Comparing regression equations

I am thinking of trying to use the logit model to determine the probability of whether patients will participate in a particular health program. My observation is the individual patient. My independent variables pertain to the characteristics of the patient (Male/Female, Age, Region, etc). My dependent variable is dichotomous (whether patients participate or not in the particular health program). There are 3 health programs total and patients can choose to participate in 0 (none), 1, 2, 3 (all) of the programs. The decision to participate in one program can be dependent on the decision to participate in another (I don’t have the data to back this statement up but this assumption makes sense); participation in one program does not exclude participation in another. In fact if one participates in one of the programs it may be easier to participate in another.

I would like to take all my observations and create 3 logit regression equations (one for each health program the patients can participate in). However I would like to then compare the three regression equations to determine if there is a statistical difference between the three. The problem is that I cannot use an ANOVA/ANCOVA type test in which I include the “health program” as a dummy variable because the data from the three different regression equations overlaps. For example Patient A who participates in Program 1 can also participate in Program 2. If I were to use data from all three regressions, I would be counting Patient A twice.

Is there any way for me to compare these three logit regressions? I am undergraduate economics major so my knowledge of advanced econometric and statistical topics is limited.

I have thought about doing a multinomial logit model with each combination of participation possibilities as a discrete choice. However the problem is that not all 3 of the programs were available to all of the patients.
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well, first of all, it seems you have interchanged the role of independents and dependents. male/female, age, etc. always come to independent variable category. reason:based on your gender or your age, you are having some perceptions. these perceptions are your outcome, hence dependent (on age, gender, etc.). your gender will not change if you change your perception so, gender is an independent variable.
now, to your main question: to compare regression equation.
i do not know about logit model, but for multiple linear regression, there is a test named Chow test. you can go for it.
i also think that you do not need to run logit regression. think again after you put gender/age as independent.
The independent-dependent variable thing was a typo. Good catch though.

I need some type of qualitative response regression model because my dependent variable is a discrete outcome (participate or not). That is why I want to use either a logit or probit model. That being said, I could use regular old multiple regression (i.e. linear probability model) but that has many flaws and I feel is inferior to the logit or probit models in this case.

I think the problem with the Chow test is that it would require me to pool the data from all three of the regression models. However this is a problem as I have mentioned because I would be double counting individuals. Is there anyway for me to compare three regressions without having to pool the data together?