I am writing the results of my thesis, and I have a few questions on how to report P-values.
When is it appropriate to give the exact P-value, instead of writing e.g. P<0.05 (also in case of non-significant P-values)?
As a general guideline you want to convey as much information about your results as possible. When you report a p-value as p=0.016 instead of as p<0.02 then the readers will have a more precise view of the results. Often it is also helpful to report the statistic along with it. For instance you can encounter sentences like: "There was a positive effect of X with a coefficient $\beta_X = 0.5$, which was significant ($t = 2.6$, $p = 0.017$)".
The reporting of only $p<0.05$ can be done when one is only interested in strict cut-off levels (but often the cut-off levels are arbitrary). It is also common in graphs or tables where significance is denoted with superscripts like $^\star: p < 0.05$, ${^\star}{^\star}: p < 0.01$, ${^\star}{^\star}{^\star}: p < 0.001$ and in that case it is used as abbreviation to prevent cluttered graphs and tables that become difficult to read.
Instead of p-values, it is also increasingly more popular to report confidence intervals instead. Then the sentence above would become "There was a significant positive effect of X with a coefficient $\beta_X = 0.5$ (95% confidence interval [0.099,0.901] )".
If I have a P-value of e.g. 0.016, should I report it as P=0.01 or P=0.02?
Rounding off p-values might give a false idea. However, you could round up and use an inequality sign like $P<0.02$.
When I have a very small P-value, e.g. 0.00000013, is it appropriate to report it as P<0.000001, or is it better to stop at e.g. P<0.001?
The precision that is appropriate will depend on the application.
Often more precision than below $0.001$ is not needed. Also a better precision can be deceptive because the computation depends on several assumptions and the p-value is an estimate that can be computed with high precision, but does not truly have that high precision because of uncertainty in the assumptions underlying the computation. The computation with some model can be performed with a high precision, but that doesn't mean that the result of that computation should be considered as precise (the potential errors due to a wrong model should be considered as well).
In exact sciences like physics where high precision measurements are possible and a strong theoretical framework is present it is more common to see high precision p-values. For instance in some fields of physics one desires the significance to be below above $5\sigma$ which is equivalent to $p < 0.00000057$.
In practice, people report the exact p-value then declare whether the hypothesis test is statistically significant. Most statisticians don't advocate this. if the goal is hypothesis testing then reporting the p-value is moot because it doesn't make the test more significant, on the other hand the exact p-value matters if the result of the analysis doesn't boil down to a decision theoretic approach. So if the audience is not statisticians, they would probably expect you to do it the wrong way.
The significant digits on a p-value matters. First, you have to follow whatever style guide there is for your thesis or publication. If there are no rules, you need to consider the variability of the analysis. In general, it's hard to say how "stable" a p-value is without simulation or bootstrapping, but you'll often find that even with a huge analysis, a p-value is only really stable out to two decimal places - hence reporting 3 decimal places conveys a level of precision that is not present in the analysis. Secondly, if there is a pre-specified "alpha" level, you need to show enough precision for the reader to know whether p < alpha. In other words, p=0.05 requires two decimal places of precision if p = 0.04 or lower, but borderline cases get awkward. Do you say p = 0.049994 or p = 0.05 but it was statistically significant.
Don't cheat the incidental precision of the analysis. Use the prespecified precision following the rules above, and report out to the smallest decimal. Other aspects of the analysis should convey how striking the result is, a "really small p-value" in and of itself means nothing. And never never report p = 0.00.
