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Adjusting Passer Rating for Era: Part V: The Results

Background reading:

Part I

Part II

Part III

Part IV

All week, I have been discussing how to adjust passer rating by era. Now that I have explained the formula, it’s time to generate the results. In a given season, ratings won’t change (unless a player moves below or above a limit as a result of the era adjustment), so the most interesting thing to do is to present career passer ratings.

To calculate career passer ratings, I first calculated each player’s passer rating in each season. Then, I created their career rating by averaging the player’s passer rating in each season, weighted of course by their number of attempts in that season. And now, the results.

The table below shows all 185 players with at least 1500 career pass attempts (this includes the 2016 season). Here is how to read the table below. Otto Graham is the career leader in era adjusted passer rating (this includes his AAFC time). He ranks 115th in career pass attempts with 2,626. Since passer rating is the sum of four variables multiplied by 100 and divided by 6, I figured we might as well present the era adjusted variables, too. In completion percentage, Graham scores a 1.40; in yards per attempt, he is at a whopping 1.53; in touchdown percent, 1.25, and in interception percentage, a remarkable 1.53. As a result, his era adjusted passer rating is 95.2.

Att RkQuarterbackAttCmp% VarY/A VarTD% VarINT% VarEA Pass RtgRk
115Otto Graham26261.401.531.251.5395.21
162Sid Luckman17441.331.451.451.4193.92
96Sammy Baugh29951.581.201.171.5692.03
63Len Dawson37411.401.221.341.2486.64
50Steve Young41491.361.291.321.2286.55
31Aaron Rodgers46571.181.201.411.2984.76
100Roger Staubach29581.231.251.211.3984.67
17Joe Montana53911.361.131.181.3583.88
179.5Frankie Albert15641.271.001.351.2981.89
105Norm Van Brocklin28951.241.291.181.1881.610
2Peyton Manning93801.271.191.321.0981.011
48Sonny Jurgensen42621.271.111.191.2680.312
5Tom Brady82241.151.131.261.2680.113
39Ken Anderson44751.281.131.041.3379.814
53Kurt Warner40701.291.271.221.0079.815
11Fran Tarkenton64671.251.081.121.3279.716
44Tony Romo43351.211.211.291.0479.217
75Bob Griese34291.251.151.251.0979.018
88.5Bart Starr31491.301.150.931.3378.719
3Drew Brees87581.281.131.211.0878.620
141.5Russell Wilson22811.121.201.231.1378.121
41Y.A. Tittle43951.281.101.051.2477.822
22Johnny Unitas51861.161.141.091.2377.223
120Bert Jones25511.131.091.141.2777.224
4Dan Marino83581.151.111.201.1676.925
118Daryle Lamonica26011.011.141.331.1176.526
15Philip Rivers59171.151.171.201.0375.927
14Ben Roethlisberger59321.151.211.161.0375.728
108Charlie Conerly28331.080.991.181.2975.729
126Chad Pennington24711.301.091.011.1375.430
16Dan Fouts56041.211.191.061.0675.431
146Frank Ryan21330.981.071.351.1075.132
27Jim Kelly47791.171.141.200.9874.933
87Neil Lomax31531.111.051.021.3274.934
174Bob Waterfield16171.201.141.101.0474.635
67Jeff Garcia36761.121.041.081.2374.336
101Danny White29501.211.101.190.9674.337
84Daunte Culpepper31991.181.181.120.9774.138
24Matt Ryan50641.171.081.081.1174.139
36Roman Gabriel44981.040.910.961.5073.640
138Greg Landry23001.151.061.001.2173.641
49Rich Gannon42061.121.011.061.2273.542
1Brett Favre101691.161.061.200.9873.343
60Ken Stabler37931.301.141.190.7773.344
165Tom Flores17151.081.061.081.1873.245
112Earl Morrall26891.001.141.181.0773.246
77Bernie Kosar33651.160.990.931.3073.047
64Trent Green37401.081.201.051.0573.048
61Craig Morton37861.071.141.101.0472.649
97Billy Kilmer29841.061.021.121.1372.350
20Dave Krieg53111.101.061.171.0172.351
70.5Joe Theismann36021.091.011.051.1972.352
46Randall Cunningham42891.021.021.151.1372.153
70.5Ken O'Brien36021.141.000.901.2872.054
33Mark Brunell46401.061.040.991.2271.955
80Matt Schaub32741.151.140.961.0571.856
21Boomer Esiason52051.031.101.151.0371.757
137Don Meredith23080.951.051.121.1771.658
179.5Tony Eason15641.131.020.941.2071.559
18Donovan McNabb53740.961.021.061.2471.360
182Kirk Cousins15561.161.151.040.9371.261
154Gary Danielson19321.101.041.001.1471.262
8Warren Moon68231.091.081.051.0471.263
29Troy Aikman47151.211.040.911.1071.164
82Neil O'Donnell32291.010.990.971.3071.165
30John Hadl46871.051.101.130.9871.066
74Brian Sipe34391.101.001.061.1070.967
37.5John Brodie44911.180.980.961.1470.968
122Billy Wade25231.161.000.921.1870.869
130Milt Plum24191.140.980.941.1970.770
73Steve Bartkowski34561.051.011.061.1270.771
32Phil Simms46470.991.051.031.1670.672
45Brad Johnson43261.160.990.971.1070.473
6John Elway72501.021.051.021.1370.474
58Terry Bradshaw39010.941.101.240.9470.375
152Bill Munson19821.110.900.901.3070.376
34Steve McNair45441.081.030.961.1570.377
66Bobby Layne37001.011.061.031.1070.178
13Carson Palmer60401.071.071.090.9670.079
135Jeff Hostetler23381.021.061.041.0770.080
141.5David Garrard22811.051.020.961.1669.981
79Steve Beuerlein33280.981.111.101.0069.882
86Marc Bulger31711.111.080.951.0569.883
56Jeff George39671.031.041.001.1169.784
119Jim McMahon25731.091.020.971.1069.785
110Brian Griese27961.181.041.040.9269.686
91Bobby Hebert31211.121.021.080.9469.487
117Mark Rypien26130.961.041.101.0669.488
128Bill Kenney24300.951.011.011.1769.189
155.5Bill Nelsen19050.951.121.061.0068.990
26Jim Everett49231.051.051.031.0068.891
55Chris Chandler40051.031.081.060.9568.792
62Joe Namath37621.041.161.010.9168.793
94Andy Dalton30601.051.021.050.9968.594
127Elvis Grbac24451.071.031.020.9968.595
51Ron Jaworski41170.910.971.031.1567.896
170Colin Kaepernick16920.881.020.971.1967.697
54George Blanda40070.990.981.180.8867.498
57Jim Harbaugh39181.080.970.871.1067.399
25Steve DeBerg50241.080.960.961.0367.2100
37.5Jay Cutler44911.031.041.070.8867.1101
76Charley Johnson33920.971.031.021.0067.0102
129Wade Wilson24281.061.051.010.9066.9103
19Matt Hasselbeck53301.010.990.961.0466.8104
47Matthew Stafford42850.990.981.001.0366.7105
102Jake Delhomme29320.971.071.040.9266.6106
81Jeff Blake32410.920.991.041.0466.5107
68Tommy Kramer36511.000.951.041.0066.4108
52Alex Smith41081.020.920.901.1466.4109
90Lynn Dickey31251.051.131.050.7566.3110
113Andrew Luck26510.861.011.120.9966.3111
23Jim Hart50760.921.010.961.0866.1112
72Steve Grogan35930.901.151.180.7365.9113
69Archie Manning36421.090.940.841.0765.8114
139Erik Kramer22991.000.971.010.9665.8115
78Babe Parilli33300.951.001.070.9265.7116
145Doug Flutie21510.871.011.001.0665.6117
7Eli Manning68250.931.011.090.9165.5118
103Cam Newton29280.831.071.050.9865.4119
106Chris Miller28920.900.941.061.0265.4120
99Aaron Brooks29630.861.011.021.0265.2121
124Stan Humphries25160.961.020.931.0065.2122
114Ryan Tannehill26371.030.960.921.0165.1123
177Charlie Batch16040.921.030.961.0165.1124
163Derek Carr17320.900.811.041.1565.1125
65Jim Plunkett37010.941.041.040.8865.1126
158Eddie LeBaron17961.011.081.120.7165.1127
9Drew Bledsoe67170.950.970.951.0365.1128
160Mike Livingston17511.000.950.801.1565.1129
28Joe Flacco47421.000.940.901.0665.0130
35Joe Ferguson45190.900.931.031.0465.0131
123Jason Campbell25180.970.920.861.1564.9132
164Jay Fiedler17170.991.031.000.8664.8133
169Steve Bono17010.880.840.941.2264.7134
133.5Scott Mitchell23460.900.991.030.9664.6135
176Byron Leftwich16050.910.930.901.1364.5136
107Sam Bradford28441.040.860.821.1664.5137
40Jon Kitna44421.050.970.950.8864.3138
83Michael Vick32170.801.010.991.0664.3139
10Vinny Testaverde67010.951.011.020.8564.0140
88.5Jim Zorn31490.940.960.871.0764.0141
104Tobin Rote29070.850.881.011.1063.9142
111Kyle Orton27120.930.910.911.0963.9143
155.5Don Majkowski19050.970.920.891.0563.9144
125Doug Williams25070.730.960.961.1863.7145
175Vince Ferragamo16151.000.991.080.7563.6146
42Norm Snead43531.021.000.910.8863.5147
144Bubby Brister22120.910.910.941.0563.5148
151Ed Brown19870.861.140.970.8463.3149
166Tim Couch17141.060.940.960.8463.2150
109Jay Schroeder28080.731.051.010.9862.9151
98Richard Todd29670.961.000.990.8062.4152
133.5Rodney Peete23460.981.040.860.8662.3153
93Jack Kemp30730.951.030.751.0062.2154
43Jake Plummer43500.930.980.930.8862.0155
92Gus Frerotte31060.821.020.940.9361.9156
183Eric Hipple15460.910.970.870.9561.6157
132Tony Banks23560.840.940.861.0461.4158
161Frank Tripucka17451.150.800.770.9661.3159
116Matt Cassel26250.890.880.940.9661.2160
143David Carr22671.000.890.751.0060.7161
12Kerry Collins62610.840.930.861.0160.6162
159Cotton Davidson17520.800.970.821.0460.5163
59Ryan Fitzpatrick38760.920.890.990.8260.3164
147Josh McCown21210.910.920.880.9160.2165
95Dan Pastorini30550.920.840.840.9959.9166
136Mike Tomczak23370.841.000.960.7959.9167
149Josh Freeman20480.840.920.930.8859.7168
148Marc Wilson20810.840.970.980.7959.5169
131Kordell Stewart23580.910.880.850.9259.2170
85Trent Dilfer31720.870.930.910.7958.2171
153Chad Henne19540.920.890.730.9157.5172
167Billy Joe Tolliver17070.770.860.890.9257.3173
168Blake Bortles17060.790.850.920.8556.7174
173Dave Brown16340.850.880.750.9156.6175
181Rex Grossman15620.760.890.890.8155.8176
140Mark Sanchez22850.780.890.890.7755.6177
121Joey Harrington25380.820.740.800.9555.3178
185Mike Pagel15090.710.790.811.0055.3179
172Jack Trudeau16440.850.820.700.9455.1180
184Kyle Boller15190.840.750.790.9355.1181
178Derek Anderson15960.650.890.910.8354.8182
171Mark Malone16480.790.790.890.7954.3183
157Mike Phipps17990.830.780.760.8253.2184
150Rick Mirer20430.770.780.720.8752.4185

Graham and Len Dawson both rank in the top 4, and both benefited to some extent by player in expansion leagues with weaker competition for the early parts of their careers. both were Hall of Famers, of course, but those numbers are arguably (at least slightly) inflated. Seeing Sid Luckman and Sammy Baugh in the top three is hardly a surprise, either.

One fun thing to do is to sort by attempts to see who stands out. Brett Favre ranks 43rd in era adjusted passer rating, Peyton Manning is 11th, and Drew Brees is 20th. Dan Marino is 25th, but like Manning, Marino is harmed a bit by ignoring his outstanding sack rate. Tom Brady ranks 13th, while John Elway is down at 74. Eli Manning is way, way down at 118 — a hair below his father, Archie Manning. That is lower than any Hall of Famer, for those curious.

The five lowest HOF QBs on the list are John Elway, Terry Bradshaw, Bobby LayneJoe Namath, and George Blanda. Namath had three terrible years when he was washed up, and was outstanding at avoiding sacks, which is why he’s underrated; given Namath’s focus on big plays and avoiding sacks, he’s never going to fare well in passer rating, but his era adjusted rating does rise to 72.9 if you remove his final three seasons.

Blanda was another guy who was incredible at avoiding sacks, and who focused on big plays rather than completion percentage (whereas a contemporary like Dawson’s game was much more about completion percentage and avoiding interceptions). He also is perhaps the weakest quarterback in the Hall of Fame. I’ve written before about the stats of Bradshaw and Elway; Layne’s stats have often underwhelmed me, but Brad Oremland has written some good words about him.

On the other side, you won’t be surprised to see Ken Anderson — whose game was well-designed for passer rating — excels here.  Other than Frankie Albert, who played most of his short career in the AAFC, he has the best passer rating of any retired player not in the HOF.  Kurt Warner is right there, too (though his stats are arguably inflated, too), with Daryle Lamonica (a player whose game was decidedly not passer rating-friendly) and Bert Jones not far behind.

What about the players with the worst passer ratings? The table is fully sortable, so you can see that Rick Mirer, Mike Phipps, Mark Malone, Derek Anderson, Jack Trudeau, Kyle Boller, and Joey Harrington have the lowest passer ratings.

What do you guys think?

  • Michael Carlson

    Anything that agrees with my ranking of Otto Graham pleases me. Did you run his stats absent his AAFC years? Dawson has always been my all-AFL QB & arguably the most underrated. But I wonder if you could put him in a grouping with say Starr and Griese style and team wise? Have you compared your list to the one Maxymuk did for QBA, basically adjusting passer rating for era but I believe he did with a straight adjustment of the rating, not breaking down the elements. Which as you pointed out early, are not aligned reasonably. Thanks for the work…

    • Thanks, Michael! Yes, good on Otto. He is at 88.9 without his AAFC years, so that’s not a huge change. Here is his yearly grade:

      Year Player Att CMP Y/A TD INT EA PR %CarAtt Rating * %CarAtt
      1946 GrahOt00 174 1.31 1.96 1.77 2.35 123.4 6.6% 8.18
      1947 GrahOt00 269 1.58 1.76 1.58 1.76 111.3 10.2% 11.40
      1948 GrahOt00 333 1.16 1.27 1.31 1.54 88.0 12.7% 11.16
      1949 GrahOt00 285 1.46 1.67 1.27 1.94 105.7 10.9% 11.47
      1950 GrahOt00 253 1.38 1.23 1.09 1.01 78.5 9.6% 7.56
      1951 GrahOt00 265 1.44 1.37 1.25 1.35 90.3 10.1% 9.11
      1952 GrahOt00 364 1.17 1.27 1.02 1.20 77.6 13.9% 10.75
      1953 GrahOt00 258 1.87 2.01 0.97 1.92 112.7 9.8% 11.08
      1954 GrahOt00 240 1.44 1.38 0.92 0.97 78.4 9.1% 7.17
      1955 GrahOt00 185 1.25 1.67 1.68 1.61 103.5 7.0% 7.29
      100.0% 95.2

      • mrh

        Great series of articles. I’m a fan of Otto so I don’t want to diminish his #1 ranking, and he’s a top 3 guy even excluding his AAFC years. Still, he never had to play against the best defense in his league – which per DSRS was always Cleveland’s. In a 32-team league, that may not be significant, but in the much smaller AAFC that would inflate his numbers. It might not be enough to make a difference…

        • The other piece of it, at least in his case, is that his numbers boost the league average significantly more in a 10-year league than in a 32-team league.

  • Adam

    Very surprised to see Kordell Stewart near the bottom. I remember him being better than that, although PR ignores his Slash capabilities.

    • For what it’s worth, I currently have him ranked 53rd since 1992 by NewTAY/P+. Right between Jim Everett and Aaron Brooks.

      • Adam

        That makes a lot more sense.

    • One of the reasons I included the four component variables in the table was to make the whole thing more accessible to the reader. So if you are surprised to see someone in a certain ranking, this table should at least make the formula more transparent.

      For Kordell, he’s extremely consistent — coming in a bit lower than average in all four variables. So maybe it’s not all that useful for him. I guess if you had asked me how I thought he would do, I would have guessed bad in completion percentage and TD%, but maybe better in Y/A and INT%. His one year in CHI hurt him — without that, his INT variable would be at 0.96, and his EAPR would be 60.9.

      But in general, his completion percentage and TD rates were not good. Obviously the TD thing is biased by the fact that Pittsburgh ran for a lot of TDs, including rushing TDs by Kordell. His CMP% was never good, and in ’00 he was 3rd worst ahead of only Leaf and Akili Smith.

      Perhaps the most surprising thing to me was his Y/A was not good, which means his Yards per Completion were not great. Kordell started only 16 games in 3 seasons. In 1997, he averaged 12.8 Y/C, which was 7th best. But In 1998, he averaged 10.2 Y/C, 29th in the NFL, and 3rd worst. And in 2001, he was at 11.7, which ranked 13th.

      And he was even worse in other years. So a mediocre Y/C average, combined with a bad comp% is going to lead to a bad Y/A. And his TD and INT rates were below average, too. Add it all up, and that’s a pretty bad passer rating.

      • Adam

        I figured his Cmp% and TD% would be pretty bad, but thought his Y/A would be average and his INT% better than average. This is where the human memory fails because I don’t remember him throwing many INT’s. Frankly I thought Kordell’s passing profile was a carbon copy of Michael Vick’s, but obviously the truth indicates otherwise.

  • AgronomyBrad

    Looking at the table you have displayed makes me wonder how MASSIVE your original spreadsheet must be, and how long this whole project must have taken. I appreciate the effort, there’s a lot of good information in here.

    • Believe it or not, it wasn’t that much work to compile the career ratings. Most of the work was in coming up with the formulas. But thanks, anyway!

  • Josh Sanford

    This probably should be posted to one of the earlier articles that focused on INTs and the ‘enjoyability’ of watching games with turnovers…but I looked at George Blanda’s 1961 season (the year that he led the league in TDs and won an AFL championship). Here are the game-by-game totals for both teams’ combined turnovers over the total of 15 games: 6-8-5-8-4-2-8-8-5-5-14-7-4-3-13. Yikes! If my numbers are right, that is 98 turnovers in 15 games, or just over 6.5 turnovers per game.

  • Winston Wood

    As someone who has done my own era-adjusted analysis to determine the best QB of all time, I found this series very enjoyable to read. My analysis arrives at a similar ranking, although I had a somewhat different approach and took into account more variables than just adjusting passer rating for era. One of the biggest differences in my analysis is that I have incorporating rushing yards and touchdowns. When deriving my methodology I asked the question: What is the objective of football and the quarterback’s role in accomplishing that objective? Of course the objective of football at the most basic level is to score more points than the other team. Since quarterback’s don’t play defense, they have little impact on the number of points that the opposing team scores, so a QB’s objective is to score as many points as possible. He achieves that by moving his team down the field in order to score points. Excluding rushing yards by a running back, which the QB only affects by executing a successful handoff, the QB moves his team down the field by either completing passes or by running the ball himself. Since successful drives can only be successful when turnovers are avoided, the lower the turnover rate, the more successful the quarterback can be in accomplishing the objective. So ultimately, I believe the best measures to compare quarterbacks are net points per play and yards per play, both of which including QB rush attempts, rushing yards and rushing TDs. Finally, to normalize for era, I calculate seasonal indeces for these metrics (and others) based on those for the average starting quarterback for each season. I should also caveat that these measures are based on regular season statistics only, which even for players like Tom Brady that have been in a relatively large number of post-season games, still accounts for the majority of career performance. My results? I too have Otto Graham #1 and likewise have Steve Young in the top 5 (#2), a guy most people erroneously leave even out of their top 10. My full top 10:

    1. Otto Graham
    2. Steve Young
    3. Aaron Rodgers
    4. Peyton Manning
    5. Joe Montana
    6. Roger Staubach
    7. Tom Brady
    8. Fran Tarkenton
    9. Dan Marino
    10. tie, Drew Brees & Len Dawson

    Of course my analysis, and yours, is purely statistical, and doesn’t take into account a lot of other factors, mostly outside of the quarterback’s control. But, having seen all but Graham and Dawson play, I feel pretty good about who is in my top 10. I would probably put Montana ahead of Manning and Rodgers. A lot of folks have Elway, Favre and Unitas in their top 10s, and although the numbers don’t support them being that high, I would have them in my next 10.

  • George Rising

    Great work, Chase! This is really useful. The passing numbers have really gone through the roof recently, so it’s hard to judge how good passers are now. I suggest you do future projects adjusting rushing and receiving stats, too. (Adjusted for era, Don Hutson was a BEAST!)

    Coincidentally, I just finished a similar project on adjusting passers, and my results are similar. A few comments:

    1) I think it’s valid to distinguish between passer evaluation and quarterback evaluation. The latter includes passing but a lot of other elements, too, especially rushing. Consequently, I ignore anyone who rejects passing stats because they are not complete evaluations of quarterbacking. I believe passer ratings in and of themselves are valid and interesting.

    2) In my rating system, I excluded AAFC and AFL stats because the numbers show that those leagues were weak (however, the AFL was pretty good by 1967 or 1968). Luckily, ignoring AAFC and AFL really only affected Dawson among the top guys. (Graham’s NFL numbers are nearly as good as his AAFC numbers.) By the way, Dawson was by far the best passer (and quarterback) in the AFL, so the fact that the HOF named him second team all-AFL for the 1960s behind Namath is a travesty. Also, George Blanda had exactly one good season as a quarterback, so to see him listed as a QB in the HOF is ridiculous.

    3) I was surprised to discover that the decade of the 1970s was such a dead-ball era for passing stats, even worse than the late 1940s and 1950s (and 1960s, of course). Consequently, guys like Staubach, Griese, and especially Anderson are underrated as passers. If they had come along 10-15 years later, their stats and reputations would rival Montana’s and Marino’s. Anderson is HOF material, no doubt. More than that, he’s one of the top ten best passers ever. (He ranks 7th in my system, just below Montana.) And even Bradshaw is not as bad as advertised; his adjusted passing stats starting in 1975 and after are actually excellent. (A surprise to me, because I always viewed him as vastly overrated.

    4) In my system, I excluded years before 1947. If you look at the yearly NFL passing statistics league-wide, the numbers are very low until about 1947 when they start to have a more modern totals/averages.

    5) However, I did a side project including 1932-1946 passing stats. Like your system, mine rated Baugh and Luckman very highly. But it also ranked Herber, Danowski, and Isbell equally highly.So I’m surprised that they didn’t make your list.

    • George Rising

      Sorry, on my point #5 above: setting a minimum of 1500 passing attempts keeps Herber, Danowski, and Isbell off the list. My apologies.

    • sacramento gold miners

      Ken Anderson’s last two postseason games were disappointing, and those were golden opportunities to help his HOF case. In SB16, Anderson was bad in the first half, as the Niners raced out to a 20-0 lead. The Bengals tacked on a very late TD, but all that did was pad his stats. The following season, the top seeded Bengals were destroyed at home by the Jets, as more Anderson mistakes helped put his team in a big hole.

      Ken Stabler was a better, more dangerous QB in the 70s, with a superior postseason resume.Anderson was a very efficient QB, but comebacks were a weakness, he has a very low total. Had Anderson done more with his playoff chances he would be in the HOF already.

  • Joseph

    Great work, Chase, and very informative! Overall the rankings just “look right” to me from a historical point of view, remembering which QBs were lauded in their own day, what unique challenges they faced depending on the teams they played for (e.g., running game or no running game/strong line or weak line/great coaching or a merry-go-round of mediocrities, etc.), the competition, changing of the rules that greatly enhanced the passing game, the different styles of play over time (e.g., “dead ball” to “west coast” style), the wearing of gloves by receivers and QBs, good stadiums vs. poor ones, etc. Perhaps the only jarring result for me is Dawson at #4, ahead of the likes of Young and Staubach, though you note the likely inflation of his numbers by playing in the expansion days of the AFL. Marino also looks a little low, all things considered.
    Do you think the rankings would change significantly if you only processed only the 5 or 7 “peak” seasons for each QB? You mention Namath as one who was pulled down by some bad years (and even further by his bad knees!). Probably Johnny U as well? Tarkenton certainly stands out for great career consistency. Jurgensen had several injury-plagued seasons which certainly cost him, and just think what Bert Jones could have done if not for his shoulder injury! Archie Manning is my favorite “bad luck” illustration, and I have little doubt that if he had landed with the Steelers and Bradshaw with the Saints, it would be Archie with the rings and HOF status.
    So many thanks for this, and for all your earlier statistical contributions on these endlessly fascinating topics.

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