An immunologist can order any test but more commonly a complete blood count, skin test for allergy, radioallergosorbent test (RAST), patch testing, and total IgE levels are tests that may be ordered.
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No a CSF examination is unlikely to be ordered by an immunologist. CSF examination is ordered by child specialists, neurologists, oncologists, and hematologists.
Any test can be ordered by any specialist and even by a general practitioner when required. However a CSF examination is rarely if ever ordered by an immunologist. Regarding the tests mentioned in the link, the tests are to assess the immune function and some of them may be ordered by other specialists as well such as CSF examination by oncologists to know the spread of cancer.