Estimation is via maximum likelihood with the count data. The kernel of the likelihood is

which generalizes the likelihood in

Baker and Lindeman (1994) from binomial to multinomial outcomes and the likelihood in

Cheng (2009) from all-or-none compliance in one group to all-or-none compliance in two groups. Let

_{CACE} denote the ML estimate of θ

_{CACE}. There are two cases for

_{CACE}:

_{CACE(PerfectFit)} if ML estimates of all parameters lie in the interior of the parameter space, and

_{CACE(Boundary)} if ML estimates of some parameters lie on the boundary of the parameter space.

A perfect fit estimate is possible with this model because the number of independent parameters equals the number of independent cell counts. There are 4

*J* − 2 independent parameters, corresponding to

*J* − 1 values for each of

*s*_{j}, b_{j}, t_{j}, and ν

_{j} plus the 2 parameters π

_{N} and π

_{A}. There are also 4

*J* − 2 independent cells counts, with 4

*J* counts for {

*n*_{raj}} and a fixed total in each randomization group. Thus the model is saturated, and one can compute

_{CACE(PerfectFit)} by setting observed counts equal their expected values. As derived in

Appendix A,

When

_{C},

_{j}, and

_{j} are each between zero and one,

_{CACE(PerfectFit)} is the ML estimate. This is because the observed counts are the ML estimates for a different saturated model in which each expected count is a parameter, there is a one-to-one function relating the parameters in the model for latent compliance classes and the aforementioned parameters corresponding to expected counts, and a theorem states that estimates derived from a one-to-one function of ML estimates are also ML estimates (

Rohatgi, 1976).

The perfect fit estimate of causal effect equals the intent-to-treat estimate,

_{ITT}, divided by the difference in the fraction receiving T1 in each group,

which parallels the perfect fit estimate for a causal effect with binary outcomes (

Baker and Lindeman, 1994) and the instrumental variables estimate for a causal effect with continuous outcomes (

Angrist Imbens and Rubin, 1996). The estimated asymptotic variance of

_{CACE(PerfectFit)} can be computed using the Mulitnomial-Poisson transformation (

Baker, 1994),

where the derivatives are readily computed using software for symbolic algebra.

If either

_{C},

_{j}, or

_{j} in

(3) is less than zero or greater than one, the perfect fit estimate is not admissible, and we require the ML estimate of θ

_{CACE} when some parameters lies on the boundary of the parameter space. In this case ML estimates can be computed using the following EM algorithm. The E-step computes the complete-data counts{

*e*_{ruj}} corresponding to the expected the number of persons in randomization group

*r* who are in latent compliance class

*u* =

*N, C, A* and have outcome

*j*. The M-step estimates the parameters from the complete data on the E-step. Let superscript (

*i*) index each iteration of the combined E-step and M-step. Based on the parameter estimates from the previous M-step, iteration (

*i*) of the E-step is

and iteration (

*i*) of the M-step is

The parameter estimates in

(7) are used for iteration (

*i* + 1) of the E-Step. The ML estimate of θ

_{CACE} is obtained by substituting the final M-step estimates at convergence into

(1). Starting values for parameter estimates equal the perfect fit parameter estimates in

(3) with any perfect fit estimate of π

_{C}, ν

_{j}, or

*t*_{j} that is less than zero set equal to zero, and any perfect fit estimate of π

_{C}, ν

_{j}, or

*t*_{j} that is greater than one set equal to one.

Imbens and Rubin (1997) also computed ML estimates for a multinomial model via an EM algorithm but did not provide details.

Consider the important special case of a binomial outcome with all-or-none compliance in one randomization group. The counts in group

*r* = 0 are {

*n*_{00j}} and the counts in group

*r* = 1 are {

*n*_{1aj}}, for

*a* = 0, 1 and

*j* = 0, 1. In this case, both ML perfect fit and ML boundary estimates can be written in closed-form,

To our knowledge, the above ML boundary estimate, which is derived in

Appendix B, is a new result. A previous ML estimate for this particular boundary scenario involved an iterative algorithm (

Cheng et al, 2009).