NCAA Tourney Bayesian Ratings and Odds

Okay, ready for the tournament?

I’ve put together some adjustments based on the work I teased yesterday.  The theory behind the adjustments may be found on STATS @ MTSU and Dr. Winner @ Florida.  Basically, I’m adjusting for teams that raise their game to a higher level against good foes (or vise-versa).  For instance, Long Island plays much better against good teams–probably because they got bored against the bad teams/pulled their starters.  Now, we can’t just take a simple linear regression; the slope doesn’t continue infinitely.  So I construct the Prediction Interval to estimate the mean and standard deviation at any point along the regression line, and regress based on the REGRESSED standard deviation of the team’s adjusted efficiency differentials.

For example, let’s see what Hampton looks like.  They raise their game more than just about anyone in the field; they’ve played a very easy schedule.  Here’s what their regressed curves look like:

To simplify the math, I presumed that all games in the tournament were against +25 level foes: I estimate that this is “the best” that each team can play with it all on the line.  Of course, a few teams have played worse with it all on the line (see Duke).

To incorporate this into the Bayesian ratings, I simply took the latest Bayesian rating and added in the projected differential at +25, minus the average efficiency differential for the team.

Next, I set about estimating the standard deviation for each team.  We can easily calculate the team’s standard deviation of performances; I estimated that the standard deviation of the Bayesian efficiency was 1 less than that.  Next, I looked at the standard deviation of the curve above at the +25 mark.  I decided the best way to combine the two was to take the geometric mean of the Bayesian-based Stdev and the stdev of the non-regressed efficiency differential curve.  I’m not totally sure how to do this here, so this is my best guess; I’m a little over my head mathematically.  Anyway, the results look about right intuitively.

So now we’ve got a new “Tourmament Power Rating” and a standard deviation… so it’s easy to calculate odds for each game.

First, here’s the ratings in tabular form:

O	Sd	Re	Team	Conf	W-L	Bayesian	Rk	25Delta	Tourney	Rk	Stdev
1	1	E	Ohio St.	B10	32-2	33.47	1	-0.22	33.25	1	14.81
2	16	E	Alabama St.	SWAC	17-17	-13.90	299	-0.77	-14.66	68	16.34
2	16	E	Texas San Antonio	Slnd	19-13	-6.58	232	4.22	-2.36	66	10.69
3	9	E	Villanova	BE	21-11	18.90	25	-1.14	17.76	37	15.51
4	8	E	George Mason	CAA	26-6	15.62	44	-1.16	14.46	46	15.71
5	5	E	West Virginia	BE	20-11	21.74	16	0.08	21.82	18	14.12
6	12	E	Clemson	ACC	22-11	20.68	19	1.94	22.62	17	13.05
6	12	E	UAB	CUSA	22-9	12.13	65	-0.95	11.18	50	14.09
7	13	E	Princeton	Ivy	25-6	6.12	100	0.54	6.66	55	13.87
8	4	E	Kentucky	SEC	25-8	27.05	6	-0.20	26.85	6	14.35
9	3	E	Syracuse	BE	26-7	25.52	9	-0.35	25.17	11	17.45
10	14	E	Indiana St.	MVC	20-13	5.52	104	-0.88	4.64	58	16.16
11	11	E	Marquette	BE	20-14	18.70	28	0.97	19.67	24	13.97
12	6	E	Xavier	A10	24-7	18.76	27	2.80	21.57	20	17.24
13	7	E	Washington	P10	23-10	22.82	15	-2.36	20.46	23	15.76
14	10	E	Georgia	SEC	21-11	13.09	61	1.87	14.96	45	14.89
15	15	E	Long Island	NEC	27-5	1.30	148	2.98	4.28	59	13.05
16	2	E	North Carolina	ACC	26-7	23.04	14	0.26	23.30	15	15.95
19	1	W	Duke	ACC	30-4	32.26	2	-3.06	29.20	3	12.23
20	16	W	Hampton	MEAC	24-8	-5.84	224	9.84	4.00	60	12.99
21	9	W	Tennessee	SEC	19-14	15.58	46	3.21	18.79	28	13.65
22	8	W	Michigan	B10	20-13	16.25	42	1.53	17.78	36	13.71
23	5	W	Arizona	P10	27-7	18.43	33	-3.17	15.26	44	13.94
24	12	W	Memphis	CUSA	25-9	10.05	73	-0.34	9.72	53	15.05
25	13	W	Oakland	Sum	25-9	10.75	70	-0.43	10.32	51	12.65
26	4	W	Texas	B12	27-7	27.42	4	1.30	28.72	4	14.29
27	3	W	Connecticut	BE	26-9	21.70	17	1.64	23.35	14	13.34
28	14	W	Bucknell	Pat	25-8	4.40	115	1.10	5.50	57	14.78
29	11	W	Missouri	B12	23-10	18.20	35	0.68	18.88	27	14.36
30	6	W	Cincinnati	BE	25-8	18.47	31	-0.42	18.05	35	17.60
31	7	W	Temple	A10	25-7	17.29	37	0.77	18.06	34	13.40
32	10	W	Penn St.	B10	19-14	15.58	45	1.38	16.96	40	17.06
33	15	W	Northern Colorado	BSky	21-10	2.87	128	-2.70	0.17	62	12.14
34	2	W	San Diego St.	MWC	32-2	24.10	12	2.61	26.72	7	14.10
36	1	SW	Kansas	B12	32-2	31.42	3	-1.42	30.00	2	15.24
37	16	SW	Boston University	AE	21-13	-0.25	160	-0.77	-1.02	64	14.23
38	9	SW	Illinois	B10	19-13	20.18	21	0.41	20.59	22	12.90
39	8	SW	Nevada Las Vegas	MWC	24-8	19.37	24	-0.84	18.53	29	14.53
40	5	SW	Vanderbilt	SEC	23-10	18.58	30	-0.12	18.45	30	13.52
41	12	SW	Richmond	A10	27-7	15.10	49	-0.81	14.29	47	15.02
42	13	SW	Morehead St.	OVC	24-9	5.97	102	4.12	10.09	52	15.10
43	4	SW	Louisville	BE	25-9	24.14	11	0.27	24.41	13	14.67
44	3	SW	Purdue	B10	25-7	25.79	8	0.40	26.19	8	14.51
45	14	SW	St. Peter's	MAAC	20-13	2.57	131	-1.67	0.90	61	16.04
46	11	SW	Southern California	P10	19-14	15.40	47	2.68	18.08	33	16.55
46	11	SW	Virginia Commonwealth	CAA	23-11	9.32	76	3.74	13.06	49	14.06
47	6	SW	Georgetown	BE	21-10	19.52	22	-0.45	19.07	26	15.06
48	7	SW	Texas A&M	B12	24-8	17.34	36	0.00	17.34	39	12.76
49	10	SW	Florida St.	ACC	21-10	16.66	39	-1.16	15.50	43	10.01
50	15	SW	Akron	MAC	23-12	4.81	112	-4.81	0.00	63	15.78
51	2	SW	Notre Dame	BE	26-6	25.37	10	0.21	25.58	10	16.26
53	1	SE	Pittsburgh	BE	27-5	27.26	5	-0.35	26.92	5	12.13
54	16	SE	Arkansas Little Rock	SB	19-17	-5.36	220	-8.35	-13.70	67	14.98
54	16	SE	NC Asheville	BSth	20-13	-0.93	167	-0.53	-1.46	65	17.12
55	9	SE	Old Dominion	CAA	27-6	14.92	50	-0.91	14.01	48	13.56
56	8	SE	Butler	Horz	23-9	16.32	41	3.03	19.35	25	17.75
57	5	SE	Kansas St.	B12	22-10	19.41	23	1.50	20.91	21	14.77
58	12	SE	Utah St.	WAC	30-3	20.35	20	1.44	21.79	19	15.45
59	13	SE	Belmont	ASun	30-4	18.60	29	-2.68	15.92	42	16.76
60	4	SE	Wisconsin	B10	23-8	26.79	7	-0.64	26.15	9	16.88
61	3	SE	Brigham Young	MWC	30-4	23.40	13	1.35	24.75	12	15.71
62	14	SE	Wofford	SC	21-12	8.09	87	-1.26	6.84	54	17.07
63	11	SE	Gonzaga	WCC	24-9	18.46	32	-2.22	16.24	41	14.68
64	6	SE	St. John's	BE	21-11	16.66	40	1.76	18.42	31	17.79
65	7	SE	UCLA	P10	22-10	14.61	52	3.58	18.18	32	14.67
66	10	SE	Michigan St.	B10	19-14	17.09	38	0.46	17.55	38	16.02
67	15	SE	UC Santa Barbara	BW	18-13	2.19	137	3.56	5.74	56	14.84
68	2	SE	Florida	SEC	26-7	21.50	18	1.74	23.24	16	16.39

A few notes:

• Teams to watch out for: Clemson, Utah St., Marquette, Missouri, Illinois, and Texas.
• Overseeded teams: Arizona, George Mason, ODU, Cincinnati, Texas A&M, and Vanderbilt.
• Hampton raises its game a lot, but it’s still not good.  San Diego does quite a bit also, and they are good.
• Duke and Arizona got their great efficiencies by beating up on the little guys.

And here are my directly calculated odds for each team to reach each level in the tournament:

O	Sd	Re	Team	64	32	16	Elite 8	Final 4	Final	Champ	1 in
1	1	E	Ohio St.	100	99.74	86.265	61.971	46.4918	31.0991	21.39532	4.67
8	4	E	Kentucky	100	92.37	59.225	22.991	13.9956	6.9810	3.68788	27.1
9	3	E	Syracuse	100	88.90	53.936	32.034	12.4445	5.6966	2.76581	36.2
16	2	E	North Carolina	100	90.42	56.027	28.646	9.7302	3.9611	1.73444	57.7
12	6	E	Xavier	100	54.82	25.495	13.004	4.0086	1.4810	0.59137	169
13	7	E	Washington	100	64.00	30.138	13.348	3.6887	1.2382	0.45524	220
5	5	E	West Virginia	100	53.59	21.430	5.842	2.8131	1.0161	0.40160	249
6	12	E	Clemson	80.0	41.91	17.492	4.957	2.4906	0.9413	0.38895	257
11	11	E	Marquette	100	45.18	18.821	8.684	2.1555	0.6591	0.22431	446
3	9	E	Villanova	100	58.36	9.079	2.929	1.1236	0.3055	0.09283	1077
14	10	E	Georgia	100	36.00	12.259	3.833	0.6438	0.1347	0.03235	3091
4	8	E	George Mason	100	41.64	4.636	1.161	0.3592	0.0741	0.01750	5714
6	12	E	UAB	20.0	4.50	0.778	0.081	0.0187	0.0026	0.00042	236328
10	14	E	Indiana St.	100	11.10	1.748	0.273	0.0163	0.0012	0.00011	872170
7	13	E	Princeton	100	7.63	1.075	0.067	0.0101	0.0008	0.00008	1261187
15	15	E	Long Island	100	9.58	1.576	0.178	0.0073	0.0004	0.00003	3684929
2	16	E	Texas San Antonio	81.4	0.24	0.017	0.000	0.0000	0.0000	0.00000	32042448855
2	16	E	Alabama St.	18.6	0.02	0.001	0.000	0.0000	0.0000	0.00000	92485855933097
19	1	W	Duke	100	97.71	78.072	46.442	30.5745	16.0242	9.54891	10.47
26	4	W	Texas	100	91.36	78.143	42.890	27.2983	13.9476	8.05711	12.41
34	2	W	San Diego St.	100	97.82	71.862	46.694	21.8962	10.0679	5.28571	18.92
27	3	W	Connecticut	100	89.76	56.499	26.854	10.0105	3.7336	1.61838	61.8
29	11	W	Missouri	100	52.07	21.844	7.877	2.1044	0.5794	0.18736	534
30	6	W	Cincinnati	100	47.93	19.623	6.978	1.9010	0.5275	0.16976	589
21	9	W	Tennessee	100	52.95	11.970	3.939	1.4547	0.3916	0.12428	805
31	7	W	Temple	100	52.85	14.729	6.102	1.4727	0.3715	0.11084	902
32	10	W	Penn St.	100	47.15	13.184	5.253	1.2949	0.3283	0.09705	1030
22	8	W	Michigan	100	47.05	9.633	2.951	1.0136	0.2518	0.07393	1353
23	5	W	Arizona	100	64.88	13.699	2.793	0.7924	0.1598	0.03829	2612
24	12	W	Memphis	100	35.12	4.599	0.562	0.1012	0.0126	0.00188	53133
25	13	W	Oakland	100	8.64	3.560	0.396	0.0681	0.0079	0.00111	89988
28	14	W	Bucknell	100	10.24	2.034	0.226	0.0148	0.0012	0.00011	935504
20	16	W	Hampton	100	2.29	0.325	0.027	0.0023	0.0001	0.00001	11921993
33	15	W	Northern Colorado	100	2.18	0.225	0.015	0.0003	0.0000	0.00000	283924556
36	1	SW	Kansas	100	98.23	74.808	53.807	34.8755	22.7012	12.19612	8.20
44	3	SW	Purdue	100	95.09	67.158	39.793	19.9643	11.1169	4.85640	20.6
51	2	SW	Notre Dame	100	94.48	69.823	38.343	18.6570	10.0540	4.27609	23.4
43	4	SW	Louisville	100	83.20	58.053	24.605	12.5245	6.3875	2.49977	40.0
38	9	SW	Illinois	100	55.97	14.822	7.233	2.9317	1.1871	0.34145	293
47	6	SW	Georgetown	100	57.52	19.716	8.116	2.6492	0.9904	0.26462	378
40	5	SW	Vanderbilt	100	61.46	24.637	6.773	2.3841	0.8385	0.20407	490
39	8	SW	Nevada Las Vegas	100	44.03	10.247	4.454	1.5950	0.5713	0.14397	695
48	7	SW	Texas A&M	100	56.37	18.024	5.986	1.6408	0.5253	0.11272	887
46	11	SW	Southern California	62.8	29.83	9.794	3.835	1.1966	0.4269	0.10962	912
41	12	SW	Richmond	100	38.54	11.894	2.391	0.6306	0.1666	0.02921	3424
49	10	SW	Florida St.	100	43.63	11.510	3.122	0.6739	0.1744	0.02766	3615
46	11	SW	Virginia Commonwealth	37.2	12.65	2.646	0.700	0.1366	0.0317	0.00470	21274
42	13	SW	Morehead St.	100	16.80	5.415	0.729	0.1338	0.0247	0.00283	35298
45	14	SW	St. Peter's	100	4.91	0.686	0.059	0.0035	0.0003	0.00001	9254589
50	15	SW	Akron	100	5.52	0.643	0.045	0.0023	0.0002	0.00001	18342534
37	16	SW	Boston University	100	1.77	0.123	0.009	0.0004	0.0000	0.00000	187954430
53	1	SE	Pittsburgh	100	97.80	72.996	44.334	28.3021	14.8112	6.69897	14.93
61	3	SE	Brigham Young	100	86.25	58.306	35.203	17.6695	8.2399	3.32266	30.1
60	4	SE	Wisconsin	100	72.83	44.945	24.059	14.5186	7.2797	3.20779	31.2
68	2	SE	Florida	100	86.84	54.880	29.759	13.8063	5.9385	2.19205	45.6
58	12	SE	Utah St.	100	52.33	24.144	10.166	5.0966	1.9999	0.66136	151
57	5	SE	Kansas St.	100	47.67	20.953	8.289	3.9709	1.4627	0.44823	223
56	8	SE	Butler	100	63.22	20.357	8.492	3.7786	1.3035	0.37840	264
64	6	SE	St. John's	100	55.32	22.533	10.194	3.6046	1.1722	0.31870	314
65	7	SE	UCLA	100	51.64	22.153	9.283	3.0833	0.9352	0.22991	435
66	10	SE	Michigan St.	100	48.36	20.196	8.230	2.6733	0.7939	0.19261	519
63	11	SE	Gonzaga	100	44.68	15.533	6.112	1.7531	0.4568	0.09473	1056
59	13	SE	Belmont	100	27.17	9.958	2.871	1.0459	0.2797	0.06073	1647
55	9	SE	Old Dominion	100	36.78	6.360	1.769	0.5415	0.1124	0.01790	5585
62	14	SE	Wofford	100	13.75	3.628	0.759	0.1046	0.0128	0.00118	84838
67	15	SE	UC Santa Barbara	100	13.16	2.771	0.460	0.0494	0.0046	0.00031	320919
54	16	SE	NC Asheville	77.7	2.17	0.286	0.021	0.0018	0.0001	0.00000	33535383
54	16	SE	Arkansas Little Rock	22.3	0.03	0.001	0.000	0.0000	0.0000	0.00000	57507213911169

That’s a bit hard to follow, so perhaps a pretty chart would be easier:

This helps us visualize which parts of the bracket are harder and which are easier.  In particular, Pitt’s bracket is really weak.  Ohio State is a solid favorite.

EDIT: Spreadsheets NOW AVAILABLE:

• Bayesian Rating Spreadsheet, including changes outlined above
• FULL NCAA Bracket spreadsheet, with all of the formulas for calculating odds.  A full matrix of odds is included.  Easy to use and manipulate for any sort of rating system.