No Evidence for a Significant Difference versus Evidence for no Significant Difference

Returning to the company employees, say again that the sample bp is 185 mmHg. We would be unable to reject the null hypothesis that the bp is higher than desired. But could we turn this around and tell the chief medical officer with confidence that the mean bp of all the employees is going to be acceptable? Not automatically, no. It could be that the mean bp is actually OK, or it could be that the mean bp is too high, but the sample was such that it gave a false impression of being OK.

There is therefore a separate estimate required for the confidence with which one can say there is no difference in mean bp from desired. This concept is extremely important:

“Lack of evidence for a significant difference between means does not constitute evidence for no significant difference.”

As usual, unfortunately, whenever we introduce a new concept we also introduce more jargon terms.

False Positive Error

The possibility of getting the estimate wrong when concluding that the sample mean is inconsistent with the population mean, in our example concluding that the bp is too high when it is not, is called a type 1, or α error and its value is annotated pα. Earlier we saw that if the sample mean bp was 190 mmHg we rejected the null hypothesis that the bp was OK with a p-value of 0.02. But this means that there is still a 2% chance we would be wrong in this rejection, even though it satisfies our p=0.05 confidence threshold. The α error is 0.02. It is the false positive error of concluding there is a discrepancy when there is none. The critical value p=0.05 means that we are accepting a false positive error of 0.05 or 5%.

False Negative Error

If, as we have discussed, the bp was 185 mmHg we would not conclude that the bp was too high. But then if we drew a different conclusion that the mean bp was OK, we would be opening ourselves to a different error. This is the type 2, or β error, and its value is annotated pβ. It is the error of falsely concluding that there is no discrepancy when there is a discrepancy, or a false negative error.


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