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Sensitivity & Specificity Confusion

By definition, Sensitivity & Specificity are:
                                 Test Result
                         Negative          Positive
                 absent     8660 (a)             960(b)
disease      present       40 (c)             340 (d)

    the probability that a symptom is present (or screening test is positive) given that the person has the disease d/(c+d) 340 / (40+340) = .895. This is also know as the true positive rate.

    the probability that a symptom is not present (or screening test is negative) given that the person does not have the disease a/(a + b) 8660 / (8660 + 960) = .90. This is also known as true negative rate.

But, in this case,

                                                    Pathological diagnosis
Clinical diagnosis           Alzheimer’s disease      Other causes of dementia
Alzheimer’s disease                     1643                                190
Other causes of dementia           127                             228

If the clinical diagnosis is considered a screening test for Alzheimer’s disease,

then the sensitivity of this test would be:

Pr (screen test is present | person has the disease)
= 1643/(1643+127) = 0.928

and the specificity of this test would be:
Pr (screening test is negative | the person does not have the disease)
= 228/(190+228) = 0.545

What I am confused is whay these two have different formula to get sensivitiy and specificity?
Am I missing something here??
Math / Science

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8/22/2022 - Mon

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The 1st one has:
Sensitivity = d/(c+d)
Specificity = a/(a+b)

                                                   Pathological diagnosis
Clinical diagnosis          Alzheimer’s disease     Other causes of dementia
Alzheimer’s disease                    1643 (d)                    190 (b)
Other causes of dementia          127 (c)                        228 (a)

So, I got it.  I didn't think that the table structure different for this....
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