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Overview

This third post focuses on the variables at play in determining the amount of damage taken against physical attacks.  The data collection and analysis in this post was completed through a collaborative effort with Seiken Valk (some of his posts here in the official forums on ARC and LNC testing), who's in the same LS as me.  Building upon the previous 2 posts, the data collection methodology relies heavily on Part I's analysis of MIN and MAX distribution; so I highly encourage you to at least read that section if you're interested in how the conclusions in this post were reached.

Special thanks to of course Seiken Valk, but also Miko Neversleeps and Katsu Kobashi for gear sets.  In addition, a good amount of credit should be given to Grain Malt's initial DEF/VIT testing as well as Stanislaw for his translation of Malt's post.

As with my other math-heavy posts, I have sectioned off the methodology and discussion sections so that you can simply skip down to the "conclusions" section if the math does not interest you.

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Methodology

Data collection was very similar to the first 2 parts.  Again, no parser was sufficient to collect the data necessary for this section so I chose to type directly to excel.  Unlike previous sections, however, I did not include a trial number.  Instead, I opted to use MIN/MAX difference similar to how testing in Part I was completed.  The only difference was the variation cutoff for physical damage was +/-8% instead of +/-3% for cures.  Essentially, I would let the mob attack me and write down the MIN and MAX values until the variation between the predicted mean and MIN was near or above 8%.  Again, this was not a trial dependent test and trial number is not even listed.

There's really not that much more to this.  We control DEF and VIT on various mobs at various level ranges and go from there.  Let's move straight into the testing...

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Initial Testing on R52 Mongrels

To first get a "feel" for things, we started with a very basic test where VIT was controlled and defense was changed to varying amounts.  We chose to use the Blotched Mongrels from the Dunesfolk for Dinner leve for a couple of reasons:

    (1) Easily repeatable way to get the same mob at the same level
    (2) Produces 4 wolves at a time to test, making data collection faster
    (3) Mongrels can only use "Threatening Growl" (hate reset move) when at 100% HP so only 1 kind of attack.


The resulting damage was then logged as MIN/MAX/PREDICT similar to how previous testing was done.  3 Different VIT sets were used at 269, 200, and 178.  We then plotted the predicted average values as 3 curves (1 for each VIT rating).  Raw data summary and graph are shown below:


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R^2 = 1.0000 for all best fit curves


There's a lot to take in here, but I think the most striking feature is that the curves all appear to linear.  In fact when we did 'best fit" curves, we find a very consistent slope around -0.295 for all 3 curves.  This implies that the relationship between defense and damage taken is directly linear and this relationship does not appear altered by VIT (range 178 to 269)

Another notable feature if we look at the table of data is that there appears to be a cap on the amount of defense one can stack before no gains are seen.  In the table, defense works up to the point where the damage range becomes 35-42 (38.5 predicted average).  Vitality values did not appear to alter this cap, but having higher VIT did lower the amount of defense required to reach this cap. 

This was our initial test.  From here we branched out to test various aspects of what we saw on the graph.  Namely these questions:

    (1) What affects the slope (ratio between defense and damage taken)?
    (2) What affects this apparent defense cap / damage floor?
    (3) What does VIT do in relation to damage taken and relative to defense?


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Repeat Testing on Mongrels of Varying Rank

We took the same test procedure and just decided to collect data for each star ranking of the leve.  This means that in addition to the R52 data, we added R40, R43, R46, and R49.  This was done just to verify that the slopes were indeed linear and that VIT did not seem to affect the slope.  We also felt this would be interesting to see if the damage floor or the slope changed.  Raw data and corresponding graph are shown below.


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R^2 for best fit curves ranged from 0.9997 to 1.0000


The 2 main points I want to make from this data are:

    (1) The slope appears to increase as the level of the mob increases.
    (2) The damage floor (minimum damage you can take) increases as the level of the mob increases.


We do not this far in the testing if the increases in the damage floor and slope (ratio of defense to damage taken reduction) with level are actually do to the level difference, an increase in stats, or both.  We only know that these 2 things changed when the level of the mongrel changed, but that they did not when VIT did (shown in the initial testing).

So the big question at this point in the testing is are these changes in the slope and damage floor stat related, level related, or both?  The next test attempts to answer this...

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Do Stats Affect the Defense to Damage Taken Ratio (Slope)?

To answer the first question stated at the end of the previous section, we conducted a set of trials aimed to control the dLVL while varying stats of the enemy.  We found the easiest way to do this would be to take 2 different classes of mobs with the same level.  We ran our first comparison on R52 Blotched Mongrels (trials already conducted) and Fellbite Peistes in the Dunesfolk for Dinner leve out of convenience.  Data and corresponding graph are shown below:


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R^2 for best fit curves ranged from 0.9996 to 0.9997


When the results are plotted and best fit curves taken, we find the slopes to be relatively similar (0.2903 vs. 0.2998). This first result was somewhat difficult to interpret since the difference between the slopes is still within a reasonable range of error; however, it could still be due to an actual different in stats between the to mobs.

We then tried a higher ranked mob (R59) since the slopes would be a bit steeper in an attempt to tease out if this was error versus an actual difference in slope due to stat differences.  We chose to use Laughing Mongrels and Sphene Doblyns (R59) just to the east of Ul'dah.  The data and corresponding graph are shown below.


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R^2 = 1.0000 for all best fit curves


We again find a small variation in the slopes in this data set comparison; however, the values are much higher (0.5671 vs. 0.5585).  Based on this and the previous result, we decided to say that the defense to damage taken ratio (aka slope) either (1) only depends on the dLVL, or (2) depends mostly on the dLVL to the point where stats play little significance

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Do Stats Affect the Damage Floor?

Based on the above test, we found that either the only, or majority contribution to the defense to damage taken ratio (slope) is dLVL.  Those tests were unfortunately not able to help us tell if the damage floor was similar since it was difficult to reach the damage floor with the available defense in game (especially on the R59 set).  We moved to an easier set of mobs at R40 by testing R40 Ashdrakes in Coerthas and tying the results to the R40 Blotched Mongrel we had already collected earlier.  Very brief data summary is shown below:


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We can see here that both mobs, despite almost certainly having different stats, had the exact same MIN MAX at the damage floor.  Although this could be luck, I can vouch that the same was found in other mob sets we tried later, though I unfortunately do not have the data sets to show you for those cases.  Based on these results, we concluded to the damage floor is only affected by dLVL, with no contribution from mob statistics.

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Effect of dLVL on Damage Floor and Defense to Damage Taken Ratio (Slope)

We've thus far determined that both the defense to damage taken ratio (aka "slope") and the damage floor both are a function of the dLVL.  We then set out to find exactly how dLVL affected both by finding mobs within the R40 to R59 range and running the same damage taken tests as before.  Because I don't want to flood this post with raw data collection tables, I'll only post the summary of ratios and damage floors that we tested over.  Summary and graphs below:


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Values in Red are only estimations based on the assumption that the increase in linear.

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Based on the data above, we can see that there is a nice steady increase from dLVL -10 to 0 in both curves, then an apparent subsequent shift from dLVL 0 to +9.  By just eyeballing the graph, it is really difficult to tell if there are actually 2 distinct linear functions - one for negative dLVL and one for positive dLVL - or alternatively if there is just 1 formula.  To get a better feel for this, the same graph was reproduced and separated into 2 distinct set of plots.  Separate linear best fit curves were them placed.  The resulting graph is shown below:


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First off, we can see that the R^2 values are still pretty good, but not nearly as tight fit as the previous graphs.  We also know that the curve for dLVL<0 must not be completely linear at the given slope since it would produce negative DEF to damage taken ratios by the time it reached dLVL=-49, which we know is not practical. 

For the purposes of practical game application, I personally do not find a need to look for ratios beyond this -10 to +10 dLVL range currently.  The vast majority of content we enter as of now will very rarely exceed R59 (for instance, Ifrit is supposedly R58; Moggle is R57).  Since we can find the values for this range fairly easy, it may not entirely necessary for us to find a best fit curve for this situation unless we wish to find the exact formula.  At this point, we stopped our dLVL testing here.

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Damage Taken From Special Attacks

Thus far, our discussions have been limited to normal "attacks" from enemies; however, we know that there are a number of different kinds of physical damage attacks and TP moves.  We decided to run a very simple test which tries to see if the defense to damage taken ratio and damage floor are unique depending on the special attack or TP move used.  We chose to use a lower level beastmen mob (specifically LNC type Ixali Strongbeak mobs near Treespeak) due to the variety of TP moves available as well as the lower defense needed to reach a cap.  Raw Data table and graph are shown below:


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Looking at the data table, we can clearly see that the damage floor is the same regardless of the physical attack, but it takes varying amounts of defense to reach the damage floor.  In this case, the damage floor was a predict of 23.5; it took a defense of <650 to cap on Full Thrust, ~675 for Light Thurst, and ~775 for Leg Sweep.

The defense to damage taken ratio conclusions are a bit more muddled.  We actually see the exact same slope for Light Thrust and Leg Sweep, but a very different slope for Pierce.  There is unfortunately a great deal of error involved here since there are only 2 data points collected in this test and we are dealing with small damage taken values (where rounding error can be pretty significant).  We still felt that the slopes should not change but needed more evidence showing this, so we ran another quick test, this time with more data points, on R41 Dormouse...


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R^2 for best fit curves ranged from 0.9999 to 1.0000


We see here that the slopes are very close when we plot for more than 2 data points each.  Based on the Dormouse and Ixali Strongbeak data, we concluded at this point that the defense to damage taken ratio (slope) does not change with varying physical attacks or TP moves.  It appears that similar to VIT, different attacks only seem to shift the curve in a left or rightward direction.

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Relationship Between DEF and VIT

Back in the initial testing on R52 Blotched Mongrels, we noted that changing VIT only appeared to shift the curves to the left or right, but that the defense to damage taken ratios themselves were unaltered.  To see exactly how VIT itself alters damage taken, we performed a very similar set of tests to those shown previously, except that Defense is controlled (at 510) while VIT is varied.  The raw data and a plots are shown below.


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First off, if you graph the raw data, you will find that the best fit curves are clearly linear.  This establishes that VIT works similarly to defense in that adding VIT generates a steady, linear decrease in the amount of damage taken.  We can also see that the VIT to damage taken ratio (slope) seems to increase as dLVL increases, again similarly to defense.  At this point, we felt that it was possible both DEF and VIT played a near exact role in the formula, except that they have different slopes or ratios associated with them.  We then took the slopes found for VIT and compared them to those we had previously found for defense and summarized them in the table and plot below:


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Similar to the previous plot comparing the effect of dLVL on the DEF to damage taken ratio, we have now added the VIT to damage taken ratio as well.  We find that the increase in the ratio with respect to dLVL also seems to follow a rough linear pattern, again similar to DEF.  If we want to find the ratio between VIT and DEF, we can now take the slopes for both curves at >R50 (0.0293 vs. 0.0189) and divide them by each other.  If we do this we end up with: 0.0189 / 0.0293 = 0.6451 ratio for VIT:DEF.  This implies that for every point in VIT you add, you get the equivalent effect of adding +0.645 defense.  This result is somewhat interesting consider previous testing in both 1.18 and 1.20 that showed a rough 2:3 ratio (0.67) for Attack:STR.  It seems from our testing that there is a rough near 2:3 ratio between DEF:VIT.

The last note on VIT I'll mention in this post is that there appears to be another damage mitigation effect to VIT other than this near 2:3 ratio of DEF:VIT.  Referring briefly to Grain Malt's testing here (translation by Stanislaw here), you can see that raising DEF does not appear to alter the damage taken from magic TP moves (for instance, Wind Claw); however, raising VIT did seem to lower the damage from these moves.  This suggests that VIT not only provides a component of physical defense, but magical/elemental defense as well.  As the focus of this post is physical damage, we won't get into this in this post, but it is an interesting side note to return to at a later time.

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Push Towards a Formula

This completes the actual testing we'll be showing regarding physical damage taken for this post.  To be honest, the results were quite surprising in terms of how simplistic this system is.  To highlight why this system is what we both considered "simplistic", I would point to these main conclusions:

    (1) Adding DEF produces a linear decrease in damage.
    (2) This linear decrease in damage due to DEF is only affected by dLVL, and not mob stats.
    (3) VIT and DEF appear to do the exact same thing for physical damage at a simple ~2:3 DEF:VIT ratio.


If we put these three key points together, we can actually start to develop a fairly accurate portrayal of the damage taken formula.  Because mob stats do not affect the rate of decrease in the damage taken we can set aside their effects and come up with this abbreviation of the damage taken concept:


   Physical Damage Taken = [ "Damage Taken at 0VIT/0DEF" ] - [dLVL modifier] * { [ DEF ] + 0.67  [ VIT ] }


where the "damage taken at 0VIT/0DEF" encompasses all the mob's attack statistics such as STR and ATK power.  If we were to find a way to bridge DEF and ATK, it would be very possible to develop a complete formula for physical damage in general.  This is a point that we will definitely come back to when we discuss the physical damage dealt portion of our testing series.

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Analysis of Increasing Returns of the DEF Stat

Up to this point, our analysis and discussion has been completely abstract.  Before completing this post, I want to briefly talk about how the results shown in this post apply to actual in game situations.  We have talked a great deal about the linear relationship between VIT/DEF and the amount of damage taken, stressing that you get a very stable return.  However, for the vast majority of players, what they will notice is an increasing return to defense.  I'll try to explain this difference.

--- Please Read ---
Let us start with a hypothetical situation where we have a R50 mob that deals 250 damage to my 0 DEF CNJ.  Looking at the DEF:Damage ratio for dLVL=0, 0.2370, we know that if I were to increase my defense to 400 (+400), I would decrease the amount of damage taken by ~95 damage, or to about 155.  By increasing DEF by 400, I got a rough 1 - 155/250 = 38% decrease in damage taken.  This is what most players care about and judge Defense by

But now let's take that same mob and increase our now 400 DEF CNJ by another +400 to now 800 DEF.  This gives us another ~95 damage taken decrease, or from 155 to now 60.  We again added the same +400 DEF, but now our return in terms of a percent decrease in damage taken is 1 - 60/155 = 61% decrease in damage taken For that same amount of added defense (+400), our efficiency has now jumped from 38% to 61%!  Again, this is what the vast majority of players notice and rate the effectiveness of all stats by.

--- Please Read ---

So what exactly happened here?

The best explanation we can provide is that the "return" you receive for adding +1 DEF to your character increases as you near the amount of DEF required to reach the damage floor for a particular mob.  In our hypothetical scenario, the initial +400 DEF provided less return than the second +400 DEF because we were beginning to near the damage floor by the second gain.  To illustrate this concept graphically, we will use the case of our initial data sets on R52 and R40 Blotched Mongrels at VIT=269:


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We can clearly see that the efficiency remains relatively low until we begin to near the DEF required to hit the damage floor.  For the case of the R52 Blotched Mongrels, this floor occurs around 750 DEF when VIT=269, and as we see, the efficiency does not start to ramp up until we near it.  For the case of the R40s, only about 600 DEF is required to reach the damage floor.  Consequently, the efficiency begins to rise earlier when stacking defense.

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Endgame Application - Explaining the "Naked Ifrit Run"

Looking at the official forums, many posters try to use the example of the "naked Ifrit run" to argue that "stats are useless".  I myself have tried to argue against this notion (admittedly unsuccessfully in the realm of official forum public opinion) a number of times.  However, with the findings in this post, we can at least now better understand what is going on at a fundamental game mechanics level and explain why this kind of run is possible.

We will start simply with an "eyeball" number.  Having CNJ tanked Ifrit a number of times, I can "eyeball" that the amount of damage I take from an Ifrit Swipe is roughly 800 damage with a 178 VIT / 550 DEF CNJ.  Based solely on this observation and the fact that Ifrit is supposedly R58 (according to YG), we can tell that:

    (1) The dLVL modifier for DEF to damage taken is roughly 0.5261 slope (dLVL=8).
    (2) For dLVL=8
, there is roughly an average damage floor of 68.5.
    (3) We would theoretically take an average Swipe damage of 1,090 at 0 DEF and 178 VIT.
    (4) At VIT = 178, it would require roughly 1,940 DEF to reach the damage floor on Ifrit.


We can also generate an efficiency curve similar to those seen above, only this time specifically for the R58 Ifrit in the primal fight...


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We can see the concept of increasing returns as we near the damage floor very, very clearly in the plot above.  The damage floor occurs around 1,940 DEF given our initial observation, which is currently an unreachable number.  The maximum amount of defense currently available in the game would cap around 1,050 (and this is with amazing Bloodwall socket gear).  If we look at the efficiencies from 0 to 1,000 current defense, we find that they barely increase since the defense required to reach the damage floor is so far away.

This is why the naked Ifrit run works from a tanking perspective.  If you go in naked, you take about 1,100 damage.  If you go in fully geared on a GLA with 1,000 Defense, you'll end up taking about 650.  Because we are so far from the damage floor, there is very little efficiency, even in upping our defense from 50 to 1,000!  It all as to do with where our current defense is relatively the defense required to reach the damage floor.  This is why defense can seem so important on R50s, but mean so little on higher end mobs.  Another way we can put this is the higher damage the mob deals to you, the less efficient defense stacking will be.

Beyond this explanation of tanking naked on Ifrit, I do not wish to get into a philosophical argument as to whether or not the defense stat is broken.  I only intend to illustrate WHY things work the way they work.  I will leave the philosophical "should be" arguments to the official forums.



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Conclusions

There's a great deal of information in this post, but I will try to keep the conclusions pretty brief.

    (1) The amount of damage one takes can be summarized in the following formula:

   Physical Damage Taken = [ "Damage Taken at 0VIT/0DEF" ] - [dLVL modifier] * { [ DEF ] + 0.67  [ VIT ] }

This formula tells us that for every 1 DEF we add, we get a static amount of damage reduction which is directly related to the dLVL of the mob we are getting hit by.  This relationship between dLVL and the amount of damage reduction per +1 DEF added can be summarized in this graph:

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    (2) VIT and DEF are related in a 2:3 DEF:VIT ratio, meaning adding 3 VIT is like adding 2 DEF.

This is also evident if we look at the formula in conclusion 1.  Testing was done specifically at a range from VIT of 252 to 345, meaning if a "tier" argument is to be made, no such "tier" appears to exist between this range.  Seeing as most R50s will fall at or below this range, we do not find any evidence of a practical VIT "tier".  This is a very interesting conclusion since previous testing in both 1.18 and 1.20 have pointed to a 2:3 ATK:STR ratio as well.

    (3) VIT, in addition to affecting physical damage taken, also decreases magical/elemental attacks.

We chose not to go any deeper into this aspect of VIT since this post focuses on physical damage.

    (4) Damage from different physical attacks (normal, WS, etc.) all carry the same damage reduction per
         adding DEF or VIT.  Meaning DEF and VIT decrease damage taken the same on everything.


This means that if we have a mob with 4 different physical attacks, say Full Thrust, Pierce, Light Thrust, and Leg Sweep - adding a certain amount of defense or VIT will decrease the average amount of damage taken by all of these attacks by the same amount.

    (5) There is a "damage floor" at which adding defense or VIT will no longer decrease damage taken.
         This "damage floor" depends on the dLVL of the enemy attacking.


The damage floor is not affected by mob or player stats and is solely dependent on the attacking enemy's dLVL relationship with the player.  The relationship between dLVL and the damage floor is summarized in the plot shown below.

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    (6) The efficiency of adding more defense increases as one approaches the required defense to reach
         the damage floor for that particular enemy.


I highly recommend reading the section of this post that focuses on efficiency.  This is ultimately what the vast majority of players will care about with respect to defensive stats ("how good is it if I add X amount of defense to my current build?").  Embedded in this analysis is a quantitative explanation as to why the "naked Ifrit run" works.

That concludes the end of part III!  There's a lot to digest here but we've really only explained a small percentage of how things work so far.  The obvious extension to this post on physical damage taken is physical damage dealt.  Being able to connect to two in a formulaic will definitely be the goal in the future.  Til next time.


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