Concomitant-Variable LCA      In the procedure summarized below, it is assumed that there are V dichotomous response variables and a single concomitant variable, Z, that has been summarized in G ordered categories indexed as g = 1, 2,... ,G (e.g., the concomitant variable, GPA, for the cheating data in C. Dayton, Latent Class Scaling Analysis (Sage Publications) has been summarized in G = 5 ordered categories). Generalization of these procedures to cases with more than one concomitant variable is straightforward in theory but can be tedious in practice. In outline, the nonlinear programming procedure to use in Excel is the following:
        1. Place cell labels for response vectors in Column A (e.g., 0000, 1000, etc.); repeat the labels in blocks down the column a total of G times. Place concomitant variable levels, Z_g, duplicated in blocks of size 2^V in Column B. These concomitant variable levels would typically be midpoints of intervals for ordered categories (e.g., 3.125 for the GPA interval 3.00 - 3.25). Place corresponding observed frequencies, n_gs, in Column B. Note that these frequencies are for the given response vector (i.e., s) at the given level of the concomitant variable (i.e., g).
        2. Place start values for the conditional probabilities in, say, column K for latent class 1 (LC1) and in Column L for latent class 2 (LC2). Note that, since p_0l^AX  = 1 - p_1l^AX ,etc., these restrictions should be included as formulae in the spreadsheet (e.g., if p_1l^AX = .2 in cell K7, then cell K8 should contain the formula "=1-K7" for the value of p_0l^AX). Place start values (e.g., 1.0 and 1.0) for the logistic parameters, b₀ and b₁, below the conditional probabilities, say, in Column K. Some trial and error with the latter start values may be necessary to find a satisfactory solution.
        3. An expected frequency of the form N_g X P(y_s |Z_s) must be computed corresponding to each observed frequency. Note that N_g is the total frequency of cases at level g of the concomitant variable where SN_g = N. This computation can be conveniently broken up among five columns. Column D (Labeled "LC1"), say, contains the computed values of
p_il^AX  X p_jl^BX X p_kl^CX  X p_ll^DX  , Column E (labeled "E(LC1)") contains the product of Column D with N_g and p^X_l|Zg (i.e., N_g X (e^{bo + biZg}/ (1 + e^{bo + biZg} ) X p_il^AX  X p_jl^BX X p_kl^CX  X p_ll^DX  which is the component of the expected frequency arising from the first latent class), Column F (labeled "LC2") contains the computed values of  p_i2^AX  X p_j2^BX X p_k2^CX  X p_l2^DX, Column G (labeled "E(LC2)") contains the product of Column F with N_g and 1 - p^X_l|Zg (i.e., N_g X (1/ (1 + e^{bo + biZg} ) X p_i2^AX  X p_j2^BX X p_k2^CX  X p_l2^DX which is the component of the expected frequency arising from the second latent class) and Column H (labeled "Expected Freq") contains the sum of Columns E and G which yields the expected frequencies, n'_gs, for the s^th response vector within the g^th level of the concomitant variable.
        4. In Column I, compute components of G² of the form:  n_gslog_e(n_gs/n'_gs). If n_gs = 0, set the component equal to 0. Place a formula for twice the sum of Column I at the end of the column; this is the G² value given the current values for the conditional probabilities and logistic function parameters.
        5. Set up the procedure, SOLVER, in the TOOLS menu as follows: the "Target Cell" field to be minimized is the column sum, G²; the "by Changing Cells" field contains the locations of the start values for the conditional probabilities and logistic function parameters; in the "Subject to the Constraints" field add constraints corresponding to  p_0l^AX <= 1, p_0l^AX >= 0, p_0l^BX <= 1, p_0l^BX >= 0etc. Standard options for SOLVER are adequate for many problems butmay 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. The solution may be sensitive to start values for the logistic parameters but does not tend to be sensitive with respect to start values for the conditional probabilities.