This package can be used to assess water quality trends for long-term monitoring data in estuaries using Generalized Additive Models and mixed-effects meta-analysis (Wood 2017; Sera et al. 2019). These models are appropriate for data typically from surface water quality monitoring programs at roughly monthly or biweekly collection intervals, covering at least a decade of observations (e.g., Cloern and Schraga 2016). Daily or continuous monitoring data covering many years are not appropriate for these methods, due to computational limitations and a goal of the analysis to estimate long-term, continuous trends from irregular or discontinuous sampling.

## Basic usage

The sample dataset rawdat is included in the package and is used for the examples below. This dataset includes monthly time series data over ~30 years for nine stations in South Bay, San Francisco Estuary. Data are available for 4 water quality parameters. All data are in long format with one observation per row.

The data are pre-processed to work with the GAM fitting functions included in this package. The columns include date, station number, parameter name, and value for the date. Additional date columns are included that describe the day of year (doy), date in decimal time (cont_year), year (yr), and month (mo as character label). These are required for model fitting or use with the analysis/plotting functions.

head(rawdat)
#>         date station param     value doy cont_year   yr  mo
#> 1 1990-02-27      18   chl 1.0333333  58  1990.156 1990 Feb
#> 2 1990-04-18      18   chl 1.6333333 108  1990.293 1990 Apr
#> 3 1990-05-30      18   chl 1.6000000 150  1990.408 1990 May
#> 4 1990-07-30      18   chl 5.2333333 211  1990.575 1990 Jul
#> 5 1990-12-06      18   chl 0.9333333 340  1990.929 1990 Dec
#> 6 1991-02-06      18   chl 1.6333333  37  1991.099 1991 Feb

One GAM model can be fit to the time series data. Each GAM fits additive smoothing functions to describe variation of the response variable (value) over time, where time is measured as a continuous number. The basic GAM used by this package is as follows:

• S: value ~ s(year, k = large)

The cont_year vector is measured as a continuous numeric variable for the annual effect (e.g., January 1st, 2000 is 2000.0, July 1st, 2000 is 2000.5, etc.). The function s() models cont_year as a smoothed, non-linear variable. The optimal amount of smoothing on cont_year is determined by cross-validation as implemented in the mgcv package (Wood 2017) and an upper theoretical upper limit on the number of knots for k should be large enough to allow sufficient flexibility in the smoothing term. The upper limit of k was chosen as 12 times the number of years for the input data. If insufficient data are available to fit a model with the specified k, the number of knots is decreased until the data can be modelled, e.g., 11 times the number of years, 10 times the number of years, etc.

The anlz_gam() function is used to fit the model. First, the raw data are filtered to select only station 34 and the chlorophyll parameter. The model is fit using a log-10 transformation of the response variable. Available transformation options are log-10 (log10) or identity (ident). The log-10 transformation is used by default if not specified by the user.

tomod <- rawdat %>%
filter(station %in% 34) %>%
filter(param %in% "chl")
mod <- anlz_gam(tomod, trans = "log10")
mod
#>
#> Family: gaussian
#>
#> Formula:
#> value ~ s(cont_year, k = 348)
#>
#> Estimated degrees of freedom:
#> 219  total = 219.93
#>
#> GCV score: 0.07280572

All remaining functions use the model results to assess fit, calculate seasonal metrics and trends, and plot results.

The fit can be assessed using anlz_smooth() and anlz_fit(), where the former assesses the individual smoother functions and the latter assesses overall fit. The anlz_smooth() results show the results for the fit to the cont_year smoother as the effective degrees of freedom (edf), the reference degrees of freedom (Ref.df), the test statistic (F), and statistical significance (p-value). The significance is in part based on the difference between edf and Ref.df. The anlz_fit() results show the overall summary of the model as Akaike Information Criterion (AIC), the generalized cross-validation score (GCV), and the R2 values. Lower values for AIC and GCV and higher values for R2 indicate better model fit.

anlz_smooth(mod)
#>       smoother      edf   Ref.df        F p.value
#> 1 s(cont_year) 218.9304 262.4482 4.796016       0
anlz_fit(mod)
#>              AIC        GCV        R2
#> GCV.Cp -3.166885 0.07280572 0.6842621

The plotting functions show the results in different formats. If appropriate for the response variable, the model predictions are back-transformed and the scales on each plot are shown in log10-scale to preserve the values of the results.

The show_prddoy() function shows estimated results by day of year with separate lines for each year.

ylab <- "Chlorophyll-a"
show_prddoy(mod, ylab = ylab)

The show_prdseries() function shows predictions for the model across the entire time series. Points are the observed data and the lines are the predicted.

show_prdseries(mod, ylab = ylab)

The show_prdseason() function is similar except that the model predictions are grouped by month. This provides a simple visual depiction of changes by month over time. The trend analysis functions below can be used to statistically test the seasonal changes.

show_prdseason(mod, ylab = ylab)

Finally, the show_prd3d() function shows a three-dimensional fit of the estimated trends across year and day of year with the z-axis showing the estimates for the response variable.

show_prd3d(mod, ylab = ylab)

## Trend testing

Statistical tests for evaluating trends are available in this package. These methods are considered “secondary” analyses that use results from a fitted GAM to evaluate trends or changes over time. In particular, significance of changes over time are evaluated using mixed-effect meta-analysis (Sera et al. 2019) applied to the GAM results to allow for full propagation of uncertainty between methods. Each test includes a plotting method to view the results.

### Evaluating changes between time periods

The anlz_perchg() and show_perchg() functions can be used to compare annual averages between two time periods of interest. The functions require base and test year inputs that are used for comparison. More than one year can be entered for the base and test years, e.g., baseyr = c(1990, 1992, 1993) vs. testyr = c(2014, 2015, 2016).

anlz_perchg(mod, baseyr = 2006, testyr = 2017)
#> # A tibble: 1 × 4
#>   baseval testval perchg     pval
#>     <dbl>   <dbl>  <dbl>    <dbl>
#> 1    9.78    5.35  -45.3 0.000376

To plot the results for one GAM, use the show_perchg() function. The plot title summarizes the results.

show_perchg(mod, baseyr = 2006, testyr = 2017, ylab = "Chlorophyll-a (ug/L)")

### Evaluating seasonal changes over time

The anlz_metseason(), anlz_mixmeta(), and show_metseason() functions evaluate seasonal metrics (e.g., mean, max, etc.) between years, including an assessment of the trend for selected years using mixed-effects meta-analysis modelling. These functions require inputs for the seasonal ranges to evaluate (doyend, doystr) and years for assessing the trend in the seasonal averages/metrics (yrstr, yrend).

The anlz_metseason() function estimates the seasonal metrics (including uncertainty as standard error) for results from the GAM fit. The seasonal metric can be any summary function available in R, such as seasonal maxima (max), minima (min), variance (var), or others. The function uses repeated resampling of the GAM model coefficients to simulate multiple time series as an estimate of uncertainty for the summary parameter.

The inputs for anlz_metseason() include the seasonal range as day of year using start (doystr) and end (doyend) days and the metfun and nsim arguments to specify the summary function and number of simulations, respectively. Here we show the estimate for the maximum chlorophyll in each season, using a relatively low number of simulations. Repeating this function will produce similar but slightly different results because the estimates are stochastic. In practice, a large value for nsim should be used to produce accurate results (e.g., nsim = 1e5).

metseason <- anlz_metseason(mod, metfun = max, doystr = 90, doyend = 180, nsim = 100)
metseason
#> # A tibble: 29 × 7
#>       yr   met     se bt_lwr bt_upr bt_met dispersion
#>    <dbl> <dbl>  <dbl>  <dbl>  <dbl>  <dbl>      <dbl>
#>  1  1991 0.264 0.354   0.417   10.2   2.06     0.0434
#>  2  1992 0.825 0.0790  5.25    10.7   7.49     0.0434
#>  3  1993 1.41  0.0866 19.4     42.4  28.7      0.0434
#>  4  1994 1.12  0.0778 10.3     20.8  14.6      0.0434
#>  5  1995 1.27  0.0816 14.6     30.4  21.0      0.0434
#>  6  1996 1.31  0.108  13.9     37.0  22.7      0.0434
#>  7  1997 1.41  0.0939 18.7     43.7  28.6      0.0434
#>  8  1998 1.96  0.0969 65.4    157.  101.       0.0434
#>  9  1999 1.58  0.151  21.5     83.9  42.5      0.0434
#> 10  2000 1.32  0.121  13.5     40.2  23.3      0.0434
#> # … with 19 more rows

The anlz_mixmeta() function uses results from the anlz_metseason() to estimate the trend in the seasonal metric over a selected year range. Here, we evaluate the seasonal trend from 2006 to 2017 for the seasonal estimate of the model results above.

anlz_mixmeta(metseason, yrstr = 2006, yrend = 2017)
#> Call:  mixmeta::mixmeta(formula = met ~ yr, S = S, data = totrnd, random = ~1 |
#>     yr, method = "reml")
#>
#> Fixed-effects coefficients:
#> (Intercept)           yr
#>     69.7151      -0.0341
#>
#> 12 units, 1 outcome, 12 observations, 2 fixed and 1 random-effects parameters
#>   logLik       AIC       BIC
#>   8.5524  -11.1048  -10.1971

The show_metseason() function plots the seasonal metrics and trends over time. The anlz_metseason() and anlz_mixmeta() functions are used internally to get the predictions. The same arguments for these functions are used for show_metseason, with the mean as the default metric.

show_metseason(mod, doystr = 90, doyend = 180, yrstr = 2006, yrend = 2017, ylab = "Chlorophyll-a (ug/L)")

To plot only the seasonal metrics, the regression line showing trends over time can be suppressed by setting one or both of yrstr and yrend as NULL.

show_metseason(mod, doystr = 90, doyend = 180, yrstr = NULL, yrend = NULL, ylab = "Chlorophyll-a (ug/L)")

Adding an argument for metfun to show_metseason() will plot results and trends for a metric other than the average. Note the use of nsim in this example. In practice, a much higher value should be used (e.g., nsim = 1e5)

show_metseason(mod, metfun = max, nsim = 100, doystr = 90, doyend = 180, yrstr = 2006, yrend = 2017, ylab = "Chlorophyll-a (ug/L)")

The seasonal estimates and mixed-effects meta-analysis regression can be used to estimate the rate of seasonal change across the time series. For any given year and seasonal metric, a trend can be estimated within a specific window (i.e., yrstr and yrend arguments in show_metseason()). This trend can be estimated for every year in the period of record to estimate the rate of change over time for the seasonal estimates.

The anlz_trndseason() function estimates the rate of change and the show_trndseason() function plots the results. For both, all inputs required for the anlz_metseason() function are required, in addition to the desired window width to evaluate for each year (win) and the justification for the window as "left", "right", or "center" from each year (justify).

It’s important to note the behavior of the centering for window widths (win argument) if choosing even or odd values. For left and right windows, the exact number of years in win is used. For example, a left-centered window for 1990 of ten years will include exactly ten years from 1990, 1991, … , 1999. The same applies to a right-centered window, e.g., for 1990 it would include 1981, 1982, …, 1990 (if those years have data). However, for a centered window, picking an even number of years for the window width will create a slightly off-centered window because it is impossible to center on an even number of years. For example, if win = 8 and justify = 'center', the estimate for 2000 will be centered on 1997 to 2004 (three years left, four years right, eight years total). Centering for window widths with an odd number of years will always create a symmetrical window, i.e., if win = 7 and justify = 'center', the estimate for 2000 will be centered on 1997 and 2003 (three years left, three years right, seven years total).

trndseason <- anlz_trndseason(mod, doystr = 90, doyend = 180, justify = 'left', win = 5)
head(trndseason)
#> # A tibble: 6 × 12
#>      yr   met     se bt_lwr bt_upr bt_met dispersion  yrcoef   pval appr_yrcoef
#>   <dbl> <dbl>  <dbl>  <dbl>  <dbl>  <dbl>      <dbl>   <dbl>  <dbl>       <dbl>
#> 1  1991 0.115 0.537   0.129  16.5    1.46     0.0434  0.0272 0.406        0.358
#> 2  1992 0.684 0.0653  4.03    7.27   5.41     0.0434  0.103  0.0386       1.80
#> 3  1993 0.689 0.0972  3.53    8.50   5.48     0.0434  0.0699 0.265        1.29
#> 4  1994 0.790 0.0870  4.67   10.2    6.91     0.0434  0.0582 0.340        1.24
#> 5  1995 0.728 0.0805  4.17    8.62   5.99     0.0434  0.0489 0.463        1.18
#> 6  1996 1.20  0.0918 11.7    26.8   17.7      0.0434 -0.0611 0.213       -1.50
#> # … with 2 more variables: yrcoef_lwr <dbl>, yrcoef_upr <dbl>

The show_trndseason() function can be used to plot the results directly, one model at a time.

show_trndseason(mod, doystr = 90, doyend = 180, justify = 'left', win = 5, ylab = 'Chl. change/yr, average')

As before, adding an argument for metfun to show_trndseason() will plot results and trends for a metric other than the average. Note the use of nsim in this example. In practice, a much higher value should be used (e.g., nsim = 1e5)

show_trndseason(mod, metfun = max, nsim = 100, doystr = 90, doyend = 180, justify = 'left', win = 5, ylab = 'Chl. change/yr, maximum')

The results supplied by show_trndseason() can be extended to multiple window widths by stacking the results into a single plot. Below, results for window widths from 5 to 15 years are shown using the show_sumtrndseason() function for a selected seasonal range using a left-justified window. This function only works with average seasonal metrics due to long processing times with other metrics. To retrieve the results in tabular form, use anlz_sumtrndseason().

show_sumtrndseason(mod, doystr = 90, doyend = 180, justify = 'left', win = 5:15)

## References

Cloern, J. E., and T. S. Schraga. 2016. “USGS measurements of water quality in San Francisco Bay (CA), 1969-2015: U.S. Geological Survey data release. https://doi.org/10.5066/F7TQ5ZPR.”

Sera, F., B. Armstrong, M. Blangiardo, and A. Gasparrini. 2019. “An Extended Mixed-Effects Framework for Meta-Analysis.” Statistics in Medicine 38 (29): 5429–44. https://doi.org/10.1002/sim.8362.

Wood, S. N. 2017. Generalized Additive Models: An Introduction with R. 2nd ed. London, United Kingdom: Chapman; Hall, CRC Press.