The Hammers are flying high in seventh place in the table but Allardyce admits their opening fixtures could have been much harder, with Arsenal the only big guns they have faced to date.
And Allardyce says the Hammers will have to up their game away at Wigan on Saturday, and in the fixtures that follow.
He said: "Every game is difficult but in our first eight fixtures we only had Arsenal who are in the top four or top six on a regular basis. So that was relatively kind to us. But we have taken full advantage of our opportunities.
"There has only been one real disappointing performance in the eight - that was Swansea away. Even though we lost to Arsenal, I thought the performance was still good against a team as good as they were.
"The rest of the performances have been excellent, followed by some excellent results as well. We have had an excellent start but we look at the next few fixtures that are coming along.
"Wigan, Man City, Man United, Chelsea, Newcastle, Stoke, all those games together, and Liverpool - and this makes this group of fixtures extremely difficult to get more points.
"At Wigan we have to look to increase the points total with a positive result - whether that is a draw or a win. We are hoping for a win but if we come away with something then we will be delighted from our point of view."
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