Optimal decision rules given a set of bimodal values

Question

Some patients have two possible illnesses ; to determine which illness a patient has, we make a test (only once) that will result in one of five outcomes, which have the following conditional probabilities : $$\begin{array}{l|c|c|c|c|c} \text{Outcome} & 1 & 2 & 3 & 4 & 5 \\\hline P(\text{outcome}\, |\,\text{illness 1}) & 0.4 & 0.3 & 0.15 & 0.1 & 0.05 \\ P(\text{outcome} \, |\,\text{illness 2}) & 0.3 & 0.1 & 0.4 & 0.15 & 0.05 \\ \end{array} $$ We want to minimize the probability of a type II error (diagnosing illness 1 to a patient with illness 2) to the constraint that the probability of a type I error (diagnosing illness 2 to a patient with illness 1) is 5% or less. What is the optimal decision rule?

My attempt

I thought maybe we could determine $n_1$, $n_2$, ..., $n_5$, such that, for example, if the result of the patient is outcome $i$ with value $v_i$ and with $v_i> n_i$, we choose the illness with the associated outcome with (conditional) probability higher than $n_i$.

To determine those $n_1$, $n_2$, ..., $n_5$, maybe we can use the chi-squared test, which will give the first constraint : $$ \frac{(n_1-0.4)^2}{0.4^2}+ \frac{(n_2-0.3)^2}{0.3^2}+ \frac{(n_3-0.15)^2}{0.15^2}+ \frac{(n_4-0.1)^2}{0.1^2}+ \frac{(n_5-0.05)^2}{0.05^2}<9.4877$$ (the last value being related to $\chi_4^2$) under which we want to minimize : $$ \frac{(n_1-0.3)^2}{0.3^2}+ \frac{(n_2-0.1)^2}{0.1^2}+ \frac{(n_3-0.4)^2}{0.4^2}+ \frac{(n_4-0.15)^2}{0.15^2}+ \frac{(n_5-0.05)^2}{0.05^2}$$ (but I don't know, up until now, how to compute such of constrained minimization).

Is the use of the chi-squared test in that situation relevant? (Knowing additionally that the chi-squared test is asymptotic, and we do to the patient a single test...)

If what I tried makes sense, how to compute the constrained minimization?

Answer 1

Let $p_i$ be the probability such that if the outcome is $i$, you diagnose illness 2 with probability $p_i$ (so if the test outcome is $i$, you draw a uniform random variable from $[0,1]$, and pronounce illness 2 if the random variable is less than $p_i$).

The probability on a type I error is $0.4 p_1 + 0.3 p_2 + 0.15 p_3 + 0.1 p_4 + 0.05 p_5$.

The probability on a type II error is $0.3(1-p_1) + 0.1(1-p_2) + 0.4(1-p_3) + 0.15(1-p_4) + 0.05(1-p_5)$.

This results in the following linear optimization problem: $$\begin{align} \min \quad & 0.3(1-p_1) + 0.1(1-p_2) + 0.4(1-p_3) + 0.15(1-p_4) + 0.05(1-p_5) \\ \text{such that} \quad & 0.4 p_1 + 0.3 p_2 + 0.15 p_3 + 0.1 p_4 + 0.05 p_5 \leq 0.05 \\ &0 \leq p_i \leq 1 \end{align}$$ This problem is readily solved with YALMIP:

p = sdpvar(5,1);
a=[0.4 0.3 0.15 0.1 0.05; 0.3 0.1 0.4 0.15 0.05];
objective = a(2,:) * (1-p);
constraints = [a(1,:)*p <= 0.05; 0 <= p; p <= 1];
optimize(constraints,objective);

the solution is $p_1=p_2=p_4=p_5=0$, $p_3 = 1/3$.