Hello,

The coefficients from the traitglm function represent the effect size of each environmental variable on each trait, after adjusting for the other variables in the model. A low coefficient means that the effect size is small, i.e., the change in the trait value per unit change in the environmental variable is small. This could indicate that there is no strong association between the trait and the environmental variable, or that there is a lot of variation or noise in the data that obscures the association.

However, the magnitude of the coefficients alone is not enough to determine the significance or importance of the associations. You also need to look at the standard errors and p-values of the coefficients, which measure the uncertainty and statistical significance of the estimates. A low coefficient could still be statistically significant if it has a small standard error and a low p-value, meaning that it is unlikely to occur by chance. Conversely, a high coefficient could be statistically insignificant if it has a large standard error and a high p-value, meaning that it is not distinguishable from zero.

Another way to assess the importance of the associations is to look at the R-squared value of the model, which measures how well the model explains the variation in the trait values. A high R-squared value means that the model captures most of the variation in the data, while a low R-squared value means that the model leaves a lot of unexplained variation. A low R-squared value could indicate that there are other factors that affect the trait values that are not included in the model, or that there is a lot of measurement error or random variation in the data.

Therefore, to interpret your results, you need to consider not only the coefficients but also their standard errors, p-values, and R-squared values. You also need to consider the biological meaning and relevance of your traits and environmental variables, and how they relate to each other in your study system. A low coefficient may not necessarily mean a weak association, but it may require more careful interpretation and explanation.

Thank you very much for your response!

Lyudmil Antonov Dear Dr. Antonov, I'm also very interested in the question brought up by Dr. Bergmark. Would you mind elaborating on how I can extract the p values from the fourth corner object calculated using traitglm() with a LASSO penalty?

Wenyi Zhou To extract the p-values from the fourth corner object, I have used the summary function with the argument test = "lrt". This performs a likelihood ratio test for each coefficient and returns the p-values along with the estimates and standard errors. Here is an example of the code I used:

```r

# Load mvabund package

library(mvabund)

# Load data

data(spiders)

# Fit fourth corner model with LASSO penalty

fit |z|)

(Intercept) 0.00000 0.00000 0 0

env 0.00000 0.00000 0 0

trait 0.00000 0.00000 0 0

env:trait 0.00000 0.00000 0 0

---

Signif. codes:

```

As you can see, all the coefficients are zero because of the LASSO penalty, and so are the p-values. This means that none of the interactions are significant or relevant for explaining the variation in abundance.

Lyudmil Antonov Thank you for the explanation. However, when I ran the summary() function with the argument test = "lrt", I could only produce the following output with three columns: Length, Class, and Mode.

Length Class Mode

coefficients 143 -none- numeric

lambda 1 -none- numeric

glm1.best 10 -none- list

all.coefficients 3575 -none- numeric

lambdas 25 -none- numeric

logL 25 -none- numeric

df 25 -none- numeric

bics 25 -none- numeric

counter 25 -none- numeric

check 25 -none- logical

phis 25 -none- numeric

y 1921 -none- numeric

X 274703 -none- numeric

penalty 143 -none- numeric

family 1 -none- character

deviance 1 -none- numeric

phi 1 -none- numeric

fourth.corner 18 -none- numeric

R.des 4 -none- list

Q.des 4 -none- list

any.penalty 1 -none- logical

L 113 data.frame list

scaling 4 -none- list

call 5 -none- call

Then I noticed that in creating the fourth corner object, I wrote:

ft1

@Wenyi Zhou When you wrote ft1

Lyudmil Antonov Thanks again Dr. Antonov. Last question, I'm still confused as to which data I should input for the argument "data="? Would you mind showing me a screenshot of your "spiders" data so I can see its structure/dimensions? Also, I assume your "spider" from the formula argument is the L matrix, correct?

The spider data come with the mvabund package and are available when you install this package. The following commands load them and show their structure:

data(spider)

head(spider)

You get the following as a complex data frame

$abund Alopacce Alopcune Alopfabr Arctlute Arctperi Auloalbi Pardlugu [1,] 25 10 0 0 0 4 0 [2,] 0 2 0 0 0 30 1 [3,] 15 20 2 2 0 9 1 [4,] 2 6 0 1 0 24 1 [5,] 1 20 0 2 0 9 1 [6,] 0 6 0 6 0 6 0 [7,] 2 7 0 12 0 16 1 [8,] 0 11 0 0 0 7 55 [9,] 1 1 0 0 0 0 0 [10,] 3 0 1 0 0 0 0 [11,] 15 1 2 0 0 1 0 [12,] 16 13 0 0 0 0 0 [13,] 3 43 1 2 0 18 1 [14,] 0 2 0 1 0 4 3 [15,] 0 0 0 0 0 0 6 [16,] 0 3 0 0 0 0 6 [17,] 0 0 0 0 0 0 2 [18,] 0 1 0 0 0 0 5 [19,] 0 1 0 0 0 0 12 [20,] 0 2 0 0 0 0 13 [21,] 0 1 0 0 0 0 16 [22,] 7 0 16 0 4 0 0 [23,] 17 0 15 0 7 0 2 [24,] 11 0 20 0 5 0 0 [25,] 9 1 9 0 0 2 1 [26,] 3 0 6 0 18 0 0 [27,] 29 0 11 0 4 0 0 [28,] 15 0 14 0 1 0 0 Pardmont Pardnigr Pardpull Trocterr Zoraspin [1,] 60 12 45 57 4 [2,] 1 15 37 65 9 [3,] 29 18 45 66 1 [4,] 7 29 94 86 25 [5,] 2 135 76 91 17 [6,] 11 27 24 63 34 [7,] 30 89 105 118 16 [8,] 2 2 1 30 3 [9,] 26 1 1 2 0 [10,] 22 0 0 1 0 [11,] 95 0 1 4 0 [12,] 96 1 8 13 0 [13,] 24 53 72 97 22 [14,] 14 15 72 94 32 [15,] 0 0 0 25 3 [16,] 0 2 0 28 4 [17,] 0 0 0 23 2 [18,] 0 0 0 25 0 [19,] 0 1 0 22 3 [20,] 0 0 0 22 2 [21,] 1 0 1 18 2 [22,] 2 0 0 1 0 [23,] 6 0 0 1 0 [24,] 3 0 0 0 0 [25,] 11 6 0 16 6 [26,] 0 0 0 1 0 [27,] 1 0 0 0 0 [28,] 6 0 0 2 0 $x soil.dry bare.sand fallen.leaves moss herb.layer reflection 1 2.3321 0.0000 0.0000 3.0445 4.4543 3.9120 2 3.0493 0.0000 1.7918 1.0986 4.5643 1.6094 3 2.5572 0.0000 0.0000 2.3979 4.6052 3.6889 4 2.6741 0.0000 0.0000 2.3979 4.6151 2.9957 5 3.0155 0.0000 0.0000 0.0000 4.6151 2.3026 6 3.3810 2.3979 3.4340 2.3979 3.4340 0.6931 7 3.1781 0.0000 0.0000 0.6931 4.6151 2.3026 8 2.6247 0.0000 4.2627 1.0986 3.4340 0.6931 9 2.4849 0.0000 0.0000 4.3307 3.2581 3.4012 10 2.1972 3.9318 0.0000 3.4340 3.0445 3.6889 11 2.2192 0.0000 0.0000 4.1109 3.7136 3.6889 12 2.2925 0.0000 0.0000 3.8286 4.0254 3.6889 13 3.5175 1.7918 1.7918 0.6931 4.5109 3.4012 14 3.0865 0.0000 0.0000 1.7918 4.5643 1.0986 15 3.2696 0.0000 4.3944 0.6931 3.0445 0.6931 16 3.0301 0.0000 4.6052 0.6931 0.6931 0.0000 17 3.3322 0.0000 4.4543 0.6931 3.0445 1.0986 18 3.1224 0.0000 4.3944 0.0000 3.0445 1.0986 19 2.9232 0.0000 4.5109 1.6094 1.6094 0.0000 20 3.1091 0.0000 4.5951 0.6931 0.6931 0.0000 21 2.9755 0.0000 4.5643 0.6931 1.7918 0.0000 22 1.2528 3.2581 0.0000 4.3307 0.6931 3.9120 23 1.1939 3.0445 0.0000 4.0254 3.2581 4.0943 24 1.6487 3.2581 0.0000 4.0254 3.0445 4.0073 25 1.8245 3.5835 0.0000 1.0986 4.1109 2.3026 26 0.9933 4.5109 0.0000 1.7918 1.7918 4.3820 27 0.9555 2.3979 0.0000 3.8286 3.4340 3.6889 28 0.9555 3.4340 0.0000 3.7136 3.4340 3.6889 $trait length colour marks Alopacce 2.110213 yellow spots Alopcune 1.909543 dark stripes Alopfabr 2.397895 yellow stripes Arctlute 2.197225 dark stripes Arctperi 1.945910 yellow spots Auloalbu 1.335001 yellow none Pardlugu 1.568616 dark stripes Pardmont 1.504077 dark stripes Pardnigr 1.599388 yellow stripes Pardpull 1.504077 dark none Trocterr 2.079442 yellow stripes Zoraspin 1.504077 yellow stripes

With this data frame in the latest mvabund package, the specification of data is more complicated. First, you fit the model with method "manyglm"

ft1=traitglm(spider$abund,spider$x,spider$trait, method="manyglm")

The first 3 arguments are L, R, Q.

Then you refit using LASSO and less variables, including row effects and only two interaction terms:

ft1=traitglm(spider$abund,spider$x,spider$trait, formula=~soil.dry:length+fallen.leaves:colour,composition=TRUE,method="glm1path")

Then you look for the coefficients of the fourth corner using

ft1$fourth.corner

With the spider data, I get

coef soil.dry:length -3.2301989 fallen.leaves:colourdark 0.7544505 fallen.leaves:colouryellow 0.5982624

Unfortunately, the likelihood ratio test "lrt" or Wald test "wald" did not work to give p. Sorry to mislead you in the previous message. I used an older version of the mvabund package there.

P.S. If you use method "cv.glm1path" instead of "glm1path" you get a plot with X axis "Number of terms in model (df) and Y axis "log-likelihood (per test observation). ft1$fourth.corner gives:

coef soil.dry:length 0 fallen.leaves:colourdark 0 fallen.leaves:colouryellow 0

Lyudmil Antonov no worries, thank you VERY much for all the explanation, Dr. Antonov.