Understanding Attack Speed and DPS
There are a lot of misconceptions when it comes to Attack Speed and DPS. For some reason most players heavily lean towards larger damage numbers when it comes to DPS optimisation and while there are reasons for it, it is not always optimal. Today we would like to spend some time diving into Dota 2 AS and DPS mechanics and clear some misunderstandings related to them.
Attack Speed (AS) = Attack Damage (AD)
A single point of AS is equal to a single point of Damage. Kind of, as it all depends on relative amounts of AS and Damage. To maximise DPS, a hero should have equal AS and Damage, so when a hero is lacking Damage, buying damage is preferable and vice versa.
Most heroes start the game with 100+ AS and ~55 Damage, hence why early on we see a high return on damage item purchases. As the game progresses, however, many right-clicking cores start scaling damage at a faster pace, compared to their AS scaling.
It could be a result of them having damage-increasing abilities, or it could be a result of itemisation, but we quite frequently see heavily skewed Damage and AS numbers even in higher level pubs.
This is a test we did to check our point. A level 30 
Alchemist with 
Reaver and 
Blades of Attack has an exactly equal amount of AS and Damage. One of them was later given a 
Divine Rapier, while the other was given three 
Moon Shards, one consumed, as well as 
Blade of Alacrity.
Essentially, starting from equal AS and Damage, one was given +350 Attack Damage and the other was given +350 Attack Speed. As can be seen from the screenshot, after some time, their DPS values on their respective dummy targets were nearly identical, with minimal discrepancy due to Attack Damage having some variance.
This is why we claim that every point of AS = every point of AD when it comes to DPS. But what are the practical applications of it?
Useful adaptations
The idea for this type of post started from a trend from several years ago, where most professional players playing core 
Tiny would rush a Moon Shard as their first item. Intuitively it felt like it didn't make sense, but it was, indeed, the most efficient way to spend gold on a farming hero that has a very strong damage steroid, but has an AS malus built-in.
Right-clicking Tiny is well and truly a thing of the past, but the realisation stayed and in many cases it can now be applied to other heroes as well. For example, 
Sniper currently has a guaranteed +110 (or +140 with level 10 talent) AD boost from 
Headshot when using 
Take Aim. With the latter effect activated, his DPS becomes skewed towards AD, so it's only natural that he starts needing extra AS to maximise efficiency.
The idea is similar to many heroes with built-in critical hits or proc effects. It is worth noting, though, that while long-term efficiency is great for farming, the burst DPS for the first several seconds of the fight favours AD over AS:
The same two Alchemists ~10 seconds into the experiment show that there is definitely a higher burst for high AD heroes, which is only natural. It is also the reason why 
Echo Sabre is so good on many STR cores.
But what about BAT?
More knowledgeable readers are probably aware of BAT (Base Attack Time) that is the basis of most Attacks/Second calculations in Dota. Most heroes have 1.7 BAT, meaning that with 100 AS they will attack once every 1.7 seconds, but there are exceptions to the rule, like 
Juggernaut, 
Anti-Mage and about ~20 other heroes.
Alchemist has a decreased BAT when using 
Chemical Rage and when using the same two Alchemists from the previous test we get:
Once again, a roughly identical DPS, with either +350 AD or +350 AS. So BAT doesn't really have an impact on this discussion, with a small caveat of a very weird interface design decision made by Valve. For some reason when calculating AS on a non-1.7 BAT hero Dota adjusts AS values in the interface. Get a 
Gloves of Haste on Juggernaut? Your AS will rise by ~24 rather than 20, because of this interfacing decision. The effect is the same, it is just expressing the 1.4 BAT hero attack speed as if they were on a 1.7 hero, and as such the values can even rise higher than 700.
This is very... counter-intuitive and serves no purpose outside of confusing players, so be aware of that when trying to make efficient item decisions: the AS values on low-BAT heroes are inflated, and are deflated on heroes with BAT of >1.7.
Efficiency isn’t everything
Finally, there is a third Alchemist:
He got +175 AD (Blue Rapier + 
Nullifier) and +175 AS ( Moon Shard + 
Blitz Knuckles). It is clear how much higher his DPS is, compared to his unbalanced brothers and it is why we feel understanding this rather trivial concept is important.
That said, efficiency isn’t everything in Dota. In fact, most of the time efficiency isn’t even a priority, since the enemy is posing particular threats and the hero needs to respond with particular tools to deal with them. It might be very efficient to buy a Moon Shard on Sniper, but most of the time you will need a 
Hurricane Pike to stay safe. Or a 
Black King Bar.
There are times where the problem to solve is the lack of DPS and this information might come in very handy in such scenarios. It can also be relevant when the patch is finally released, as there could be new items or reworked heroes, who will need their builds reevaluated.
Thank you for reading and let us know whether you found this blog post useful in the comment section below! Your feedback will allow me to make better content plans in the future.
understanding how can such room temperature m o t h e r f u c k e r s exist in this game
Little mistake there
"so when a hero is lacking AS, buying damage is preferable and vice versa."
I guess you meant the exact opposite of that
Thanks for the article though
Oh, yeah, absolutely! Should be fixed now, thanks!
I’d pick brood here
More posts like this!
Probably a decent article for newer players or those who never got into the inner mechanics of the game.
Very theoretical though.
Stuff like procs with cooldowns (basher and mjollnir), added or converted magic damage (brooch, mkb and parasma). I think any of those would shift the formula a little, some by a lot even.
Simple example - mjollnir has internal cooldown of 0.25s, so having any attack speed above that (while rare, still possible) would be slightly less valuable since you go on two-cycle for chance of proc.
In the end, right click damage is just not a simple thing in dota.
interesting
I wonder how Attack modifiers could affect the outcome? For example If I depend on getting a bash quickly (or constitutive bashes), it seems intuitive that AS would bolster the relative time between my bashes (to a limit bc cooldown). Throw crits into the equation and it could get interesting.
I think another factor is heros in Dota don't typically get over 3k health in a game. So a DPS comparison over a long time like in your example doesn't reflect the reality of a Dota game. Seeing what combo of AS+DMG gets to that avg health threshold quickest would be insightful. Or for an extreme case what balance should I build for an extremely tanky hero (7K health).
fix yo site add 7.36
yeah...and in team you have 4 feeders....you can buy 7 Divine and 50 moonshards....they are still 0
@El Huron you probably believe the person in math problem actually have 50 watermelons.