Changes: Stat comparison

Revision as of 19:00, 2 August 2008

Among the most interesting decisions to make in World of Warcraft are those concerning gear. Should a warrior prefer +Stamina or +Dodge%? Is +heal better or mana/5? Of course there can be no "right" answers to such questions. It's always a matter of "it depends...", but sometimes reasonable upper and lower limits can be found (e.g. 5 mana/5 will for all casters always be better than 10 spi). Another helpful insight is the Item Values article, just to see which relative values Blizzard gives to different stats.

Healer mana

There are many aspects to the problem how to optimize a healers endurance. Lets look at the various item stats which can affect healer mana and/or efficiency.

Intellect

Int simply increases the mana pool and spell crit chance. It's the benchmark for the other stats.

Spirit

When considering spi, it is important to understand how mana regeneration works (see 5 second rule). There are mods which collect the data on how much time is spent inside the 5 second rule, and how much mana each point of spirit regenerated (like Spirit versus Intellect). For "average" combats it's safe to assume that 1 spi = 1 int.

Mana/5

At first glance mana/5 is quite similar to spi - it regenerates mana. In combat though mana/5 is usually "better", as a rule of thumb a factor of 3 can be assumed (1 mana/5 = 3 spi)

+heal

Comparing +heal to the other stats is a little tricky. It is necessary to consider current mana efficiency (HP/mana and HP/time) and its change due to +heal. For any given combat thus the saved amount of mana can be found. The effect of additional +heal becomes less after a certain point, because increasing an already high efficiency yields less of an effect than increasing a low efficiency. The various sources agree that a factor of about 8 is appropriate to convert +heal to mana/5 (1 mana/5 = +8 heal).

Spell crit%

Similar to +heal, this increases efficiency (with the added problem that crits may easily result in overheal). One percent crit in theory increases total HP healed by 0.5%, which in turn could be translated to an increase of the available mana by the same amount.

Summary

Reducing all stats to Int leads to the following:

1 Spi    = 1 Int
1 Mana/5 = 3 Int
+8 heal  = 3 Int

In longer fights +heal and mana/5 become more important, when grinding spi and int are preferrable. For PvP, Int is probably the most important stat, because PvP encounters tend to be short but intense, and the increased critrate is important there too.

A very thorough discussion can be found at WoWHealers

Caster DPS Equivalence

The equivalence between damage stats for casters is greatly simplified by using the following general damage formula:

E = hq(m+rd)(1+bc)

Where...

E = expected damage/cast
h = chance to hit
q = flat multipliers on damage
m = average base damage
r = +damage coefficient
d = +damage
b = critical strike bonus, not multiplier
c = critical strike chance

Some notes:

This model assumes a two-roll system for hit/crits.
Chances to hit and crit are decimal numbers: i.e. 30% chance to crit -> c = 0.30
b is the critical strike bonus, not the multiplier. Untalented crits are 150% of normal damage; this implies b = 0.50.
m is the average base damage before multipliers. Using values from WoWhead of Thottbot is usually sufficient.
q represents all multipliers active on a spell. This includes...
- Talented multipliers (like a Fire Mage's [Fire Power] talent, which increases the damage of Fire spells by 10%),
- Buffs (like a Priest's [Power Infusion], which increases all damage and healing by 20%), or...
- Debuffs on a target (like a Warlock's [Curse of the Elements], which increases all Fire and Frost damage done by 10%).
Multipliers are taken to multiply on each other: thus, the three multipliers above would not be a net +40% damage, but rather 1.1*1.2*1.1 = 1.452 = +45.2% increase.
This is only valid for long-run damage/DPS equivalence. Calculations for, say, damage per mana pool are more complicated.

Marginal Benefit of Stats

To compute the equivalence between stats, it's convenient to figure out just how much damage is gained by increasing stats. Once this is done, the gains can be set equal and solved for one stat or the other. In short, it's easiest to calculate how much damage or DPS is added by +damage, +spell crit, +spell hit, and so on, as these values can then be compared to figure out how much of one stat is equivalent to another.

Given the expected damage/cast formula above, the following expressions can be derived:

∆E = hq(1+bc)r∆d
   = hq(m+rd)b∆c
   = q(m+rd)(1+bc)∆h

Where ∆E represents the change in E (the change in expected or average damage/cast), ∆d the change in +damage, and so on.

With the marginal benefits calculated above, stat equivalence can now be computed.

Example: Spell Damage and Spell Crit

This example how equivalences in general are calculated, using the particular case of spell damage versus spell crit.

For adding ∆d +damage to be considered equivalent to ∆c crit chance, the benefit (∆E) of both must be equal. Thus, their individual expressions for ∆E can be set equal.

That is...

∆E = ∆E

Which is equivalent to...

hq(1+bc)r∆d = hq(m+rd)b∆c
  (1+bc)r∆d = (m+rd)b∆c
  (1/b+c)∆d = (m/r+d)∆c

From here, one can solve for either ∆d (which would tell you how much +damage ∆c crit chance is worth) or ∆c (which would tell you how much crit chance ∆d +damage is worth).

These expressions would look like...

∆d = (m/r+d)∆c/(1/b+c) (1)
∆c = (1/b+c)∆d/(m/r+d) (2)

From these expressions, some generalizations can be made. First, the value of +damage in terms of crit decreases as you add more +damage; conversely, the value of +crit in terms of +damage decreases as you add more +crit. Thus, there is not so much a breakeven "point" so much as there is a breakeven "line."

It is, however, convenient to look at the value of 1 +damage in terms of crit, or 1 crit rating in terms of +damage.

Given expression (1) above and substituting ∆c = 1/2208 (the value of 1 crit rating), the result is...

1 spell crit rating ≡ 1/2208*(m/r+d)/(1/b+c) +damage

Or, given expression (2) above and substituting ∆d = 1, the result is...

1 +damage ≡ 2208*(1/b+c)/(m/r+d) +spell crit rating

General Stat Equivalences

The following formulas result from the same methodology applied to four primary stats--+damage, +spell crit, +spell hit, and +spell haste. In order to accommodate spell haste, consider an expected DPS function D:

D = (1+z)*E/t

Where...

z = haste
t = unhastened casting time

The method continues as before, however, yielding the following results.

1 spell hit rating ≡ 1/1262*(m/r+d)/h +damage ≡ 2208/1262*(1/b+c)/h +crit ≡ 1576/1262*(1+z)/h +haste
1 spell damage ≡ 1262*h/(m/r+d) +hit ≡ 2208*(1/b+c)/(m/r+d) +crit ≡ 1576*(1+z)/(m/r+d) +haste
1 spell crit rating ≡ 1262/2208*h/(1/b+c) ≡ 1/2208*(m/r+d)/(1/b+c) ≡ 1576/2208*(1+z)/(1/b+c) +haste
1 spell haste rating ≡ 1262/1576*h/(1+z) ≡ 1/1576*(m/r+d)/(1+z) ≡ 2208/1576*(1/b+c)/(1+z) +crit

These formulas allow for some blanket conclusions:

Two properties contribute to the high value of spell hit rating: its good conversion to percentage and the cap on hit chance at 99%. That is...
1. 1 spell hit rating is always better than 1 spell haste rating.
2. 1 spell hit rating is always better than 1 spell crit rating.
For 1 +damage to be worth more than 1 spell hit rating, m/r+d must be less than 1262*0.99 = 1249. This is actually quite restrictive. A talented Starfire has the lowest m/r value at around 500. This means that +spell hit rating is more valuable than +spell damage at any point above +800 spell damage, provided you're under the cap, of course.
1 spell haste rating starts (at the 0% crit and 0% haste level) 40% more valuable than crit. Additional crit only makes haste more valuable.

Spell damage vs Spell crit

The following reasoning will focus on DPS alone. Taking other factors (like procs off crits) into consideration would be possible, but since there are many such effects, and they are very dependent on class abilities and talents, they will be ignored here. Additionally, spell crits are assumed to increase damage by 100% (not by the normal 50%, but classes interested in this discussion should have taken their talent).

The following abbreviations will be used:

crat : crit rating (the crit percentage is the crat / 21)
pdmg : +damage, the spell damage value added from gear
bdps : base dps (dps prior to adjustment for crits, including talents and pdmg)
bep : break even point (the bdps value at which crat and pdmg yield an equal increase of total DPS)

pdmg and crat are basically multiplied to yield final DPS. Contrary to popular belief, pdmg does *not* really result in a "linear" DPS increase, it's rather just another factor in the product. In principle it's (bdps + pdmg) * crat = final DPS, which means that the pdmg is also multiplied by crat. If we want a product to become as large as possible, but the sum of the factors needs to be constant, it's best if the two factors are equal. Thus in theory, both factors should be increased in parallel.

Assuming a crat of 0, one point of crat increases total DPS by 1 / 2100 of bdps (this equals 0.048%), and one point of pdmg results in an increase of bdps by 1 / 3.5 = 0.286. If we want the two to be equal, we need to do the following calculation:

bep / 2100 = 0.286
bep = 0.286 * 2100 = 600

So at 0% crit chance, we need 600 DPS base damage before crat starts to produce more DPS than pdmg. If we do have some prior crat, the above calculation becomes:

bep / 2100 = 0.286 * (1 + old crat / 2100)
bep        = 600 * (1 + old crat / 2100)
bep        = 600 + (old crat * 600 / 2100)
bep        = 600 + (old crat * 0.286)

This means that for each point of crat we already have, we need 0.286 more DPS in order to make crat and pdmg equal. These 0.286 DPS are exactly what one pdmg yields. In other words, for each point of crat, we also need a point of pdmg to make it viable (and vice versa). The optimal compromise is to have enough +damage gear to bring base DPS to 600, and from this point on an equal amounts of crit rating and plus damage.

In effect this means that +damage will usually result in a higher DPS increase than +crit (especially considering that Blizz values +damage lower than +crit rate, see Formulas:Item Values, spell damage is at 0.85 and crit rate at 1). Spell crit rating is expensive, but the "price" for crit rating decreases as the amount of +damage increases.

Examples (assuming a caster with 320 DPS from his spells prior to adding +damage):

+damage	base DPS	crit %	total DPS	increase
500	462.86	10%	509.14
501	463.14	10%	509.46	0.32
500	462.86	10.048%	509.37	0.23
1000	605.7	10%	666.29
1001	606	10%	666.6	0.31
1000	605.7	10.048%	666.56	0.27
1500	748.57	25%	935.71
1501	748.86	25%	936.07	0.36
1500	748.57	25.048%	936.07	0.36
2000	891.43	25%	1114.29
2001	891.71	25%	1114.64	0.35
2000	891.43	25.048%	1114.71	0.42

The first two groups show that even at +1000 damage and with a rather low crit rating, +damage is still better than + crit rating, although the difference is smaller than at +500 damage. Break-even (at 10% crit) would occur at 660 base DPS, which would mean +1190 damage. So while levelling or still wearing blue/green gear, crit rating should not be a sought-after stat (if interested in a high average DPS output).

As shown in the lower two groups, +1500 damage is just enough to compensate a spell crit rating of 25% (the exact bep would be at +1505 damage, but this doesn't really matter). With even more +damage gear, increasing crit rating will be better than further +damage. At high levels with real endgame gear, crit rate is a much more interesting stat (again, strictly considering DPS only).

If we consider that crat is 17% more expensive than pdmg we get that the break even point is ~700 and not 600 and would show that you will mostly never want to go for crit heavy gear unless you have over 1k spelldamage and no critrating, which will never happen.

STR vs AGI

When considering the effects of str and agi on pure white DPS, Str increases DPS by a linear amount (1 Str yielding about .14 DPS), while Agi increases the chance to crit (and thus DPS) by a factor. In practice this means that depending on the current damage output, there's an equilibrium point at which the DPS increase from Agi is higher than the increase from Str.

In summary however, it is safe to assume that for most melee characters STR adds more DPS than equivalent AGI, because the equilibrium point is at a rather high value.

Strength

Hunters/Rogues gain 1AP per str (= approx. 0.07 DPS per Str).
All other classes gain 2AP per str (= about 0.14 DPS per Str).

Note that this is strictly melee attack power; the calculations for ranged attack power (RAP) are different. Also note that this is base "white" dps, and does not consider changes to other abilities.

Value of Agility Depending on level and class, agility increases white DPS by increasing the chance to critically hit. Additionally,

Rogues/Hunters gain 1AP or .07 DPS per AGI.

Equilibrium Point

At this DPS point, the DPS gained by 1 STR and 1 AGI are equal. Past this point 1 AGI will always yield more DPS.

The formula to find the equilibrium point is:

X = ((DpsPerStr - DpsPerAgi)*(1 + CurrentCrit))/CritPerAgi

where

X is the equilibrium point in DPS.
DpsPerStr = .0714 for Rogues/melee Hunters and .1428 for all other classes.
CritPerAgi = Critical Hit percentage in decimal form gained by 1 AGI point, changes based on level and class, see section on Agility.
CurrentCrit = Your current Critical Hit chance in decimal form (eg: 10% crit = 0.1)
DpsPerAgi = .0714 for Rogues/Hunters/Cat druids, 0 for other classes
These constants are subject to alteration by talents. (such as Heart of the Wild)

*Because DpsPerStr and DpsPerAgi are equal for rogues and hunters, the equilibrium point is zero. There is never a point where a rogue or hunter should take a point of STR over a point of AGI.

For the rest of us... Consider a level 60 (or lower) druid (not in cat form) with 10% crit chance.

DpsPerStr = .1428
CritPerAgi = .0005 (that is; 1 agi = 0.05% crit chance, 20 agi = 1% crit)
DpsPerAgi = 0
X = (0.1428 - 0)*(1 + 0.1)/0.0005
X = (0.1428*1.1)/0.0005
X = 314.16

Until the shown DPS (what is shown when you hover your mouse over your damage display in the character menu) is equal to 314.16 it is MORE beneficial from a white DPS standpoint to put 1 point into STR than it is to put one point into AGI. After the equilibrium it is always better to put points into AGI. Note however that this will change your critical hit chance, and therefore the equalibrium point.

Now consider for a moment that same druid in cat form (assuming same crit chance).

DpsPerStr = .1428
CritPerAgi = .0005 (that is; 1 agi = 0.05% crit chance, 20 agi = 1% crit)
DpsPerAgi = .0714
X = (0.1428 - 0.0714)*(1 + 0.1)/0.0005
X = (0.0714 * 1.1)/0.0005
X = 157.08

Based on the values given in the section for Agility; For a level 70 paladin or bear druid, with a 20% crit chance the equalibrium point is 514.29 DPS. For a level 70 warrior with 20% crit chance it is 678.86. Obviously these values are insanely high.

IMPORTANT: Remember that this calculation only considers the 100% increased damage on crit when calculating DPS from agi. In cases in which you gain a secondary advantage from a crit (such as rogues or feral druids) this would result in your favoring agility over strength sooner then your calculated equilibrium point (due to the secondary advantages of the increased crit chance). Also when in PVP spike damage is usually considered preferable to sustained damage, as such when choosing PVP gear you may place mildly higher preference on agility for its spike damage from crits, even if it means sacrificing average sustained DPS.

Other considerations

Str and Agi both can also help to reduce damage taken. Agi increases the chances to dodge, Str increases the amounts blocked with a shield. Additionally, there are talents like [Flurry], [Deep Wounds], and Vengeance which can proc off critical hits, and complicate these calculations further.

See the discussions page for further insights.

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