The Hammers twice lost the lead at Upton Park, with Nicolas Anelka and Saido Berahino earning West Brom a share of the spoils in a thrilling encounter on Saturday
Allardyce's men went ahead inside the first four minutes at Upton Park thanks a Joe Cole strike, only for Nicolas Anelka to turn things around with a first-half double.
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"The game was very frustrating," Allardyce told BBC Sport. "There is a reason why that happened, none of our centre-backs are fit.
"Losing James Tomkins was a big blow for us. Our coping mechanism was playing four full-backs across the back four. That has cost us three points.
"They did their very best but got caught out. We didn't have enough aerial threat defending corners."
Allardyce did credit his team for the character displayed in fighting back from 2-1 down, but was quick to criticise their lack of defensive concentration.
"We showed great fighting spirit in the second half," he added.
"We practice on the training ground how to hold onto leads. But the players forgot how to deal with West Brom's kick-off and stop them scoring.
"You can blame anxiety or fatigue, but you can't forget how to deal with them."