(
Edit #2: added information for Backhand Blow and Blade: Jin, and another source for Rampage.)
(
Edit #1: added another source for Drakesbane.)
This is an attempt to summarize any evidence following attempts to determine the critical hit rate bonus at or around 100 TP (if any) for weapon skills whose "chance of critical varies with TP."
I am not aware of any (non-anecdotal) evidence for the following weapon skills: Ascetic's Fury, Vorpal Blade, Power Slash, Sturmwind, Keen Edge, Vorpal Scythe, Vorpal Thrust, Skewer, Blade: Rin, True Strike, Hexa Strike, Sniper Shot, Heavy Shot, Dulling Arrow, and Arching Arrow (17 weapon skills). That leaves only six: Backhand Blow, Evisceration, Rampage, Raging Rush, Drakesbane, and Blade: Jin.
For now, "convenient" determination of critical hit rate is possible only for the first hit. Most of the testing done concerns the first hit, and conclusions are based on the assumption that the bonus (where it exists) is additive.
Backhand Blow (hand-to-hand, 2 hits)
Source: dex/crit relation, WS crits, WS gorgets discussion (Blue Gartr forums)
Comparing the sample proportions 22/50 (.44) at 9% baseline critical rate and 37/50 (.74) at 30% baseline (with 6% from Destroyers), it is obvious that there is some kind of innate critical rate bonus for at least the first hit of Backhand Blow.
But with Backhand Blow TP varying between 100 and 120 TP, it seems likely that the critical rate was not fixed for each sample. The consequences of this on the allocation of Type I error and coverage probability of the corresponding interval estimate are explored for Blade: Jin bonus estimation (later in the post), as data for that was obtained by the same person, but for now I will just describe briefly how to go about estimating the bonus for Backhand Blow.
Assume that the innate bonus is additive and constant (meaning it's independent of whatever the baseline critical rate is). Also assume that the critical rate bonus from Destroyers (6%) increases the critical hit rate of Backhand Blow by an additional 6% (starting from 24%).
Let
X1 be the number of critical hits observed at 9% baseline,
n1 the total number of hits observed at 9%,
X2 the number of critical hits observed at 30% baseline, and
n2 the total number of hits observed at 30%. A natural "pooled" estimator for Backhand Blow's critical hit rate bonus is
and its standard error is
The sample proportion is .395 and a corresponding 95% confidence interval for the WS bonus is
(30.32%, 48.68%).
Conclusion: there is a critical hit rate bonus for Backhand Blow at 100 TP. A bonus of 40% would be consistent with the given data.
Evisceration (dagger, 5 hits)
Source: Evis crit rate testing (Allakhazam forums)
At ~100 TP and given 24% base critical hit rate, the pooled sample gives a sample proportion 248/696 = .3563. A 95% confidence interval for the critical hit rate bonus is
(8.61%, 15.32%).
Conclusion: there is a critical hit rate bonus for Evisceration at 100 TP, with +10% being a possibility.
Rampage (axe, 5 hits)
Source (1): ランページとDEXの関係There are two sets of estimates: one for DEX 68, and one for DEX 124, with Gigantobugard as the target mob in both cases. I'm not much interested in calculating base AGI and confirming that the Megalobugard's level range is 40-43, so I ignored the estimates for DEX 68. DEX 124 ensures a 24% base critical hit rate.
At 100 TP, the sample proportion of critical hits is 35/130 = .2692. A 95% confidence interval for the critical hit rate bonus is
( -4.48%, 11.40%). But suppose there actually is a 10% critical hit bonus. For a sample size of 130, the probability that the sample is sufficient to show a statistically significant bonus is about .7388 (power calculation).
At 200 TP, the sample proportion of critical hits is 68/150 = .4533. A 95% confidence interval for the critical hit rate bonus is
(3.20%, 29.66%).
Source (2): dex/crit relation, WS crits, WS gorgets discussion (Blue Gartr forums)
I did say I wasn't interested in calculating a mob's AGI, but a Clipper's AGI is either 18 or 21 regardless of the levels reported on FFXIclopedia, and either AGI value doesn't affect the actual crit rate for the DEX 57 case, which is indeed 13%. (
See this for details about critical hit rate as a function of your DEX - mob AGI.)
Using the same "pooled" estimator rationale I used for Backhand Blow (earlier in the post), the sample proportion for Rampage's crit bonus at 300 TP is .465 and a corresponding 95% confidence interval for the rate bonus is
(31.80%, 61.20%). For the sake of completeness, estimates for the bonus at 100 TP and 200 TP are
(-9.26%, 25.40%) and
(3.20%, 48.80%), respectively.
Conclusion: if there is a critical hit rate bonus for Rampage at 100 TP, the known evidence is insufficient to show that, but if the bonus were 10%, for
n = 130 the power to reject the null hypothesis of no bonus is fairly high (.7388). Given all the data, it is relatively unlikely that the bonus is 10%, but a smaller bonus cannot be ruled out with such small samples.
Unsurprisingly, there is a bonus at 200 TP and 300 TP.
Raging Rush (great axe, 3 hits)
Source (1): レイグラのクリティカル率につい て その１The sample proportion is 20/40 given the usual 24% base. The "control" data for base critical rate (which is a good idea to have by the way), however, gives the sample proportion 44/130 = .3384, which is somewhat unusual, but I write that off merely as that, not a sign of dubious experimental error. This data alone gives the tentative impression that there is a bonus.
Source (2): RagingRush Critical rate test (Killing Ifrit forums)
The raw data (showing damage values) are in a spreadsheet, but you don't need to download it.
At 100 TP and given 24% base critical hit rate, the proportion of critical hits is 155/373 = .4155. A 95% confidence interval for the critical hit rate bonus is
(12.50%, 22.74%). This is strong evidence that the critical hit rate bonus is
not 10%. Possible candidates are 15% and 20%.
More interesting to me is that the damage for 1 TP return (2o occurrences) was also noted, providing an opportunity to determine whether a critical hit rate bonus also applies to off-hand hits (despite there being no way to tell the difference between a double attack hit and a regular off-hand hit). Assuming a 24% base critical hit rate, with 9 observed critical hits out of 20, the corresponding
p-value is .03614, which suggests a critical hit rate bonus.
Conclusion: there is a critical hit rate bonus for Raging Rush at 100 TP, with +15% and +20% being possible candidates. The small sample for critical hits from off-hand hits suggests a critical hit rate bonus for off-hand hits of Raging Rush as well.
Drakesbane (polearm, 4 hits)
Source (1): drakesbane native crit% (FFXIclopedia forums)
The first sample is 38/100 and the second, 24/100 (given 106 TP).
38/100 is a fairly extreme observation given 24% base critical hit rate (if there were no bonus). On the other hand, 24/100 is not that extreme an observation given a 34% rate. Since there is no good reason to think the conditions changed between the two samples, pool the data and crank out an interval estimate for the rate bonus, which is
(0.66%, 13.91%).
Source (2): 雲蒸竜変の検証There are four samples: three for 100 TP and one for 300 TP.
For 100 TP, the sample proportions are 12/49, 15/45, and 15/41 (given 24% base critical hit rate). The pooled estimate is 42/135 = .3111 and a 95% confidence interval for the bonus is
(-0.57%, 15.64%). While this interval covers 0, 0 is again close to the left endpoint (in the other case the 0 being on the "right" side based on expectations).
As for 300 TP, the sample proportion is 16/30 and a 95% confidence interval for the rate bonus is
(10.32%, 47.66%), which rules out 50% (tentatively).
Conclusion: there is suggestive evidence for a critical hit rate bonus at 100 TP, with +5% and +10% being possible candidates. At 300 TP, a +50% bonus appears to be an "unlikely" possibility.
Blade: Jin (katana, 3 hits)
Source: dex/crit relation, WS crits, WS gorgets discussion (Blue Gartr forums)
The sampling was done in the same fashion as for Backhand Blow, with observed critical hit proportions 3/30 at 9% baseline crit rate and 8/30 at 30% baseline (with Senjuinrikio's 6% bonus) at 100 TP. Using the same estimator that I used for Backhand Blow, the "pooled" sample proportion for Blade: Jin's critical bonus is -0.01167, and a corresponding 95% confidence interval is
(-10.73%, 8.39%).
Taking the confidence interval at face value, if there is a critical bonus for Blade: Jin at 100 TP, it is unlikely that it's 10% or higher, especially considering the "sloppy" manner in which the data was likely collected (with TP not being held fixed, the critical hit rate could have varied), which further supports that contention. If the bonus were 10%, obviously, the probability that a 95% confidence interval wouldn't cover 10% at the right endpoint of the interval would be near .025 (half the Type I error). The consequences of experimental "error" are explored in a simulation study described at the end of this post.
Conclusion: if there is a critical hit rate bonus for Blade: Jin at 100 TP, it is unlikely that the bonus is as high as 10%.
Simulation study: is a 10% critical hit rate bonus that unlikely for Blade: Jin?
Consider the following simulation study based on
hypotheticals: if there actually were a 10% bonus at 100 TP, with a 1% increase for every 5 TP, then with TP varying between 100 and 119 TP, the critical rate varies between 10% and 13%.
Given that "TP overflow" is inevitable with dual wield, and that extra hits occurring beyond TP were quite possible because data collection was reported to be boring, suppose that each of the critical rates between 10% and 13% (inclusive) are equally likely to be "chosen" for Blade: Jin.
The purpose of the study is to show how likely it is that the "pooled" large-sample confidence interval covers 10% given the above conditions.
A histogram of the simulated sampling distribution of the critical hit rate bonus shows that it's obviously not normal, with the mean (about 11.5%) higher than 10%, which is supposed to be the "actual" bonus at 100 TP for this simulation. (The shape of the large-sample approximation of the sampling distribution is traced with the solid curve.)
On the other hand, the margin of error for all simulated sample proportions is higher than 9.56%, the margin of error for the actual sample, about 97.7% of the time. (The mean margin of error is 11.19%.) Also, the "actual" (in the context of the simulation) Type I error is about .059, with about .040 allocated to the right tail (meaning there is a probability of .0402 that the null hypothesis of .10 is rejected because the estimate is higher than .10 based on the criterion of statistical significance) and about .019 allocated to the left tail (meaning the null is rejected with probability .019 because the observed estimate is significantly lower than .10). By comparison, the nominal left-tail error is .025.
Repeating this exercise under the condition that there is no bonus, the margin of error for all simulated sample proportions is higher than 9.56% only 58.0% of the time, and the probability that a confidence interval's right endpoint is higher than 8.39% is less than 0.1%.
If Blade: Jin's critical hit rate bonus at 100 TP were actually 10%, considering TP overflow and additional hits occurring beyond TP overflow, it would be very unlikely that a given 95% confidence interval would not cover 10%. The margin of error would also be very likely to be higher than 9.56%. Therefore, it is more plausible that its critical rate bonus is significantly less than 10%, if it even exists.The following is some code for the simulation, but the inner loop should probably be expanded so that it finishes faster.
n = 100000
ci.lower = numeric(n)
ci.upper = numeric(n)
p.pool = numeric(n)
for (i in 1:n) {
X1 = 0
X2 = 0
for (j in 1:30) {
X1 = X1 + rbinom(1,1,sample(seq(.19,.22,by=.01),1))
X2 = X2 + rbinom(1,1,sample(seq(.40,.43,by=.01),1))
}
p.pool[i] = (X1 + X2 - .39*30)/60
ci.upper[i] = p.pool[i] + qnorm(.975)*sqrt((X1/30*(1-X1/30) + X2/30*(1-X2/30))/120)
ci.lower[i] = p.pool[i] - qnorm(.975)*sqrt((X1/30*(1-X1/30) + X2/30*(1-X2/30))/120)
}
mean(p.pool)
me = (ci.upper - ci.lower)*.5
mean(me>sqrt((3/30*(1-3/30)+8/30*(1-8/30))/120)*qnorm(.975))
mean(ci.upper<.10) mean(ci.lower>.10)
mean(ci.upper<.10) + mean(ci.lower>.10)
