I tested my main hypothesis with a multiple regression, which (unfortunately) turned out to be not significant at all.
I am not sure in how to report the results now.
Which parameters should I include and how do I phrase the findings?
Solved – How to report non-significant multiple regression
multiple regressionreportingstatistical significance
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The multiple regression model controls for other sources of variability in the DV, whereas in the t-test, all of that variability is lumped into the error term. Thus, the t-test has lower statistical power to detect the effect. Under the assumption that the effect is real, however, the t-test would show 'significance' with a sample that was large enough.
My eyebrows go up when numbers are reported with far too much precision, too, but there's more going on in a regression setting than we might expect. So much, so, that I won't venture a thorough answer; but I think I can share some useful principles and examples.
Principle 1: Meaningfulness
Rounding $R^2$ to the nearest whole number (thereby producing only $0$ or $1$), for instance, is not of much use. Often two decimal digits suffice, as in $R^2 = 0.69.$ But sometimes more--even much more--precision is called for. The issue is that $1-R^2$ is arguably the more meaningful value. When the response variable is a vector $(y_i),$ the variance of those numbers times $1-R^2$ is the residual variance, so the square root gives us information about the typical sizes of the residuals in your fit. Your readers will likely want some sense of how closely your fit reflects the data and this value is a good statistic for that.
Thus, when you report $R^2=0.99$ (which often occurs in scientific studies, engineering studies, where errors are positively correlated, or when explanatory variables are varied over wide ranges), all one can say about $1-R^2$ is that it lies somewhere between $1-0.985 = 0.015$ and $1-0.995 = 0.005.$ That range covers a factor of three--half an order of magnitude, or just 1/2 significant digits of precision.
Moral: Report $R^2$ with enough precision that $1-R^2$ is also precisely known.
Principle 2: Reproducibility
Others might want to reproduce your calculations. Present your statistics with enough precision that they can be confident little errors are not sneaking into their (or your) computations. For instance, if you have $n=1000$ data then one possible source of discrepancy could be the use of $1/(n-1)$ instead of $1/n$ in computing variances. That would be a relative error of about $1/n,$ or just one part in a thousand. In such a case you ought to consider reporting all results with at least three decimal places of precision (plus perhaps one or two more).
Moral: Use more precision in reporting statistics associated with large datasets.
Principle 3: Applicability
Regression coefficients are plugged into a model to produce predicted (fitted) values and from them, via subtraction, residuals and other quantities. In some models these quantities of interest are hugely smaller in magnitude than the coefficients or the values of variables used to compute them.
One striking example is regression of quantities against a date variable. When a recent date is represented (as many often are) in seconds elapsed since Jan 1, 1970, internally it is a quantity in the billions. The intercept (which often is meaningless) may be huge, requiring many more significant digits for accurate calculation.
Here, for instance, is a synthetic dataset of eight daily temperature readings at a weather station in January 2020: about as plain and simple a situation as one can conceive of.
The least-squares estimates look innocuous--but notice how large the intercept is compared to the slope:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.358e+04 1.640e+03 -8.280 0.000168 ***
Date 8.607e-06 1.039e-06 8.282 0.000168 ***
I computed the predicted value for the first day by rounding both coefficients to 1, 2, ..., 8 significant figures (sf) and finally by using the full double-precision values (about 15.6 sf), producing these possible values for the first day:
1 sf 2 sf 3 sf 4 sf 5 sf 6 sf 7 sf 8 sf DP
4200 -430 -14.10 1.18 0.396 0.0591 0.0523 0.0531 0.0533
Notice how ridiculous the values are even when using three sf and how they don't produce even the first digit of the correct value until 6 sf are used (two more than the default output shown above!). Even when using 8 sf, the third digit of the value is still incorrect.
Moral: Beware the potential for linear combinations of numbers to produce results of much smaller magnitude. Report all values with enough significant figures to reproduce the linear combinations.
Principle 4: Watch out for transformations
The questions mentions using logarithms of data. Logarithms and other "strong" non-linear transformations can change the precision in varying ways. For example, when you round a positive result to two decimal places, then the value $0.001$ (whose logarithm is $-3$) is rounded to $0,$ so all your readers will know is the original value was between $0$ and $0.01$ and its logarithm is anywhere between $-\infty$ and $-2.$ That's truly poor precision! In some situations this won't matter, but in others it can make your results useless.
Moral: When your readers might need to transform your results, report your results with enough precision so that the results of those transformations are adequately precise.
Summary
A common thread to these principles is the need to be aware of how people might use the values you report. The choice of significant figures in reporting regression results is especially subtle because the values have such versatile uses, ranging from reproducing your calculations to forecasts, extrapolation, post hoc testing, and more. It might be prudent, then, to use several more significant figures of precision in your regression summaries than you might otherwise be inclined to.
Best Answer
This is a profound question. At some time, it may be worth reading the American Statistical Association discussion on p-values: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5187603/
It gets more complex when you are modelling. With observational data, it is possible to try a vast combination of including / excluding predictors, adding interactions and so on. You can also have confounding whereby omitting predictors can mask an import effect. A lot of work is done in terms of model search, with techniques such as Lasso. However, you say you had a specific hypothesis (which you may have in an experimental setting). Failure to reject the null hypothesis here could be due to the null hypothesis being true, you having insufficient data, or any one of several technical assumptions failing. So I think you report it as saying you did not reject the null hypothesis. Especially if your sample size is small, I would report the summary statistics, and it really depends whether you think what you have could be important. If I did a study on poison, with two groups of three animals, I would not have a significant result. But I would have three dead animals in the treatment group. That is important, so I would report that. It is up to others to read this report and decide whether further investigation is appropriate.
I don't know what you mean by testing a main hypothesis. Are you looking at the F-value for the overall model fit, or at the Wald tests on the individual parameters?