>our comparison of survey results with official election truth indicates that the use of a so-called secret ballot yields more accurate, valid self-reports of voting on socially sensitive ballot issues, such as supporting taxes for elderly services.

https://academic.oup.com/poq/article-abstract/59/4/568/19401...

>regression analysis shows that social forces appear to have a greater effect on vote choices among people who doubt the formal secrecy of the ballot.

https://www.jstor.org/stable/23526132