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Cost per click is defined by the cost per impression and the click through rate. Given a period of time, divide number of clicks by the sum of the cost per impression for all ads in the same time period.

Cost per click is equivalent to click through rate multiplied by cost per impression (which are assumed to be independent).

I already know how to report significance for changes in click through rate and for cost per impression. How can I report a significance value for cost per click?

cpi = cost / impressions 
ctr = clicks / impressions
cpc = cost / clicks

cpc / clicks = cost
cpc / clicks / impressions = cost / impressions
cpc = (clicks / impressions) * (cost / impressions)
cpc = ctr * cpi
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    $\begingroup$ What do you mean by "significance value" - is it related to a statistical hypothesis test or is it something about materiality? $\endgroup$
    – Peter Ellis
    Commented Feb 16, 2013 at 23:08
  • $\begingroup$ Hypothesis testing - people are separated into experimental groups with some thing changed - and I want to know whether the change affected several different output variables by more than would be predicted by chance. $\endgroup$
    – user20916
    Commented Feb 17, 2013 at 4:02

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I don't see any conceptual difference in judging whether there is statistically significant evidence of differences between groups for this variable rather than any of your others.

The main practical difference is that each of these variables is likely to have a different distribution, none of them likely to be Normal. So for each variable you choose as the variable of interest you will need to carefully check for the appropriate method and test any assumptions that the test relies on. Of course, this applies as much for cost per impression as it does for cost per click.

Complications arise if you intend to test all three hypotheses. Just make sure you don't regard these as independent checks on eachother - rather, they are all tests of different aspects of a single underlying unstated idea.

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  • $\begingroup$ So for CTR I can use a test of proportions, for CPI I can look at the distribution of all of the costs of individual bids, but what I'm not sure about is CPC - it's an average of CTR times CPI for a given (arbitrary) time period. So I'm not sure what the "distribution" is that I can look at. $\endgroup$
    – user20916
    Commented Feb 17, 2013 at 4:37
  • $\begingroup$ Start by drawing a histogram (same as you should with the others). $\endgroup$
    – Peter Ellis
    Commented Feb 17, 2013 at 18:37

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