i am no antenna guy, but i believe the antenna you are trying to measure is "electrically large", and the anechoic chamber is physically small, so you are only measuring the radiation pattern of "the near field". If you put that same antenna outdoors, and measured it again in the "far field" you would see a big difference in the two patterns measured.
Some antenna guys are able to do a conformal transform to convert the measured near field into a reasonable facsimile of the predicted far field pattern, but there are limitations that may have to be imposed on how the measurement is done.