2.4.2 | Structural Equation Modelling
All statistical analyses were performed using R 4.1.2 (R Core Team,
2021). To estimate the causalities among host body conditions, parasite
infections, and other possible factors, we used piecewise structural
equation modelling using package “piecewise SEM ” version 2.1.2
(Lefcheck, 2016) based on the hypothetical scheme shown in Figure 2.
Piecewise SEM allowed us to test the effects of parasite numbers and
host body conditions on several parameters in subsequent months
simultaneously and to use mixed effects. The whole model (i.e. basis
set) is composed of several generalised linear (mixed) models (Shipley,
2009; Lefcheck, 2016). The goodness of fit of the whole model (i.e.
basis set; Lefcheck, 2016) was evaluated by Shipley’s test of direct
separation using Fisher’s C value (Shipley, 2009; Lefcheck, 2016). If
that value did not fall below a significant level (p <
0.05), the model was fitted and explained our datasets well. Since
unexpected correlations, such as temporal correlations between parasite
numbers and body conditions among months (Figure 2), severely reduced
the model fitting due to collinearity, we treated these correlations as
correlated errors and removed them from the basis set (Shipley, 2009).
All linear mixed models in piecewise SEM were constructed using the R
package “lme4 ” version 1.1 (Bates et al., 2015), and all
responses and explanatory variables were standardised before the
analysis. We analysed our datasets for each of two separate seasons
(i.e. from June 2020 to July 2020 and from July 2020 to Oct. 2020).
Based on the hypothetical schema (Figure 3), we constructed four linear
mixed models. We expected that host body conditions could affect
parasite numbers, and vice versa, and there should be time lags across
seasons (Figure 3). Fish with higher growth rates may show higher body
conditions in the post month (Figure 3). Therefore, in the first model,
the response variable is host body condition in the post month, and an
explanatory variable is parasite numbers and host growth rate in the pre
month to check if parasite infections reduce host body condition (i.e.
parasite are cause of host body condition; Figure 3). In the second
model, the response variable is parasite numbers in the post month, and
the explanatory variable is body condition in the pre month to check if
the prior host body condition affects parasite numbers (i.e. parasite
are a consequence of poor condition; Figure 3). Host density could
affect individual host body condition, and body condition also affects
host growth rate (e.g. Gabelhouse, 1991), which is generally affected by
initial body size in salmonids (e.g. Morita, 2001). Therefore, we
constructed two more models (Figure 3): a model that included body
condition in the post month as the response variable, with its
explanatory variable being host density in the pre month, and another
model that included growth rate as the response variable, with its
explanatory variables being body condition and body size in the pre
month. The study sections were included as random effects in all
constructed models.