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I am estimating a a finite mixture model to identify proportions of four behavioral types using an experimental dataset. This dataset has data for 500 individuals and each individual has 30 tasks. In each task an individual chooses lottery A or lottery B.

I am assuming there are 4 behavioral types. 1. Rational Type 2. Type A that chooses A all the time 3. Type B that chooses B all the time 4. Type R that chooses A or B with equal probability.

The rational type chooses option A or B based on utility difference of A and B. If utility difference between A and B is EU then probability of choosing A is given by

$P(A) = \frac{1}{(1+e^{-EU})}$

EU depends on a parameter $\theta$. Let Y_{ij} = 1 if individual $i$ chose option A in task $j$ and Y_{ij} = 0 if individual $i$ chose option B in task $j$.

Then likelihood of $\theta$ for a rational type person is

$l(\theta|Y) = \frac{1}{(1+e^{(1-2Y_{ij})EU_{ij}(\theta)})}$

I have parameterized 4 proportions of types as follows in my likelihood function

$ p_1 (rational) = \frac{1}{1+e^a+e^b+e^c}$

$ p_2 (type A)= \frac{e^a}{1+e^a+e^b+e^c}$

$ p_3 (type B)= \frac{e^b}{1+e^a+e^b+e^c}$

$ p_4 (typeR)= \frac{e^c}{1+e^a+e^b+e^c}$

So likelihood function for entire data is

$L(\theta, a,b,c|Y) = \prod_i \prod_j \big[(1-p_2-p_3-p_4)\frac{1}{(1+e^{(1-2Y_{ij})EU_{ij}(\theta)})} + p_2 I_A(Y_{ij})+ p_3I_B(Y_{ij}) + p_4\frac{1}{2} \big]$

Where $I_A(x) = 1$ if x=1, 0 otherwise. $I_B(x) =1$ if x =0, 0 otherwise.

I am maximizing log likelihood of the data using unconstrained optimization and have estimates of a,b,c and their standard errors.

I am not sure how to compute standard errors of $p_2, p_3, p_4$ using these and then computing their p-values.

Previously, I have done estimations where I was using transformations like $e^a$ or $\frac{1}{1+e^{-a}}$. In these cases it was straightforward to use delta method to compute standard errors of the transformed variable using the standard error of $a$ that is obtained from maximum likelihood estimation. However, when I am estimating a number of proportions like above which are parameterized in a complex way using variables $a,b,c$ as above, I am not able to think how to proceed.

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  • $\begingroup$ Can you give the expression of the likelihood function or explain from which model or distribution the data come from? $\endgroup$
    – javlacalle
    Apr 30, 2015 at 20:39
  • $\begingroup$ In the estimations I am doing, I am estimating parameters of different beahvioral types from experimental data. In this data, individuals choose between two options A, B. For each individual we have 20 observation which A and B different for each of the tasks. I have four behavioral types: 1. rational type which has a parameter $\theta$ using which it makes a decision between A and B 2. biased type A which always chooses option A 3. biased type B which always chooses option B 4. random type which chooses one of the options with equal probability.I am estimating proportions of these types. $\endgroup$
    – Chenna Cotla
    Apr 30, 2015 at 21:03
  • $\begingroup$ Can you give the expression of the likelihood function that you are maximizing? I have some idea in mind, but I would like to do some checks before giving an answer. $\endgroup$
    – javlacalle
    Apr 30, 2015 at 21:24
  • $\begingroup$ I have edited my question with more details. Please let me know if you want more detail. Thanks. $\endgroup$
    – Chenna Cotla
    Apr 30, 2015 at 22:09
  • $\begingroup$ I couldn't look into it in depth. I think that the idea that you employed for a single parameter can be generalized. The standard errors for $p_i$ could be obtained as the square root of the diagonal in this product of matrices: $g(\varphi) V(\theta) g(\varphi)$, where $g(\varphi)$ contains the partial derivatives of $p_i$ with respect to $a$, $b$, $c$; $V(\theta)$ is the covariance matrix of the maximized likelihood function with respect to $a$, $b$, $c$ (the matrix that you use to get the standard errors of $a$, $b$ and $c$). I didn't check it, so take this just as a guess. $\endgroup$
    – javlacalle
    May 1, 2015 at 16:23

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