I work mainly with non-statisticians in fields such as medicine, social sciences and education.
Whether I am consulting with graduate students, helping researchers with articles or reviewing articles for journals, I often have the problem that someone (client, author, dissertation committee, journal editor) wants to use some relatively well-known technique when it is either entirely inappropriate or when better but lesser-known methods exist. Often, I will explain the alternative technique but then be told "everybody does it the other way".
I'd be interested in how others deal with this sort of difficulty.
@MichaelChernick suggested I could share some stories, so I will
Currently I am working with one person who is duplicating a previous paper and adding one independent variable to see if it helps. The previous paper is, frankly, terrible. It treats dependent data as if they were independent; it is tremendously overfit and there are other problems too. Yet he (my client) submitted an earlier version as a dissertation and not only got his degree but was widely praised for the research.
Many times I have tried to convince people not to dichotomize variables. This comes up very often in medicine. I patiently point out that dicohotomizing (say) birthweight into low and normal (usually at 2,500 g) means treating a 2,499 g baby as just like a 1,400 g one; but treating the 2,501 gram baby quite differently. The clinician agrees with me that this is silly. Then says to do it that way.
I had a graduate student client long ago whose committee insisted on a cluster analysis. The student did not understand the method, the method did not answer useful questions, but that's what the committee wanted, so that's what they got.
The entire field of statistical graphics is one where, for many, "this is how grandpa did it" is enough.
Then there are people who seem to just push buttons. I remember one presentation (not by someone I helped!) who had taken an entire questionnaire and factor analyzed it. One of the variables she included was ID number!