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  1. #81
    Community Member MrWindupBird's Avatar
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    Interesting thread, terrible grasp of control testing and statistics.

    I'm going to summarize what I feel are the important questions. I'll also propose some ways of testing- I don't have time right now to do any testing myself, but may get around to it in the next few weeks.

    #1. What is the order in which AC/Doge/Incorporeal/Concealment is checked?
    As has been hashed into the ground, it's not actually relevant in which order things are checked, in terms of total damage mitigation. However, it is relevant in terms of perceived utility. If AC is indeed checked last, then we will see fewer "misses" than might be expected, and AC will be undervalued. Example: I run around with 25% dodge, 25% incorp, 50% displace and 120 AC in EE. Let's presuppose that 120 AC gives me 20% misses in EE content (I'd be happy with that, but I doubt it). If AC is checked first, in 100 swings I'd expect to see 20 misses- that's pretty noticeable as you're playing, and you associate that with AC having value. But if AC is checked last, in 100 swings I'd expect to see 100*(0.75*0.75*0.5)*0.2 = 5.6 misses. I might perceive this as being 1/4th as valuable as seeing 20 misses, but they're actually of equal value, as order of operation is irrelevant to overall mitigation. So, order doesn't actually matter, but it definitely does matter in the visual feedback we get for seeing AC as being worthwhile, and I'd like to see it confirmed. The OP's data and conclusions in this thread are neither conclusive nor rigorous.
    How to test, without enormous sampling sizes: An easy way to test: get a toon with high AC (lets say >100), unequip all blur/incorp items, maximize dodge as much as possible. Step into normal Butchers Path. Record the results of a hundred hits (large sample size is not needed here). Let Y be the characters %dodge chance. If dodge is tested first, we expect to see Y dodges. If AC is tested first, we expect to see a much lower number: we have 100 AC in a lvl 2 quest, and there will be many many misses. If AC is checked first, we expect to see some number much smaller than Y, likely less than the order of Y/10. This test will order AC/dodge.
    Next we need to order AC/concealment. As above, but use a setup with high AC, displacement, and minimal dodge/no incorp. Same logic applies.
    AC/incorp may be ordered the exact same way. Setup with high incorp (25% shadowfade would be ideal), high AC, no concealment, minimal dodge.
    If we don't care about the incorp/dodge/concealment ordering, and it doesn't seem that anyone does, we're done now. If we do care, then we need to setup a displacement/incorp scenario with minimal AC and dodge (here it would be good to use a highlevel quest to minimize misses due to AC). Subsequently we'd need a displacement/dodge scenario, with max dodge and displacement, and minimal AC and no incorp. That ought to complete the ordering (perhaps redundantly, based on the x/AC pairing results, we're trying to minimize sampling sizes not to minimize trials).

    #2. Is high but attainable (let's say 150-200) AC effective in EE damage mitigation?
    How to test:This is very easy to test as well. Bring a 160 AC toon into various EE quests with a healbot. Minimize the dodge chance, do not wear concealment or incorp. Let beasties swing at you, count the misses from 100 attacks. If the miss % is appreciable, it'd be nice to test versus several different monsters/areas, ie giants in EE Storm Horns, archers in EE storm horns, hobgoblin warlords in EE GH, to see what sort of heterogeneity there is in monster to-hit. Personally, I've seen very few misses to AC in EE content, but I tend to run with very high dodge/displace/incorp, so if AC is indeed checked last, that may be why. It sounds like others in the thread have used this testing method and found AC to be sorely lacking, but verification would be nice.

    #3. I haven't seen this question explicitly, but I think it's relevant: does AC function differently in elite difficulty vs other difficulties?
    I suspect that the actual to-hit formula has built-in biases depending on the difficulty. If we think that the monster to-hit formula actually is hit% = AB+10.5/AC*2, then the hypothetical result that 160 AC results in 20% misses for a mob means that mob has 246 tohit. That seems a little high, but it's possible that monster stats scale exponentially with CR. This one is a little bit harder to get at, but I have an idea. If we think that monster to-hit is just a factor of CR, then we might expect AC to be as effective in heroic elite High Road quests (~CR24 trash) as in epic normal (~CR24 trash), for the same quests and monsters. So: hop into heroic elite Stay at the Inn with something like 80 AC, no incorp/conceal/dodge (ideally would like to see ~50% hit rate), kill everything in the first spawn but the archers or the melee, and record 100 attacks. Recall out, reset, and reenter on epic normal, keep the same monsters alive, same procedure. If roughly the same number of misses due to AC are observed, then monster to-hit calculations are probably working the same across difficulties. If they aren't, then maybe we've hit on something about why 160 AC feels so lackluster in EE. NB: I chose the above example because I think the heroic elite/epic normal CRs ought to match up well. If they don't pick a different chain: either heroic elite/epic norm Wheloon prison or heroic elite/epic norm Storm Horns are potentially good candidates. Additionally, I am making the huge assumption that monster CR is the primary input of tohit, for a given monster type. Finally, I honestly don't even know what the current glancing blow implementation is, so some information here would be nice. I think glancing blows work differently on heroic elite vs epic norm, and this will complicate things: classifying glancing blows as misses should be fine as a first approximation.

  2. #82
    Community Member Therigar's Avatar
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    Quote Originally Posted by RedHost View Post
    EDIT: As to the discussion branch about 5% auto-miss/hit, I really wish that I could effectively search the old pre-beta MotU stuff. I can not find any of the relevant posts, but I had thought that the developers said that Players were going to retain the effect of 'natural 1 always misses/natural 20 always hits' rolls, but NPCs were not because they were not doing any kind of d20 roll facade (the number rolled on the screen means less than most players realize) for monsters? My understanding was that they use straight percentages.
    If you go to the link leading to the MotU discussion you'll see a discussion that strongly implies that this is the case. I reference this is a post earlier in this thread and conclude from it that monsters are not using the auto hit/miss but players are. We seem to be in agreement but to have come to the conclusion from different directions.

    OTOH, I'm glad to have someone post something supporting my hypothesis that monsters are just using the % chance to miss and that after all the checks, if it is not a miss, it becomes a hit.

  3. #83
    Community Member Therigar's Avatar
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    Quote Originally Posted by MrWindupBird View Post
    Interesting thread, terrible grasp of control testing and statistics..
    I'm not sure why you make this observation and then restate every point that I have been making in the thread up to this point.

    I do not need to do a Butcher's Path test because I already know definitively that dodge is checked first and armor is checked last. And, you are right that with only 2 elements to check there is no need for a particularly large sample size.

    However, you should be aware that the larger the sample size the closer the results will draw towards the expected values.

    To illustrate this I'll make reference to a test I ran this afternoon (which I have not yet posted about). With 10% incorporeality over the course of the first 44 attacks not a single miss was recorded for that reason. Expectation would be that at least 3 misses would occur, but there were zero. Yet, by the time the test was over at ~120 attacks there were 8 misses due to incorporeality. After adjusting for order of precedence this comes close to expectations.

    So, sample size is important in order to give time for "on average" events to occur.

    As to high and attainable AC in epic elite, this is my current project. It requires a healbot and hirelings are not sufficient for this. I have to get a healbot to L20 in order to get into the epic quest before I can conduct the test. As I noted elsewhere in the thread this will take me a few days -- but I'll get to it.

    As to your last question, I think this is tin-foil hat thinking. It could be tested but I don't see a big value in it. Let's just say it is a low priority for me.

  4. #84
    Community Member Levonestral's Avatar
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    Just did a quick test to mostly get a personal "feel" for the different numbers everyone has been discussing.

    All tests were done against one of the first melee EE Cabal.

    (CR 61 mob)
    139 AC, no incorporeal, no blur/displace. 13% dodge (tried to remove what I could, but couldn't get lower due to feats)
    52 swings. 13 were missed, 8 of them were dodged, leaving 31 hits.

    (CR 58 mob)
    55 AC, no incorporeal, no blur/displace. 13% dodge
    50 swings. 8 dodged. 1 Missed. 41 hits.

    I also ran my sorc in there with no AC (sub 23 I believe), dodge, incorporeal or blur/displacement. I did see one "miss" reported there also. This suggests to me that the mobs do fail on a roll of a 1 but by no means am I submitting this as proof.


    Then I went back in to get a feel for how often "other" levels of mitigation are being checked.

    I went in with 25% dodge, incorporeal and 20% blur. I then tried two different tests. One with 146 AC and another dropping down to only 63.

    When I was at 146 with everything else above I was able to easily keep myself healed the entire time using just cacoon and never once felt I couldn't.

    However, once I dropped back down to only 63AC, I had to get help from a healer to cover the losses. (It wasn't insane, but was more than just my cacoon could handle).

    We also tried around 109AC and though it felt better than the 63AC, it wasn't nearly as "pleasant" as when I was at 146.

    So overall, it certainly "felt" like the higher AC did provide additional mitigation when being attacked. It's a small data set I know, but just from that alone I feel there is some benefit to AC.


    Combined with blur/incorporeal & a 25%+ dodge, your chances of that AC being tested against is much lower (as others have mentioned). This does lead into the "Not worth as much" because it is the last thing checked on the list and quite often it's never reached. I observed this behaviour also. The higher my dodge got, the more times I saw it come up, even with the AC numbers being higher. Adding in incorporeal also reduced the number of "missed" I saw in the logs, but I also saw more "incorporeal" messages over my head.

    I do understand that this is entirely based on the mob's to-hit, which from what I've been able to see so far, isn't readily available for us to run calculations against and we're left "guessing" mostly. It has been suggested that it's based on the mob CR, but still seems to be inconsistent. Sadly this one variable is a significant portion of the success rate of AC in the various EE content available and is probably why we're seeing such a large variation in data results from everyone.

    To everyone, on both sides of this discussion, keep up the good work and let's see if we can come to some kind of final result that everyone can agree on.
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  5. #85
    Community Member hucka's Avatar
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    Quote Originally Posted by Levonestral View Post
    13% dodge (tried to remove what I could, but couldn't get lower due to feats)
    you could equip heavy armor to cap dodge at 4% though that might hurt more then it helps if youre a monk

  6. #86
    Community Member Bilger's Avatar
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    Quote Originally Posted by Levonestral View Post
    Just did a quick test to mostly get a personal "feel" for the different numbers everyone has been discussing.

    All tests were done against one of the first melee EE Cabal.

    (CR 61 mob)
    139 AC, no incorporeal, no blur/displace. 13% dodge (tried to remove what I could, but couldn't get lower due to feats)
    52 swings. 13 were missed, 8 of them were dodged, leaving 31 hits.

    (CR 58 mob)
    55 AC, no incorporeal, no blur/displace. 13% dodge
    50 swings. 8 dodged. 1 Missed. 41 hits.

    I also ran my sorc in there with no AC (sub 23 I believe), dodge, incorporeal or blur/displacement. I did see one "miss" reported there also. This suggests to me that the mobs do fail on a roll of a 1 but by no means am I submitting this as proof.


    Then I went back in to get a feel for how often "other" levels of mitigation are being checked.

    I went in with 25% dodge, incorporeal and 20% blur. I then tried two different tests. One with 146 AC and another dropping down to only 63.

    When I was at 146 with everything else above I was able to easily keep myself healed the entire time using just cacoon and never once felt I couldn't.

    However, once I dropped back down to only 63AC, I had to get help from a healer to cover the losses. (It wasn't insane, but was more than just my cacoon could handle).

    We also tried around 109AC and though it felt better than the 63AC, it wasn't nearly as "pleasant" as when I was at 146.

    So overall, it certainly "felt" like the higher AC did provide additional mitigation when being attacked. It's a small data set I know, but just from that alone I feel there is some benefit to AC.


    Combined with blur/incorporeal & a 25%+ dodge, your chances of that AC being tested against is much lower (as others have mentioned). This does lead into the "Not worth as much" because it is the last thing checked on the list and quite often it's never reached. I observed this behaviour also. The higher my dodge got, the more times I saw it come up, even with the AC numbers being higher. Adding in incorporeal also reduced the number of "missed" I saw in the logs, but I also saw more "incorporeal" messages over my head.

    I do understand that this is entirely based on the mob's to-hit, which from what I've been able to see so far, isn't readily available for us to run calculations against and we're left "guessing" mostly. It has been suggested that it's based on the mob CR, but still seems to be inconsistent. Sadly this one variable is a significant portion of the success rate of AC in the various EE content available and is probably why we're seeing such a large variation in data results from everyone.

    To everyone, on both sides of this discussion, keep up the good work and let's see if we can come to some kind of final result that everyone can agree on.
    I was there on my FVS helping him test this.

    This is from the healers perspective.

    When he was at higher numbers cocoon was all I needed could of fell asleep, because his cocoon was enough to keep himself going at that time.

    Now when he dropped it to lower ac 50-60 ish I needed to hit my cocoon, heal, and scrolls to keep him going so could get his small test done.

    When he added the other mitigation and ac it was a snooze fest. With mitigation and no ac was back to scroll, cocoon, heal spell not as bad as before because some was mitigated, but huge difference from when had ac.

    So from the healer perspective in this instance I would say that AC made quite a difference.

    Now from no tests just self observations on my own characters with high ac I see a difference and am on Therigar side in the fact that AC is not useless at all.

    As far as misses on a one sorry Therigar no screenie or anything, but it is there and seen it with these tests did with him earlier. That's one point will disagree with you on, but on the rest I am on your side on the point of AC usefulness.
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  7. #87
    Community Member Therigar's Avatar
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    Quote Originally Posted by Bilger View Post
    Now from no tests just self observations on my own characters with high ac I see a difference and am on Therigar side in the fact that AC is not useless at all.

    As far as misses on a one sorry Therigar no screenie or anything, but it is there and seen it with these tests did with him earlier. That's one point will disagree with you on, but on the rest I am on your side on the point of AC usefulness.
    You know what, I'd gladly lose the battle over miss on a 1 if I can win the war on AC.

  8. #88
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    I would thing testing of this kind would be easier on Lam, since you can more quickly build characters to specifications. Just a thought.
    http://myaccount.turbine.com

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  9. #89
    Community Member Therigar's Avatar
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    Quote Originally Posted by Levonestral View Post
    J(CR 61 mob)
    139 AC, no incorporeal, no blur/displace. 13% dodge (tried to remove what I could, but couldn't get lower due to feats)
    52 swings. 13 were missed, 8 of them were dodged, leaving 31 hits.

    (CR 58 mob)
    55 AC, no incorporeal, no blur/displace. 13% dodge
    50 swings. 8 dodged. 1 Missed. 41 hits.

    I also ran my sorc in there with no AC (sub 23 I believe), dodge, incorporeal or blur/displacement. I did see one "miss" reported there also. This suggests to me that the mobs do fail on a roll of a 1 but by no means am I submitting this as proof.
    I appreciate you all running this test. I'm glad to have sparked enough interest to get other active in confirming/debunking the notions being tossed around about the value of AC. So, thank you.

    Let's go back for a moment to the miss on 1.

    In your first test with 52 attacks you would expect 2 or 3 auto misses. You observed 13 misses so there certainly could have been some auto misses in there.

    In your second test with 50 attacks you would also expect 2 or 3 auto misses. You observed only 1. Now, it is a small sample size but there is a lot of difference in % terms between and expected 2 or 3 and an actual 1.

    Go back to my earlier post about how we need longer tests to get "on average" results. Each occurrence is a separate event and it is possible to get strings of unexpected results. Over time those will even out and the results over larger sample sizes will tend back towards the expected.

    At 50 attacks it is actually very significant that you observe only 1 miss as this is half, or less than half, of the expected results.

    As to the sorcerer, without knowing how many attacks there were it is hard to draw any hypothesis. But, if the number of attacks is particularly small then the % check might have been even less than 1% and still registered as a miss.

    I think this is important to understand. Player hits are rounded to the nearest 5% and any value below 5% is automatically a miss. Monsters don't do that. With monsters a 3% could be a hit and a 2.9% could be a miss. It is all about the miss threshold for that mob given the particular AC.

  10. #90
    Community Member Therigar's Avatar
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    Quote Originally Posted by Tscheuss View Post
    I would thing testing of this kind would be easier on Lam, since you can more quickly build characters to specifications. Just a thought.
    Yes, but I only have 1 Lamannia client loaded and access to Lamannia is therefor limited. I have access to 5 different live accounts and already have the live client loaded on 3 different machines (and 3 times on my main machine).

    So, I can wait hours to load Lamannia multiple times or I can spend hours leveling a character. Lamannia might end up faster, but it won't be nearly as much fun and I'll have a much harder time explaining to my wife why her children can't do homework because their computers are busy.

  11. #91
    Community Member Therigar's Avatar
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    I want to refer readers once again to the Summary of Combat Changes in Menace of the Underdark.

    Notice this statement in that posting:

    A general rule with the new formula is that every doubling of Armor Class pretty much doubles your mitigation. A character with 30 Armor Class will be hit approximately half as often by a specific monster as one with a 15 Armor Class, and one with a 60 Armor Class will be hit approximately one quarter as often as the 15 Armor Class character.

    I've been testing with a character running 112 AC. His "at level" defense is 67%.

    The formula for monsters is (AB+10.5)/(AC*2). I can use this to calculate (AB+10.5)/(112*2)=.33 and I can solve for AB. If I do this I end up concluding that my "at level" enemy has an attack bonus of 63.42.

    I can test whether doubling or halving my armor class affects the chance to be hit by altering my AC value now that I have an attack bonus.

    (63.42+10.5)/(56*2)=.66. So, halving my armor class pretty much doubles my chance to get hit.

    (63.42+10.5)/(224*2)=.16. Again, doubling my armor class pretty much halves my chance to get hit.

    So, since this statement from Turbine appears to be correct and since my tests indicate that observed results run close to expectations, what happens if I drop AC all the way down to 10?

    (63.42+10.5)/(10*2)=3.6. That armor class should never see a miss, ever, at all. Unless, of course, there is some result that is always a miss.

    So, maybe there is a rule that treats greater than .95 as a miss regardless.

    Like I said, I'll be happy to be wrong on this point if we turn out to be correct on the usefulness of AC overall.

  12. #92
    Community Member Therigar's Avatar
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    Then again, maybe there isn't a limitation.

    Notice this statement from the same posting:

    The Epic Lord of Blades now acquires a chance to miss at 66 Armor Class, and the curve doesn’t reach a 95% miss chance until 1244 Armor Class.


    Does this say that if your AC is 65 that epic Lord of Blades doesn't have a miss chance?

    And then notice the next bit in that post:

    While these curves dramatically assist characters with Armor Class lower than the Attack Bonuses of their opponents, this isn’t sufficient to keep high Armor Class characters “tanking” as well as we would like. There are two additional changes that we’re planning to help them out there, and one for the lightly armored dexterous classes out there.

    The post goes on to introduce the changes in protection values of higher level armors, PRR, and changes to dodge.

    I have to say that this part of the post makes me think that there is no auto miss. Auto hit is pretty much irrelevant with a 1244 AC target....

  13. #93
    Community Member AtomicMew's Avatar
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    In your second test with 50 attacks you would also expect 2 or 3 auto misses. You observed only 1. Now, it is a small sample size but there is a lot of difference in % terms between and expected 2 or 3 and an actual 1.

    Go back to my earlier post about how we need longer tests to get "on average" results. Each occurrence is a separate event and it is possible to get strings of unexpected results. Over time those will even out and the results over larger sample sizes will tend back towards the expected.

    At 50 attacks it is actually very significant that you observe only 1 miss as this is half, or less than half, of the expected results.
    There is no statistical significance between 1 and 2 and 3 hits in a test of 50 trials. The reason being, the ratio between the variance and expected value (average) is large at low probabilities. For example, if we hypothesis an auto miss on a roll of 1, the expected value is: n * p = 2.5, variance: np(1-p) = 2.375. The ratio then goes by 1-p.

  14. #94
    Community Member Therigar's Avatar
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    Quote Originally Posted by AtomicMew View Post
    There is no statistical significance between 1 and 2 and 3 hits in a test of 50 trials. The reason being, the ratio between the variance and expected value (average) is large at low probabilities. For example, if we hypothesis an auto miss on a roll of 1, the expected value is: n * p = 2.5, variance: np(1-p) = 2.375. The ratio then goes by 1-p.
    There is a reason I don't like statistics.

    n is 50 in this example. p is .05. 50*.05=2.5. 2.5(1-.05)=2.5(.95)=2.375. 1-.05=.95 -- I have that right, yes?

    Variance helps me to identify standard deviation. Standard deviation is the square root of the variance value. In this case it works out to ~1.5 so results from 1 to 4 would be within the standard deviation and thus considered "normal."

    Alright, I'll accept the correction and the explanation. The small sample size makes it difficult.

    Had the sample been 100 the numbers would have been 100*.05=5; 5(.95)=4.75; standard deviation is 2.18. Results from 3 to 7 would be considered "normal."

    So, if the observation had held true with the 1 in 50 continuing on to become 2 in 100 then we would say that the result is unexpected and outside of "normal."

    Am I understanding that correctly?

  15. #95
    Community Member Therigar's Avatar
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    Alright, taking the info from our more talented math contributors and the testing done by Levonestral w Bilger I'm ready to concede on the auto miss issue.

    Using the CR61 test the approximate attack bonus for that hobgoblin is 184. If I give the same attack bonus to the CR58 hobgoblin then 55AC should never be missed. Never, ever, ever.

    For it to get missed the attack bonus at CR58 would have to drop off from 184 to around 95 in order to give any miss chance.

    I don't believe that a difference of 3 in the CR is going to have that dramatic of a drop off in attack bonus.

    So, it seems most logical to me that there is an auto miss for monsters.

    OTOH, their test shows that the nonsense spouted by several posters about 150AC being hit on 2+ is also incorrect. The 139AC gave the equivalent of 28-30% protection against a CR61 monster. I think that is worth noting.

    So, I'll lose that battle on auto misses because it looks like the evidence is piling up to win the war regarding armor class.

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    I know what I'm talking about AND
    I'm not going to gimp my monk to make additional tests, I slotted ACout of him entirely because it didn't work on EE, at all
    I've done enough testing on it

    Now let it go
    Kev
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  17. #97
    Community Member Soulfurnace's Avatar
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    Quote Originally Posted by keveniaftw View Post


    Kev
    Sorry to derail this, but 6 hotbars? What? How do you have so few?

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    Quote Originally Posted by Soulfurnace View Post
    Sorry to derail this, but 6 hotbars? What? How do you have so few?
    I am not the person that you are asking, but will respond anyway - how can you have more then 6 on most chars? serisouly, I had 6 on my FvS life, 6 on my artie life and.... I can hardly imagine filling ANOTHER two bars. One? yes, would not use it much, but yeah. Two? No way.
    that said, lets focuse on AC.

  19. #99
    Community Member Therigar's Avatar
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    Quote Originally Posted by keveniaftw View Post
    I am not sure what it is that you are trying to demonstrate with your screenshot.

    I think that it is to validate that you have a character with 160AC.

    That is great, and I suppose it is in response to me noting that a lot of people saying that AC doesn't work have not really done testing but are just responding based on what they think has happened in the game.

    I don't see how the photo changes that observation.

    Playing a character with that AC is not the same as running tests to see the way in which AC helps with misses. It does not show that AC is not helping.

    To the contrary, it shows nothing at all about the usefulness of AC.

    The picture does show one thing that is very interesting, however. I notice your PRR.

    I have not gotten into PRR because it does not actually stop attacks from hitting. Others have tried to discuss it from time to time and I've always put them off.

    But, with your PRR value you are basically taking the full brunt of every hit. Even if your armor is protecting you 30% of the time, you are taking so much damage on each hit that does land it is going to seem like you have no armor on at all.

    In my tests I run with 105 PRR which reduces the damage by 47%. I still get hit for 130+ damage when I am being hit. Each time a monster hits you in eElite content you are taking probably twice that much damage.

    Again, perceptually it is going to seem to you that your armor value is giving you no protection. While my armor is letting 9 of 10 hits through I am taking ~1200 damage. With your PRR letting 7 of 10 hits through is going to leave you with ~1750 damage. You have more AC but are dying faster.

    And so, it seems to you like your armor is doing nothing for you. But, you are wrong -- or at least, that is what the testing seems to show.

    I appreciate what you are trying to communicate. But I am equally certain that the evidence does not back up your claims.

  20. #100
    Community Member Therigar's Avatar
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    Quote Originally Posted by Soulfurnace View Post
    Sorry to derail this, but 6 hotbars? What? How do you have so few?
    Hey, every thread suffers a few derails. The impetus for this thread was a derail.

    I find it interesting when people post pictures to see just how their screen is set up. My son and I have dramatically different set ups. I think that it is remarkable just how individualized the pictures are. It shows that we all process the data differently so we set up our screens to feed us the data in a way that lets us process it and turn it into information.

    In a way it relates to an important discussion in this thread, that of perception.

    One of my contentions is that people perceive that AC is not working because of the feedback they get from their screens. They have not really tested to see that it does or doesn't work. But, they strongly believe that it does not work because of how they are receiving and processing the data they see on their screens.

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