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I am fairly new to statistics and would wish to find out if there is a statistical test that tells me the magnitude/contribution/importance of different causes over a perceived effect.

For example, let's say that we have different flaws, and these different flaws all contribute to an error count. The flaws are the causes, and the error count is the effect of those causes. Each flaw contributes more or less to the total error count.

Is there any statistic tool that computes, with a certain degree of confidence, what would be the magnitude of each flaw in the final error count? Which flaw contributes the most to the error?

The data looks like this (header and one example row):

flaw1 flaw2 flaw3 error_count
 30    20    60        8

Thanks

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This instrument is called regression. In your case you have to specify whether the relationship between flaws and error count is linear, multiplicative (various flaws interacting with each other), or something completely different (an avalanche of errors upon the flaws crossing some threshold), and what sources and characteristics of discrepancy are.

For this particular scenario quite a bit depends on whether you observe cases with error_count equal to zero. If you do, look up censored regressions, if you don't - the relevant model is called truncated regression.

Since the dependent variable is a count, the (likely) Poisson regression model has to be combined with the complications of truncation/censoring. Of course, we cannot tell from your post whether Poisson distribution for the dependent variable is appropriate, and whether discrepancies (errors) in different observations are independent of each other. This makes a whole lot of difference for the model and for estimation methods as well.

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