# Supreme Court Reversal Rates

## The Tweet:

This tweet unleashed a tempest in a teapot, with a host of inane "fact checks." The purpose of this article is to show that everybody was wrong and nobody asked the right question in the first place. But first, we'll look at the questions they asked.

Is the 9th circuit "the most reversed?" If you are talking about number of reversals, the answer is yes, since the 9th circuit has a lot more cases than any other circuit, and would need to have a ridiculously low reversal rate for that NOT to be true. It is indeed the most reversed circuit. But it is also the most affirmed circuit:

 "Yes," said Sancho, "but I have heard say that there are more friars in heaven than knights-errant." "That," said Don Quixote, "is because those in religious orders are more numerous than knights." — Miguel de Cervantes, Don Quixote, Vol. II Ch. VIII

Supreme Court Reversal Rates, 2007-2019
(As of 2019-08-14. See notes section below for source.)
Court Total Cases Affirmed Reversed % Reversed
First Circuit 25 13 12 48.00%
Fourth Circuit 44 18 26 59.09%
Tenth Circuit 32 13 19 59.38%
Second Circuit 65 25 40 61.54%
D.C. Circuit 39 15 24 61.54%
Seventh Circuit 46 17 29 63.04%
U.S. District Court 21 7 14 66.67%
Fifth Circuit 65 20 45 69.23%
Federal Circuit 51 15 36 70.59%
Third Circuit 41 12 29 70.73%
Eleventh Circuit 61 17 44 72.13%
State Court 135 31 104 77.04%
Ninth Circuit 181 41 140 77.35%
Eighth Circuit 42 9 33 78.57%
Sixth Circuit 66 11 55 83.33%
*****TOTAL 914 264 650 71.12%
For a cheap thrill, click a circuit name in the table above to see the binomial distribution of probabilities for various numbers of reversals, given the circuit's total cases and reversal probability of % Reversed. When looking at the calculator output, remember that "successes" means "reversals."

After Trump apologists had finished observing that the 9th Circuit is indeed the most reversed (but without observing that it is also the most affirmed), a horde of "fact checkers" began observing that the 9th Circuit reversal percentages were below this or that other circuit. Using observations from cherry-picked periods of time was a favorite. Some even looked at individual terms, as if reversal percentages based on tiny numbers of cases are even worth talking about. They slavered and barked over trivial differences in percentages, apparently unaware that, probability theory aside, dividing up an integer number of reversals among circuits as fairly as possible will almost invariably result in differing reversal percentages (a fact discussed in the notes section below). Yes, it was THAT bad.

## Confidence intervals

So can we say say with confidence that any circuit's reversal percentage differs by more than chance from any other's? Good question! We'll estimate the standard error of the reversal percentage for each circuit from its total cases and number of reversals, then estimate confidence intervals for each circuit's reversal percentage.

Doing this, we can see that confidence intervals do not overlap for the First Circuit and apparent miscreants like the 6th, 9th, and State Courts, indicating that the First Circuit's reversal percentage differs significantly from theirs. We also see that many confidence intervals overlap. In particular, the overlap of the 6th, 8th, 9th, and State Courts is very strong. This might suggest that babbling about the difference in reversal rates for these courts is a waste of time.

<Digression> While non-overlapping confidence intervals imply statistical significance, overlapping confidence intervals do not necessarily imply a lack of statistical significance. Not that it matters, since we will be testing for this next. The width of the confidence interval decreases as sample size (number of cases in the circuit) increases, and as the reversal percentage varies from 50%, which is where the standard error of a proportion for any given sample size has its maximum. Thus, the Ninth Circuit has the narrowest confidence interval, and the First Circuit and U.S. District Court, the widest. <End Digression>

## An Exercise in Data Dredging

All well and good, but what we really want to know is: what is the probability that the difference in reversal rates between Circuit A and Circuit B is due to chance? We can compute this value for all 105 (15 choose 2) pairs of circuits, and get this interesting table, with the p values for the 105 pairs shown below the main diagonal of 1's:

First Circuit Second Circuit Third Circuit Fourth Circuit Fifth Circuit Sixth Circuit Seventh Circuit Eighth Circuit Ninth Circuit Tenth Circuit Eleventh Circuit D.C. Circuit Federal Circuit State Court U.S. District Court 1.00000 0.24413 1.00000 0.06479 0.33352 1.00000 0.37333 0.79755 0.26178 1.00000 0.06120 0.35664 0.86978 0.27573 1.00000 0.00064 0.00521 0.12291 0.00470 0.05759 1.00000 0.22032 0.87205 0.44764 0.70061 0.49574 0.01470 1.00000 0.00996 0.06463 0.41142 0.05168 0.28848 0.53456 0.11078 1.00000 0.00176 0.01360 0.36955 0.01353 0.19362 0.30726 0.04700 0.86402 1.00000 0.39222 0.83739 0.31030 0.98015 0.33540 0.00970 0.74324 0.07347 0.03121 1.00000 0.03300 0.20748 0.87791 0.16194 0.72091 0.12816 0.31755 0.45966 0.40910 0.21124 1.00000 0.28679 1.00000 0.38474 0.82011 0.42156 0.01242 0.88652 0.09346 0.03980 0.85276 0.26804 1.00000 0.05511 0.30876 0.98802 0.24056 0.87438 0.10011 0.43002 0.38126 0.31903 0.29296 0.85715 0.36680 1.00000 0.00282 0.02223 0.41055 0.02042 0.23549 0.30250 0.06334 0.83551 0.94801 0.04143 0.45962 0.05321 0.36315 1.00000 0.20334 0.67253 0.74247 0.55713 0.82578 0.10055 0.77417 0.30615 0.27633 0.59220 0.63501 0.69419 0.74265 0.30305 1.00000

Which pretty much confirms that all that yammering over who has the worst reversal rate was a waste of time. The probability that the Ninth Circuit's reversal rate differs by chance alone from the 6th, 8th, and State Courts is 30%, 86%, and 95%, respectively.

But the table above is hard to read. We would more likely just want to see the 105 pairs sorted in descending or ascending order by p value, which is a lot easier to digest. (Which court is denominated Court A or Court B in the pair is arbitrary; order is irrelevant.)

Court A Court B p value
First Circuit Sixth Circuit 0.00064
First Circuit Ninth Circuit 0.00176
First Circuit State Court 0.00282
Fourth Circuit Sixth Circuit 0.00470
Second Circuit Sixth Circuit 0.00521
Sixth Circuit Tenth Circuit 0.00970
First Circuit Eighth Circuit 0.00996
Sixth Circuit D.C. Circuit 0.01242
Fourth Circuit Ninth Circuit 0.01353
Second Circuit Ninth Circuit 0.01360
Sixth Circuit Seventh Circuit 0.01470
Fourth Circuit State Court 0.02042
Second Circuit State Court 0.02223
Ninth Circuit Tenth Circuit 0.03121
First Circuit Eleventh Circuit 0.03300
Ninth Circuit D.C. Circuit 0.03980
Tenth Circuit State Court 0.04143
Seventh Circuit Ninth Circuit 0.04700
Fourth Circuit Eighth Circuit 0.05168
D.C. Circuit State Court 0.05321
First Circuit Federal Circuit 0.05511
Fifth Circuit Sixth Circuit 0.05759
First Circuit Fifth Circuit 0.06120
Seventh Circuit State Court 0.06334
Second Circuit Eighth Circuit 0.06463
First Circuit Third Circuit 0.06479
Eighth Circuit Tenth Circuit 0.07347
Eighth Circuit D.C. Circuit 0.09346
Sixth Circuit Federal Circuit 0.10011
Sixth Circuit U.S. District Court 0.10055
Seventh Circuit Eighth Circuit 0.11078
Third Circuit Sixth Circuit 0.12291
Sixth Circuit Eleventh Circuit 0.12816
Fourth Circuit Eleventh Circuit 0.16194
Fifth Circuit Ninth Circuit 0.19362
First Circuit U.S. District Court 0.20334
Second Circuit Eleventh Circuit 0.20748
Tenth Circuit Eleventh Circuit 0.21124
First Circuit Seventh Circuit 0.22032
Fifth Circuit State Court 0.23549
Fourth Circuit Federal Circuit 0.24056
First Circuit Second Circuit 0.24413
Third Circuit Fourth Circuit 0.26178
Eleventh Circuit D.C. Circuit 0.26804
Fourth Circuit Fifth Circuit 0.27573
Ninth Circuit U.S. District Court 0.27633
First Circuit D.C. Circuit 0.28679
Fifth Circuit Eighth Circuit 0.28848
Tenth Circuit Federal Circuit 0.29296
Sixth Circuit State Court 0.30250
State Court U.S. District Court 0.30305
Eighth Circuit U.S. District Court 0.30615
Sixth Circuit Ninth Circuit 0.30726
Second Circuit Federal Circuit 0.30876
Third Circuit Tenth Circuit 0.31030
Seventh Circuit Eleventh Circuit 0.31755
Ninth Circuit Federal Circuit 0.31903
Second Circuit Third Circuit 0.33352
Fifth Circuit Tenth Circuit 0.33540
Second Circuit Fifth Circuit 0.35664
Federal Circuit State Court 0.36315
D.C. Circuit Federal Circuit 0.36680
Third Circuit Ninth Circuit 0.36955
First Circuit Fourth Circuit 0.37333
Eighth Circuit Federal Circuit 0.38126
Third Circuit D.C. Circuit 0.38474
First Circuit Tenth Circuit 0.39222
Ninth Circuit Eleventh Circuit 0.40910
Third Circuit State Court 0.41055
Third Circuit Eighth Circuit 0.41142
Fifth Circuit D.C. Circuit 0.42156
Seventh Circuit Federal Circuit 0.43002
Third Circuit Seventh Circuit 0.44764
Eleventh Circuit State Court 0.45962
Eighth Circuit Eleventh Circuit 0.45966
Fifth Circuit Seventh Circuit 0.49574
Sixth Circuit Eighth Circuit 0.53456
Fourth Circuit U.S. District Court 0.55713
Tenth Circuit U.S. District Court 0.59220
Eleventh Circuit U.S. District Court 0.63501
Second Circuit U.S. District Court 0.67253
D.C. Circuit U.S. District Court 0.69419
Fourth Circuit Seventh Circuit 0.70061
Fifth Circuit Eleventh Circuit 0.72091
Third Circuit U.S. District Court 0.74247
Federal Circuit U.S. District Court 0.74265
Seventh Circuit Tenth Circuit 0.74324
Seventh Circuit U.S. District Court 0.77417
Second Circuit Fourth Circuit 0.79755
Fourth Circuit D.C. Circuit 0.82011
Fifth Circuit U.S. District Court 0.82578
Eighth Circuit State Court 0.83551
Second Circuit Tenth Circuit 0.83739
Tenth Circuit D.C. Circuit 0.85276
Eleventh Circuit Federal Circuit 0.85715
Eighth Circuit Ninth Circuit 0.86402
Third Circuit Fifth Circuit 0.86978
Second Circuit Seventh Circuit 0.87205
Fifth Circuit Federal Circuit 0.87438
Third Circuit Eleventh Circuit 0.87791
Seventh Circuit D.C. Circuit 0.88652
Ninth Circuit State Court 0.94801
Fourth Circuit Tenth Circuit 0.98015
Third Circuit Federal Circuit 0.98802
Second Circuit D.C. Circuit 1.00000

Some pretty low p values there, well below .05. But there is a problem. We tested 105 hypotheses. With that many tests, you would naturally expect some, say about five, to turn up significant at the .05 level, by chance.

There are many ways to adjust for this. We will use the Benjamini-Hochberg procedure. This allows us to specify a percentage of false "discoveries" that we are willing to tolerate. Setting this level to 10% makes the first three tests (First Circuit versus Sixth Circuit, Ninth Circuit, and State Courts) significant. These are highlighted in violet. Setting the tolerable false discovery level to 20% adds another ten, through Second Circuit - State Court, at a p value of .02223. These are highlighted in lavender.

## Conclusions

• The reversal rates of the Sixth, Eighth, and Ninth Circuits, and State Courts, do not differ at any reasonable level of significance. The difference in their reversal rates could easily be due to chance and is not worth talking about, unless you need to write an article about it.

• The Sixth Circuit is implicated in six of the significant effects, the Ninth Circuit in three, State Courts in three, and the Eighth Circuit in one. For whatever reason, the Sixth and Ninth Circuits, and State Courts, seem to have high reversal rates.

• Conversely, the First Circuit is implicated in four significant effects, the Second and Fourth Circuits are each implicated in three, and the D.C., Seventh and Tenth Circuits each implicated in one. For whatever reason, the First, Second, and Fourth Circuits seem to have low reversal rates.

## Why it all means nothing

The major premise in the teapot tempest seemed to be, "If a circuit has a high reversal rate, it is lazy, stupid, biased, or feckless, and must have a high error rate in its decisions." Or something like that.

The problem is that the reversal rates do not tell us much of anything, other than that the Supreme Court is more likely to reverse than affirm. Given that the main role of an appellate court is to correct errors, this is unsurprising.

The Supreme Court gets about 8,000 petitions for certiorari each year, and hears about 80 of them. Petitions for certiorari are not a random sample of cases. And the Supreme Court does not randomly sample petitions for certiorari. It does a purposive sample based on criteria that may fluctuate over time. If petitions for certiorari were a random sample of cases, and the Supreme Court took a random sample from the petitions, then a circuit's reversal rate might indeed reflect its error rate in deciding cases.

While such randomization might tell you error rates by circuit, it would not allow you to use those error rates to compare the precision of legal reasoning between circuits. Since circuits differ notably in the types of cases they hear, some might be getting more difficult (and thus error-prone) cases than others. To correct for that, you would need to assign cases to circuits randomly from the entire pool of U.S. cases. Once you did all these randomizations, you could make inferences about each circuit's error rate, and use the error rates to compare the precision of legal reasoning between circuits.

But that grand experiment will never happen, so in the meantime, we can muck around with some logistic regressions and misinterpret them as implying causality.

How about dividing a circuit's reversals by the total number of cases it handles? People have tried that. It would put a lower bound on the circuit's overall error rate. Other than that, it would not tell you anything about the circuit's overall error rate, which might be much higher. It is analogous to estimating the overall manufacturing defect rate for widgets from a handful of hard cases presented to the chief widget inspector.

So the real question — how correctly does each circuit decide its cases? — remains unanswered. And that is the question people should care about.

 "It's like Shakespeare. Sounds well enough, but it doesn't really mean anything." — Bertie Wooster Quasi pannus menstruatae universae justitiae nostrae... — Isaiah 64:6

## NOTES

### Why variation in reversal rates is (nearly) unavoidable

Given the data, it is impossible to avoid some variation in reversal rates between circuits, even if we try to avoid it by apportioning reversals as closely as possible between them. The fourth-grade arithmetic reason for this is that the overall reversal percentage is computed as 650/914. You can reduce that to 325/457, but not further, because 457 is a prime number. You can only assign an integer number of reversals to each circuit. It is impossible to find an integer i such that i/(circuit number of cases) = 325/457 for any circuit, given the total number of cases in each. If you try to assign the integer that comes closest to solving this for each circuit, you will end up with something that looks like this:
Would somebody from Vox or the Daily Kos write an article about this, excoriating the 1st circuit? Probably. The point here is that the discrete nature of the data makes it extremely unlikely that we will ever have equal reversal rates in all circuits, even if the true probability of reversal were exactly the same in all circuits. Discrete data is a great boon to journalists. For example, if you expect 1.77 landfall hurricanes in the U.S. per year, the number of hurricanes in a given year will always be above or below average, and you will always have something to write an article about.

### State Courts, Federal Circuit, District Court

In the article we use the term "circuits" to refer collectively to all listed categories of court. It would probably be advisable to exclude State Courts, Federal Circuit, and District Court from the analysis and concentrate solely on roughly comparable federal circuit courts, but if we did that, people would complain bitterly about the exclusion. Besides, there is a lot of variation between circuits in the proportions of cases of different types, and going down such a rathole was not the purpose of this article. Actually, I'm not sure what the purpose was.

### A more interesting question

Frankly, viewing reversals as a binary variable is rather silly. It's one thing for a close decision of a lower court to be reversed by a close decision of the SCOTUS. Might be a matter on which reasonable minds could differ. It's quite another for a unanimous decision of a lower court to be unanimously reversed by the SCOTUS (as in R.A.V. v. City of St. Paul, 505 U.S. 377 [1992]). A more refined measure of disagreement between lower courts and the SCOTUS would be useful. I'm sure people have done that, but don't have a reference. Washington University's Supreme Court Database has a variable called "lcDisagreement," but the variable lacks sufficient detail to be of use in such an analysis.

It would also be interesting to analyze whether slight changes in the composition of lower courts — circuit courts of appeal and state supreme courts — might radically alter reversal rates for those courts. You could construct something like a ROC curve for that.

"Affirmed in part, reversed in part" offers another rathole.

### P value for comparison of proportions

More than one way to do this, with slightly different results in this case, but we ended up using the more common one. Not that it matters much one way or the other.