings in terms of increasing cumulative number of discordant pairs vis a vis the computer rankings, we plot the percentage of concordant pairs for each computer ranking against each composite.

1-ISR 2-KLK 3-MAS 4-DII 5-FMG 6-MGS 7-RT 8-SAG 9-DOK 10-MOR 11-NOL 12-RPI

The first five and next three ratings seem to relate to the composites in about the same way and it looks like there are two or three more groups. When we compare the ratings to each other we can order them by their average correlation to the others, and for each we can order the others by most-like to least-like the rating.

Cond Borda Mcomp ISR KLK MAS DII FMG MGS RT SAG DOK MOR NOL RPI

BMaj 9898 9862 9846 9743 9740 9705 9676 9653 9529 9481 9442 9305 9227 9079 9057

Cond 9908 9893 9761 9766 9733 9706 9660 9526 9503 9458 9319 9241 9096 9062

Borda 9975 9729 9724 9695 9684 9649 9495 9542 9452 9331 9273 9128 9082

Mcomp 9720 9711 9693 9676 9640 9488 9543 9447 9325 9272 9143 9083

ISR 9681 9711 9592 9552 9561 9339 9525 9159 9077 9031 9104

KLK 9624 9622 9608 9505 9403 9379 9292 9174 8985 9013

MAS 9594 9492 9469 9384 9497 9148 9086 9112 8986

DII 9526 9401 9346 9367 9426 9214 8948 8989

FMG 9516 9428 9236 9304 9259 8943 8901

MGS 9184 9238 9109 9073 8851 8879

RT 9161 9226 9328 9134 8933

SAG 8933 8837 8951 9119

DOK 9403 8699 8659

MOR 8832 8546

NOL 8735

And here we show the computer ranking order based upon that table, from most-like to least-like.

Best% Avg% Worst% Rating Most
like Least
like

97.11 93.94 90.31 ISR MAS KLK DII MGS FMG SAG RT DOK RPI MOR NOL

96.81 93.89 89.59 KLK ISR MAS DII FMG MGS RT SAG DOK MOR RPI NOL

97.11 93.73 89.86 MAS ISR KLK DII SAG FMG MGS RT DOK NOL MOR RPI

96.22 93.66 89.48 DII KLK MAS ISR FMG DOK MGS SAG RT MOR RPI NOL

96.08 93.42 89.01 FMG KLK ISR DII MGS MAS RT DOK MOR SAG NOL RPI

94.28 92.61 89.33 RT FMG KLK MAS DII ISR MOR DOK MGS SAG NOL RPI

95.61 92.53 88.51 MGS ISR FMG KLK MAS DII SAG RT DOK MOR RPI NOL

95.25 92.04 88.37 SAG ISR MAS KLK DII MGS FMG RT RPI NOL DOK MOR

94.26 91.23 86.59 DOK DII MOR FMG KLK RT ISR MAS MGS SAG NOL RPI

94.03 90.75 85.46 MOR DOK RT FMG DII KLK MAS ISR MGS SAG NOL RPI

91.34 89.29 86.99 NOL RT MAS ISR KLK SAG DII FMG MGS MOR RPI DOK

91.19 88.97 85.46 RPI SAG ISR KLK DII MAS RT FMG MGS NOL DOK MOR

The "other" ratings that are more like the given rating than its average "like-ness" are highlighted.

Other rank-correlations

Above I used the cumulative total distance (number of discordant pairs) over all the computer rankings for the initial ranking order, and the percentage of concordant pairs for the measure of "like-ness." There are other correlations based upon the notion of concordant and discordant pairs and here are two that I also calculate.

Kendall's τ (tau)

τ =	#Concordant pairs − #Discordant pairs

	½ × #Ranked × (#Ranked−1)

Goodman and Kruskal's γ (gamma)

γ =	#Concordant pairs − #Discordant pairs

	#Concordant pairs + #Discordant pairs

These give -1 ≤ { τ, γ } ≤ 1 with |τ| ≤ |γ|. Both will be -1 if the teams are in exactly reverse order, 0 if the relationship is perfectly random (whatever that means!) and +1 if the rankings are identical. The τ and γ are the same if there are no ties (but notice that ties in the Majority Consensus rank are to be expected, in which case τ will be closer to zero than γ.)

We see that had I used average tau instead of cumulative distance to order the rankings we'd have had the Condorcet consensus ahead of the Bucklin majority and RT ahead of MGS.

1-ISR 2-KLK 3-MAS 4-DII 5-FMG 6-MGS 7-RT 8-SAG 9-DOK 10-MOR 11-NOL 12-RPI

The same changes result from the gamma correlation.

1-ISR 2-KLK 3-MAS 4-DII 5-FMG 6-MGS 7-RT 8-SAG 9-DOK 10-MOR 11-NOL 12-RPI

I report the extra "composite" rankings as Computer Ranking Composites including a table of distances between pairs of ranking after the report by team.