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I’m using glm to see whether there is an association between zooplankton biomass (response) with two variables: 1) hours from high tide (high tide is zero, hours before are negative at one hourly intervals and hours after are positive at one hourly intervals) and 2) fish behaviour, which has three levels (feeding (F), not feeding (NF) and absent (A), where A is the reference category).

To account for the tide's circular nature (∼12-hr cycle), I have transformed the variable using a truncated Fourier series (a harmonic function of sines and cosines). I used a gamma error structure and log-link function because zooplankton biomass is not normally distributed (while transforming zooplankton biomass using log10() did give me a normal distribution, allowing the use of a Gaussian distribution, the model residuals were not normally distributed, so I decided it would be best to stick with gamma as I think they are acceptable)

A reproducible example is given below.

My questions are:

  1. Is it necessary to transform the tide variable using a truncated Fourier series?
  2. If yes to question 1, how should I test interactions between behaviour and the tide? For example, should I test the interactions of behaviour with all possible combinations of the sin and cos tide variables and select the most appropriate model using AICc?
  3. If the answer to question 1 is no, I have included an interaction effect between behaviour and tide in model3 below. I have used exp(coef(model3))[2] for the significant association between biomass and behaviourF, which gives me 4.35. Is it correct to say that zooplankton biomass is 4.35-fold higher when fish are feeding than when they are absent? If so, how do I interpret the interaction effect between tide and behaviourF? exp(coef(model3))[5] for behaviourF:tide is 0.98.

#dataframe

tide_values <- c(1,1,-1,4,4,4,4,3,-4,-3,-2,4,5,1,-3,-6,-6,3,4,2,2,3,1,0,6,7,0,1,0,0,5,5,4,5,-2,2,2,-3,2,3,4,5,-3,-3,-2,-2,0,0,1,5,-3,5,3,-2,-1,-1,0,0,0,-3,6,5,6,5,0,0,2,-4,-4,1,3,-1,-4,5,-5,6,4,1,-4,-4,6,-6,0,4,2,1,-4,-2,-1)

biomass_values <- c(22.92,21.27,13.4,12.52,42.81,34.99,53.65,25.83,22.37,8.81,8.99,15.35,17.04,8.54,5.75,1.39,5.84,4.42,2.98,16.92,6.14,6.55,66.52,7.74,62.18,6.88,31,19.6,9.29,11.93,19.81,18.56,34.93,41.67,43.8,131.97,58.36,84.86,36.81,14.18,111.83,26.51,214.2,34.1,200.32,60.31,14.59,20.26,43.68,17.83,325.28,23.04,21.81,18.81,8.58,15.79,25.54,3.87,1.8,4.42,58.61,7.63,9.94,11.51,11.89,8.71,22.29,9.71,2.45,2.17,20.18,28.92,99.06,1.93,25.71,1.15,8.44,5.79,3.28,2.18,4.75,8.61,1.54,3.92,2.19,1.85,11.45,8.75,3.86)

behaviour<-c("A","A","F","F","F","F","F","F","A","A","F","F","F","A","NF","A","A","A","A","F","NF","NF","F","A","F","A","A","F","A","A","A","A","NF","NF","F","F","F","F","F","F","A","F","F","A","F","F","NF","F","F","F","F","F","F","NF","A","A","F","A","A","A","F","A","A","A","A","F","F","A","A","F","F","A","F","A","A","A","A","A","A","A","A","A","A","A","A","NF","F","F","F")
data <- data.frame(BIOMASS = biomass_values,tide = tide_values, behaviour = behaviour )

head(data)

#transform tide variable
# Define the number of harmonics to include
num_harmonics <- 3
# Create the Fourier series terms
for (k in 1:num_harmonics) {
  data[[paste0("sin_", k)]] <- sin(2 * pi * k * data$tide / 12)
  data[[paste0("cos_", k)]] <- cos(2 * pi * k * data$tide / 12)
}
head(data)

# Fit the GLM model with Gamma error structure and log link

#full model with tide transformation 
model1<-glm(BIOMASS ~ behaviour + tide + sin_1 + sin_2 + sin_3 + cos_1 + cos_2 + cos_3, data = data, family = Gamma(link = "log"))

#model with interaction effect with tide transformation
model2<-glm(BIOMASS ~ behaviour * tide + sin_1 + sin_2 + sin_3 + cos_1 + cos_2 + cos_3, data = data, family = Gamma(link = "log"))

#model without tide transformation and intercation effect
model3<-glm(BIOMASS ~ behaviour * tide, data = data, family = Gamma(link = "log"))

summary(model3)

exp(coef(model3))[2] # zooplankton biomass is 4.35-fold higher when fish are feeding?

exp(coef(model3))[3] # = 0.98, how do I interpret this interaction effect
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  • $\begingroup$ These look like questions a PhD student might bring to a meeting, within a context of known project and dataset. There are conversely many assumptions here about what we might understand despite the detail you give. From a quick look at the data: The difference between NF and A has biological meaning but doesn't seem to have an impact. I wouldn't call using sines and cosines here a transformation (even though it is) but rather a means to parameterise a circular predictor. Which terms you use depends on some biology that I can't impute,:what kind of dependence on the tide level do you expect? $\endgroup$
    – Nick Cox
    Commented Jul 19 at 14:27
  • 1
    $\begingroup$ A generalized linear model with log link seems bang on to me, but this isn't such a large dataset that you can easily afford too complicated a model. I can easily imagine that the data collection was long and hard and wet. $\endgroup$
    – Nick Cox
    Commented Jul 19 at 14:29
  • $\begingroup$ @NickCox Good guess. I'm a postdoc and a masters student has asked for assistance. The student is replicating a peer-reviewed publication (where fish were shown to only feed at high tide, when biomass it highest) and is keen to match the analysis as closely as possible. However, after a few extra days of mulling it over in the back of my head, it is not as relevant to their study location as it was in the original paper. For context, I almost always use BRTs as my datasets are usually >10k observations, so I have been working to refresh my memory of GLMs. Thank you for your helpful feedback $\endgroup$
    – Jo Harris
    Commented Jul 21 at 7:07
  • $\begingroup$ @NickCox Thank you for your feedback; it was very helpful $\endgroup$
    – Jo Harris
    Commented Jul 21 at 7:07

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