2.4 | Statistical analyses
The percentage of structures meeting performance targets (ranking
< 3) for integrity, erosion, and deposition was evaluated
annually. If a structure received a failed (4) rating for integrity and
function, it was excluded from the statistical analysis in subsequent
years. The maximum ranking across years was used to summarize the
overall effectiveness of instream structures for the project. The
percentage of structures meeting the performance criteria was evaluated
by year to investigate performance over time. The maximum ranking was
also used to calculate the proportion of structures by type that failed
to meet performance criteria for integrity, erosion, and deposition.
Percentages were used for comparison to the project goal that 90% of
instream structures be stable and functional three years after the
completion of construction.
Cumulative link mixed models (CLMM) were fit to the ordinal rankings for
integrity, erosion, and deposition to evaluate performance over time and
between structure types using the ‘ordinal’ package (Christensen, 2022)
in RStudio (Posit team, 2022). Integrity was treated as categorical
response variable with four levels (Table 2), while rankings for erosion
and deposition were collapsed into three levels: minor (ranking = 0-1),
moderate (ranking = 2-3), and severe (ranking = 4-5). Structure type was
included as a categorical predictor with eight levels (Table 1), and
year was included as a categorical predicator with six levels. As
structure type did not change across years, an interaction between
structure type and year was not included in the models, but structure
number was included as a random effect. Likelihood ratio tests were
performed for each model to test for significant effects, and the
‘emmeans’ package (Lenth et al., 2023) was used to obtain estimated
marginal means and perform pairwise comparisons for any significant
effects. We defined statistical significance for all test as p< 0.10 to protect against Type II error.
Ordinal regression was also used to investigate the effects of log-vane
slope and channel morphology on the maximum rankings for log-vane
integrity, erosion, and deposition across years. Cumulative link models
(CLM) were fit to the log-vane data using integrity, erosion, and
deposition as ordinal response variables with the ‘ordinal’ package in
RStudio (Christensen, 2022). Log-vane slope and angle of departure were
treated as continuous factors, while preexisting morphology was included
as a categorical factor with two levels (riffle or pool). Linear
regression was used to test for correlation between log-vane slope and
angle of departure. As the relationship between slope and angle of
departure was significant (β = -129.4, p = 0.049, multiple
R-squared = 0.095), only slope and morphology were included as additive
factors in the CLMs for log-vane integrity, erosion, and deposition.
Similar to the CLMMs, likelihood ratio tests were used to test for
significant effects, and threshold coefficients were reported for all
CLMs and CLMMs.
Residual pool depths were analyzed by structure type, years since
construction, and preexisting morphology (riffle or pool). As the
distribution for RPD values did not meet the assumption of normality,
the change in RPD between each survey was calculated for each pool. We
then used the ‘lme4’ package (Bates, Mächler, Bolker, & Walker, 2015)
in RStudio to fit linear mixed-effect models with the change in RPD as
the continuous response variable. As the variance in the response
variable was not equal for riffles and pools (F = 0.327, p< 0.0001), separate models were used to evaluate the change in
RPD for riffles and pools. The linear mixed-effects models included two
categorical fixed effects: structure type (n = 7) and years since
construction (n = 5). The two-way interaction between the independent
variables was included in the models, and pool number was included as a
random effect due to the repeated measures design. ANOVA was used to
test for significant interaction and main effects, and the ‘emmeans’
package (Lenth et al., 2023) was used to perform pairwise comparisons
for any significant effects.
Structure types for the RPD analysis included boulder clusters, boulder
vanes, log toe, log vanes, and no structure, in addition to structure
combinations of boulder vane/boulder toe and log vane/log toe. As
construction occurred over two years (2013-2014), the years since
construction was used for the RPD analysis rather than the actual year
of the topographic survey. The changes in RPD between pre-construction
and as-built conditions were represented by year 0, then the first
post-runoff survey for each pool became year 1, and so on and so forth.
Pools that were developed during construction in 2013 had five years of
subsequent surveys (2014-2018 or years 1-5), while pools built during
2014 had four years of subsequent surveys (2015-2018 or years 1-4). As
not all pools had five years of subsequent surveys following
construction, year 5 was excluded from the models to improve balance in
the dataset. Additionally, the structure types of boulder vane, boulder
vane/boulder toe combination, and log vane/log toe combination all had
sample sizes equal to or less than one for riffle habitats (Table 3). As
such, those structure types were excluded from the model for riffles.