The Christmas Bird Count (CBC) is a valuable source of information about midwinter populations of birds in the continental U.S. and Canada. Analysis of CBC data is complicated by substantial variation among sites and years in effort expended in counting; this feature of the CBC is common to many other wildlife surveys. Specification of a method for adjusting counts for effort is a matter of some controversy. Here, we present models for longitudinal count surveys with varying effort; these describe the effect of effort as proportional to exp(B effortp), where B and p are parameters. For any fixed p, our models are loglinear in the transformed explanatory variable (effort)p and other covariables. Hence we fit a collection of loglinear models corresponding to a range of values of p, and select the best effort adjustment from among these on the basis of fit statistics. We apply this procedure to data for six bird species in five regions, for the period 1959-1988.