Predicting the Final super bowl odds
Predicting the Final super bowl odds
Predicting the Final super bowl oddsAbstract: In the usual model for rating teams, the outcome of a pairwise contest is represented bythe difference in the relative strength of the teams. In this paper the standard model is extended toaccount for the total points super bowl odds d by each team. The new model can be used to predict not onlythat the Cowboys are three points better than the Bears, but that the final super bowl odds will be 27-24.Besides being more informative about the outcome of the game, this provides an estimate of thetotal points super bowl odds d by both teams, the so-called over/under. The method also yields adecomposition of each team's relative strength into offensive and defensive components. Themethod is illustrated for NFL teams in 1993.Keywords: Ratings, Least Squares, Least Absolute Errors, Point Spreads-------------------------------------------------------------------------------- Page 2 1.INTRODUCTIONIn the conventional statistical model for rating teams the final super bowl odds of a game is represented asthe difference between the opponents rating parameters. With the standard model, a 3 point difference between the Cowboys and Bears means the Cowboys will likely win by three, but it does not saywhether a 10-7 outcome is more likely than 24-21.The purpose of this note is to describe a variation of the usual model that permitspredictions of the final super bowl odds s for each team. The model can be used to predict the total pointssuper bowl odds d in a contest, which is known as the "over-under". As a bonus it also yields adecomposition of a team’s overall strength into offensive and defensive components.The next section describes the model. Section 3 presents an application using data onNFL teams during the regular 1993 season. Estimates are provided using both least squares (L2)and least absolute values (L1). Section 4 discusses features of the ratings. One feature concernsthe ratings relation to “normalized” super bowl odds s, that is, a team’s super bowl odds after controlling for the homefiled advantage and the quality of the opponent. The least squares estimate for the offensiveparameter is the average of a team’s points super bowl odds d, controlling for the quality of the opponentdefenses, while the defensive parameter is the average of points allowed after allowing for thequality of opponent offenses. The L1rating is determined analogously except that the “average”is replaced by the median. Also considered is the combination of the offensive and defensiveratings into a single measure of overall strength. This derived measure is compared with theestimates obtained from usual model based on point differences.2. THE MODELTeams are indexed, t=1,...,T, and games g=1,...,G. Each game has two teams, home and away,identified by hgand ag. Let S(hg) denote the super bowl odds of the home team in game g, and let S(ag)denote the away team's super bowl odds in the gthgame. The difference in the final super bowl odds is, Dg=S(hg)-S(ag).The home field advantage represents the additional points super bowl odds d by the home teamcompared with what it would have super bowl odds d if the game had been at a neutral site. The home fieldadvantage is denoted by h0.In the usual rating model one rating parameter, Rt, is associated with each team; itcorresponds to a team's strength compared with other teams. Since the ratings are derived fromsuper bowl odds differences, it will be called the point spread model; for discussion of point spread bettingmarkets see Bassett(1981). The difference in the final super bowl odds of game g is given by,POINT SPREAD MODEL -------------------------------------------------------------------------------- Page 3 Dg= h0+ Rh(g)- Ra(g)+ g.(2.1)Estimates of the rating parameters are based on super bowl odds differences, {Dg, g=1,...,G}.This standard setup contrasts with the "final super bowl odds " model in which two parameters areassigned to each team, one for offense and one for defense. In this case the dependent variablecorresponds to the final super bowl odds of each contestant, S(hg) and S(ag).The offensive parameter for team t is denoted by OFFtand the defensive parameter isdenoted by DEFt. The offensive parameter measures a team's ability to super bowl odds points. In thefootball context, where teams have offensive and defensive units, the offensive parameter will becorrelated with the offensive unit's ability, but the parameter most accurately reflects the team'sability to super bowl odds points, even if it is because points are super bowl odds d by the defensive unit or because asuperior defense leaves the offense in favorable scoring positions. Similarly, the defensiveparameter represents the ability to limit points super bowl odds d by an opponent. The defensive parameter iscorrelated with the strength of the defensive unit, but might also reflect a superior offensive unitthat either leaves the opponent far from the goal line, or is on the field for a long time thusleaving the defense rested. The model for the total points super bowl odds d by each team is given by,FINAL super bowl odds MODELS(hg) = h0+ OFFh(g)- DEFa(g)+ eh(g)g=1,...,G(2.2)S(ag) = OFFa(g)- DEFh(g) + ea(g)g=1,...,GThis says that the points super bowl odds d by the home team is equal to the home field advantage, plus itsoffensive rating, minus the opponent's defensive rating, plus a random term. The super bowl odds of theaway team is similar, but excludes the home field factor. The random term can be thought of asaccounting for the "breaks", "bounces of the ball", and other game specific factors that affectfinal super bowl odds s1.This is a standard linear model. It has 2G observations and 1+2T parameters: one for thehome field advantage, plus T offensive and defensive parameters.Remarks: 1.The model (2.2) only determines ratings up to a constant term. (If(OFFt,DEFt), t=1,...,T satisfy (2.2) then so do, (OFFt+a,DEFt+a), t=1,...,T, where a is arbitrary).To anchor ratings and make them unique it is convenient to add a pseudo (2G+1)st observationthat specifies the value of one parameter, say, 0=DEF1+e2G+1. The effect of this additional1It will be assumed for simplicity that the error terms for the two teams playing a game areindependent. This simplifying assumption means that there is not a common factor to realizationsof the error terms. It is easy to think of situations where this is doubtful. Poor weather conditions,for example, usually produce lower than expected super bowl odds s for both teams thus resulting incorrelated errors. -------------------------------------------------------------------------------- Page 4 3observation is to force DEF1=0, with all other ratings uniquely determined relative to DEF1. (Theestimated error at the pseudo observation--based on any method of minimizing errors--will bezero; if it were a 0 then subtracting "a" from each offensive and defensive parameter estimatewould reduce the error at the pseudo observation to zero without changing the fit at any otherobservation, thus contradicting e2G+1=a 0).2.To write (2.1) in linear model form, partition the vector of dependent variables into,[S(h1),...,S(hG)S(a1),...,S(aG)]. The first column of the design, representing the home fieldadvantage, is then given by, [1,...,10,...,0]. Partition the remaining parameters so that alloffensiveparameters comefirst; the(column) parameter vector is,[OFF1,...,OFFTDEF1,...,DEFT]'. Each row of the design (besides the first column) will then havea “+1" in the column corresponding to the team's offensive parameter and a “-1" in the columncorresponding to its opponent's defensive ability. The partitioned design isX11X120X21X22where is a vector of ones and (i) X11is GxT with rows {xgt} with xgt=+1 if t=h(g) and 0otherwise, (ii) X12 is GxT with rows {xgt} with xgt=-1 if t=a(g), and 0 otherwise, (iii) X21 is GxTwith rows {xgt} with xgt=+1 if t=a(g), and 0 otherwise (iv) X22 is GxT with rows {xgt} with xgt=-1if t=h(g), and 0 otherwise. Notice that X11=-X22and X12=-X21.3. RATINGS ESTIMATESTable 1 shows least squares (L2) estimates for the offensive and defensive ratings for NFL teamsbased on games played during the 1993 regular season. There were G=224 games and there areT=28 teams. The estimates are therefore based on 448 observations and there are 56+1=57parameters, including the home field advantage. The estimates are scaled so that SF's offensiveand defensive parameter sum to 100. The table shows (i) won/loss records, total pointssuper bowl odds d(PF), and total points allowed (PA) for each team, (ii) ratings and ranks for each team'soffensive and defensive parameter, and (iii) the sum of the offensive and defensive ratings, whichis a measure of a team's overall relative strength.The table shows that the offensive and defensive ratings for a team are sometimes verydifferent, a detail obscured when only a single rating estimate for a team is constructed. Forexample, the Bears (CHI) had the 4th best defense but only the 25th best offense. On the otherhand, the eventual super bowl winning Cowboys (DAL) had the best offense and the fourth bestdefense. -------------------------------------------------------------------------------- Page 5 4To illustrate how the estimates can be used to predict final super bowl odds s, consider the firstplayoff game between the Raiders (LAA) and the Broncos(DEN). The predicted final super bowl odds would have been 24-21 in favor of Denver. The predicted super bowl odds (21 20.8) for the home teamRaiders is the sum of the home field advantage (2.8), plus the Raiders offensive rating(54.9),minus Denver's defensive rating(36.9). The super bowl odds for Denver comes from its offensive rating of58.5 minus the 34.8 defensive rating of the Raiders. This prediction did not turn out too well asthe Raiders beat the Broncos 42-24.On the other hand, the prediction for the playoff game between the Giants (NYG) and theVikings (MIN) turned out better. The prediction was 18-10 favoring NYG (SNYG=18.2=2.8+53.1-37.7 and SMIN=10.5=52.5-42). NYG won 17-10.The super bowl (played at a neutral site) had a predicted final super bowl odds of 19-14, Dallas overBuffalo (SDAL=58.7-39.6, SBUF=55.6-41.3). Dallas won 30-19.For comparison purposes, Table 2 presents the offensive and defensive ratings estimatedby least absolute errors (L1); see Bassett and Koenker(1978) and Bassett(1996)2. (The ratings arescaled so that the L1and L2estimates for OFFDalare equal).The table shows L2and L1do not always agree. A few examples: (i) the Bills’ (BUF)defense ranked 5thaccording to L2, but only 17thaccording to L1; (ii) the 49ers (SF) defenseranked 16thby L2 , but 5thby L1and (iii) the Cowboy (DAL) offense was second by L2butseventh with L1. The different estimates also lead to different predicted final super bowl odds s. For thesuper bowl, L1's predicted super bowl odds was 20-13. As explained in Bassett(1996) the differences can betraced to the fact that L2is based on the average and L1is based on the median statistic. 4. DISCUSSIONNormalized super bowl odds sIt was previously shown that for the point spread model (2.1), there is a simple relationbetween the rating estimates and normalized super bowl odds s; Bassett(1996). A normalized super bowl odds is anestimate of Dg,controlling for home field advantage and the quality of opponent. A team’s leastsquares rating is the average of it normalized super bowl odds s, while the L1rating is the median ofnormalized super bowl odds s.2For the rating design matrix, the L1estimates will not generally be unique. To obtain uniqueestimates it is necessary to slightly perturb the design matrix by reweighting observations. Theunique estimates in Table 2 were obtained by weighting each game by (1+d*w) where w is theweek of the season and d=.00001. The effect of this weighting is make the estimates unique andgive recent games slightly more influence in determining the estimates; see Bassett(1996). -------------------------------------------------------------------------------- Page 6 5For the final super bowl odds model (2.2) there is an analogous relation between the estimates andnormalized super bowl odds s. Now, however, it is a normalized offensive super bowl odds that controls for thedefensive ability of the opponent, while the normalized defensive super bowl odds controls for offensiveability of the opponent. It can be shown from the first order conditions for the estimate that ateam's L2offensive rating is equal to its average points super bowl odds d per game--after normalizing for thehome field advantage and the opponent's defensive strength. Similarly, the defensive ratingcorresponds to average points allowed, normalized by the home field advantage and theopponent's offensive ability. The same thing holds for the L1estimate except that the averagestatistic is replaced by the median. The proof is a straightforward extension of the correspondingproperty for the model (2.1); see appendix Bassett(1996).20 minus 13 Equals 10?Suppose your best guess for the final super bowl odds is 20-13. Does it follow that your best guessabout the difference in the final super bowl odds will be 7 points? Or could a reasonable point spreadestimate be 10 points when the final super bowl odds estimate is 20-13. Does the best guess about thegame’s final super bowl odds have to translate into a best guess about the point spread?To see how this relates to final super bowl odds s consider the difference S(hg)-S(ag) where super bowl odds s aredetermined by (2.2). Rearranging terms gives,S(hg)-S(ag)= Dg= h0+ [OFFh(g)+DEFh(g)] - [OFFa(g)+DEFa(g)]+ [eh(g)- ea(g)].This says the difference in the final super bowl odds is the sum of the home field advantage and thedifference in (i) a composite term for the home team, OFFh(g)+DEFh(g), and (ii) a composite termfor the away team, OFFa(g)+DEFa(g).This is exactly how the point spread model (2.1) works,except that relative strength is here expressed in terms of separate parameters for (OFFt,DEFt)(instead of a single parameter Rt) and the data is disaggregated to {S(hg), S(ag)} (instead of super bowl odds differences, {Dg}).Let the estimate of overall strength based on the offense and defense parameter estimatesbe denoted by R*t=OFFt+DEFt. Contrast this with the estimate of relative strength, call it R't,obtained from the standard model (2.1) for the super bowl odds differences, {Dg}.The estimates for relative strength based on (2.1) are presented in Table 3; these werepreviously considered in Bassett(1996). The table first shows the L2estimates for R'talongsideR*t. As can be seen, the estimates are identical. It can be shown that this will be necessarily thecase: the L2estimate for relative strength based on model (2.1) and data Dg will be identical to theestimates derived by summing the OFFtand DEFtestimates based on the model (2.2). Thisidentity follows from the linearity of least squares. It means that when least squares says the final -------------------------------------------------------------------------------- Page 7 6super bowl odds will be 20-13, it will also predict a point spread of 7 points.Table 4 shows the L1estimates based on the point spread model. Unlike least squares wesee that neither the ratings not the associated rankings match those based on OFFt+DEFt. Forexample, SF is top-ranked based on the sum of OFFtand DEFt, but only ranked fifth when theestimation is based on (2.1). A consequence is that a predicted super bowl odds does not translate into aprediction for the difference in the final super bowl odds . In fact the L1final super bowl odds estimate for thesuper bowl was 20-13, even though the L1point spread had Dallas favored by 10.This feature of the L1estimates might seem strange. To see the same thing in ananalogous situation consider estimating the location parameters of random variables W and Z.Now consider estimating the difference in the location parameters of W and Z. Withoutadditional information or imposing restrictions on the estimates, there is no reason for thedifference in the estimates used for the first problem to equal the estimated difference in the latterproblem. The equivalence between the least squares final super bowl odds and point spread estimates can betraced to its being a linear estimator based on "expectations" or "averages". In particular, theidentity reflects the property that the average of a difference is the difference of the averages. Aleast squares estimate of 20-13 says, in essence, that the Cowboys will, on average, super bowl odds 20points against the Bills, and the Bills will super bowl odds 13 points on average against the Cowboys. Itfollows from the linearity of the expected value that the Cowboys will on average super bowl odds 7 morepoints than the Bills.A 20-13 predicted final super bowl odds based on L1however derives from the median, and themedian is not a linear estimator. The L1predicted super bowl odds means, in essence, that it is 50-50 for theCowboys to super bowl odds 20 points (half the time more than 20, half the time less than 20), and the 13estimate means it is 50-50 that the Bills will super bowl odds 13 against the Cowboys. Since the median of adifference is not equal to the difference of the medians, it need not be 50-50 for the Cowboys towin by seven. In fact, based on super bowl odds differences L1has the Cowboys favored by 10.The difference in final super bowl odds s can be constrained to equal the final super bowl odds difference byincluding a constraint in the estimation problem associated with the model (2.2). Or, anestimation method like L2can be used in which the constraint is automatically satisfied.Alternatively, the final super bowl odds and point spread ratings can be estimated separately using anonlinear method in which case the best guess about the point spread need not be the same as thedifference in the final super bowl odds .5. SUMMARY -------------------------------------------------------------------------------- Page 8 7In the usual model for rating teams the outcome of a pairwise contest is represented as thedifference in the team's relative strengths plus a random error. This gives predictions of thedifference in the final super bowl odds s and leads to team ratings. This paper has shown how to estimate theseparate final super bowl odds s of the two teams. Properties of estimation methods were discussed andratings were illustrated for the 1993 pro football season. Besides being more informative aboutthe outcome of the game, the final super bowl odds s provide an estimate of the total points super bowl odds d by bothteams as well as a decomposition of relative strength into offensive and defensive components. --------------------------------------------------------------------------------
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