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.