For the case of two latent classes with responses to four dichotomous variables for the s^th response vector, {i,j,k,l}, n'_s is a product of the form a_i1b_j1c_k1d_l1 + a_i2b_j2c_k2d_l2 where the a, b, c, d values are positive (actually, non-negative) constants computed such that: (a) n'_s =< n_s for all cells and (b) the sum, S n'_s, is maximized across the cells. The a, b, c, d values are proportional to conditional probabilities, p_il^AX, p_jl^BX, etc. as in equation 3.2 of C. Dayton, Latent Class Scaling Analysis (Sage Publications), and can be rescaled if desired. In outline, the procedure to use in Excel is the following:

        1. Place cell labels in Column A (e.g., 0000, 1000, etc.) and place corresponding observed frequencies, n_s, in Column B.
        2. Place start values (e.g., 1's) for the 4 X 2 = 8 values of a, b, c, d for latent class 1 in, say, column G and for latent class 2 in column H.
        3. Compute the  n'_s value corresponding to each n_s and place in Column C (this process is somewhat tedious but mechanical). The sum of the n'_s column, S = S n'_s , should be included as a formula at the bottom of Column C.
        4. Set up the procedure, SOLVER, in the TOOLS menu as follows: the "Target Cell" field to be maximized is the column sum, S; the "by Changing Cells" field contains the locations of the 16 start values for the a, b, c, d values; in the "Subject to the Constraints" field add constraints corresponding to  n'_s < n_s and to n'_s > 0 (these can be done by highlighting appropriate fields in the spreadsheet; sometimes the alternate constraints  n'_s => .001 seem to work better). Standard options for SOLVER are adequate  for many problems but may be worth modifying at times (e.g., using the "quadratic" option under "estimates") and, in general, some trial and error may be necessary to arrive at the optimal solution. For example, the optimizer may converge to non-optimal values because the step-to-step change in the sum, S, can be very small. Initially changing the "Target Cell" to the expected frequency corresponding to the maximum observed frequency may be useful for forcing the a, b, c, d values into an appropriate region of the parameter space. Note that fit index, p^*, is equal to 1 - S/N once the optimal solution has been found.