Interpretation of p-value near alpha level If the alpha is set to 0.05 I have encountered many scientific publications saying that a "tendency for an effect" is present based on a p-value of 0.05 < p-value < 0.1. On the other hand, I have had statisticians who criticize me for doing so, because there is only "reject" or "not reject". Therefore, it does not make sense to distinguish between a p-value of 0.08 or 0.97. Also, some statisticians have criticized me for reporting p-values as p<0.05 because it is not precise.
My question is: how to deal with a p-values that are not below but close to my alpha?

One consideration that is missing from the discussion is the problem when you have more than one test. As your p-value defining significance increases there is a concomitant increase in the chance that one or more of your statistical tests will have a p-value below that cut-off by chance alone given that the null-hypothesis is true. While there are standard methods for controlling for this, these are seldom applied to entire manuscripts.
If given a choice, I would discuss outcomes with p-values < 0.05 and give the p-value. I will not discuss the hugely awesomely gargantuanly significant p-value < 0.0000001 any different than it was significant at 0.037. I don't shift my significance level based on wanting one more significant variable to discuss.
Lastly, please consider the p-value as an estimate. Visualize a 95% confidence interval about your p-value. Given the variability in your data and your sample size do you really feel that your p-value of 0.000001 is accurate to 6 decimal places? Did you report all means and standard errors to that level of accuracy?
