Dear Amy,
You don't have survival times, you have waiting times, all observed more or less contemporaneously. This distinction is central in choosing a method. Regression methods useful for survival times might be useful, but only as a convenience, not because the measurements require survival methods (see below). Similarly, you don't have count data, so regression methods (Poisson, negative binomial) for count data might be useful, but again, only as a convenient kind of regression.
You didn't say anything about the source of the list of clinics. If these are unrelated to each other, then information from the clinics who are willing to make an appointment (waiting times) doesn't seem to provide any information about the clinics who didn't make an appointment. In other words, treating them as censored in a survival-like model seems uninformative and inappropriate.
If these clinics are from a single organization with all or most policies the same, then the clinics that made appointments provide some information about the rest of the clinics in their organization. Examples of organizations might be veterans affairs (VAMCs) or a large provider such as Kaiser-Permanente. Also, in the UK there are national policies on waiting times for an appointment, so all NHS providers could be included, but these statistics are tracked by the NHS and violations of policies are often in the news.
There are few such organizations that seem large enough to qualify, so I think that treating the clinics that refused to make an appointment as censored would not be informative and would be confounding. By this, I mean that they can influence the parameter estimates in any model for waiting times though the actual causes of refusal are different.
I conclude that it would be more informative to treat the refusal to make an appointment as a separate issue from the waiting time. That is, model the willingness to make an appointment using one method (for binomial data) and the waiting time for scheduled appointments with another method (for more or less continuous data).
There is no survival as such and there isn't the usual idea of a hazard here (all the wait times are observed more or less simultaneously) so other regression methods may be useful. I am thinking of a transformation of the wait times or a different distributional link function. In that vein, a transformation used in survival, such as the weibull, might work quite well, but with no interpretation of the waiting times as survival times. Some regression methods for count data might also be informative, but with no such interpretation. In view of this, nonparametric proportional hazards methods appear to be inappropriate.
Since appointments are usually filled for at least the next few days, except for some time that might be reserved for emergencies, then short waiting times might actually be lower or reduced, that is, deflated. Short wait times might not be informative at all about you main questions of interest, since they might be haphazard.
There is also a question of seasonality in wait times, as any parent who tried to make an appointment with an orthodontist knows. Many doctors are much busier during school vacations. This raises the question of how long it took to collect the data and whether seasonality might be important in the analysis.
Graphical methods should not be neglected. They might provide useful information for the selection of a regression model for waiting times.
Can you divulge the number of clinics, calls per clinic, and how long the data collection took?
Regards,
David Smith