Both “a causal relationship with” and “a causal relationship to” are used, often in a medical context. There seems to be a slight preference for 'with' in the UK, and 'to' in the US.
Unfortunately Ngram can't graph the frequency of their use because it returns phrases we're not interested in (such as "He cited a causal relationship to explain..." and "She mentioned a causal relationship with a groan") mixed in with phrases we ARE interested in! Nor can it search for the phrase
causal relationship with _NOUN_ as you can see from my attempt here.
Nonetheless there are plenty of examples to be found on Google, though they have to be sifted carefully. Here are a few.
"relationship to" :
The Code of Federal Regulations of the United States of America, Revised as of January 1, 1981:
- Thus, for a given final cost objective to qualify for special
treatment, the difference in its beneficial or causal relationship to
G&A expenses as compared with the relationship of other final cost
objectives to G&A expenses should be one which is apparent and capable
of being supported.
H.M. Blalock: Causal Inferences in Natural Experiments. 1985:
- If it could be assumed that [they] had the same causal relationship
Stephens' Detection of New Adverse Drug Reactions, edited by John Talbot, Patrick Waller, 2004:
- All cases judged [. . .] as having a reasonable suspected causal
relationship to the medicinal product qualify as ADRs.
"relationship with" :
Essential Community Medicine, edited by R.J. Donaldson, 1983:
- ...a number of diseases have a causal relationship with cigarette
Communicable Diseases, 5th Edition. Roger Webber, 2016:
- ...there is also a causal relationship with malaria.
Caroline Trouet: Clinical Trials in Belgium, 2004
- Any untoward medical occurrence in a patient [. . .] administered with a pharmaceutical product does not necessarily have a causal
relationship with this treatment.