Appendix B — How to quote statistics in scientific manuscripts

Students frequently ask me how to quote statistics in manuscripts. The question is a fair one, but before I give you some examples please allow me to explain why I can’t give you a definitive answer to this. (1) Don’t do as I say, do as your advisor says. Your advisor knows more about your field, and its norms (see below), than either of us does. Thus their opinion is the one that counts the most. When you have more experience, feel free to (dis)agree with them or me, of course! (2) Follow the norms of your field. I find my sub-field of eco-phylogenetics’ obsession with certain numbers (“were there $9,999$ null permutations or $10,000$? Answer me!”) absurd, but readers expect them and so I give them what they want. Presenting information in the form your readers expect is often more important than being “correct”. (3) There is no single best way to present statistics. You shouldn’t lie, or lie through omission, but other than that you’re basically free to do whatever you wish.

One caveat: I always tell my students to follow two general rules: (1) write structured methods that map onto the results, and (2) hide statistics in figure legends. (1) By writing your methods section around the questions you are answering, and the statistical results that will get at them, you can write results sections whose structures mirror the sub-headings of your methods but are only a few sentences long. You can also write results like “in support of our hypothesis that XXX” if you have done this, because you outlined how your methods would test those hypotheses. It makes papers much clearer. (2) I put all my statistics in figure and table legends, so I can write results sentences like “heating a vial denatured the enzymes more effectively than shaking it (figure 1)”. This is much easier to read, and allows your reader to focus on the biology not the statistics. Show, don’t tell: statisticians are much more impressed by this sort of thing (we know where to look to get the methods), and biologists are much less intimidated.

$t$-test.

“Heating a vial denatured the enzymes more than shaking ($t_{65} = 10.98$, $p < 0.001$)”. Notice I’ve put the degrees of freedom in subscript, and not given an exact $p$ (who needs beyond that level of detail) and $t$ to two decimal places. You know I don’t love $p$-values, but for a $t$-test they’re appropriate.

(Pearson’s) correlation test.

“Temperature and precipitation are positively correlated ($r_{98} = 0.15$, $p = 0.012$)”. As above. Notice that, for simple statements like this, I probably would put the statistic in the results text (and the figure legend as well, if necessary). If I felt like it, I might write “significantly correlated”, but it’s not necessary.

ANOVA.

“Feeding rates did not significantly differ accross day, morning, and night ($F_{67,2} = 1.32$, $p = 0.43$)”. Notice that I’ve given the two degrees of freedom for the F-test as a subscript. I would feel under absolutely no obligation to put in an ANOVA table (i.e., the output from summary.aov, which I have not focused on in this course because it’s pretty useless), even if the results were significant. I would, however, consider putting in coefficient tables (see below).

Multiple regression.

Summarizing a multiple regression is hard, because you need to find a way to express to the reader that the top entry is the reference and everything else is a contrast. There is no easy way to do this; some people simply cheat by providing ANOVA tables (see above) but these are useless as they don’t actually tell you what the effects of your explanatory variables are! The solution is to describe, in words, in your results section what you’ve found, and then reference a table where you explain it all. The best way to learn about this is by example: see Weedop et al. Animal Conservation (in press) DOI: 10.1111/acv.12495 for two examples.

Generalized linear models.

Summarize these in exactly the same way as you would a multiple regression, but remember you may need to give null deviances and residual deviances. See, for example, Pearse et al. Global Ecology & Biogeography (in press) DOI: 10.1111/geb.12938.

Mixed-effects models.

Report them as you would a Generalized Linear Model, but make sure you also give the variances of the random effects. See, for example, the supplement to Pearse et al. Ecosphere 9(2):e02105 (2018).

Bayesian hierarchical models.

These don’t follow the normal rules and table heading of the other models because you’re reporting posterior distributions. Thus you need to give the quantiles (as well as the means and standard errors). Frustratingly, you often need to also give mini-explanations of what Bayesian posterior distributions are, often couching them in terms of “Bayesian p-values”. I don’t like holding my nose and doing it, but it does allow you to write some reasonably powerful prose. For an example, take a look at the closing section in Pearse et al. Nature Ecology & Evolution 1(12):1876 (2017). Take a look at the model table (note I report ‘pooling’ estimates, which I don’t cover in depth but I do report the $\sigma^2$ values as I suggest you do for mixed-effects models) and also the text where I talk about relative probabilities (something you cannot do with a frequentist approach).

# How to quote statistics in scientific manuscripts Students frequently ask me how to quote statistics in manuscripts. The question is a fair one, but before I give you some examples please allow me to explain why I can't give you a definitive answer to this. (1) *Don't do as I say, do as your advisor says.* Your advisor knows more about your field, and its norms (see below), than either of us does. Thus their opinion is the one that counts the most. When you have more experience, feel free to (dis)agree with them or me, of course! (2) *Follow the norms of your field.* I find my sub-field of eco-phylogenetics' obsession with certain numbers ("were there $9,999$ null permutations or $10,000$? Answer me!") absurd, but readers expect them and so I give them what they want. Presenting information in the form your readers expect is often more important than being "correct". (3) *There is no single best way to present statistics.* You shouldn't lie, or lie through omission, but other than that you're basically free to do whatever you wish. One caveat: I always tell my students to follow two general rules: (1) write structured methods that map onto the results, and (2) hide statistics in figure legends. (1) By writing your methods section around the questions you are answering, and the statistical results that will get at them, you can write results sections whose structures mirror the sub-headings of your methods but are only a few sentences long. You can also write results like "in support of our hypothesis that XXX" if you have done this, because you outlined how your methods would test those hypotheses. It makes papers much clearer. (2) I put all my statistics in figure and table legends, so I can write results sentences like "heating a vial denatured the enzymes more effectively than shaking it (figure 1)". This is much easier to read, and allows your reader to focus on the biology not the statistics. Show, don't tell: statisticians are much more impressed by this sort of thing (we know where to look to get the methods), and biologists are much less intimidated. **$t$-test.** "Heating a vial denatured the enzymes more than shaking ($t_{65} = 10.98$, $p < 0.001$)". Notice I've put the degrees of freedom in subscript, and not given an exact $p$ (who needs beyond that level of detail) and $t$ to two decimal places. You know I don't love $p$-values, but for a $t$-test they're appropriate. **(Pearson's) correlation test.** "Temperature and precipitation are positively correlated ($r_{98} = 0.15$, $p = 0.012$)". As above. Notice that, for simple statements like this, I probably would put the statistic in the results text (and the figure legend as well, if necessary). If I felt like it, I might write "significantly correlated", but it's not necessary. **ANOVA.** "Feeding rates did not significantly differ accross day, morning, and night ($F_{67,2} = 1.32$, $p = 0.43$)". Notice that I've given the two degrees of freedom for the F-test as a subscript. I would feel under absolutely no obligation to put in an ANOVA table (*i.e.*, the output from `summary.aov`, which I have not focused on in this course because it's pretty useless), even if the results were significant. I would, however, consider putting in coefficient tables (see below). **Multiple regression.** Summarizing a multiple regression is hard, because you need to find a way to express to the reader that the top entry is the reference and everything else is a contrast. There is no easy way to do this; some people simply cheat by providing ANOVA tables (see above) but these are useless as they don't actually tell you what the effects of your explanatory variables are! The solution is to describe, in words, in your results section what you've found, and then reference a table where you explain it all. The best way to learn about this is by example: see Weedop *et al.* *Animal Conservation* (in press) DOI: 10.1111/acv.12495 for two examples. **Generalized linear models.** Summarize these in exactly the same way as you would a multiple regression, but remember you may need to give null deviances and residual deviances. See, for example, Pearse *et al.* *Global Ecology & Biogeography* (in press) DOI: 10.1111/geb.12938. **Mixed-effects models.** Report them as you would a Generalized Linear Model, but make sure you also give the variances of the random effects. See, for example, the supplement to Pearse *et al.* *Ecosphere* 9(2):e02105 (2018). **Bayesian hierarchical models.** These don't follow the normal rules and table heading of the other models because you're reporting posterior distributions. Thus you need to give the quantiles (as well as the means and standard errors). Frustratingly, you often need to also give mini-explanations of what Bayesian posterior distributions are, often couching them in terms of "Bayesian p-values". I don't like holding my nose and doing it, but it does allow you to write some reasonably powerful prose. For an example, take a look at the closing section in Pearse *et al.* *Nature Ecology & Evolution* 1(12):1876 (2017). Take a look at the model table (note I report 'pooling' estimates, which I don't cover in depth but I do report the $\sigma^2$ values as I suggest you do for mixed-effects models) and also the text where I talk about relative probabilities (something you cannot do with a frequentist approach).