Stat comparison between different AA turrets:
Hacksaw[Spoiler]
* Cost: 220M
* Health: 580
* Range: 480 -> Coverage: 724
* Damage: 600X2
* Reload speed: 15s
* DPS: 80
* DPS per metal: 0,36
Razor[Spoiler]
* Cost: 280
* Health: 3000 (12000 in armored form)
* Range: 1040 -> Coverage: 3398
* Damage: 14,9
* Reload speed: 10
* DPS: 149
* DPS per metal: 0,53
Thresher[Spoiler]
* Cost: 450
* Health: 3000
* Range: 1000 -> Coverage: 3142
* Damage: 132
* AoE: 64
* Reload speed: 0,5
* DPS: 264
* DPS per metal: 0,59
Chainsaw[Spoiler]
* Cost: 900
* Health: 2500
* Range: 1800 -> Coverage: 10179
* Damage: 225
* Reload speed: 1
* DPS: 225
* DPS per metal: 0,25
Artemis[Spoiler]
* Cost: 2400
* Health: 3200
* Range: 2400 -> Coverage: 18095
* Damage: 1601
* AoE: 120
* Reload speed: 1,8
* Metal cost to make ammo: 80
* Time to make ammo: 20s
* DPS: 890
* DPS per metal (ignoring metal cost and manufacturing time): 0,37
The way I counted the coverage was pii*Range^2/1000 to get more readable numbers, otherwise the numbers would be 1000 times larger, but that is not important, what is important however, is that when range doubles, the coverage gets 4 times larger in comparison.
Time for some stat analysis:
Hacksaw: relies on high alpha damage, after which it is vulnerable for further attacks, but is at least able to make cost against two swifts or one raven. Low coverage and health means that it performs badly on frontline due to constant danger of enemy ground attacks and lack of necessary reach on possible advancing friendly forces. Still the missiles are able to chase and take down an plane that enters only briefly in its range.
Razor:Out of all the AA turrets, Razor is clearly the one that is most suitable for frontlines, since it has great range and its armored form lets it survive decently even ground forces assaulting it in comparison to other AAs, that can easily get destroyed by artillery and other means of destruction.
Thresher: Has worse range than Razor but slightly better DPS per metal. Is also be able to hit multiple targets thanks to its 64AoE. Has same health as Razor, but lacks the armored form which makes it much more vulnerable in comparison despite being more expensive.
Chainsaw: Large range and great coverage, but weak health and low DPS per metal. However, with the cost of single Chainsaw, one could buy 3 razors which would result in coverage of 10054 (9426 with just 3 Razors) for twice the DPS per metal cost. Still has the advantage of being positioned so far in the back that enemy artillery will barely reach it, but is extremely vulnerable once reached by enemy forces.
Artemis: This one is really tricky unit. While it possesses the greatest DPS and range out of all the AA turrets, it will take really long time to pay itself back by taking out enemy air units. Due to its rockets costing 80metal, it will need to take out 11 ravens or 35 Swifts to make its cost. With time investment alone to manufacture those rockets, it will take long before Artemis will be able to pay itself back, and against cheap fliers such as drones, Artemis may even bleed metal since the enemy it took down doesn't even make up the cost of the rocket. Against more specialized targets such as Krow or Likho it will pay itself quickly, but I suppose players tend to avoid throwing heavier air units at Artemis anyway.
Conclusion: Overall I would say that Razor, Thresher and Hacksaw are clearly the most effective AAs with Razor clearly in the lead thanks to its decent coverage, damage per metal and outstanding survivality while Thresher is better options when enemy ground units and artillery aren't such a threat and Hacksaw is able to make cost as long as it gets its hands on enemy aircraft.
Chainsaw is clearly popular option as AA, but effectively it could be replaced with spread out Razors for similar or even better effect other than the fact that they will need to be closer to enemy frontlines.