DK Stat Weights: A Derivation (Now with more spreadsheets!)

♦ Darkside · 02/14/09, 3:24 AM

Since there have been no definitive stats weights developed for the Death Knight, I took it upon myself to calculate these values. Due to the natural dynamic nature of the system, there is no universal stat weighting for all Death Knights, or even for each spec. The weightings for an individual character are specific to the gear and talents of that character.

GENERAL DERIVATION

Stat weighting is fairly simple to determine, once you realize the formula for how damage is calculated. Let’s look, for example, at the simplest example (for a Death Knight): an instant cast spell, such as Icy Touch or Death Coil (variable assignments are given at the end of the post).

$(b + AP \cdot k)(1 + c \cdot m) = d$

From here, it is easy to calculate what the change in damage will be if we add any amount of a given stat. For example, a change in Attack Power produces the following change in Damage.

$(b + (AP + \Delta AP)k)(1 + c \cdot m) = d + \Delta d$

$(b + AP \cdot k + \Delta AP \cdot k)(1 + c \cdot m) = d + \Delta d$

$\Delta AP \cdot k(1 + c \cdot m) = \Delta d$

We can just as easily use this formula to determine the change in damage if we increase the critical strike chance:

$(b + AP \cdot k)\Delta c \cdot m = \Delta d$

Setting these two equations equal to each other, we get:

$\Delta AP \cdot k(1 + c \cdot m) = (b + AP \cdot k)\Delta c \cdot m$

Which can then be solved for delta_AP in terms of delta_c:

$\Delta AP = \frac{{(\frac{b}{k} + AP)}}{{(\frac{1}{m} + c)}}\Delta c$

Finally, exchanging Crit Rating for Crit Chance yields the following conversion:

$\Delta AP = \frac{1}{{4590}}\frac{{(\frac{b}{k} + AP)}}{{(\frac{1}{m} + c)}}\Delta c_r$

This is the relationship between attack power and crit rating. This formula makes sense at first glance: as the non-critical damage of an attack increases, the value of crit rating increases. Likewise, as crit chance increases, the value of additional crit rating is decresed. This relationship holds for all Death Knight instant attack spells, though it changes a bit for anything that uses weapon damage to the following:

$\Delta AP = \frac{1}{{4590}}\frac{{(b\frac{{14}}{} + AP)}}{{(\frac{1}{m} + c)}}\Delta c_r$

The only thing left to do is carry out calculations for white damage, which are nearly identical, except they include haste and must be done with respect to DPS, not overall damage done.

$(DPS + \frac{{AP}}{{14}})(1 + h)(1 + m \cdot c) = dps$

Applying the same ideas as above, we get the following weightings:

$\Delta AP = \frac{1}{{4590}}\frac{{(14 \cdot DPS + AP)}}{{(\frac{1}{m} + c)}}\Delta c_r$

All that is left now is to determine the weighting of Strength and Agility and combine the weightings for each attack into an overall weight scale.
Strength is really simple, you simply take the multipliers from talents and abilities to derive the final Str -> AP conversion (where FC uptime is the percent uptime of the buff):

$1Str = 1.1 \cdot (TalentBonuses)(1 + 0.3 \cdot FCUptime)AP$

(For most people, this will come out to about 2.6 or so)

Agility is a little tougher, since it only benefits melee attacks, so we must first convert it to crit rating and then multiply it by the percent of total damage dealt as melee:

$1Agi = \frac{{45.91}}{{62.5}}\% Melee$

COMBINING STAT WEIGHTS

Since the above formulas only work for individual attacks, they are not very useful for overall analysis. We must therefore combine all the various weights through a weighted average:

$\sum\limits_{i = 0}^N {w_i } \cdot p_i = W$

SUMMARY

Stat weights for spells:

$\Delta AP = \frac{1}{{4590}}\frac{{(\frac{b}{k} + AP)}}{{(\frac{1}{m} + c)}}\Delta c_r = \frac{1}{{2623}}\frac{{(\frac{b}{k} + AP)}}{z}\Delta z_r$

Melee weights:

$\Delta AP = \frac{1}{{4590}}\frac{{(b\frac{{14}}{\lambda } + AP)}}{{(\frac{1}{m} + c)}}\Delta c_r = \frac{1}{{3280}}\frac{{(b\frac{{14}}{\lambda } + AP)}}{z}\Delta z_r$

Auto attacks:

$\Delta AP = 14 \cdot \Delta DPS = \frac{1}{{3280}}\frac{{(14 \cdot DPS + AP)}}{{(1 + h)}}\Delta h_r = \frac{1}{{4590}}\frac{{(14 \cdot DPS + AP)}}{{(\frac{1}{m} + c)}}\Delta c_r = \frac{1}{{3280}}\frac{{(14 \cdot DPS + AP)}}{z}\Delta z_r$

b = Base Damage
AP = Current Attack Power
k = Attack Power Coefficient
c = Current  Critical Chance (in percent)
m = Critical Damage Bonus (1, w/o CSD and talents)
d = Damage Done
DPS = Base Weapon DPS
 = Normalization Constant (3.3 for 2H, 2.4 for 1H)
z = Hit Percent/Expertise
w = Stat weight for an individual attack
p = Percent of total damage for an individual attack
W = Overall Stat weight

ToDo:

1) Add values for Armor Penetration (it is horribly non-linear)
2) Clean up the page format
3) Create a spreadsheet for auto-calculation of weights

EDITS
1) Uploaded Beta version of the spreadsheet.

• Lanky · 02/14/09, 5:32 PM

Way to go Kroot! Look at you cleaning up this forum like a champ!

Garkbit · 02/14/09, 8:22 PM

Is this more or less what that DrDamage addon does? How can the average player benefit from this?

everwatch · 02/14/09, 8:49 PM

Wow...I have no idea what any of that means. And as such, it probably will not benefit the "average player". However, something to remember is that it will benefit any math guru who can come up with stat weights to begin with. Whether they correct mistakes in the formulas, add something to them, or show other applications to determine overall weights more accurately and quickly. I hope this thread continues on, and that the players who have so far modeled weights can benefit from this.

Kaveli · 02/14/09, 9:01 PM

well this could help in the development of an addon for determining dps or item value. I stared at the wall of math equations and just feel a bit dumber. But good job on the hard work!

♦ Darkside · 02/14/09, 9:06 PM

Right now I'm working on a spreadsheet that will automagically calculate stat weights for you if you input your character values. If all goes well, it should be up in a few hours.

UPDATE: This is taking longer than I thought it would, since I messed up the hit calculation. Spreadsheet is still on the way, though.

Kaveli · 02/14/09, 9:22 PM

Originally Posted by Darkside

Right now I'm working on a spreadsheet that will automagically calculate stat weights for you if you input your character values. If all goes well, it should be up in a few hours.

wow... your the man! can't wait to see this.

Mugaaz · 02/15/09, 12:17 AM

Wow, hope you're still working on that project. Sounds great.

huntcaudata · 02/15/09, 1:26 AM

A few questions on the project, that you've probably already thought of, but provide me something to do while up with a sick kid.

How are you going about weighting the abilities, i.e. the values for Icy Touch vs. the Values for Scourge Strike?
Are you considering the effect of haste on the GCD of spells?
Are you bothering with computing the effect on the ghoul, gargoyle, or DRW?
I imagine any spreadsheet would also take into consideration talents, and gear where appropriate?

Is there any point at which the results would change the formula? One extreme example that I think of is Loatheb. I often consider that going to a diseaseless rotation would be best in that circumstance. Could this also be the case for extreme values of AP, Haste, or Armor Pen?

Norther · 02/15/09, 4:39 PM

Please move the meaning of the variables to the top so I can stop feeling stupid for trying to figure out what they ment for 10 minutes...then seeing the bottom of the post XD.

I'm assuming C/H/Z sub r is just C/H/Z?

hammysink · 02/16/09, 12:42 PM

Originally Posted by Kaveli

well this could help in the development of an addon for determining dps or item value. I stared at the wall of math equations and just feel a bit dumber. But good job on the hard work!

Such an addon exists for this very thing. Once we have the spreadsheet (Thanks so much for doing the hard work!) we can plug in our gear/specs and out comes weightings that we plug in to PAWN. Then, loot is given a PAWN score in the tooltip that we can use to easily compare 2 pieces of gear.

bathoz · 02/17/09, 7:09 AM

Heh. Just need the spreadsheet to automatically create the pawn string.

What? I'm lazy, okay.

Dirich · 02/19/09, 10:06 AM

There's a tricky part: the attack table system.
That formula works, but the value of c need to be modified if you are not using spells.
Spells apply a double table, so the crit percentage applies after something hit. Which means c in such case is exactly the crit percentage showed by our character sheet.

When using a single table, like physical attacks do (a part from some rogue attack I think), we divide 100% into dodge, miss, parry, hit and crit.
So only if we are hit capped, and expertise capped (not the soft cap, the hard cap we can't really be reaching), than c is what is shown in our character sheet.

I.e. (numbers are chosen to make counts easy)
If we have 5% chance to miss, 10% chance to be dodged and 10% chance to parried, and our crit chance is 25%, than we have 50% hits and 25% crits.
Which means that c = 1/3, while without thinking about attack tables we would use c = 1/4.
Infact on 1000 strikes we would see 250 miss+dodge+parry, 500 hit and 250 crits. But that formula only accounts for hits and crits, which means we are evaluating 500 hits and 250 crtis, thus the higher "effective crit chance" (aka " c ").

From another point of view, since miss/dodge/parry affects only hit, we could modify " 1 " in the formula to account for such thing (while keeping c the value showed in our character sheet).
Using the above example 0.75 should be used in place of 1.

It may seem strange, but the higher our combined miss/dodge/parry chance is, the most crit affects our dps since the effective crit chance is higher, due to the fact that crits can't be missed/dodged/parried.

Conclusion
Because of the attack table system, the last formula seems to me wrong (the formula you used for the melee attacks I mean). Since you applied misses/dodges/parries to crits too.

♦ Darkside · 02/19/09, 12:19 PM

Originally Posted by Dirich

There's a tricky part: the attack table system.
That formula works, but the value of c need to be modified if you are not using spells.
Spells apply a double table, so the crit percentage applies after something hit. Which means c in such case is exactly the crit percentage showed by our character sheet.

When using a single table, like physical attacks do (a part from some rogue attack I think), we divide 100% into dodge, miss, parry, hit and crit.
So only if we are hit capped, and expertise capped (not the soft cap, the hard cap we can't really be reaching), than c is what is shown in our character sheet.

I.e. (numbers are chosen to make counts easy)
If we have 5% chance to miss, 10% chance to be dodged and 10% chance to parried, and our crit chance is 25%, than we have 50% hits and 25% crits.
Which means that c = 1/3, while without thinking about attack tables we would use c = 1/4.
Infact on 1000 strikes we would see 250 miss+dodge+parry, 500 hit and 250 crits. But that formula only accounts for hits and crits, which means we are evaluating 500 hits and 250 crtis, thus the higher "effective crit chance" (aka " c ").

From another point of view, since miss/dodge/parry affects only hit, we could modify " 1 " in the formula to account for such thing (while keeping c the value showed in our character sheet).
Using the above example 0.75 should be used in place of 1.

It may seem strange, but the higher our combined miss/dodge/parry chance is, the most crit affects our dps since the effective crit chance is higher, due to the fact that crits can't be missed/dodged/parried.

Conclusion
Because of the attack table system, the last formula seems to me wrong (the formula you used for the melee attacks I mean). Since you applied misses/dodges/parries to crits too.

Yeah, I noticed that a few days ago. The revised complete damage formula should read something like:

$(b + kAP)(z + cm)(1 + h) = d$

Or for white damage:

$(DPS + {\raise0.7ex\hbox{${AP}$} \!\mathord{\left/ {\vphantom {{AP} {14}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${14}$}})(z + cm)(1 + h) = dps$

Where the variables are the same as those in the initial post.

This reduces down to the original formula (minus the constant 'z') if your chance to hit is 100%.

Sh4d0wfury · 02/19/09, 12:40 PM

It seems to me you didn't take into account the pet's damage. It would, for example, increase the value of strengh, as it was noted in the unholy thread.

♦ Darkside · 02/19/09, 12:46 PM

That's true, I haven't accounted for the ghoul in the strength weightings, largely because I'm not sure how his damage is calculated. For the time being, I've chosen to ignore his damage and anyone specced into the perma-ghoul should add a bit onto the strength weighting to compensate for this.

♦ Darkside · 02/19/09, 3:58 PM

All right, I've uploaded my first version of the stat weight spreadsheet. In it's current incarnation, it's a bit unintuitive, so I'll try and give a basic explanation.

In the 'Stats' tab, you need to input your character's stats as they are shown in the character pane. In the 'AP from gear' box, input the green number that is displayed when you mouse over your character's AP. From there, you'll notice that values will automagically be calculated for the 'In Raid' portion of the sheet. This is done by taking values from the 'Buffs' page and applying them to your standard character stats.

On the 'Buffs' page, are listed all the major raid buffs that affect a Death Knight. To turn a buff off, simply set it's value to '0' instead of '1'. There are also listed the various talents that factor into stat weight calculation, simply enter the number of points spent into each.

Finally, there's the weights page. Here, nearly everything is automated. All you need to do is enter in what percent of your overall damage is from each attack, and the final stat weights will be displayed on the far right of the page.

Some additional notes: I haven't yet calculated values for ArPen, as it's a bitch. As such, overall weights for Blood will be somewhat off. I also need to add in values for base weapon damage, which is fairly trivial, so I'll get that done as soon as I have some downtime.

Sekke · 02/19/09, 6:16 PM

Originally Posted by Darkside

All right, I've uploaded my first version of the stat weight spreadsheet. In it's current incarnation, it's a bit unintuitive, so I'll try and give a basic explanation.

You have Shadow of Death as a 2 point talent, when it's only 1 point. Leaving it at 2 gives you the correct value for the talent, but someone might not notice that and just put it to 1, figuring it for a bug or something.

EDIT: Lest you think I'm only criticizing your work, this is amazing, thank you very much for your contribution!

Hitit · 02/19/09, 6:19 PM

Thanks for putting in the time and effort it takes to make this spreadsheet, it is much appreciated.

As I was trying out the spreadsheet, I stumbled upon something that seemed odd to me. I am, however, unsure if it is an error or if it is just me being stupid.

It's the amount of strength/agility you get from HoW/SoE and Imp SoE that doesn't make sense to me. Using the default stats, I tried setting all the buffs to '0' and the Fallen Crusader uptime to 0% so that the 'In Raid' Strength and Agility matches the 'Outside Raid'. Then I toggled HoW/SoE and Imp. SoE seperately and combined and got these results:

Buffs	Strength	Agility	Increase
No Buffs	982	275	N/A
HoW/SoE	1034	327	52
Imp SoE	1137	430	155
HoW/SoE + Imp SoE	1189	482	207

After doing this table I realized that it seems that when you toggle on HoW/SoE you're in fact toggling on MotW instead, which is right above in the table, and when trying to toggle on Imp. SoE you instead receive the unimproved buff from right above in the spreadsheet.

♦ Darkside · 02/19/09, 6:20 PM

Originally Posted by Sekke

You have Shadow of Death as a 2 point talent, when it's only 1 point. Leaving it at 2 gives you the correct value for the talent, but someone might not notice that and just put it to 1, figuring it for a bug or something.

EDIT: Lest you think I'm only criticizing your work, this is amazing, thank you very much for your contribution!

All right, I've fixed this for the next version of the sheet (will be up sometime in the near future). Don't worry about the criticism, I knew it was going to happen as there is now way I could have caught everything myself.

Originally Posted by Hitit

Thanks for putting in the time and effort it takes to make this spreadsheet, it is much appreciated.
...
After doing this table I realized that it seems that when you toggle on HoW/SoE you're in fact toggling on MotW instead, which is right above in the table, and when trying to toggle on Imp. SoE you instead receive the unimproved buff from right above in the spreadsheet.

Looking into this right now.

[e]: As it turns out, all the buffs are messed up. Fixing now. Should be done, I'll be reupping the sheet after my raid tonight.
[e2]: I just noticed that I've not included any flask/food buffs. I'll be adding those as well.

• sp00n · 02/19/09, 8:09 PM

Originally Posted by Dirich

There's a tricky part: the attack table system.
That formula works, but the value of c need to be modified if you are not using spells.
Spells apply a double table, so the crit percentage applies after something hit. Which means c in such case is exactly the crit percentage showed by our character sheet.

When using a single table, like physical attacks do (a part from some rogue attack I think), we divide 100% into dodge, miss, parry, hit and crit.
So only if we are hit capped, and expertise capped (not the soft cap, the hard cap we can't really be reaching), than c is what is shown in our character sheet.

It may seem strange, but the higher our combined miss/dodge/parry chance is, the most crit affects our dps since the effective crit chance is higher, due to the fact that crits can't be missed/dodged/parried.

As far as I know, all special abilities use a two roll system. At least that is how the rogue mechanics are (all specials two roll, melee white hit single roll), and I haven't heard of it being different for any other physical DPS class yet, so I assume it is the very same for death knights. I could have missed other classes' findings of course, as I haven't been reading their forums since WotLK hit.
Please correct me if I'm wrong.

And to carry on your last statement, for white hits this is true as long you actually have any hit to be 'eaten' by crit. If you are sitting on 60% hit (say you're attacking from the front and are parried a lot), any crit above that 60% (think Loatheb) will be lost.
For special attacks, misses/dodges/parries do reduce your crit rating, since only if an attack lands, it can crit. So each % of 'non-hit' decreases your chance to crit by (crit chance/100)%.

I.e. at 90% hit and 50% crit, you will see 45 crits over the course of 100 attacks (90*0.5), for an effective crit chance of 45%.
With 80% hit, this decreases by 10*(50/100) = 5% to 40 crits over 100 attacks (80*0.5), for an effective crit chance of 40%.

And out of curiosity, and because I'm really really bad at reading formulae, have you taken crit depression versus boss level mobs into account?
The most stable formula being found is

Crit reduction versus boss level mobs is 4.8%: 0.6% per level and an additional 3% due to a weapon skill to defense difference of >10

♦ Darkside · 02/19/09, 8:41 PM

Originally Posted by sp00n

As far as I know, all special abilities use a two roll system. At least that is how the rogue mechanics are (all specials two roll, melee white hit single roll), and I haven't heard of it being different for any other physical DPS class yet, so I assume it is the very same for death knights. I could have missed other classes' findings of course, as I haven't been reading their forums since WotLK hit.
Please correct me if I'm wrong.

And to carry on your last statement, for white hits this is true as long you actually have any hit to be 'eaten' by crit. If you are sitting on 60% hit (say you're attacking from the front and are parried a lot), any crit above that 60% (think Loatheb) will be lost.
For special attacks, misses/dodges/parries do reduce your crit rating, since only if an attack lands, it can crit. So each % of 'non-hit' decreases your chance to crit by (crit chance/100)%.

I.e. at 90% hit and 50% crit, you will see 45 crits over the course of 100 attacks (90*0.5), for an effective crit chance of 45%.
With 80% hit, this decreases by 10*(50/100) = 5% to 40 crits over 100 attacks (80*0.5), for an effective crit chance of 40%.

And out of curiosity, and because I'm really really bad at reading formulae, have you taken crit depression versus boss level mobs into account?
The most stable formula being found is

Crit reduction versus boss level mobs is 4.8%: 0.6% per level and an additional 3% due to a weapon skill to defense difference of >10

I'm not sure if the DK specials are on a two-roll system or not, I'll do some dummy testing sometime in the near future and take a look. If anyone else would like to test as well, that would be a huge help.

I was taking crit depression into account at one point or another, but it might have been deleted in the chaos of revision; I'll add it to list of things to look into.

Winco · 02/19/09, 11:02 PM

This is an excellent tool indeed. Have you considered putting this into the form of an addon?

♦ Darkside · 02/19/09, 11:37 PM

I'd do that if I had any idea how to code Lua.

If anyone wants to take up the project, though, I'd be more than willing to help them with the math.

Winco · 02/20/09, 12:31 AM

That is a problem. Until then I suppose I can try out Pawn. What do you suggest for gem/meta slots, if you do not mind me asking? I was assuming value of strength X amount on a gem (so in my case, 2.66x16)?