Computing all possible transition scenarios

Question

thanks for reading. I have the following problem:

We have data on the progression of the health status of 52 patients over two time periods (Table 1). The patients only progressed to a worse health status, but we do not know if during this period of time the patients have progressed 1, 2 or 3 health statuses. What are all the transition posibilities?

Table 1.

|Health state| Time 0 | Time 1 |
|------------|--------|--------|
|Best        |20      |10      |
|Good        |10      |1       |
|Bad         |20      |19      |
|Death       |2       |22      |

As result, I'm looking for all scenarios that fit in Table 2.

Table 2.

|↓Time 0\Time 1→|Best    |Good    |Bad    |Death |Total| 
|---------------|--------|--------|-------|------|-----|
|Best           |    a   |    b   |   c   |   d  |20   |
|Good           |    -   |    e   |   f   |   g  |10   |
|Bad            |    -   |    -   |   h   |   i  |20   |
|Death          |    -   |    -   |   -   |   j  |2    |
|Total          |   10   |    1   |  19   |  22  | 52  |

All numbers must be integers and positive and, as you can see, a=10. Is there any general method to solve this? Is there any software that can help me with this? Thank you! :)

Answer 1

With $11$ unknowns and $8$ data points, there isn't enough information given to decide what values $a,\ldots, k$ have.

$$\begin{pmatrix}1&1&1&1&0&0&0&0&0&0&0\\0&0&0&0&1&1&1&0&0&0&0\\0&0&0&0&0&0&0&1&1&1&0\\0&0&0&0&0&0&0&0&0&0&1\\1&0&0&0&0&0&0&0&0&0&0\\0&1&0&0&1&0&0&1&0&0&0\\0&0&1&0&0&1&0&0&1&0&0\\0&0&0&1&0&0&1&0&0&1&1\end{pmatrix}\begin{pmatrix}a\\b\\c\\d\\e\\f\\g\\h\\i\\j\\k\end{pmatrix}=\begin{pmatrix}20\\10\\20\\2\\10\\1\\19\\22\end{pmatrix}$$

Usually one would use the method of least squares, but in this case this gives negative values for the unknown values. Instead, perform Gaussian elimination to get

$$\begin{pmatrix}1&0&0&0&0&0&0&0&0&0&0\\0&1&0&0&0&-1&-1&0&-1&-1&0\\0&0&1&0&0&1&0&0&1&0&0\\0&0&0&1&0&0&1&0&0&1&0\\0&0&0&0&1&1&1&0&0&0&0\\0&0&0&0&0&0&0&1&1&1&0\\0&0&0&0&0&0&0&0&0&0&1\\0&0&0&0&0&0&0&0&0&0&0\end{pmatrix}\begin{pmatrix}a\\b\\c\\d\\e\\f\\g\\h\\i\\j\\k\end{pmatrix}=\begin{pmatrix}10\\-29\\19\\20\\10\\20\\2\\0\end{pmatrix}$$

Keeping the positivity constraints, this implies $$\begin{matrix}a=10,& b=0 \textrm{ or }1,& c+d=9\textrm{ or }10,& e=0\textrm{ or }1,\\ f+g=9\textrm{ or }10,& h=0\textrm{ or }1,& i+j=19\textrm{ or }20,& k=2.\end{matrix}$$ with the additional constraints $f+i\le 19$, $g+j\le 20$.