odds ratio for two variants of covid I need to calculate the odds ratio for mechanical ventilation and death in relation to
different early COVID-19 symptoms for two COVID variants - delta and omicron. How should I do it? I want to showcase that delta held higher risk of mechanical ventilation and death than omicron.
So far, I have calculated it from such table (example fo cough):
                      |Delta|Omicron|
mechanical ventilation|   a      c  |
no mech.ventilation   |   b      d  |

odds ratio: $\frac{a/b}{c/d}$
Is it correct?

I think you got the odds ratio right (check also the definition on Wikipedia)
Note that if you use software like R to run a Fisher test, you will get a slightly different odds ratio relative to the formula you have (see Why do odds ratios from formula and R's fisher.test differ? Which one should one choose?). For example:
mat <- matrix(c(10, 2, 10, 8), nrow= 2, byrow=FALSE)
mat
     [,1] [,2]
[1,]   10   10
[2,]    2    8

fisher.test(mat)

    Fisher's Exact Test for Count Data

data:  mat
p-value = 0.2353
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
  0.5567896 45.8529162
sample estimates:
odds ratio 
  3.823229 
  ^^^^^^^^

Manual calculation:
(10/2)/(10/8)
[1] 4
   ^^^



I want to showcase that delta held higher risk of mechanical ventilation and death than omicron.

This may be beyond the scope of the question, but I would be careful in interpreting the odds ratio directly as the difference between variants. If this were a randomized control trial you could assume that the only difference between groups is the variant (delta or omicron) and other differences are random. In such case, you can interpret the odds ratio as a consequence of the variants. However, presumably you have observatioanl data and the two groups differ for things other than the variant (for one thing, omicron came after delta).
