# Problem understanding the result of multinomial regression applied to clinical data.

So, I have a dataset of 240 patients that are diagnosed with COVID-19, where for each patient the outcome is recorded ( recovered / hospitalised / dead) and the rest of the parameters are demographic info, additional health issues ( arrhythhmia, diabetes, anemia and so forth) and applied therapy (non-invasive ventilation, ECMO [Extracorporeal membrane oxygenation]) etc. The applied therapy parameters only apply to those with outcomes hospitalised / dead. I'd like to interfere which health issues significantly worsen the prognosis of outcome.

Let's say the reference factor is 1 - recovered (without being hospitalised) and 2 - hospitalsied, 3 - dead.

Approx. 83 out of 149 that recovered have a condition x, 45 out of 79 that have been hospitalised, has a condition x, and 9 out 13 that died has a condition x.

When I apply multinomial regression in R, for the condition x, it gives coefficients: -2.68 for log(P(X = recovered)) / log(P(X = hospitalised)), i.e., 0.07 = P(X = recovered) / P(X = hospitalised), it means, that if the person has a condition x, then he is 0.07 times less likely to recover [can I express it in terms of being hospitalised, if person has a condition x, then he is $$100/7 \approx 14$$ times more likely to be hospitalised]? The coefficient has a p-value of $$\approx 0.01$$. That means that the condition x contributes significantly to the outcome. As for the second outcome - dead, the coefficient: log(P(X = recovered)) / log(P(X = dead)) equals to -1.46, i.e., P(X = recovered) / P(X = dead) = 0.23. But, the p value for this coefficient is equal to 0.233, the st. error equals 1.22. What could contribute to this significant st. error in case the predictor is a factor? What in general could contribute to large p-values for the coefficients for one factor and not for the other ( I mean recovered / dead factors)?

Also, can I simply use two binomial regressions with response variable as a factor with levels - recovered / hospitalised and recovered / dead?