= Multiple Frame Surveys =

'''Multiple Frame Surveys''' was written by H.O. Hartley in 1962. It was part of the proceedings of the American Statistical Association ''Social Statistics Section''. 

Compensation for overlaps, adjusting [[Statistics/SurveyWeights|weights]] to reflect the probability of selection.

Given two sampling frames (''G'' and ''H''), consider the discrete domains to be ''a'', ''b'', and ''ab'' (i.e., the overlap). (Note: the author uses ''A'' and ''B'' to refer to sampling frames, which leads to expressions like ''N,,A,,/n,,a,,''. Not great. I am substituting in ''G'' and ''H'' here.) Clearly the population total ''Y'' of ''y,,i,,'' is equal to ''Y,,a,, + Y,,ab,, + Y,,b,,''. The author introduces the attribute ''u,,i,,'' to all cases, defined in frame ''G'' by:

{{attachment:ua.svg}}

and in frame ''H'' by:

{{attachment:ub.svg}}

where ''p + q = 1''.

The important consequence is that ''Y = Y,,a,, + pY,,ab,, + qY,,ab,, + Y,,b,,''.



== 'Case 2': known domain sizes ==

The author first considers the case where domain sizes (''N,,a,,'', ''N,,b,,'', and ''N,,ab,,'') are known. The appropriate estimator for ''Y'' is:

''Ŷ = N,,a,,y̅,,a,, + N,,ab,,(py̅,,ab,,^G^ + qy̅,,ab,,^H^) + N,,b,,y̅,,b,,''

where ''y̅,,ab,,^G^'' is ''y̅,,ab,,'' computed from sampling frame ''G'', and so on.

The variance of ''Ŷ'' is in terms of population variances (''σ,,a,,^2^'', ''σ,,b,,^2^'', and ''σ,,ab,,^2^'') as well as proportions of overlap in either sampling frame (''α = N,,ab,,/N,,G,,'' and ''β = N,,ab,,/N,,G,,'').

{{attachment:var1.svg}}

where ''N,,G,,'' is the size of sampling frame ''G'', ''n,,G,,'' is the number of cases selected from sampling frame ''G'', and so on.

The author then discusses [[Statistics/SurveySampling|sampling fractions]] which cost optimize this point estimate variance.



== 'Case 3': unknown domain sizes ==

The author next explores the case where domain sizes are unknown. The appropriate estimator for ''Y'' is:

{{attachment:est1.svg}}

where ''y,,a,,'' is the total of ''y,,i,,'' for ''i'' in domain ''a'', and so on.

The variance of ''Ŷ'' is in terms of population variances, proportions of overlap in either sampling frame, and the difference between mean responses (''Y̅'') across domains.

{{attachment:var2.svg}}

The author then discusses sampling fractions which cost optimize this point estimate variance.



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