Appendix B — Cheat sheet

B.1 Formula syntax

Table B.1: Formula syntax

Formula	Interpretation
y ~ x	Intercept + slope
y ~ 1	intercept only
y ~ x - 1 or x + 0	only slope, for categorical
log(x) ~ sqrt(x)	predictor and response variable can be transformed
y ~ x1 + x2	multiple predictors can be added
y ~ x1 * x2	interaction and main effects
y ~ x1:x2	only interaction
y ~ (x1 + x2)^2	all 2nd order interactions, works also with 3rd order ^3 etc
y ~ I(x^2)	I() forces mathematical interpretation of whatever is in the parenthesis
y ~ s(x)	spline (works only for mgcv)
y ~ te(x1, x2)	tesnsor spline (works only for mgcv)
y ~.	include all variables as main effects
y~.^2	include all variables as main effects and their 2nd order interactions

B.2 Random effect syntax

The following formulas are used by many R packages (e.g. glmmTMB and lme4). For a detailed overview, see Ben Bolkers glmmFAQ

Formula	Interpretation
(1|group)	Random intercept within group
(x|group)	Random intercept within group and random slope for x within group; correlation between random intercept and random slope
(0+x|group) or (x-1|group)	Random slope for x within group without random intercept for group
(0+x|group) + (1|group) or (x||group)	Random intercept and slope but uncorrelated
(1|group:site)	Random intercept for sites within group (nested random effect)
(1|group) + (1|site)	Random intercept within groups and within sites (crossed random effects)

B.2.1 Correlation structures:

B.2.1.1 glmmTMB

glmmTMB supports several correlation structures but we will show here only two common representatives. See also the glmmTMB vignette about correlation structures.

Type	Formula	Interpretation
Spatial	coords = numFactor(cbind(Latitude, Longitude)) y~x + exp(0+coords|group)	Use glmmTMB::numFactor() function to represent your coordinates as a factor. Conditional autoregressive random intercepts within group based on the coords (sites).
Temporal	yearFactor = factor(years) y~x + ar1(0+yearFactor|group)	Change your time variable to factor. Conditional autoregressive random intercepts within group based on years.

B.2.1.2 lme4

lme4 does not support correlation structures

B.2.1.3 nlme::gls

Type	Formula	Interpretation
Spatial	gls(y~.., correlation = corExp( ~ Lat + Long|group)	Spatial correlation structure (no random effects) between sites within group (group is optional)
Temporal	gls(y~.., correlation = corAR1( ~ years | group)	Temoral correlation structure (no random effects) between years within group (group is optional)
Phylogeny	library(ape)gls(y~.., correlation = corBrownian(phy = pyhTree, form ~ species | group)	Pyhlogenetic correlation structure (no random effects) between species within group (group is optional)