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)

Why Love Halberd is underrated... for dragoon

While I personally have yet to determine the virtue stone consumption rate for virtue weapons other than Fortitude Axe (so far, I'm assuming it's 55% across all virtue weapons given the limited evidence thus far), how exactly the normal double attack trait interacts with the virtue weapon's "occasionally attacks twice" (OAT) property seems to be described correctly. With a reasonable level of confidence, one can draw conclusions about how effective the other virtue weapons are compared to their "peers."

I can't say the likes of Hope Staff and Prudence Rod are worth discussing, but Love Halberd has some properties relevant for dragoon and samurai that seem to be misunderstood and even dismissed out of hand, the inconvenience of acquiring virtue stones notwithstanding. I go through them in order of importance and then compare Love Halberd to its competing options for DRG.

Is Love Halberd's delay undesirable?

Love Halberd has 396 delay, so with current quantities of Store TP available, it's possible and reasonable to achieve an "8-hit setup" with 23 Store TP (12.5/10.2 = 1.22549, which rounds up to 1.23).

People act like this this is a bad thing. But so what if it takes Love Halberd 8 hits to get to 100 TP? Noting how many hits it takes to get to 100 TP is trivial and irrelevant especially because of Love Halberd's OAT property. Instead, one should ask, how many attack rounds does it take for Love Halberd to get to 100 TP, given that 8 hits are required to get there?

It may help to show a graph illustrating, for both a virtue weapon (singly wielded) and a weapon without any multi-hit property (also singly wielded) but under 9% double attack rate and 95% hit rate, the relationship between the nominal number of hits to get to 100 TP and the "actual" (in a long-run, "missing the first hit of a WS 5% of the time," weapon skill-spamming context), average number of attack rounds it takes to get to 100 TP:


First, look for the average number of attack rounds it takes for a weapon without any multi-hit property to get to 100 TP in 6 hits. On the graph, the average number of attack rounds appears to be 5, and the actual value is 4.9526 rounds. This figure is reasonable because even though 5% of the time, the first hit of a WS misses (most of the time it takes 5 hits to get to 100 TP) , the 9% double attack rate results in the average value falling slightly below 5.

Now, look for the average number of attack rounds it takes for a virtue weapon to get to 100 TP in 8 hits. "Wait a second," you observe, "isn't the corresponding average number of rounds below 4.9526?" In fact, on average it takes a virtue weapon only 4.7305 rounds to get to 100 TP in 8 hits, so an 8-hit virtue weapon setup ideally has a higher weapon skill frequency than a 6-hit setup with a non-multi-hit weapon.

Is the average attack round argument unconvincing? Let's instead examine the probability distributions of the number of attack rounds it takes for a virtue weapon, a weapon without a multi-hit property, and, for comparison's sake, a "Trial of the Magians" OAT weapon (for dragoon, Bradamante) to get to 100 TP:


These probability distributions were obtained via Markov chain methods.

For a weapon without a multi-hit property, the probability of getting to 100 TP in 5 attack rounds is .580, and the probability for fewer than 5 attack rounds is higher than the probability for greater than 5 attack rounds, which is consistent with the average attack round figure of 4.9526.

In comparison, while the probability of getting to 100 TP in 5 attack rounds is lower for a virtue weapon (.403), the higher probability of getting to 100 TP in 4 attack rounds (.373) contributes to the average number of attack rounds to get to 100 TP being lower (4.7305).

And for the sake of comparison, it takes about 3.783 rounds for a Magian OAT weapon to get to 100 TP in 6 hits. This breaks down such that, most of the time, there is a high probability that a Magian OAT weapon takes either 3 or 4 attack rounds to get to 100 TP.

Note that for all three types of weapons, the probability that it takes 7 or more attack rounds to get to 100 TP is, at most, about .028 (for both the virtue weapon and the non-multi-hit weapon), which underscores the fact that, at least given 95% hit rate, it's not like the virtue weapon "needs" 7 or more attack rounds to get to 100 TP with any significant probability just because 8 landed hits are required to generate 100 TP.

In short, delay for virtue weapons, and the corresponding nominal number of hits it takes to get to 100 TP, is relatively unimportant because of the OAT property. In the case of the 8-hit Love Halberd setup, this property results in a lower average number of attack rounds to get to 100 TP than that for a 6-hit setup for a weapon without a multi-hit property (assuming a 55% virtue stone consumption rate).

Is the Love Halberd's base damage rating too low?

Love Halberd's 60 base damage is only 4 lower than Fortitude Axe's 64, which has 504 delay, so I'd say dragoons and samurai are relatively "spoiled" with access to a weapon with such high attack frequency and low delay.

Also, with a low base damage, the relative damage gap between Love Halberd and a higher-damage weapon decreases with additional fSTR.

Does Love Halberd's DEX +7 matter?

This is relatively unimportant, but with DEX +8 generally guaranteeing a 1% increase in critical hit rate when the target's AGI is not obscenely higher than your DEX, one can expect, effectively, a +1% critical hit bonus most of the time with DEX +7, which is not bad. DEX +7 is also a nice amount of DEX in the weapon slot that could help to ramp up one's critical hit rate if the opportunity presents itself (yeah, yeah, Greater Colibri...).

At least you can say it counters the loss of any attack (or accuracy) bonus associated with equipment for the ammo slot, Smart Grenade, Tiphia Sting, or whatever it is that DRG uses.

An additional +5 or +6 accuracy, if actually realized from the DEX bonus, is nothing to ignore, either.

Finally, a comparison of polearm options

All the features of Love Halberd described culminate such that Love Halberd is better than "conventional wisdom" allegedly holds.

Earlier, I did a write-up of how to model (approximately) the effect of Jump on damage rate as a preliminary step to doing a comparison of polearms that accounts for the increased WS frequency that Jumps provide. As usual, this comparison is done in terms of a long-run, WS-spamming, Jump-spamming situation so that one gets a decent idea of the relationship among the weapons in terms of maximum potential.

The weapons to be compared are

• Valkyrie's Fork (6 hits to 100 TP)
• Bradamante (with 75 base damage and 6 hits to 100 TP)
• Love Halberd (8 hits to 100 TP).

Some of the conditions I specified are

• fSTR 6 (+5 for Drakesbane)
• 42 additional WS "base" damage from the STR 50% modifier
• 95% hit rate
• 0% Zanshin rate
• base double attack rate of 9%
• ATK/DEF ratio of 1.5 and base critical hit rate of 9%, corresponding to an (approximate) average pDIF of 1.599 across all weapons (the critical hit rate bonus of Love Halberd treated as though it offsets the use of virtue stones at the expense of any attack bonus from the ammo slot)

Also, for Drakesbane, I am assuming a critical hit rate bonus of +10% and basing WS damage on 100 TP (ignoring excess TP effects, if they even exists). For Jumps (when accounted for), I treat the damage of Jumps as equivalent to normal hits (yet another simplification).

Let's start with a high quantity of haste, say, 64%, which accounts for Hasso (10%), double March (20%), Haste spell (15%), and haste from equipment (19%), which would relatively favor Valkyrie's Fork, a weapon with fundamentally lower WS frequency than the others, because of weapon skill delay (2 seconds).

Without accounting for the effect of Jumps, the summary of relevant numbers comes out as follows:

Weapon	Avg. TP dmg	Avg. WS dmg	Time per WS	Dmg/sec	TP:WS dmg
Valkyrie's Fork	832.01	1041.54	16.29 s	114.98	444:556
Bradamante	701.52	894.93	13.78 s	115.83	439:561
Love Halberd	793.79	789.77	13.23 s	119.61	501:499

These figures are merely a point of comparison to the more "realistic" figures that account for the effect of Jumps. But first, as an aside, I have to point out that the OAT effect of virtue weapons doesn't proc on Jumps and discuss the major implication for using Jumps with Love Halberd.

In general, Jumps can be considered an attack round that occurs "on demand." Moreover, Jumps generally delay the start of the following attack round by 2 seconds (a consequence of job ability or weapon skill delay in general), so Jumps, in effect, help to decrease the time between weapon skills except when the time between auto-attack rounds falls below 2 seconds. This is the primary effect of Jumps as slight increases in Jump damage per hit compared to auto-attack damage per hit are minor in comparison.

But since Jumps with Love Halberd are effectively normal attack rounds, they do not generate TP (on average) as much as auto-attack rounds. Therefore, there is a critical value of haste after which jumping with Love Halberd is unproductive.

Given the above conditions, Love Halberd averages about 1.579 landed hits per attack round, and "normal" jumps average exactly .95*1.09 = 1.0355 landed hits per "attack round" or 0.51775 landed hits per second (if spammed, so this is the upper limit for Jumps). It follows that it's counterproductive to jump with Love Halberd (in a long-run sense, not in a "need damage on demand" sense) when haste is above 53% (an approximate critical value). Therefore, for the following table, the effect of Jumps is considered only for Valkyrie's Fork and Bradamante:

Weapon	Avg. TP dmg	Avg. WS dmg	Time per WS	Dmg/sec	TP:WS dmg
Valkyrie's Fork	832.01	1041.54	16.00 s	117.08	444:556
Bradamante	701.52	894.93	13.51 s	118.13	439:561
Love Halberd	793.79	789.77	13.23 s	119.61	501:499

As stated previously, the primary effect of Jumps is to decrease the time per weapon skill. Given 64% haste, the effective increase in damage per second is at most around 2%. (At lower levels of haste, the contribution of Jumps to increasing the rate of damage is higher.) Even when Jumps are accounted for, Love Halberd is still slightly better than either Valkyrie's Fork or Bradamante. (The TP:WS damage ratios are my usual check on how well the calculations represent what is observed in the game, but I have no idea if these are typical ratios.)

Certainly, virtue stone consumption is a strike against Love Halberd for everyday, humdrum situations, and it's possible Bradamante can be further augmented after future updates, but can Bradamante be enhanced to the point where formerly top-end polearms (like Valkyrie's Fork) are completely outclassed after accounting for human "inefficiency"? It remains to be seen, but now let's consider the viability of these weapons in a zerg-like situation with 80% haste:

Weapon	Avg. TP dmg	Avg. WS dmg	Time per WS	Dmg/sec	TP:WS dmg
Valkyrie's Fork	832.01	1041.54	9.94 s	188.47	444:556
Bradamante	701.52	894.93	8.55 s	186.80	439:561
Love Halberd	793.79	789.77	8.24 s	192.08	501:499

As discussed in a previous post, the benefit of increasing haste is higher for weapons with lower WS frequency than weapons with higher frequency, a consequence of weapon skill delay. Unsurprisingly, Bradamante falls behind Valkyrie's Fork, yet Love Halberd still has a slight advantage over Valkyrie's Fork even at maximum haste, lending actual credence to the use of Love Halberd for high-haste zergs (and discrediting the idea of using Bradamante for such, at least when compared to Valkyrie's Fork).

Conclusions

Love Halberd's delay in conjunction with its OAT property can give it a weapon skill frequency lower than weapons without any multi-hit property. For example, an 8-hit Love Halberd setup has a higher WS frequency than a 6-hit setup for a polearm without any multi-hit property. This, along with its relatively high base damage (for a multi-hit weapon) and DEX +7 make it a "peer" to the likes of Bradamante, the latest fashionable polearm. At 80% haste, Bradamante is a relatively poor weapon compared to Love Halberd.

How do you account for the effect of Jump?

Modeling the effect of Jump on damage rate isn't too bad provided that you invoke the following major simplifications: let both Jump and High Jump have the same amount of merit upgrades, and treat TP from Jumps as accumulating toward a weapon skill independently of TP from auto-attack. In this way, we can estimate the proportion of attack rounds that Jumps contribute to the average number of attack rounds required to accumulate 100 TP (for a weapon skill). We need this proportion to estimate the time savings from using Jumps that contribute to increasing WS frequency.

Suppose that there are 5 merits both in Jump and High Jump. This means that in a 150-second time frame, two Jumps and one High Jump can occur, for a total of three jumps. Also, do not (yet) assume that attack rounds from Jumps are equivalent to those from auto-attack in terms of multi-hit "capability" (from double attack, multi-hit weapons, etc.). It is then possible to obtain a general expression for the denominator required to obtain the respective proportions of attack rounds that auto-attack, Jump, and High Jump contribute to the average number of attack rounds to 100 TP:


The implied units for this denominator are rounds per WS (with spamming of TP after 100 TP is achieved).

The first term (factors specific to it denoted with the subscript 1) in the expression accounts for how many weapon skills from auto-attack can occur in 150 seconds when accounting for a weapon skill delay of two seconds. T₁ denotes the time per attack round at 0% haste, and H denotes the haste level as an integer. E[R] in general denotes the average number of attack rounds to 100 TP, and usually, E[R₁] = E[R₂] = E[R₃] except in the case of virtue weapons, apparently (the only reason the equality wouldn't hold because virtue weapons apparently do not work with Jumps).

The second term accounts for how many weapon skills from Jump (two Jumps in 150 seconds, remember) can occur in the previously specified 150-second time frame (necessarily a fraction), and the third term accounts for how many weapon skills from High Jump can occur in 150 seconds.

With this denominator expression, it should then be obvious how to obtain the actual proportions of attack rounds that each of auto-attack, Jump, and High Jump contribute to the average number of attack rounds to 100 TP. For example, the proportion of attack rounds that High Jump contributes to the average number of attack rounds to 100 TP is


These proportions can then be used to obtain an estimate of the adjusted average of the number of attack rounds to 100 TP accounting for Jump effects (this is a weighted average). Of course, if E[R₁] = E[R₂] = E[R₃] = E[R], then the weighted average simplifies to E[R].

However, the adjusted average of attack rounds cannot be multiplied by a simple "time per attack round" conversion factor to get the average time to 100 TP. Recall that TP from Jumps is treated as independent of TP from auto-attack as a simplifying assumption. Instead, the aforementioned proportions must be used to obtain a weighted average of the time "per cycle" of 100 TP generated, with 2E[R₂] seconds for Jump and 2E[R₃] seconds for High Jump (ignoring stacking of Jump and High Jump; the units of E[R] are attack rounds "per cycle" of 100 TP generated) and E[R₁]T₁(100-H)/100 seconds for auto-attack.

Mechanistically, we should already recognize before doing modeling that the dominant effect of Jumps is to increase WS frequency by reducing the time required to generate 100 TP, except when T₁(100-H)/100 < 2 seconds. With modeling, it is possible to estimate the reduction (both absolute and relative) in average time to generate 100 TP from Jumps. From modeling, it is also possible to account for differences in damage between auto-attack hits, Jump, and High Jump (you don't use haste equipment for Jumps, right?), but this effect is slight compared to the effect on WS frequency and will not be accounted for in future posts.