1 Productivity is continuous and weakly increasing in talent and effort.
2 The sum of efforts allocated to all activities is bounded, and this bound is similar across people.
3 Families and hobbies take some effort, thus less is left for work. (For this assumption to hold, it may be necessary to focus on families with children in which the partner is working in a different field. Otherwise, a stay-at-home partner may take care of the cooking and cleaning, freeing up time for the working spouse to allocate to work. A partner in the same field of work may provide a collaboration synergy. In both cases, the productivity of the top person in question may increase.)
4 The talent distribution is similar for people with and without families or hobbies. This assumption would be violated if for example talented people are much better at finding a partner and starting a family.
Under these assumptions, reasonably rational people would be more productive without families or hobbies. If success is mostly determined by productivity, then people without families should be more successful on average. In other words, most top people in any endeavour would not have families or hobbies that take time away from work.
In short, if responsibilities and distractions cause lower productivity, and productivity causes success, then success is negatively correlated with such distractions. Therefore, if successful people have families with a similar or greater frequency as the general population, then success is not driven by productivity.
One counterargument is that people first become successful and then start families. In order for this to explain the similar fractions of singles among top and bottom achievers, the rate of family formation after success must be much greater than among the unsuccessful, because catching up from a late start requires a higher rate of increase.
Another explanation is irrationality of a specific form – one which reduces the productivity of high effort significantly below that of medium effort. Then single people with lots of time for work would produce less through their high effort than those with families and hobbies via their medium effort. Productivity per hour naturally falls with increasing hours, but the issue here is total output (the hours times the per-hour productivity). An extra work hour has to contribute negatively to success to explain the lack of family-success correlation. One mechanism for a negative effect of hours on output is burnout of workaholics. For this explanation, people have to be irrational enough to keep working even when their total output falls as a result.
If the above explanations seem unlikely but the assumptions reasonable in a given field of human endeavour, then reaching the top and staying there is mostly not about productivity (talent and effort) in this field. For example, in academic research.
A related empirical test of whether success in a given field is caused by productivity is to check whether people from countries or groups that score highly on corruption indices disproportionately succeed in this field. Either conditional on entering the field or unconditionally. In academia, in fields where convincing others is more important than the objective correctness of one’s results, people from more nepotist cultures should have an advantage. The same applies to journals – the general interest ones care relatively more about a good story, the field journals more about correctness. Do people from more corrupt countries publish relatively more in general interest journals, given their total publications? Of course, conditional on their observable characteristics like the current country of employment.
Another related test for meritocracy in academia or the R&D industry is whether coauthored publications and patents are divided by the number of coauthors in their influence on salaries and promotions. If there is an established ranking of institutions or job titles, then do those at higher ranks have more quality-weighted coauthor-divided articles and patents? The quality-weighting is the difficult part, because usually there is no independent measure of quality (unaffected by the dependent variable, be it promotions, salary, publication venue).