What does each SPECIAL stat in Fallout Shelter do?

Besides its workplace uses, like Charisma in Living Quarters, Strength in Power Generators, Agility in Diners.

Do we know if Endurance increases HP in the Dwellers, or that strength helps in melee damage / survivability in the wastelands? And Perception?

Solution 1:

Do note that nothing has been made official, and this is a compilation of observations/hypotheses.

Strength

• Vault:
  • Decreased timer while working in power plants
• Wastes:
  • Needed to pry open lockers and boxes.
  • May also affect damage done by guns.
  • May increase a Dwellers capacity.

Perception

• Vault:
  • Hit chance in combat
  • Decreases water production timer of the Water Treatment and Water Purification rooms
• Wastes:
  • Find stuff like abandoned buildings
  • Critical attack cross hairs expand and contract slower, making it easier to deal maximum damage (×5) with a critical attack.

Endurance

• General:
  • Health points gained on level up (does not increase hp retroactively, max out endurance to 10+7 outfit and then start leveling to get maximum health)
  • Resist radiation and damage (11 Endurance = immune to radiation)
• Vault:
  • Food and water production timers of the Nuka-Cola bottler room.

Charisma

• Vault:
  • Improved fertility. (Reduced time to get pregnant.)
  • Decreases production timer in Radio room.
• Wastes:
  • Also chance of befriending people in the wastes (Gives the xp but no damage taken)
  • Interaction with groups of slaves

Intelligence

• Vault:
  • Med-bay (Stims and Radaway) production timers
• Wastes
  • Heal/help people

UPDATE:

• Vault/Wastelands:
  • also increases Exp gain (for level and Special stat increasing), extinguishing fire

Agility

• Vault:
  • Food production timer of the Diner and Garden rooms.
  • Increases attack speed.
• Wastes:
  • Run away from unwinnable fights taking less damage

Luck

• Vault:
  • Decreases chance of rush failure.
  • Increases amount of caps earned when resources collected from a production room (also increases chance to earn those caps)
  • Increases critical bar fill up rate.
• Wastes
  • Amount of loot found outside vault

Further reading and sources:

• Reddit: /r/foshelter What does each attribute do?
• Reddit: /r/foshelter Guys, power armor doesn't give any defensive stats in the wastes. ...
• Loading screen tips from Fallout Shelter for iOS

Tip:

• I set my shelter that way, to have the medbay spread in the vault to handle fire faster.
• the difference in having 1 or 2 intelligence is that a dweller with 2 intelligence levels up in half the time
• queuing my new matured dwellers to go to school so they get intelligence 2 or 3 then endurance for max HP and after that their main stat - super efficient

UPDATE: If you want to nerd out on more specifics, I recommend Section 7 of this Fallout Shelter Efficiency Data.

Also, here someone suggests that skills above 10 continue to benefit the dweller/explorer:

• Room efficiency data thread on Reddit

Solution 2:

Effects during quests only:

• Strength: Increased attack damage
• Perception: Reduced critical hit speed
• Endurance: Reduced radiation damage
• Charisma: Better dialogue choices
• Intelligence: Increased "this is the right/wrong way" frequency
• Agility: Reduced time between attack rounds
• Luck: Increased chance of critical hit

Solution 3:

These are the other uses for S.P.E.C.I.A.L. stats:

• Strength: Damage increased everywhere.
• Perception: Damage increased in-Vault; better Loot in the Wasteland, and better Critical lock-on on Quests.
• Endurance: Increased health.
• Charisma: Helps charming Wasteland Creatures. (And eventually better results in dialouges on Quests.)
• Intelligence: Increased medical capacities in the Wasteland.
• Agility: Better Damage in-Vault, better Hit Speed on Quests, and less Damage from losing a combat in the Wasteland.
• Luck: Also gives better Loot on Quests and in the Wasteland.