Restraint revisited

According to the version update notes on December 14, 2011, the max Restraint damage bonus was reduced. After checking it with Spirits Within at 300 TP with 1249 HP and a 240-delay sword and lots of normal attacks (too lazy to count), the max bonus seems to be 30% (599 vs 778; 778/599 = 1.2988). This was repeated and I got the same result. For comparison here are the test server findings (a reminder that the final update may differ from that seen on the test server).

I also checked observed WS bonus per normal attack (taking the observed WS damage bonus and dividing by the number of normal attacks landed) versus weapon delay for both single-wield and dual-wield cases for linearity. WS bonus per normal attack appears to be linear with respect to total delay after accounting for dual wield delay reduction (acknowledging truncation of damage values). I'm just gonna throw the following data in here just to show I didn't make this stuff up:



BTW in the process of accounting for Fencer the TP bonus for tier V Fencer seems to be +50 instead of +55.

From regression I get -0.00001437022 + 0.00002596923*(delay). The intercept term is close enough to 0 that I would drop it, so each landed normal attack will add approximately an additional delay/385 to the WS bonus multiplier up to a maximum of 1.3. This assumes the bonus is independent of the number of normal attacks contributing to the total WS bonus. My understanding is that Ravager's Mufflers +2 doubles the bonus but the 1.3 cap cannot be exceeded (these findings suggest this).

Remember that if you have the average number of attack rounds per WS, you need to multiply this by 0.95 to get the average number of normal attacks that actually landed (missed attacks do not contribute to the Restraint bonus) to estimate the average Restraint WS bonus.

Restraint: tentative findings

(Edit 07/09/2010: edited some figures to account for a Critical Attack bonus of 5%.)

This time I'm going to start with the current legitimate claims about the effect of Restraint, a level 78 warrior job ability that "enhances your weapon skill power with each normal attack you land, but prevents you from dealing critical hits" per the help description. Then I will go over the support for those claims, one at a time, and then discuss some implications for the effectiveness of Restraint.

Claims

• Restraint's enhancement seems to manifest as a damage multiplier distinct from other factors such as TP bonus, pDIF, and "TP modifier" (fTP).
  
• Restraint's enhancement is not exact but actually has some variability, controlling for the number of attacks landed. (The damage multiplier is effectively a random variable.)
• The damage multiplier of Restraint appears to have a maximum of 1.5 (+50% bonus).
  
• Restraint's enhancement is dependent on weapon delay. Generally speaking, the higher the weapon delay, the higher the damage increase per normal attack landed.
• The damage multiplier appears to increase linearly with the number of landed attacks up to a maximum of 1.5 (controlling for weapon delay).
  

Is Restraint's effect really a simple damage multiplier?

Using the weapon skill Spirits Within, whose damage function I described when discussing how I determined Fencer's TP bonus, it is straightforward for anyone to show that Restraint doesn't provide a TP bonus like Fencer does.

But first, why use Spirits Within? Its damage is completely deterministic and can be calculated exactly given your current HP and current TP. If there is then any deviation from the predicted value, that deviation can be attributed to whatever factor you had changed. Of course, this doesn't say anything about weapon skills one would actually want to use (for reasons to be discussed later), but getting a general idea of how Restraint appears to work should help focus further investigation (in theory because no one really gives a shit about doing it).

Anyway, ruling out a TP bonus is easy enough as soon as you observe a damage return from Spirits Within under Restraint that is impossible were it the result of a TP bonus. To go over briefly how I determined this, I whacked a Zvahl Fortalice with a Trainee Sword/Trainee's Needle combination (5.1 TP/hit with Dual Wield II) until I got 107.1 TP, then used Spirits Within. (Actually this basically is the general experimental procedure performed to reach some of the conclusions about Restraint I described earlier.) Given my current HP of 1148, the predicted damage, given 107.1 TP, is 147, but the observed damage was 164. It is impossible to obtain 164 damage from a TP bonus (the damage equation I provided is exact and has yet to fail), so a TP bonus can be ruled out.

As for pDIF, obviously pDIF doesn't enter into Spirits Within damage, so one cannot really speak of any kind of pDIF bonus, whether additive or multiplicative.

Ruling out an fTP bonus like that from an elemental gorget is not as straightforward. A conceptually simple method is to determine, using the same weapon(s) (holding weapon delay and therefore TP/hit constant), whether the damage of one weapon skill scales by approximately (accounting for flooring) the same factor as the damage of another weapon skill with a different fTP "profile" given the same TP and the same number of landed attacks under Restraint.

If the scaling factors are dramatically different, this could be considered evidence of an additive fTP bonus and one can rule out the idea of a damage multiplier. However, I chose not to do this for the following reason.

The damage increase from Restraint has some variability...

Continuing to whack on a Fortalice, I made the unpleasant discovery that, even though Spirits Within damage is supposed to be completely deterministic (knowing only two pieces of information, TP and current HP, means being able to calculate the damage exactly), I observed some variability of damage return holding TP constant under the effect of Restraint. The predicted damage values (based on 1148 current HP) and the observed damage values (which have a relationship to the number of attacks landed with Restraint active) are given in the following text table as I was too lazy to use my inelegantly constructed table markup:

Attacks landed     TP        Predicted damage     Observed damage (Restraint)
------------------------------------------------------------------------------
19                 102       143                  160, 158, 160, 160, 158, 158
20                 107.1     147                  164, 163, 160, 166, 163, 164
21                 112.2     147                  167
22                 117.3     152                  174, 170, 179
23                 122.4     156                  177, 179
24                 127.5     161                  183, 180
25                 132.6     165                  184, 189
26                 137.7     170                  192, 197
27                 142.8     170                  195, 202
28                 147.9     174                  201, 201
29                 153       179                  209, 209, 209
30                 158.1     183                  214, 219
31                 163.2     188                  218, 223
32                 168.3     188                  225, 229, 218, 218
33                 173.4     192
34                 178.5     197                  232, 236

To me, the fact that the "designers" apparently decided to make the Restraint enhancement a random variable is extremely obnoxious. (What the fuck is the point? Or was this unintentional?) If this is not merely an "anomaly" specific to Spirits Within, it makes it that much more annoying to pin down the effect of Restraint using weapon skills whose damage is normally variable. But at least now people should be aware of this.

Perhaps this is a glitch specific to dual wielding? I also checked for single wield and also observed variability of Spirits Within damage. I also attempted to check for damage variability with a magical WS whose damage is also "deterministic" (controlling for resist), but after getting a 87-damage quarter-resist and then a 171-damage half-resist with Seraph Blade (given 117.3 TP), I got very annoyed and switched back to Spirits Within for the purposes of exploring Restraint's effect further. Note that 171 is less than twice that of 87 (174), which can be considered evidence of variability fundamental to Restraint.

Restraint's maximum damage increase appears to be 50% (1.5 damage multiplier

For one iteration of Restraint, I wanted to see how much of a damage increase to Spirits Within I could get by accumulating as many landed attacks as possible within the 5-minute duration. I managed to get 110 hits in before using Spirits Within, which gave 807 damage, which happens to be exactly 1.5 times 538, the usual damage at 300 TP given 1148 HP.

Studio Gobli's version update notes also suggest that 50% might be the upper bound for the weapon skill damage bonus. But more important, the update notes indicates that Restraint's effect appears to be dependent on weapon delay.

The effect of Restraint depends on weapon delay

Reiterating Studio Gobli's notes, given 20 landed attacks, the weapon skill bonus (the weapon skill used is not stated) is highest (+21%) for the weapon with the highest delay (444), lower (+17%) for the weapon with the second-highest delay (264), and lowest (+13%) for the weapon with the lowest delay (218). Again, I used a Trainee Sword/Trainee's Needle combination (187 delay per weapon given Dual Wield II), and referring back to my text table, the damage increase given 20 hits was observed to vary from +8.84% (160/147) to +12.9% (166/147), so my results are consistent with Studio Gobli's claims (seemingly unsourced by the way). Moreover, my use of dual wield suggests that Restraint is affected by effective weapon delay.

Therefore, the effect of Restraint on Spirits Within cannot be generalized to other weapon skills because of this dependence on weapon delay. But findings from my Spirits Within investigation can be considered a kind of lower bound on Restraint's WS damage bonus (not that I would actually check for anything below 187 weapon delay).

For a given weapon delay, Restraint's (apparent) damage multiplier may increase linearly with the number of landed attacks

Given the above data in the text table, as well as some other observations toward the "extremes" (based on the number of landed attacks below 19, the minimum number to get to 100 TP after Spirits Within, which never misses, and above 34), I just performed OLS regression mainly to see visually if it is "safe" to assume that the Restraint damage bonus scales linearly with the number of landed attacks:


It appears that linearity is a valid enough assumption, and it could be said that the Restraint damage bonus increases by about 0.00588515 (0.058%) for every additional landed attack, up to a maximum bonus of 1.5 (50%). Also note that my 110-attack observation is well off the trend line. Extrapolation here is not fatal; I predict that either 85 or 86 is the minimum number of landed attacks to reach the maximum bonus.

Implications for Restraint use

Here it may be useful to reflect on snap judgments about Restraint's potential utility (or lack thereof).

First, it has been shown that Mighty Strikes is unaffected by Restraint (but is Restraint unaffected by Mighty Strikes?), so there shouldn't be any disadvantage to using Restraint for the purposes of zerging, whatever the bonus is (or isn't).

Second, suppose that it is even desirable to achieve the maximum Restraint bonus to start a zerg off (say, for a 300 TP Steel Cyclone). There aren't that many situations where this is feasible, due to unavailability of mobs to "power up" Restraint and/or time limitations. (The "stored" WS damage potential disappears when Restraint wears off.)

Other than that, let's look at the use of Restraint from the long-run, "optimal" perspective of dealing damage, which means WS spamming and whatnot. Losing critical hit damage for increased weapon skill damage may not seem like a good trade-off, but whether the trade-off is acceptable is determined primarily by what the actual bonus is, which no one has yet to determine for the "usual" range of landed attacks before using a great axe weapon skill. (I would say the range is between 5 and 9.)

Of course, this doesn't mean one can't estimate how much of a WS damage increase there needs to be to offset the loss of critical hit damage during Restraint. To do this, consider that for two-handed weapons, the loss of critical hits in the auto-attack phase is relatively more "severe" for a low attack/defense ratio than a high one. Also consider that the loss of critical hit damage is relatively more severe if your critical hit rate is high than when it is low. Yet another factor to keep in mind that the more hits that end up being landed in the process of getting to 100 TP (think multi-hit weapons and being lazy), the greater the Restraint bonus must be to offset the loss of critical hits. Finally, since Restraint has often been mentioned with King's Justice (because it is thought that Raging Rush is adversely affected by Restraint), I will base my estimation on the basis of improving KJ damage.

With that in mind, I estimate that, for a high attack/defense ratio (such that the maximum average pDIF is attained without any level correction involved), the Restraint damage bonus (as a percent increase) needed to offset the loss of critical hit damage is between 3.1% and 4.6% given a 9% critical hit rate and between 8.2% and 12.3% given a 24% critical hit rate (on average). It may be that the actual WS damage bonus (for 5, 6, ... landed attacks) exceeds the above estimates given 504 delay, although it remains to be determined.

For a relatively more modest attack/defense ratio (corresponding to an average non-critical pDIF of 1.5 and average critical pDIF of 2.6; these are rough estimates based on someone's empirical observations, which I will not go over at this time), the Restraint damage bonus needed to offset the loss of critical hit damage is between 6.0% and 9.0% given a 9% critical hit rate and between 16.0% and 24.0% given a 24% critical hit rate. (I give ranges based on the some of the current "final upgrade" Magian great axes. But the low estimates are based on the horrible "occasionally attacks 2-3 times" great axe.)

The above suggests a place for Restraint where attack is high or where critical hit rate is low (or a combination of both, which might be experienced in a merit party), but more work needs to be done to justify that contention.

TP bonus of Fencer

(Edit: This is for WAR at level 75. I didn't consider the possibility of "increasing levels of mastery" bullshit.)

Saw this dumb shit so I thought I would act a dumb shit too by wasting my time figuring this out. (Figuring out the critical hit rate bonus did not waste that much time as I was sleeping while the data was being collected...)

One way to characterize the TP bonus of Fencer is to see how (and whether) the damage of the weapon skill Spirits Within varies with TP in the presence of Fencer and then compare the results to the damage-TP relationship of Spirits Within without Fencer. (Then you assume the findings can be generalized to all weapon skills and hope your observed damage with other weapon skills is consistent with the findings from Spirits Within testing.)

Some preliminary considerations

The problem is that the latter has not been fully characterized to account for flooring, so after retrieving Spirits Within damage observations between 100 and 300 TP without Fencer (using a Trainee Sword with store TP +5 for 6.7 TP and given 1000 current HP), I came up with a formula that matched the observations exactly.

Let D denote Spirits Within damage, H denote current HP and T denote current TP. Then Spirits Within damage appears to follow the piecewise function


This function describes the TP modifier (the fraction) increasing with TP in increments of 1/256 (other increments, such as 1/128 and 1/1024, result in calculated damage values that disagree with the set of actually observed damage values), so perhaps the TP modifiers at 100, 200, and 300 TP are better described as 32/256, 48/256, and 120/256, respectively. (Note: the inner bracket is there to ensure TP values are floored for the purposes of damage calculation, as TP values, while discrete, need not be integers.)

Now, with the same 1000 current HP and 6.7 TP, we can then observe how Fencer affects Spirits Within damage in terms of modifying base TP. We assume the TP bonus is additive and hope it is constant.

Actual TP bonus determination

The actual TP bonus (assuming it's additive and constant) was determined by a step-wise process of elimination by identifying "candidates" for the TP bonus as follows:

Step 1: For 100.5 TP, the predicted Spirits Within damage is 125. The observed damage is 148. The TP bonus could be 38, 39, 40, 41, 42, or 43. (At this point, 40 is the most plausible candidate as one would expect SE to make the TP bonus a mutiple of 5 or 10.)

Step 2: For 120.6 TP, the predicted Spirits Within damage is 136. The observed damage is 160. The TP bonus could be 37, 38, 39, 40, 41, or 42, but only 38, 39, 40, 41, or 42 are consistent with both observations.

Step 3: For 147.4 TP, the predicted Spirits Within damage is 152. The observed damage is 175. The TP bonus could be 35, 36, 37, 38, 39, or 40, but only 38, 39, or 40 are consistent with all three observations.

Step 4: For 140.7 TP, the predicted Spirits Within damage is 148. The observed damage is 171. The TP bonus could be 35, ..., 41, but, again, only 38, 39, or 40 are consistent with all four observations.

Step 5: For 154.1 TP, the predicted Spirits Within damage is 156. The observed damage is 183. The TP bonus could be 40, 41, 42, 43, 44, or 45, but only 40 is consistent with all five observations. Assuming the TP bonus is additive and constant, Fencer adds +40 TP to the current TP for WS damage calculation.

At this point, we should make sure adding 40 TP to the current TP allows us to predict correctly Spirits Within damage when the "net" TP exceeds 200 TP (so that damage is calculated based on the other part of the function).

For 167.5 TP, the predicted Spirits Within damage, based on 207.5 TP, is 207, which is also the observed value.

For 174.2 TP, the predicted Spirits Within damage, based on 214.2 TP, is 226, which is also the observed value.

For 180.9 TP, the predicted Spirits Within damage, based on 220.9 TP, is 246, which is also the observed value. (Note that you cannot floor the current TP to 180 and then add 40, which would give a predicted value of 242 based on 220 TP, which is wrong.) At this point, it seems reasonable to conclude that there is a 40 TP bonus from Fencer between 100 and 200 TP.

Now what about between 200 and 300 TP?

For 201.0 TP, the predicted Spirits Within damage, based on 241.0 TP, is 300, which is also the observed value.

For 227.8 TP, the predicted Spirits Within damage, based on 267.8 TP, is 375, which is also the observed value.

Finally, to make sure the actual TP for damage calculation is actually min(TP + 40, 300), for 300 TP, the predicted Spirits Within would be 578 given 340 TP, but the observed damage is 468, which is consistent with the 300 TP maximum.

Conclusion

Fencer gives a constant TP bonus of 40 TP for weapon skills independent of what the current TP is.

Critical hit rate bonus of Fencer

(Edit: now with information on Fencer with dual wield.)

Fencer is a new job trait from the July 21, 2010 version update that is available to the Warrior job at level 45 and the Beastmaster job at level 80. It has the following help description: "Increases rate of critical hits when wielding with the main hand only. Grants a TP bonus to weapon skills." The critical hit rate bonus was estimated using the following procedure.

Methods (brief)

An estimate of the critical hit rate bonus was obtained by auto-attacking overnight a level 69 Ul'hpemde, which has AGI 65 (source). WAR75/MNK01 was used. The following equipment was used to obtain STR 57, DEX 64, and an accuracy score of 276:

• Trainee Knife (240 dagger skill)
  
• Walahra Turban
• Dusk Gloves
• Snow Ring (STR -2)
  
• Swift Belt (Accuracy +3)
  
• Aurum Sabatons (DEX +3, accuracy +5)
  

STR 57 ensures 0 damage to any Ul'hpemde, and DEX 64 ensures (with 4/4 critical hit rate merits) a 9% critical hit rate before the effect of Fencer (source). kparser was used for automated data collection.

The level of the targeted Ul'hpemde was inferred by comparing the predicted hit rate for a level 69 Ul'hpemde (.92) against a point estimate of the hit rate of 5628/6133 = .9176 , with 95% confidence interval (.9105, .9244). The observed hit rate is consistent with the prediction.

Estimation of Fencer's effect with dual wield was also done with a Trainee Knife/Trainee's Needle combination, but the Ul'hpemde was level 68. (Critical hit rate is "directly" independent of level, but not AGI, which depends on level to some extent. But for both level 67 and level 68 Ul'hpemdes, the AGI is 65.) The following image summarizes the final base attribute values for this particular trial:

Results

Single wield: A point estimate of 802/5628 - .09 = .0525 was obtained for the critical hit rate bonus, with a 95% confidence interval (.0434, .0619).

Dual wield: A point of estimate of 464/4983 - .09 = .0031 was obtained for the critical hit rate bonus, with a 95% confidence interval (-.0048, .0115).

Interpretation and conclusion

Since critical hit rate has statistically been shown to take only integer percent values, assuming that the bonus is additive, the critical hit rate bonus of Fencer is either 5% or 6% with 95% confidence.

For the dual-wield case, suppose there were a 5% bonus for the main hand and none for the off hand. The effective bonus would then be 2.5%. Yet the observed estimate is much less than 2.5%, which should be taken as evidence that Fencer has no effect when dual wielding.

Weapon skill critical hit rate bonus: summary of evidence

(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１

The 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)

You think these new abilities and traits for WAR actually matter?

When was the last time there was a good version update teaser? I can't even remember since I'm usually apathetic toward version updates in general, but the latest teaser concerning "Adjustments of the Job Persuasion!" has provided a lot of fodder for blabbing and speculation, and here are my two Zimbabwean cents on the warrior job abilities and traits.

Restraint (level 77)

Ability delay: 10 minutes
Effect duration: 5 minutes
Description: "Enhances your weapon skill power with each normal attack you land, but prevents you from dealing critical hits."

Would use this for zerging when you have Mighty Strikes? No. I can't imagine a situation where the WS bonus would exceed the substantial benefit of 100% critical hit rate.

Consider the implications of 0% critical hit rate for the purposes of great axe WS spam without MS. Then, consider the implications of 0% critical hit rate for Raging Rush (doubtful that the loss of critical hits wouldn't apply to the WS), leaving King's Justice as the only choice for WS spam. Will the trade-off be worth it? I doubt it. If anything, I wouldn't be surprised if the bonus applied only to the first hit of any WS, and I doubt you can cancel Restraint and keep the WS bonus.

The loss of critical hits and actual WS bonus notwithstanding, this naturally favors one-handed weapons relative to two-handed weapons in theory, except that dual-wielding generally sucks relative to using great axe (or polearm when applicable) and one-handed weapon skills not named Rampage generally suck, too.

Critical Attack Bonus (level 78)

(Also Thief level 78, Dancer level 80)
Description: "Improves power of critical hits."

If this is like a permanent Brave Grip (estimated 3.5% increase in critical hit damage), it's better than nothing. Technically it's "good," but it's not to going to change anything about how to "play" warrior.

Of course this bonus has to be introduced alongside Restraint, which is likely to be a spurious JA.

Fencer (level 45)

(Also Beastmaster level 80, Ranger level 80)
Description: "Increases rate of critical hits when wielding with the main hand only. Grants a TP bonus to weapon skills."

The description and jobs for which this is available seems to imply that the rate increase applies to one-handed weapons only without dual-wield. Now why would anyone want to do that? Tanking!?

It's way more relevant for jobs subbing WAR at level 90 and beyond. But watch the rate increase be like 1%.

Shield Defense Bonus (level 80)

(Also Paladin level 77)
Description: "Reduces damage taken when blocking an attack with a shield."

Joke.

A hierarchy of great axes?

This is a rehash of a previous post comparing Bonesplitter and the good Luchtaine, two "Magian" great axes, to that old standby Perdu Voulge and Fortitude Axe, the presumptive weapon of choice for Campaign (even though Waltz recast ends up being the rate-limiting factor for curing yourself), but new evidence, both for Fortitude Axe (see first relevant BG post and second relevant BG post for details that I won't go over here) and Luchtaine (to be discussed later, perhaps), show that I underrated Fortitude slightly and overrated Luchtaine significantly.

In particular, evidence indicates Luchtaine behaves similarly to Joyeuse such that regular DA and Magian OAT are "directionally" exclusive, which is different than mutually exclusive. Suppose that the DA rate were 20%. Then, mutually exclusive would mean P(OAT) = .40, P(DA) = .20, and P(OAT and DA) = 0. On the other hand, directionally exclusive would mean that either P(OAT|not DA) = .40 and P(DA) = .20 OR P(DA| not OAT) = .20 and P(OAT) = .40. Consequently, given 20% DA and 40% OAT rate, the effective DA rate would be .20 + .80*.40 = .40+.60*.20 = .52.

Also, I decided to repeat the previous analysis using Raging Rush. Even if Raging Rush's three base hits (in other words, those not arising from double attack) are the only ones that have a chance to be critical hits, it's still generally better than King's Justice. One consequence: because RR's STR modifier is lower than KJ's, the relative difference in damage between a Perdu RR and Fortitude RR is more than that between a Perdu KJ and Fortitude KJ, so the relative difference between Perdu and Fortitude "overall" would be less with RR than KJ "all other things being equal."

I find it is worth including Rune Chopper in the discussion, too, along with Hephaestus with STR +4 and attack +15 as a basis of my pontificating about what kind of effort is warranted to get "good enough" (not the most). Since I have 19% haste normally, I will use that as a haste baseline before Rune Chopper, so the full haste bonus of Rune Chopper is not fully realized. On the other hand, I will also consider having Rune Chopper with only 1 MP refresh such that the latent is active one out of every two rounds (as it appears to be anyway). I will also consider the situation of having a "typical" double March (~20% haste with March +2 instrument and 8/8 merits in both wind and singing skill), Haste spell (~15%), and Hasso (~10%).

I will also account for the concept of time delay between the initiation of a weapon skill and the start of the next attack round, as it apparently is fundamental to the game and not associated with human reaction time or laziness (not that I really noticed or cared), kind of like how the delay associated with Curing Waltz screws up Drain Samba actually working properly (something that is easy to notice and that I find very annoying). This could also be considered the time delay associated with execution of a weapon skill that must elapse before the start of the following auto-attack round, or "WS delay" for short. It's something worth considering because this delay is unavoidable, but since I don't know what is actually the so-called WS delay, I will do this comparison for 0, 1, 2, 3, and 4 second delays.

Finally, I find it really unnecessary to go into excruciating detail about what goes in the calculations, so I will just report something I call "relative efficiency" ratios relative to the baseline of Perdu Voulge, which are merely ratios of damage rates. In the end, one should focus only on the gross differences, rather than whether something is really 2.15% more as opposed to 2.2% more, for example.

Relative efficiency of great axes relative to Perdu Voulge (in terms of damage rate)

Weapon	No WS delay	1s delay	2s delay	3s delay	4s delay
Rune Chopper (latent active always)	1.097	1.082	1.070	1.059	1.049
Fortitude Axe	1.037	1.008	0.984	0.964	0.947
Hephaestus (6 hits to 100 TP)	1.032	1.029	1.026	1.023	1.021
Bonesplitter	1.017	1.017	1.017	1.017	1.017
Perdu Voulge	1	1	1	1	1
Rune Chopper (latent 1/2 active)	0.997	0.991	0.986	0.981	0.977
Luchtaine	0.985	0.972	0.961	0.952	0.944
Hephaestus (7 hits to 100 TP)	0.953	0.961	0.968	0.974	0.980

Again, these ratios are based on 19% equipment haste before Rune Chopper, along with double March (~20% march), Hasso, and Haste spell.

The ratios under the hypothetical situation with zero WS delay can represent the "intrinsic" relative efficiency of weapons that have a higher WS frequency that Perdu Voulge (notice that Bonesplitter has the same relative efficiency regardless of WS delay because it has the same WS frequency as Perdu), but intrinsic doesn't mean actual or true. The higher the WS delay, the more disproportionately affected are weapons with higher WS frequency compared to Perdu.

(Note that the concept of WS delay can be generalized to job abilities that interrupt or postpone attack rounds, but I did not account for that here.)

Implications for Fortitude Axe: wonder why Fortitude Axe doesn't actually appear to be better than Perdu Voulge in practice? WS delay could explain it. In particular, if you plan to use Fortitude Axe in a maximum haste situation (~80% haste), spamming weapon skills might be relatively counter-productive (for Fortitude compared to Perdu) because of WS delay, but WS frequency is pretty much the only benefit of using Fortitude Axe (aside from TP gain without using WS), so why not just use a high-damage great axe (Perdu or even Berserker's Axe)? Using Fortitude Axe for a zerg basically means having hope that you get more hits per round in a small time frame compared to the long-run average, e.g., stringing together several 3-attack rounds. Having 80% haste is usually the decisive factor in a max-haste zerg because you probably have max attack and accuracy as well. Maybe if you had a BLM land Choke and got some STR etudes... things you could do to compensate for the low base damage.

Implications for Rune Chopper: On the other hand, Rune Chopper with latent always active (if you somehow manage to achieve this; not a trivial thing) is still substantially better than Perdu even with significant WS delay (4 seconds), and this is under the situation where the 9% haste bonus isn't fully realized (it could be if you switched out other haste equipment to increase other damage-related factors), albeit under the double March/Hasso/Haste spell situation. On the other hand, Rune Chopper with only 1 MP refresh is rather pointless. If you had an Ares Cuirass lying around and RDM accommodating you, it would be good.

Implications for Luchtaine and other Magian great axes: SE really needs to allow Luchtaine to attack 3 times or even 4 times in the future or increase the base damage dramatically. At least there is hope that SE might do this later, whereas with Fortitude Axe, SE will never allow 4 attacks per round. You don't get much out of the others (as the final forms currently are) considering the time investment required, compared to spending IS on a Perdu Voulge. Hephaestus 6-hit is not terribly reasonable because of the 29 store TP requirement alone.

Didn't I say something about a hierarchy? Stick with Perdu Voulge in general...