... Correlation coefficients were calculated for each of the 16 outcome measures with each of the 33 environmental variables, which resulted in a total of 16 × 33 = 528 correlations. Radin reported that the 16 × 33 matrix produced 44 correlations that were associated with p < .05. He then used the binomial probability distribution to compute the probability of obtaining that many, or (presumably) more, correlations associated with p < .05. He reported a value of p = .0004. I have been unable to reproduce this number ... In any event, Radin’s reported result is statistically significant.
However, the binomial distribution assumes independence for each of the measurements. But the correlations were clearly not independent. For instance, the environmental variables included background X-ray flux and log of background X-ray flux, humidity and precipitation, sunspot number and sunspot number for the day before.... Readers who have some familiarity with statistics may wish to ponder the implications.
The answer to his first concern was due to a simple mistake. I based the calculation on 45 rather than 44 significant correlations. I thank Hansen for spotting this error.
I addressed his second concern in the original article*, as follows:
It might be argued that some of the excess significant correlations in the experimental data might have been due to the fact that some of the environmental variables tested were intercorrelated with each other....
I explored this possibility by, among other things, forming a control correlation matrix using the same environmental variables in their originally recorded order, but randomly scrambling the chronological order of the MMI variables. (Under the null hypothesis, the former and latter variables should not be related in any way.) This control matrix resulted in a nonsignificant number of correlations, supporting the idea that the original matrix contained some meaningful relationships.
Hansen then failed to report that one of the points of this study was to test the idea that one reason MMI is highly variable in laboratory experiments may be due to fluctuations in environmental factors and differences in psychological variables like mood and confidence. Through use of an artificial neural network (Brainmaker, based on a backpropagation design), I demonstrated a genuine relationship between six environmental variables, two psychological variables, and one mind-matter interaction variable. After training the network on half the available data, the correlation between the MMI variable in the other half of the data and the neural network's prediction of the MMI effect was a highly significant r = 0.405.
The conclusions of this study were that (a) the environment appears to modulate MMI performance, (b) there are intriguing hints of space-like and time-like rebounds [in the MMI results], and (c) there is reason to believe that fairly good predictive models of MMI performance are realistically attainable.
To paraphrase Hansen's concern, readers who have some familiarity with his trickster theory may wish to ponder the implications of Hansen failing to report the rest of the story. While I think the trickster concept and lore are interesting, and that Hansen's own book on the trickster is an excellent exposition on that topic, I disagree that psi is forever doomed to a marginal existence.
The reason I don't agree is because similar pessimistic complaints have been voiced throughout history whenever we've been faced with seemingly incomprehensible effects in medicine, physics, astronomy, chemistry, biology, etc. In other words, whenever imagination fails, someone will invariably assert that we'll never be able to understand [fill in the blank], and so they come up with trickster-like theories to allow us to place our ignorance into a mysterious netherworld lying somewhere beyond our understanding. Failures of imagination are common, but promoting theories based on those failures is tantamount to glorifying an anti-scientific position.