Calculating stratified mean biomass/abundance

Source: vignettes/calc_strat_mean.Rmd

Background

The survdat package is designed to work with the Northeast Fisheries Science Center’s (NEFSC) bottom trawl surveys. The NEFSC has been conducting standardized bottom trawl surveys in the fall since 1963 and spring since 1968. The surveys follow a stratified random design. Fish species and several invertebrate species are enumerated on a tow by tow basis [@Azarovitz_1981]. The data are housed in the NEFSC’s survey database (SVDBS) maintained by the Ecosystem Survey Branch.

The get_survdat_data function will query the NEFSC survey database (SVSDBS) and apply the appropriate calibration factors associated with gear or vessel differences throughout the time series. However, end users are usually interested in more derived estimates of biomass or abundance that the tow by tow data retrieved via get_survdat_data. The stratified random design of the survey allows for the calculation of a stratified mean biomass or abundance for a species or group of species.

To facilitate the calculation of these stratified means, the survdat package has the function calc_stratified_mean. This function is actually a wrapper of several intermediate functions.

Work flow of the `calc_stratified_mean` function with associated intermediate functions.

Work flow of the calc_stratified_mean function with associated intermediate functions.

strat_prep

The first function called is strat_prep. As the name implies, strat_prep prepares the data set so that the stratified means can be calculated. Nested within this function are the post_strat and get_area functions. The post_strat function is used if not following the stratified design of the survey. It is automatically called if the user inputs an sf object rather than the specified default ‘NEFSC strata’. When this occurs, the R object poststratData generated by post_strat will replace the survdat data object and is ultimately passed through to the prepData object. This will also turn on the poststratFlag which is used by the strat_mean function.

At this point you can also filter the data using the filterBySeason or filterByArea arguments. The first will allow you to select one of three options: ‘FALL’, ‘SPRING’, or ‘all’. The default option is ‘all’. Note that selecting both the spring and fall survey will combine the two surveys. So if you want to have separate means for both surveys you need to run them independently. The second filter will allow you to select a subset of strata to use in your analysis.

The rest of the prep work done by strat_prep is counting the number of tows per strata and the proportional weight of the strata. This adds the columns ntows and W.h to the data set. The number of tows (ntows) is simply the length of unique station records per stratum:

# Count the number of stations in each year for each Region
  data.table::setkey(stations, YEAR, STRAT)
  stations[, ntows := length(STATION), by = key(stations)]

Proportional weights of the strata are based on their area. The area of the strata are calculated using the get_area function which uses a Lambert Conformal Conic projection. The get_area function creates a list of strata and their corresponding areas (A) which strat_prep uses to calculate the relative weight (W) of each strata h as:

\[\begin{equation} \label{PropArea} \begin{split} W_h = \frac{A_h}{\sum{A_h}} \end{split} \end{equation}\]

After counting the number of tows and calculating the relative weight of each strata, the resulting prepData is then passed to the strat_mean function.

strat_mean

Once the data has been prepared using strat_prep the real calculations are carried out in the function strat_mean. This function will remove duplicate catch records that may be present due to length data, merge sexed species (or leave separate if specified), calculate the total number of stations for the year, subset the species list if desired, and calculate the stratified mean biomass and adundance along with the associated variance.

The first few actions are straightforward. Length data is removed to ensure each catch is accounted for once.

#Remove length data if present
  data.table::setkey(stratmeanData, CRUISE6, STRATUM, STATION, SVSPP, CATCHSEX)
  stratmeanData <- unique(stratmeanData, by = key(stratmeanData))
  stratmeanData[, c('LENGTH', 'NUMLEN') := NULL]

Similarly, catch from sexed species such as spiny dogfish are merged or kept separate. The default is to merge them but a user may specify to keep them separate by setting the mergesexFlag to FALSE. Sexed species are kept separate by changing their group designation to a concatenation of their group name and catch sex.

#Merge sex or keep separate
  if(mergesexFlag == F) stratmeanData[, group := paste(group, CATCHSEX, sep = '')]

  data.table::setkey(stratmeanData, CRUISE6, strat, STATION, group)
  stratmeanData[, BIOMASS   := sum(BIOMASS),   by = key(stratmeanData)]
  stratmeanData[, ABUNDANCE := sum(ABUNDANCE), by = key(stratmeanData)]
  stratmeanData <- unique(stratmeanData, by = key(stratmeanData))

The total number of tows per year is the sum of ntows for all of the strata sampled for a given year.

#Calculate total number of stations per year
  data.table::setkey(stratmeanData, strat, YEAR)
  N <- unique(stratmeanData, by = key(stratmeanData))
  N <- N[, sum(ntows), by = 'YEAR']
  data.table::setnames(N, 'V1', 'N')

Species or group list can be subsetted using the filterByGroup argument.

Stratified mean and variance calculations

Finally, the stratified mean biomass is calculated as:

\[\begin{equation} \label{Stratifed} \bar{B} = \sum_{h = 1}^{L}{W_h\bar{B_h}} \end{equation}\]

where \(\bar{B}\) is the stratified mean biomass, \(W_h\) and \(\bar{B_h}\) the relative weight and mean biomass from stratum \(h\), and \(L\) the total number of strata. \(\bar{B_h}\) is calculated as:

\[\begin{equation} \label{mean biomass} \bar{B_h} = \frac{\sum_{i = 1}^n{B_i}}{n_h} \end{equation}\]

where \(B_i\) is the biomass at station \(i\) and \(n_h\) the number of stations in stratum \(h\).

Additionally, variance is calculated as:

\[\begin{equation} \label{stratified variance} s_{\bar{B}}^2 = \sum_{h = 1}^L{\frac{W_{h}^{2} s_{h}^{2}}{n_{h}}} \end{equation}\]

where \(s_{\bar{B}}^2\) is the variance of the stratified mean biomass and \(s_h^2\) the variance of the mean for stratum \(h\). \(s_h^2\) is calculated as:

\[\begin{equation} \label{strata variance} s_h^2 = \frac{{(B_i - \bar{B_h})}^2}{n_h - 1} \end{equation}\]

If the poststratFlag was set to TRUE because a different stratification was used rather than the survey design, the variance calculation becomes:

\[\begin{equation} \label{penalty variance} s_{\bar{B}}^2 = \frac{\sum_{h = 1}^L{s_{h}^{2} W_h}}{N} + \sum_{h = 1}^L{\frac{(1 - W_h)s_h^2}{N^2}} \end{equation}\]

where \(N\) is the total number of stations for the year.

The final calculation for the calc_stratified_mean function is the standard error of the mean which is:

\[\begin{equation} \label{standard error} s_{\bar{B}} = \sqrt{s_{\bar{B}}^2} \end{equation}\]

Similar calculations are carried out for abundance with numbers replacing biomass in the equations above.

Tidy data

The default output from the calc_stratified_mean function is a wide table with a row for every year/group combination. You can elect a tidy format (long) instead by specifying the argument tidy = TRUE. This will melt the table down so that each year/group combo will now have a seperate row for stratified mean biomass, stratified mean abundance, variance of the stratified mean biomass, and variance of the stratified mean abundance. The tidy data will also append a units column to the data set. The units for the stratified mean biomass is \(kg tow^-1\) while the stratified mean abundance is \(numbers tow^-1\). Note that the tidy data set does not include the standard errors.