Any "value" in using a big 4 SOC auditing firm vs a smaller one?

I have been doing SOC audits for years, over this time period with two companies I have used two firms. The experience has always generally been about the same between the two. They would be considered smaller firms compared to the big 4 (Deloitte, PwC, Ernst & Young, or KPMG).

I'm wondering what if any value there would be if I used one of these. Is there any kind of price difference? is there some kind of clout when someone …

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