The following steps were taken during the development of the two structural equation models using in the study ‘Personal traits predict conservationists’ mixed optimism about outcomes for nature’. This walkthrough has the following main sections:

Nationally-focused situational optimism

To develop the nationally-focused situational optimism instrument, the first of the ten imputed datasets was randomly split into training (70%) and test (30%) data. Using the training dataset, the polychoric correlation between the ten nationally-focused situational optimism items were examined, suggesting a positive correlation between items (Figure 1).

Figure 1. The polychoric correlation between the ten nationally-focused situational optimism items.

Figure 1. The polychoric correlation between the ten nationally-focused situational optimism items.

The Ordinal Alpha was then examined, shown below, suggesting good internal consistency between items.

# Ordinal Alpha 
round(alpha(poly_cor$rho)$total,2) # The Ordinal Alpha, using polychoric correlation matrix

Parallel analysis of the ten nationally-focused situational optimism items within the training dataset, using polychoric correlation and WLS, was performed. This analysis suggested the extraction of three factors (Figure 2).

Figure 2. Parallel analysis of the ten items within the training data.

Figure 2. Parallel analysis of the ten items within the training data. 

## Parallel analysis suggests that the number of factors =  3  and the number of components =  NA

The root mean square error of approximation across exploratory factor analysis with increasing numbers of factors was examined. The three factor model had a very high score, whereas the five factor model (the number of factors expected to be extracted) appeared to have a better fit, as shown in the table below.

Factors RMSEA lower upper
1 0.167 0.160 0.174
2 0.125 0.117 0.133
3 0.098 0.089 0.108
4 0.048 0.035 0.061
5 0.021 0.000 0.044

The factor loadings for the five factor exploratory factor analysis was inspected, shown in the table below. When repeating the analysis with differing random seeds, seven items were consistently loaded onto three factors, but three items were unstable, cross-loaded, or were the only item loading on a single factor (‘SO_7’, ‘SO_9’ and ‘SO_10’). Consequently, these three items were removed.

Factor 1 Factor 2 Factor 3 Factor 4 Factor 5
Public support for conservation will grow over the next ten years (SO_1) 0.64
Government spending on conservation will grow over the next ten years (SO_2) 0.80
The harmful impact of people on nature will be less in ten years’ time than it is now (SO_3) 0.61
Human society will be more environmentally sustainable in ten years’ time than it is now (SO_4) 0.91
There will be more wildlife in ten years’ time than there is today (SO_5) 0.92
There will be more natural areas and habitats in ten years’ time than there are today (SO_6) 0.72
People will spend more recreational time in nature in ten years’ time than they do now (SO_7) 0.83
Nature will be able to provide the same benefits to people in ten years’ time as now (SO_8) 0.52
There will be more local participation in conservation in ten years’ time than now (SO_9) 0.39
Conservationists will have better tools and knowledge in ten years’ time than now (SO_10) 0.84

A graded response model was used to explore the association between nationally-focused situational optimism and each item. The item response categories characteristic curves and an item information curve are shown in the following figures. These curves suggest reasonable discrimination across most of the latent construct. However, floor effects among some items suggested the instrument may have had poor discriminatory power among those with extremely low nationally-focused situational optimism.

The analysis sought to extract a single latent variable, representing respondents’ nationally-focused situational optimism. Consequently, correlated error terms were included for those items loaded on the same factor, using the test data (Figure 3).

# The test dataset 
test_DF2_ord <-  mice.imp.SO[[1]][-train_ind.SO, ]

# The model
model_SO_simple <- '
SO =~ SO_1 + SO_2 + SO_3 +SO_4 + SO_5+ SO_6+ SO_8  

# Correlated error terms 
SO_1 ~~ SO_2
SO_3 ~~ SO_4 
SO_5 ~~ SO_6 
SO_5 ~~ SO_8 
'

# Fitting the SEM
fit_SO_simple <- lavaan::cfa(model = model_SO_simple,  estimator = "WLSMVS", data=test_DF2_ord[,SO_row_name], ordered = c("SO_1" , "SO_2" , "SO_3" , "SO_4", "SO_5", "SO_6",   "SO_8"))
Figure 3. The structure of the model used to estimate latent nationally-focused situational optimism.

Figure 3. The structure of the model used to estimate latent nationally-focused situational optimism.

The chi-square test statistic of this model was 0.034, the comparative fit index was 0.998, the mean root mean square error of approximation was0.037, the mean Tucker-Lewis index was 0.996, and the standardized root mean square residual was 0.029. The chi-square test statistic becomes an increasingly poor indicator of model fit as sample sizes increase. As such, excluding the chi-square test statistic, all other measures suggested the model had good fit.

Dispositional optimism

The structure of the Life Orientation Test-Revised used here has been evaluated in other studies. However, we sought to explore how well our data fit this structure. First, the polychoric correlation between the Life Orientation Test-Revised items within the training dataset were inspected (reversing the coding of negatively worded items), suggesting a positive association between items (Figure 4).

## Converted non-numeric input to numeric
Figure 4. The polychoric correlation between the six Life Orientation Test-Revised items.

Figure 4. The polychoric correlation between the six Life Orientation Test-Revised items.

The Ordinal Alpha was then examined, shown below, suggesting reasonable internal consistency between items.

# Ordinal Alpha 
round(alpha(poly_cor$rho)$total,2) # The Ordinal Alpha, using a polychoric correlation matrix

As above, a graded response model was used to explore the association between nationally-focused situational optimism and each item. The item response categories characteristic curves and an item information curve are shown below. This model suggested reasonably discriminatory power across most of the latent construct, but some ceiling effects at extremely high levels of dispositional optimism.

We constructed a structural equation model following the factor structure recommended in previous literature (Figure 5). With this structure, all items are loaded onto one factor, assumed to represent dispositional optimism, and the three positively worded items were loaded onto a second factor, representing the method effect of the positive wording. Conceptually, these factors are orthogonal, and so their covariance was fixed to zero. This model was fit using all data from the first imputed dataset.

model_OP_method <- '
###### Dispositional optimism 
# Dispositional optimism 
DO =~ LOTR_1 + LOTR_2 + LOTR_3 +  LOTR_4 + LOTR_5 + LOTR_6

# The method effect 
method =~  LOTR_1 + LOTR_3 + LOTR_6

# DO and the method effect are orthogonal  
DO ~~0*method
'

# Fitting the SEM
fit_OP_meth <- lavaan::sem(model=model_OP_method,  estimator = "WLSMVS",  data=mice.imp.SO[[1]] , ordered = c("LOTR_1" , "LOTR_2" , "LOTR_3" , "LOTR_4", "LOTR_5", "LOTR_6") )
Figure 5. The structure of the model used to estimate latent dispositional optimism.

Figure 5. The structure of the model used to estimate latent dispositional optimism.

The chi-square test statistic of this model was 0.000, the comparative fit index was 0.995, the mean root mean square error of approximation was 0.064, the mean Tucker-Lewis index was 0.989, and the standardized root mean square residual was 0.027. Again, given the limitations of the chi-square test statistic, these indicators suggested good model fit.

Model development

The below shows a series of structural equation models. These start with a basic model, including only personal-level characteristic (1), then also including national-level biodiversity and conservation indicators (2), and finally the full model presented in the main text.

1) Simple model - individual characteristics

Figure 5. The association between nationally-focused situational optimism and locally-focused situational optimism and individual characteristics. Coefficients are in standardised units, meaning a one-unit change in continuous explanatory variables are associated with a given standard deviation (SD) change in the response variables. Estimate uncertainty is presented in 95% confidence intervals. Levels representing unknown or other responses are not shown.

Figure 5. The association between nationally-focused situational optimism and locally-focused situational optimism and individual characteristics. Coefficients are in standardised units, meaning a one-unit change in continuous explanatory variables are associated with a given standard deviation (SD) change in the response variables. Estimate uncertainty is presented in 95% confidence intervals. Levels representing unknown or other responses are not shown.

The mean comparative fit index of the nationally-focused situational optimism model shown above (Figure 6) was 0.964, and the mean root mean square error of approximation was 0.060 (95% CI = 0.057 - 0.063). The mean Tucker-Lewis index was 0.980, and the mean standardized root mean square residual was 0.072.

The mean comparative fit index of the locally-focused situational optimism model shown above (Figure 6) was 0.941, and the mean root mean square error of approximation was 0.078 (95% CI = 0.073 - 0.083). The mean Tucker-Lewis index was 0.977 and the mean standardized root mean square residual was 0.096.

2) Simple model - individual characteristics and national-level biodiversity and conservation indicators

Figure 7. The association between nationally-focused situational optimism and locally-focused situational optimism and (a.) individual characteristics and (b) biodiversity state, conservation effort and focal environment. Coefficients are in standardised units, meaning a one-unit change in continuous explanatory variables are associated with a given standard deviation (SD) change in the response variables. Estimate uncertainty is presented in 95% confidence intervals. Levels representing unknown or other responses are not shown.

Figure 7. The association between nationally-focused situational optimism and locally-focused situational optimism and (a.) individual characteristics and (b) biodiversity state, conservation effort and focal environment. Coefficients are in standardised units, meaning a one-unit change in continuous explanatory variables are associated with a given standard deviation (SD) change in the response variables. Estimate uncertainty is presented in 95% confidence intervals. Levels representing unknown or other responses are not shown.

The mean comparative fit index of the nationally-focused situational optimism model above (Figure 7) was 0.956, and the mean root mean square error of approximation was 0.051 (95% CI = 0.049 - 0.054). The mean Tucker-Lewis index was 0.985, and the mean standardized root mean square residual was 0.078.

The mean comparative fit index of the locally-focused situational optimism model above (Figure 7) was 0.944, and the mean root mean square error of approximation was 0.055 (95% CI = 0.051 - 0.058). The mean Tucker-Lewis index was 0.988 and the mean standardized root mean square residual was 0.089.

3) Primary analysis

Figure 8. The association between nationally-focused situational optimism and locally-focused situational optimism and (a.) individual characteristics, (b) biodiversity state, conservation effort and focal environment, and (c) regional grouping variables. Coefficients are in standardised units, meaning a one-unit change in continuous explanatory variables are associated with a given standard deviation (SD) change in the response variables. Estimate uncertainty is presented in 95% confidence intervals. Levels representing unknown or other responses are not shown.

Figure 8. The association between nationally-focused situational optimism and locally-focused situational optimism and (a.) individual characteristics, (b) biodiversity state, conservation effort and focal environment, and (c) regional grouping variables. Coefficients are in standardised units, meaning a one-unit change in continuous explanatory variables are associated with a given standard deviation (SD) change in the response variables. Estimate uncertainty is presented in 95% confidence intervals. Levels representing unknown or other responses are not shown.

The mean comparative fit index of the nationally-focused situational optimism model above (Figure 8) was 0.958, and the mean root mean square error of approximation was 0.044 (95% CI = 0.042 - 0.046). The mean Tucker-Lewis index was 0.989, and the mean standardized root mean square residual was 0.077.

The mean comparative fit index of the locally-focused situational optimism model above (Figure 8) was 0.947, and the mean root mean square error of approximation was 0.046 (95% CI = 0.043 - 0.049). The mean Tucker-Lewis index was 0.992 and the mean standardized root mean square residual was 0.087."