When does replication reveal fraud? Found this question here, and I couldn't answser it and thought it would be a wonderful question for this site.

Here's a little thought experiment for your weekend pleasure. Consider the following:
Joe Scientist decides to conduct a study (call it Study A) to test the hypothesis that a parameter D > 0 vs. the null hypothesis that D = 0. He designs a study, collects some data, conducts an appropriate statistical analysis and concludes that D > 0. This result is published in the Journal of Awesome Results along with all the details of how the study was done.
Jane Scientist finds Joe's study very interesting and tries to replicate his findings. She conducts a study (call it Study B) that is similar to Study A but completely independent of it (and does not communicate with Joe). In her analysis she does not find strong evidence that D > 0 and concludes that she cannot rule out the possibility that D = 0. She publishes her findings in the Journal of Null Results along with all the details.
From these two studies, which of the following conclusions can we make?

*

*Study A is obviously a fraud. If the truth were that D > 0, then Jane should have concluded that D > 0 in her independent replication.


*Study B is obviously a fraud. If Study A were conducted properly, then Jane should have reached the same conclusion.


*Neither Study A nor Study B was a fraud, but the result for Study A was a Type I error, i.e. a false positive.


*Neither Study A nor Study B was a fraud, but the result for Study B was a Type II error, i.e a false negative.
I realize that there are a number of subtle details concerning why things might happen but I've purposely left them out. My question is, based on the information that you actually have about the two studies, what would you consider to be the most likely case? What further information would you like to know beyond what was given here?

 A: Similar to Stijn's comment, there isn't sufficient information to make a conclusive judgment.  There could be fraud (in either of the papers, not just the first), or there could have been a Type I / Type II error in one of the papers.
Out of interest if the two studies were identical and independent, so had identical power functions which factored, would it make sense to compare (ruling out fraud) the scenario for $D=0$:
$P(A=false positive, B=true negative)=\alpha(1 -\alpha)$
Against $D>0$
$P(A=true positive, B=false negative)=(1 -\beta)\beta$
And going with the more likely scenario?
A: The information provided does not indicate fraud by either party.  Assuming 95% confidence as the acceptance point then 2 in 20 tests could be expected to produce a type 1 or 2 error. This is a major reason for doing meta analyses.
Furthermore, to check for fraud, one would want to review the record keeping for the two experiments and perhaps corroborative evidence from colleagues.
As well as natural variability, a major reason for obtaining different results is confounding, where for one reason or another the conditions or sample were not identical, or perhaps there were equipment or calibration problems.
