The authors of the game have derived the system to make it as levelled as possible, and in that sense, the odds should be the same irrespectively of the order one starts with.
Having played the game a few hundred times, I have never really experienced any difference in the outcome based on starting order, so if an effect exists, it is probably small.
But then again, it might be impossible to estimate such an effect, because there are too many confounding factors that you cannot realistically control for.
I believe it is really hard to isolate an effect because even if you were able to conduct a proper experiment, where each player would play the game many times with 3 or 5 other players, and then evaluate how many times they won given the starting order, I suspect that such results would only be valid for that set of players.
Depending on the set of players that you might be playing with, say players with an aggressive versus a defensive strategy, or players that are fond of progress cards vs players that are fond of ports, or whatever, you would end up with a different result.
Worse, even for a given set of players, over time folks will adjust play based on tactics other players tend to use, and so any advantage a given strategy could have in the beginning might change over time, and then the effect of starting order would change over time too.
Without really knowing of a proper study that looks into it, I’m willing to bet that this will not have been looked at in an experimental setup, simply because it would be really hard to so in practice. It seems like this would be possible to evaluate this via a simulation where each player has clearly defined constant strategies, but whether that would be of any use in a real-life scenario is a different question. Maybe it would be a fun project for an MSc in Operational Research or some such!