What's the proc rate for virtue weapons? How do you know?

One "line" of evidence: checking with Justice Sword

Taken at face value, the estimate 555/1000 indicates the "occasionally attacks twice" (OAT) rate of Justice Sword is significantly higher than 50% and could be considered 55% (source). But is the OAT property the same for all so-called "virtue weapons"?

Another line of evidence: checking with Fortitude Axe... and WAR

The rest of this post discusses how to estimate the OAT rate of Fortitude Axe in the presence of the double attack trait from WAR. But first, I needed a good idea about how Fortitude Axe OAT actually interacts with the DA trait. In the past, I blabbed a lot about how Fortitude Axe might interact with double attack, but my "conclusion" was based on very weak evidence. After collecting some more count data with kparser under 12% DA (source), which ruled out my previous weak hypotheses about the DA/OAT interaction, I got a better idea about how to explain these results (assuming kparser was working correctly...).

It appears (not exactly "proof") that OAT can process on both the normal hit (which is guaranteed to occur for a given attack round, if not actually land) as well as the hit from the possible DA proc (with a major caveat to be discussed soon). More specifically, the hit from the DA proc occurs independently of whether an OAT proc occurs. (Not probabilistically, of course, but "mechanistically.")

It is worth noting that conceptually the order of DA and OAT could easily be reversed, such that the hit from the OAT proc occurs independently of whether a DA proc occurs, but I will just say the resulting probability calculations are not supported by the data when OAT is mechanistically independent of DA.

Anyway, one way to show pictorially all "possible" outcomes where DA and OAT can interact is with the following "tree":


There are six hypothetically "distinct" outcomes, but it is very inconvenient to monitor the equipment menu for virtue stone expenditure. More important, though, is the fact that Fortitude Axe cannot quadruple attack, so the case of expending two virtue stones is impossible. (This makes sense, noting that triple attacks are impossible with zero DA rate.)


So what "happens" to this 2-virtue stone attack round that is impossible? It appears that even if a DA proc occurs, only one virtue stone can be expended anyway, so the "tree" simplifies further:


The resulting probability model of the number of hits in an attack round (ignoring the distinction between hits and misses) is specified as follows. Let X denote the number of hits in a given attack round, d the probability of a double attack proc, and π the probability of an OAT proc. Then,

Now that we have a reasonable probability model describing the interaction between double attack and the OAT property of Fortitude Axe ("reasonable" based on chi-square goodness of fit to the data given 12% DA rate and posited virtue weapon proc rates of 50% and 55%), we can now estimate the OAT rate. Proceeding with maximum likelihood estimation is not really necessary when an obvious unbiased estimator can be based off the observed number of single hits (denoted as X₁) in n attack rounds:


It follows that the unbiased estimator is


with variance


Note that when d = 0, the variance reduces to that for the estimator for a simple binomial proportion (marginal in the context of the multinomial distribution). (Note to self: from simulation, this estimator is only very slightly less efficient, from an MSE standpoint, than the MLE, which I would bet is UMVUE even if an analytical expression for the MLE and the CRLB is annoying to obtain.)

The estimated proportion of single hits (per attack round) is 1 - 595/1425/.88 = .5255183, with corresponding 95% confidence interval (.4964218, .5546148). Given the specified probability model (which cannot be "proven" to be true at this time) and the data, it is not possible to conclude that the OAT rate of Fortitude Axe is either 50% or 55% (both are plausible given the confidence interval), unfortunately. But it should be possible to rule out one or the other with further data collection (with the hope that the probability model is correct), using the estimator specified above.

A third way: Faith Baghnakhs

Among all virtue weapons, it would be fastest to determine the OAT rate of Faith Baghnakhs by counting the number of triple attacks and quadruple attacks. It would be easier to do this on ninja because you wouldn't have to pay attention to kick attacks (because you want to use a parser instead of counting manually). If the OAT rate for Faith Baghnakhs can be shown to be 55%, that, along with the observed proc rate for Justice Sword, could be used as evidence for a common OAT rate of 55% across all virtue weapons.