FBS vs FCS Score Differentials Equated to FBS vs FBS Score Differentials

Every matchup is unique. Either team could win. Although related to the outcomes of the other matchups of either team, the outcome of any one matchup is somewhat independent of the others. This notion underlies the nature of competition, the allure of sports betting, and the precedence for retold stories of unlikely winners. For football, because of its small sample size relative to other games, this notion underlies the complexity of numerating many activities on the gridiron and is, to some extent, the topic of this post. 

A recent undertaking at work portends a new analytic technique: observed-score linking and equating. I will undoubtedly seek guidance from our expert colleagues, but, of course, I prefer to be informed before that day is upon us. Linking and equating have distinct definitions, applications, and procedures but I will refer to these casually as equating. Equating allows us to generate uniform score-ranges between sections or items belonging to different versions of a single assessment, two unique assessments, or an old and a new version. 

More practically, consider the ACT, for example. Let us imagine that ACT Inc (the ACT developer) develops 20 versions of the ACT Reading Section. ACT Inc needs the scores for each version to be equitable so that a 36 is always a 36. Of the imaginary 20 versions, let us focus on Versions 6 and 12, or V6 and V12, for short. So, to test these versions, ACT Inc has 200 freshmen in college complete both versions. Say, 100 freshmen completed V12 in the first test session and V6 in the second whereas the other 100 freshmen completed V6 in the first and V12 in the second session. Afterwards, ACT Inc realizes that the average score for V6 is 18.5 and the average for V12 is 20.5, whoops. However, the average for all tests completed in the first session is 20.4 and all tests completed in the second session is 20.3, so ACT Inc knows that the disparity in Version-scores is not due to sequence of test administration. Likewise, because the same freshmen completed both versions, the 2-point disparity in Version-scores is not due to differences in the test-takers. ACT Inc must conclude that the disparity is due to differences in V6 and V12. Then, ACT Inc could use equating procedures to develop uniform scores to ensure little Johnny sets realistic standards for his future based on an ACT Reading Version 12 score of 30 instead of the inflated 36 it would have been without equating.

Here, I use equating to generate equivalency score-differentials for interdivisional college football games. That is, a 35-point win (or, +35 score differential) by a FBS team over a FCS team, for instance, is equivalent to what differential in an FBS versus FBS matchup. Let us relate this to the above example. This analysis would get restrictively complex if we sought to equate scores between all FBS and FCS teams—ACT Reading V6 and V12 would be tantamount to FBS Teams 1, 2, 3, …, 128! However, for FBS and FCS programs alike, most matchups each season are versus FBS and FCS foes, respectively. Like many FBS teams face a smattering of inferior opponents with FCS status, many FCS teams face a few inferior opponents with DII or NAIA memberships. So, we can consider two types (or versions) of games: [i] intradivisional games and [ii] interdivisional games. Intradivisional games are FBS vs FBS or FCS vs FCS whereas interdivisional games are FBS vs FCS or FCS vs non-DI. Thus, if the distributions for score differentials of FBS-FBS and FCS-FCS games are similar, and the same is true for FBS-FCS and FCS-non-DI, we can generate FBS-FCS scores that equate to FBS-FBS scores.

Chart 1: Distributions of Score Differentials

First, I obtained all Division I NCAA football game scores for 2012-2016 from this vast resource hosted by Kenneth Massey, that includes 8,349 games in which either an FBS or FCS team played. Second, I specified whether the home team won each game because home advantages are well-documented (here, here, here, but cf. here). Third, I specified one of four classifications for each game, the first two of which are intradivisional and the second two, interdivisional:

•    FBS vs FBS,
•    FCS vs FCS,
•    FBS vs FCS, or
•    FCS vs non-DI teams.

Fourth, I removed all games in which both teams did not play in an interdivisional game in that season, leaving 6,397 games for the analysis. For example, in 2012, neither UCLA nor USC played an FCS team so, the UCLA vs USC game was excluded from the analysis. However, the 2012 USC versus Washington game was included because Washington played a FCS team (Portland St.). The data was prepared in this manner because I only want to analyze score differentials of teams that played both types of games. That is, although it is only one FCS game, we know about FBS-FBS and FBS-FCS games that involve ’12 Washington whereas we only know about FBS-FBS games that involve ’12 USC.

Chart 2: D1 Teams Ranked by Win% and Mean Score Diff.

Fifth, I prepared Chart 1. It shows the distributions of score differentials for the four categories of games. Chart 1 demonstrates that FBS vs FBS scores (green) differentials are distributed almost identically to FCS vs FCS (brown) score differentials. Likewise, the score differentials are similarly distributed for the interdivisional games, but with some distinct dissimilarities. I attribute the dissimilarity in interdivisional distributions to the similar talent levels of lesser-FBS/better-FCS teams and lesser-FCS/non-DI teams while, concurrently, more better-FBS teams play FCS opponents (green) than better-FCS teams play non-DI opponents (orange). Hence, there are more 35-point blowouts in FBS-FCS games. This is evident in the ad hoc chart below, which was the sixth thing I did. 

Anyhow, because the distributions for FBS vs FBS and FCS vs FCS are nonetheless similar, we will consider in the analysis only home field advantage and whether a game was intra- or inter-divisional (i.e., we will ignore whether a team was FCS or FBS). I do this for simplicity—mostly for me, but maybe also for you. 

Seventh, the equating procedure was performed using a nonequivalent-groups design with one anchor, a home team win. Here, the anchor informs the equating procedure that differences in these games might be due to home-field advantage. The influence of including home team victory is evident in Chart 3. The black line represents the intradivisional score differential and the other lines are the corresponding interdivisional scores with or without home advantage. Some descriptive statistics appear in the table below. A table with unadjusted and adjusted score differentials and SEs appears at the close of the post.

Table 1. Descriptive Statistics for NCAA 1 D1 Intra- & Inter-Division Games, 2012-16

	mean	sd	skew	kurt	min	max	n
Intradivisional	17.49	13.5	0.96	3.53	1	78	5493
Interdivisional	30.52	19.69	0.36	2.29	1	86	904
Intra- Home Wins	0.54	0.5	-0.15	1.02	0	1	5493
Inter- Home Wins	0.87	0.34	-2.16	5.67	0	1	904

Chart 3: Equated Interdivisional Score Differentials

Controlling for home-advantage—the green line—produces equated scores which are more sound, in my estimation. Notice how the green line equals the black line in the bottom left corner. The green line diverges at the 7-point differential. So, with this equating procedure, if an FBS team wins by 7 or fewer points over an FCS, it is the same differential as an FBS-FBS victory. To this author, this validly reflects in the score differential the competitiveness of an FBS-FCS game decided by one touchdown or less. Without adjusting for home winning, there are inflated point differentials in this range. Also, compared to the orange and the black lines, there is less of a difference between the green and black lines as the score differential increases (if such a feat were meaningful, Baylor). Likewise, Iowa St. is not additionally penalized for succumbing to a last-second field-goal whereas the orange line equates a 7-point FBS-FCS victory to 16 FBS-FBS points and a field-goal lead at 00:00 in the 4th quarter to 5 points. 

Now, there are of course shortcomings to this study, primarily one. Recall in the verbose example I provided earlier that the same 200 college freshmen completed both V6 and V12 of the ACT Reading sections. By doing so, we could be relatively certain that any disparity in V6 and V12 averages was not due to the test takers.  In the analysis, however, I included only games involving at least one team that played in intra- and inter-division games in the season. Thus, this analysis rests on the potentially fallible assumption that all intra- or inter-divisional opponents to these teams are identical—which is patently untrue. Hence, the reason we considered the distribution of different classifications of games in Chart 1.

Summarily, an equating procedure was used to generate score-differential equivalencies for FBS-FCS games to FBS-FBS games. This author concluded that adjusting for well-documented home field advantages provided more valid equivalencies. Secondarily, an ad hoc analysis demonstrated that upper echelon FBS teams more frequently play FCS opponents than upper echelon FCS teams play non-DI teams.

	Adjusted			Unadjusted
FBS Scr Diff	Est.	SE		Est.	SE
1	0.974	0.2		1.358	0.175
2	1.743	0.285		2.672	0.21
3	2.81	0.208		5.106	0.779
4	3.762	0.756		7.546	0.79
5	5.054	0.896		9.956	1.156
6	6.127	0.8		12.3	1.166
7	7.359	0.891		15.918	1.072
8	10.068	1.461		19.907	1.127
9	10.968	1.578		20.872	0.771
10	13.186	1.415		22.593	1.161
11	14.47	1.015		24.394	0.867
12	15.391	1.017		25.328	0.954
13	16.529	1.173		26.464	1.014
14	18.299	1.438		28.122	1.024
15	20.713	1.223		30.582	0.942
16	21.174	1.141		31.068	0.82
17	23.077	1.324		32.161	0.962
18	24.61	1.284		34.025	0.9
19	26.413	1.334		35.059	1.109
20	27.782	1.262		36.855	1.117
21	30.308	1.197		38.215	0.688
22	31.551	0.981		39.434	0.943
23	32.615	1.131		40.542	1.061
24	34.24	1.375		41.97	1.075
25	37.279	1.458		44.231	1.228
26	38.21	1.093		45.077	1.165
27	39.011	1.182		45.894	1.214
28	41.563	1.188		47.865	1.147
29	42.478	1.111		49.033	1.238
30	44.187	1.182		49.589	1.415
31	45.563	1.132		51.843	1.423
32	47.876	1.165		53.61	1.266
33	48.85	1.181		54.631	1.102
34	50.254	1.454		55.331	0.775
35	52.735	1.534		56.108	0.747
36	54.72	1.304		56.876	1.063
37	55.346	1.134		58.072	1.234
38	56.116	1.085		59.217	1.359
39	57.228	1.251		61.627	1.452
40	58.761	1.543		62.474	1.195
41	59.231	1.681		62.889	1.011
42	61.718	1.732		63.532	1.122
43	62.719	1.574		64.919	1.179
44	62.99	1.426		65.649	1.247
45	63.557	1.384		66.142	1.177
46	65.6	1.338		66.812	1.522
47	65.788	1.319		67.265	1.673
48	65.991	1.261		68.845	1.777
49	66.467	1.093		69.941	1.934
50	67.188	1.348		71.796	2.045
51	68.502	1.659		72.619	2.149
52	69.592	1.949		73.615	1.891
53	70.118	2.188		74.109	1.831
54	70.359	2.166		74.335	1.861
55	72.129	2.096		75.157	1.77
56	73.833	2.102		76.604	1.884
57	74.595	1.891		77.468	1.617
58	75.772	1.74		77.78	1.724
59	77.016	1.535		78.302	2.118
60	77.7	1.329		78.892	2.151
61	77.876	1.252		79.221	2.14
62	78.095	1.368		79.633	2.24
63	78.405	1.623		80.209	2.572
64	78.877	1.726		81.62	2.747
65	79.142	1.818		81.785	2.716
66	79.672	1.996		82.114	2.65
67	80.336	2.151		83.525	2.608
68	81.733	2.146		83.772	2.547
69	82.131	2.244		84.019	2.447
70	83.794	2.309		84.43	2.217
71	84.192	2.288		85.677	1.879
72	84.325	2.255		85.759	1.872
73	85.59	2.218		85.924	1.778
74	85.855	2.275		86.089	1.748
75	85.988	2.341		86.171	1.739
76	86.116	2.343		86.253	1.714
77	86.244	2.251		86.335	1.633
78	86.372	2.17		86.418	1.613