Question

and without HIV. Given in the statement of the problem: Prevalence of HIV=25.9%=0.259 P (Elisa test positive ∣ HIV) =99.7%=0.997 (this is the specificity of the test) P (Elisa test negative ∣ don't have HIV) =92.6%=0.926 (this is the sensitvity of the test) Use this information to get the following contingency table. First decide on the size of the total hypothetical population - the lower right box. Then fill in the lower left box - which is the total that have the disease. This is based only on the prevalence and the hypothetical total population (and does not use any information about the screening test). Once you have the total that have HIV, you can fill in the total that don't have HIV. (either you have it or you don't, so those numbers add to the total population). Once you have these totals, you can use the conditional probabilities given to fill in other boxes. Conditional probabilities are not based on the total population, but based on the sub-group you are conditioning on - the "given that" or "of group". See calculations in the boxes.

Answer 1

The contingency table for the given information is as follows:

```

| HIV Positive | HIV Negative | Total

-----------------|--------------|--------------|-------

Elisa Positive | 0.259 * P | 0.001 * P | 0.26P

Elisa Negative | 0.074 * P | 0.666 * P | 0.74P

-----------------|--------------|--------------|-------

Total | 0.333 * P | 0.667 * P | P

```

Explanation:

To construct the contingency table, we first decide on a hypothetical total population size denoted as
`\(P\)`. The lower right box represents the total population size. Given the prevalence of HIV as 25.9%, we calculate the lower left box, which is the total number of individuals with HIV, as
`\(0.259 * P\).`

The total number of individuals without HIV is then obtained by subtracting the HIV-positive individuals from the total population, resulting in
`\(0.741 * P\)`. With these totals, we use the conditional probabilities provided to fill in the other boxes.

The Elisa Positive and HIV Positive cell is filled with
`\(0.259 * P * 0.997\),` representing the probability of testing positive given that an individual has HIV.

Similarly, the Elisa Positive and HIV Negative cell is filled with
`\(0.259 * P * 0.997\)` ,
`\(0.259 * P * 0.997\),` representing the probability of testing positive given that an individual does not hav
`\(0.741 * P\)`e HIV. The Elisa Negative and HIV Positive cell is filled with
`\(0.074 * P * 0.997\),`and the Elisa Negative and HIV Negative cell is filled with
`\(0.666 * P * 0.926\).`

In summary, the contingency table is constructed by utilizing the prevalence, conditional probabilities, and the chosen hypothetical total population size. This table serves as a valuable tool for understanding the relationship between HIV status and Elisa test results in a given population.