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 the 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.
"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."
After winning five of their opening 11 league matches since earning promotion through the play-offs Allardyce believes the rest of the Premier League are now starting to sit up and take notice of the Hammers.
"I think so. I think that, at the end of the day, when you continue to get results then people cannot ignore what you're doing, both as a manager and a group of players and coaching staff," he said.
"From our point of view, most of us have been here (in the Premier League) before - certainly I have for a long time and we know the position we're in is a very good one.
"But we know only a quarter of the season has gone. It's early days and a lot can happen between now and the end of the season.
"If we're around this position going into the last six or seven games, then that's a great season. At the moment, it's a great start and that's all it is."
Much has been made of West Ham's tough run of fixtures up until Christmas but, after taking a point against champions Manchester City and beating Newcastle last weekend, Allardyce is hoping to keep the momentum going.
He said: "The tough phase has already begun for us and, after Stoke on Monday we go into a run against the other big boys. We've coped very well so far and I hope we continue to cope on Monday night because that's a big game for us.
"We want to continue our home form and home results and get a very important three points against a well-experienced and well-organised team with a really good manager."