Estimating changes in magic hit rate with skill

For the purposes of estimating melee hit rate, the functional relationship among accuracy, dexterity, combat skill, mob level, and mob evasion has long been established, thanks to the clever use of the check function. Sadly, no such relationship has really been justified for magic "hit rate" (or resist rate), but that doesn't mean we are condemned to flail in the dark.

Having wondered myself about the utility of meriting elemental magic skill for the purposes of reducing the frequency of resists on "hard stuff," I looked for some information on the relationship between magic skill and resist rate, but solid evidence was hard to come by. Fortunately, after wading through senseless conjecture on BG, I managed to come across an interesting data set for which the "success" rates of casting magic on Ebony Puddings were recorded, given specific levels of elemental skill, magic accuracy, and INT. Even better, this data all but invites me to take a swing at it using some kind of linear regression analysis.

But first, if the factors that go into "magic hit rate" (rate of success or rate of no resists) are similar to those that go into melee hit rate, there are several issues that immediately come to mind when trying to suss out some kind of relationship, such as the relationship between magic accuracy and magic resistance (or evasion?). (Dec. 15: I wrote "Is a ratio involved, as is the case with melee accuracy and melee evasion?" which is incorrect. I probably was thinking of MAB/MDB, but that would be analogous to melee attack and defense.) Furthermore, even if magic resistance/evasion were constant among the flans on Mount Zhayolm, there is a range of levels for Ebony Puddings (supposedly 75-80 on Mount Zhayolm), and if a "magic hit rate" calculation involves a level correction, there is no practical way to account for that.

Still, looking specifically at the nuke data (tests II, III, IV), there appears to be some evidence of a linear association between magic skill alone (holding other relevant factors constant) and success rate. You can do your own plot if you're not convinced.

But as far as magic accuracy is concerned, there are only three combinations of magic accuracy and elemental skill where the success rate was measured. One may argue that magic accuracy seems to be less effective at 242 elemental skill than at higher levels of skill, which may seem persuasive (random variability and unaccounted sources of variability notwithstanding). Really, though, it's a reach to conclude that elemental skill and magic accuracy are correlated with the limited data here.

Finally, INT seems to have no effect at 242 elemental skill, yet has some effect in large quantities at 274 skill. Maybe it's not all that far-fetched to say that the effect of INT on magic hit rate is dependent on magic skill level, which can compromise the estimates associated with a regression analysis. Even worse, perhaps the relationship between INT and magic hit rate (holding other factors constant) is not strictly linear but follows some weird piecewise function depending on your target mob's INT. This calls attention to the need for more data at other levels of INT, macc, and elemental skill (or perhaps a better choice of target whose level and magic resistance value is known to be fixed, but in practice this will be extremely difficult to achieve).

At any rate, using linear regression (with unresisted magic hit rate as the binary response) on the above observations (ignoring the middle rows of test II because they contribute to a poor model fit) gives the following parameter estimates (I truncated output to save space):

                     Standard   Wald 95% Confidence
Parameter  Estimate     Error         Limits         Pr > ChiSq

Intercept   -1.9393    0.1872   -2.3062     -1.5724      <.0001
skill        0.0095    0.0007    0.0082      0.0109      <.0001
macc         0.0147    0.0022    0.0103      0.0190      <.0001
int          0.0028    0.0010    0.0009      0.0047      0.0038

I included both INT and magic accuracy in the model just for the heck of it even though the parameter estimates associated with them aren't all that reliable. Certainly, including more observations with varying levels of INT and magic accuracy may improve those estimates (assuming magic hit rate is linear over some range of either factor), and they should be included in a model for the sake of a comprehensive view of magic hit rate. But for now, we can see that the data suggest that magic hit rate increases by about 1% for every one-point increase in elemental magic skill (holding INT and magic accuracy fixed). The range of elemental magic skill considered is between 242 and 295.

One can also perform a similar analysis with the Sleep trials (tests V and VI), but note that the "success" rate encompasses partial resists also:

                     Standard   Wald 95% Confidence
Parameter  Estimate     Error         Limits         Pr > ChiSq

Intercept   -1.3636    0.5853   -2.5108     -0.2164      0.0198
skill        0.0056    0.0018    0.0021      0.0091      0.0016
macc         0.0085    0.0025    0.0035      0.0134      0.0008

It seems that the effects of magic skill (enfeebling in this case) and magic accuracy are weaker for sleeping than for nuking. (Granted, the interval estimates are rather wide.) The range of enfeebling magic skill is between 307 and 333. It's possible that the acts of sleeping and nuking are just not comparable (unlikely) with respect to resist rates. It's also possible that the effects of general magic skill and accuracy on magic hit rate are diminished past the 300 level of general magic skill. Either way, this complicates understanding of magic hit rate somewhat and steps can be taken to rule out either explanation.

It hasn't escaped my attention that magic accuracy seems to increase magic hit rate more than magic skill, ignoring the wide interval estimates. If this is really the case, the difference is so slight and direct competition between the two attributes so rare that it's not worth caring about. Even comparing Oracle's Robe (magic accuracy +6) to Igqira Weskit (elemental magic skill +5), I would first argue the benefits of using Oracle's Robe to replace both Errant Houppelande (like anyone cares about the elemental enfeebling line) and Igqira Weskit. The HP+20 for Sorcerer's Ring activation can be useful, too.

It also occurred to me that one may try to argue, in analogy to melee accuracy and melee hit rate, that this data support the contention that magic skill increases magic hit rate by 0.9% above the 200 skill level (1% at or below 200), although it is ludicrous to distinguish between 0.9% and 1% based on random data without excessive sample sizes.

But, if all you cared about was estimating the change in magic hit rate for every one-point increase in elemental skill, you might as well focus on the change in magic hit rate between two levels of elemental skill that are relatively far apart, assuming the rate of change is constant (in other words, a linear relationship between hit rate and skill), an assumption that is borne out by the previously considered data.

The regression analysis for the nuke data used 1,400 total trials; these trials could be allocated equally between, say, 242 skill and 292 skill. Then you'll have an easier time showing that the increase in magic hit rate is less than 50% (less than 1% per point of elemental skill). (Use a test for two proportions.)