Allardyce welcomes Tony Pulis' side to Upton Park on Monday night with the Potters sitting 14th in the table having collected only two wins this term. The Hammers have picked up six more points than tomorrow's visitors and, although Allardyce knows his side face a challenge, he reckons Pulis will feel Stoke should be further up the table.
"Tony Pulis' team are well drilled and disciplined in the way they want to play," Allardyce said. "I think that they get the defensive side of it right, as I try to do as well. I think the improvement in the quality of their players has made them an established Premier League side now."
He added: "I think Tony probably thinks they are under-achieving a bit at the minute with the team he has built over the last few years.
"He had a more difficult start to the season in terms of fixtures than we did, so I'm sure they'll get back to winning ways, but I hope it's not on Monday."
Pulis has earned plaudits for the way he has kept Stoke in the top flight on a small budget since guiding them to promotion in 2008.
Allardyce feels the way Pulis works alongside chairman Peter Coates has been key to the continued success at the Britannia Stadium.
He said: "I think it's true that Tony Pulis has had a great relationship with his owner and that relationship has borne fruit, particularly because they have backed him and know that they can trust him.
"I think that trust has given him the finance to build year in, year out and that has made them what they are today. They have improved every year in terms of the quality of player that they have brought in.
"I think that's why they've been successful, along with the team spirit they have and how organised they are. That relationship, buying wisely and maximising resources has borne fruit for Stoke City."