# Risk Analysis

## Introduction

This document presents relative risk and attributable risk analysis of a GRTS survey design. The resource employed in the analysis is lakes in the 48 contiguous United States. Data was obtained from the National Lakes Survey (NLA) that was conducted in 2007 by the U.S. Environmental Protection Agency [U.S. Environmental Protection Agency (2009). Relative risk measures the strength of association between stressor and response variables that can be classified as either “good” (i.e., reference condition) or “poor” (i.e., different from reference condition). Attributable risk measures the percent reduction in the extent of poor condition of a response variable that presumably would result from eliminating a stressor variable. Discussion regarding relative risk in the context of aquatic resource surveys is provided in Van Sickle et. al. (Van Sickle et al. 2006) and Van Sickle and Paulsen (Van Sickle and Paulsen 2008).

## Preliminaries

The initial step is to use the library function to load the spsurvey package. After the package is loaded, a message is printed to the R console indicating that the spsurvey package was loaded successfully.

Load the spsurvey package:

library(spsurvey)
library(sf)

## Load the survey design and analytical variables data set

The original NLA data file contains more than 1,000 records. To produce a more manageable number of records, lakes located in the western U.S. were retained in the data that will be analyzed, which produced a data set containing 236 records.

The next step is to load the data set, which includes both survey design variables and analytical variables. The data function is used to load the data set and assign it to a data frame named NLA_2007. Thenrowfunction is used to determine the number of rows in theNLA_2007data frame, and the resulting value is assigned to an object namednr. Finally, the initial six lines and the final six lines in theNLA_2007data frame are printed using theheadandtail functions, respectively.

Load the survey design and analytical variables data set:

data(NLA_2007)
nr <- nrow(NLA_2007)

Display the initial six lines in the data file:

head(NLA_2007)
#>          siteID     xcoord  ycoord       wgt Lake_Origin   Chla      OE5 PTL
#> 1 NLA06608-0001 -1327628.1 3012181  7.594532     Natural  0.240 0.504031   6
#> 2 NLA06608-0004 -1084415.8 1668316  9.171940    Man-Made  4.600 1.032252  18
#> 3 NLA06608-0005 -1497348.8 2475338 15.027385     Natural  1.205 0.988630   4
#> 4 NLA06608-0015 -1044530.9 1166122  6.920957    Man-Made 20.000 0.918628 109
#> 5 NLA06608-0033 -1901234.0 2956669 32.549373     Natural  8.920 0.673385  67
#> 6 NLA06608-0042  -874392.3 2436245  9.832508    Man-Made  2.208 0.860663  15
#>   NTL Turbidity Chla_cond OE5_cond PTL_cond NTL_cond Turbidity_cond
#> 1 151     0.474      Good     Poor     Good     Good           Good
#> 2 344     3.810      Poor     Good     Good     Good           Good
#> 3  85     0.475      Good     Good     Good     Good           Good
#> 4 470    32.700      Good     Good     Good     Good           Poor
#> 5 835    12.200      Poor     Good     Poor     Poor           Poor
#> 6 213     0.791      Good     Good     Good     Good           Good

Display the final six lines in the data file:

tail(NLA_2007)
#>            siteID   xcoord  ycoord       wgt Lake_Origin   Chla      OE5 PTL
#> 231 NLA06608-3121 -1599693 2614663  4.035550    Man-Made 14.640 0.709114  46
#> 232 NLA06608-3153 -1970907 3130822  7.938297     Natural  1.499 0.737076   1
#> 233 NLA06608-3157 -1581199 2449359  4.035550    Man-Made  2.208 0.922396   8
#> 234 NLA06608-3265 -1595910 2964913 21.498248     Natural  1.768 0.648352   7
#> 235 NLA06608-3313 -1294482 2232798  3.399432    Man-Made  7.728 0.592139  41
#> 236 NLA06608-3329 -1543474 2998349  3.664951     Natural  3.704 0.991219  10
#>     NTL Turbidity Chla_cond OE5_cond PTL_cond NTL_cond Turbidity_cond
#> 231 455     5.720      Poor     Good     Poor     Poor           Poor
#> 232 116     0.420      Good     Good     Good     Good           Good
#> 233  70     1.790      Good     Good     Good     Good           Good
#> 234 338     0.561      Good     Good     Good     Good           Good
#> 235 316     5.670      Good     Poor     Good     Good           Good
#> 236 374     1.050      Poor     Good     Good     Good           Good

The location of lakes that were sampled in the western United States is displayed in the figure below. The sample sites are displayed using a unique color for the two values of lake origin (natural and manmade).

Relative risk analysis will be investigated by examining two response variables and three stressor variables. The response variables are chlorophyll-a concentration for each sample site, which is a mesure of trophic condition, and an index of macroinvertebrate taxa loss that is based on modeling the ratio of observed and expected loss. The stressor variables are total nitrogen concentration, total phosphorus concentration, and turbidity for each site.

The relrisk.analysis function will be used to calculate relative risk estimates. Four data frames constitute the primary input to the relrisk.analysis function. The first column (variable) in the four data frames provides the unique identifier (site ID) for each sample site and is used to connect records among the data frames. The siteID variable in the NLA_2007 data frame is assigned to the siteID variable in the data frames. The four data frames that will be created are named as follows: sites, subpop, design, and data.risk. The sites data frame identifies sites to use in the analysis and contains two variables: (1) siteID - site ID values and (2) Use - a logical vector indicating which sites to use in the analysis. Since we want to include all sampled sites, the rep (repeat) function is used to assign the value TRUE to each element of the Use variable. Recall that nr is an object containing the number of rows in the NLA_2007 data frame. The subpop data frame defines populations and, optionally, subpopulations for which estimates are desired. Unlike the sites and design data frames, the subpop data frame can contain an arbitrary number of columns. The first variable in the subpop data frame identifies site ID values and each subsequent variable identifies a type of population, where the variable name is used to identify type. A type variable identifies each site with a character value. If the number of unique values for a type variable is greater than one, then the set of values represent subpopulations of that type. When a type variable consists of a single unique value, then the type does not contain subpopulations. For this analysis, the subpop data frame contains three variables: (1) siteID - site ID values, (2) Western_US - which will be used to calculate estimates for all of the sample sites combined, and (3) Lake_Origin - which will be used to calculate estimates for each of the two classes of lake origin (natural and manmade). The rep function is used to assign values to the Western_US variable, and the Lake_Origin variable in the NLA_2007 data frame is assigned to the Lake_Origin variable in the subpop data frame. The design data frame consists of survey design variables. For the analysis under consideration, the design data frame contains the following variables: (1) siteID - site ID values; (2) wgt - final, adjusted, survey design weights; (3) xcoord - x-coordinates for location; and (4) ycoord - y-coordinates for location. The wgt, xcoord, and ycoord variables in the design data frame are assigned values using variables with the same names in the NLA_2007 data frame. Like the subpop data frame, the data.risk data frame can contain an arbitrary number of columns. The first variable in the data.risk data frame identifies site ID values and each subsequent variable identifies a response or stressor variable. For this analysis, the response variables are Chlorophyll_a and MacroInvert_OE, and the stressor variables are Total_Nitrogen, Total_Phosphorus, and Turbidity, which are assigned, respectively, variables Chla_cond, OE5_cond, NTL_cond, PTL_cond, and Turbidity_cond in the NLA_2007 data frame.

Conduct a relative risk analysis. Create the sites data frame, which identifies sites to use in the analysis:

sites <- data.frame(siteID=NLA_2007$siteID, Use=rep(TRUE, nr)) Create the subpop data frame: subpop <- data.frame(siteID=NLA_2007$siteID,
Western_US=rep("Western_US", nr),
Lake_Origin=NLA_2007$Lake_Origin) Create the design data frame: design <- data.frame(siteID=NLA_2007$siteID,
wgt=NLA_2007$wgt, xcoord=NLA_2007$xcoord,
ycoord=NLA_2007$ycoord) Create the data.risk data frame: data.risk <- data.frame(siteID=NLA_2007$siteID,
Chlorophyll_a=NLA_2007$Chla_cond, MacroInvert_OE=NLA_2007$OE5_cond,
Total_Nitrogen=NLA_2007$NTL_cond, Total_Phosphorus=NLA_2007$PTL_cond,
Turbidity=NLA_2007\$Turbidity_cond)

Names of the response and stressor variables for which relative risk estimates are desired must be specified. The values “Chlorophyll_a” and “MacroInvert_OE” are assigned to resp_vars. The values “Total_Nitrogen”, “Total_Phosphorus”, and “Turbidity” are assigned to stress_vars.

Assign response and stressor variable names:

resp_vars <- c("Chlorophyll_a", "MacroInvert_OE")
stress_vars <- c("Total_Nitrogen", "Total_Phosphorus", "Turbidity")

The relrisk.analysis function is used to calculate relative risk estimates. In the call to function relrisk.analysis, arguments response.var and stressor.var provide the names of columns in the data.risk data frame that contain response variables and stressor variables, respectively. The rep function is used to repeat each of the response variable names in resp_vars once for each of the stressor variable names in stress_vars and the result is assigned to the response.var argument. Similarly, the rep function is used to repeat the set of stressor variable names in stress_vars once for each of the values in resp_var and the result is assigned to the stressor.var argument. The result is that relrisk.analysis will calculate a relative risk estimate for each combination of response and stressor variables.

Calculate relative risk estimates:

RelRisk_Estimates <- relrisk.analysis(sites, subpop, design, data.risk,
response.var= rep(resp_vars, each=length(stress_vars)),
stressor.var=rep(stress_vars, length(resp_vars)))

The relative risk estimates are displayed using the print function. The object produced by relrisk.analysis is a data frame containing twentyone columns. The first five columns identify the population (Type), subpopulation (Subpopulation), response variable (Response), stressor variable (Stressor), and number of response variable (or stressor variable) values used to calculate the relative risk estimate (NResp). The next six columns provide results for the relative risk estimate: the estimate (Estimate), numerator of the estimate (Estimate.num), denominator of the estimate (Estimate.denom), logarithm of the standard error of the estimate (StdError.log), lower confidence bound (LCB95Pct), and upper confidence bound (UCB95Pct). Argument conf for relrisk.analysis allows control of the confidence bound level. The default value for conf is 95, hence the column names for confidence bounds contain the value 95. Supplying a different value to the conf argument will be reflected in the confidence bound names. Confidence bounds are obtained using the logarithm of standard error and the Normal distribution multiplier corresponding to the confidence level. Results are then backtransformed to the original scale to obtain the confidence bound estimates. The next column in the data frame contains the sum of the survey design weights (WeightTotal). The next four columns provide cell counts for the two-by-two table of response variable categories and stressor variable categories and are named CellCounts.rc, where r indicates row number in the table and c indicates column number. Rows contain the response variable categories and column contain the stressor variable categories. By default, number 1 is the “Poor” category, and number 2 is the “Good” category. The final four columns in the data frame contain the cell proportion estimates for the two-by-two table, where columns are named CellProportions.rc using the same convention as the cell count columns. Note that the cell proportion estimates are weighted estimates obtained using the survey design weights.

Print the relative risk estimates:

print(RelRisk_Estimates)
#>           Type Subpopulation       Response         Stressor NResp  Estimate
#> 1   Western_US    Western_US  Chlorophyll_a   Total_Nitrogen   236 2.7274819
#> 2  Lake_Origin      Man-Made  Chlorophyll_a   Total_Nitrogen   152 2.4104712
#> 3  Lake_Origin       Natural  Chlorophyll_a   Total_Nitrogen    84 1.8529034
#> 4   Western_US    Western_US  Chlorophyll_a Total_Phosphorus   236 2.2817129
#> 5  Lake_Origin      Man-Made  Chlorophyll_a Total_Phosphorus   152 2.4506118
#> 6  Lake_Origin       Natural  Chlorophyll_a Total_Phosphorus    84 1.4260122
#> 7   Western_US    Western_US  Chlorophyll_a        Turbidity   236 1.8703252
#> 8  Lake_Origin      Man-Made  Chlorophyll_a        Turbidity   152 1.0599859
#> 9  Lake_Origin       Natural  Chlorophyll_a        Turbidity    84 4.7199940
#> 10  Western_US    Western_US MacroInvert_OE   Total_Nitrogen   234 2.7177105
#> 11 Lake_Origin      Man-Made MacroInvert_OE   Total_Nitrogen   151 1.4635342
#> 12 Lake_Origin       Natural MacroInvert_OE   Total_Nitrogen    83 3.0081542
#> 13  Western_US    Western_US MacroInvert_OE Total_Phosphorus   234 1.5114042
#> 14 Lake_Origin      Man-Made MacroInvert_OE Total_Phosphorus   151 0.8614559
#> 15 Lake_Origin       Natural MacroInvert_OE Total_Phosphorus    83 3.8828510
#> 16  Western_US    Western_US MacroInvert_OE        Turbidity   234 5.1694904
#> 17 Lake_Origin      Man-Made MacroInvert_OE        Turbidity   151 2.9228876
#> 18 Lake_Origin       Natural MacroInvert_OE        Turbidity    83 4.6573413
#>    Estimate.num Estimate.denom StdError.log  LCB95Pct  UCB95Pct WeightTotal
#> 1     0.5869872     0.21521213    0.4309409 1.1720452  6.347160    4890.777
#> 2     0.7072461     0.29340576    0.2913936 1.3616556  4.267137    2049.445
#> 3     0.3371698     0.18196839    0.7970782 0.3884889  8.837450    2841.333
#> 4     0.5381828     0.23586790    0.4265909 0.9888859  5.264727    4890.777
#> 5     0.7786874     0.31775225    0.2763906 1.4256417  4.212488    2049.445
#> 6     0.2688624     0.18854142    0.7579254 0.3228315  6.298984    2841.333
#> 7     0.5582292     0.29846639    0.3918041 0.8677865  4.031079    4890.777
#> 8     0.5278154     0.49794564    0.3971241 0.4867069  2.308515    2049.445
#> 9     0.9277937     0.19656671    0.4505363 1.9518403 11.414020    2841.333
#> 10    0.3015402     0.11095377    0.4505836 1.1237397  6.572652    4882.983
#> 11    0.3753646     0.25647817    0.4703811 0.5821217  3.679527    2045.360
#> 12    0.1476541     0.04908460    0.7608096 0.6771702 13.362950    2837.623
#> 13    0.2225565     0.14725151    0.5029961 0.5639357  4.050715    4882.983
#> 14    0.2902637     0.33694546    0.6017883 0.2648439  2.802052    2045.360
#> 15    0.1467371     0.03779108    0.7215134 0.9440553 15.969968    2837.623
#> 16    0.5656631     0.10942338    0.3707848 2.4993963 10.692034    4882.983
#> 17    0.5866627     0.20071339    0.4241173 1.2729250  6.711528    2045.360
#> 18    0.2924337     0.06278985    0.8528069 0.8754424 24.776992    2837.623
#>    CellCounts.11 CellCounts.12 CellCounts.21 CellCounts.22 CellProportions.11
#> 1             41            33            22           140        0.188105511
#> 2             29            20            17            86        0.365103884
#> 3             12            13             5            54        0.060437097
#> 4             39            35             8           154        0.175320038
#> 5             26            23             5            98        0.319782522
#> 6             13            12             3            56        0.071119667
#> 7             22            52            16           146        0.077115692
#> 8             19            30            15            88        0.160770986
#> 9              3            22             1            58        0.016775371
#> 10            28            26            33           147        0.096304303
#> 11            23            17            22            89        0.193412993
#> 12             5             9            11            58        0.026308304
#> 13            17            37            30           150        0.072616404
#> 14            11            29            20            91        0.119440241
#> 15             6             8            10            59        0.038865755
#> 16            20            34            16           164        0.077364465
#> 17            19            21            14            97        0.177880988
#> 18             1            13             2            67        0.004912098
#>    CellProportions.12 CellProportions.21 CellProportions.22
#> 1          0.14624540        0.132353799          0.5332953
#> 2          0.14193999        0.151129239          0.3418269
#> 3          0.14935088        0.118811130          0.6714009
#> 4          0.15903087        0.150442929          0.5152062
#> 5          0.18726135        0.090886158          0.4020700
#> 6          0.13866831        0.193401056          0.5968110
#> 7          0.25723522        0.061027730          0.6046214
#> 8          0.34627289        0.143826029          0.3491301
#> 9          0.19301261        0.001305557          0.7889065
#> 10         0.07551795        0.223070334          0.6051074
#> 11         0.12432343        0.321854037          0.3604095
#> 12         0.04033894        0.151866985          0.7814858
#> 13         0.09920585        0.253666542          0.5745112
#> 14         0.19829618        0.292048553          0.3902150
#> 15         0.02778149        0.226000766          0.7073520
#> 16         0.09445779        0.059403274          0.7687745
#> 17         0.13985543        0.125327293          0.5569363
#> 18         0.06173515        0.011885203          0.9214676

The write.csv function is used to store the relative risk estimates as a comma-separated value (csv) file. Files in csv format can be read by programs such as Microsoft Excel.

Write results as a comma-separated value (csv) file:

write.csv(RelRisk_Estimates, file="RelRisk_Estimates.csv", row.names=FALSE)

## Attributable risk analysis

The attrisk.analysis function will be used to calculate attributable risk estimates. The four data frames used to calculate relative risk estimates can be used for attributable risk estimation. Arguments for the attrisk.analysis function are identical to arguments for the relrisk.analysis function

Calculate attributable risk estimates:

AttRisk_Estimates <- attrisk.analysis(sites, subpop, design, data.risk,
response.var= rep(resp_vars, each=length(stress_vars)),
stressor.var=rep(stress_vars, length(resp_vars)))

The attributable risk estimates are displayed using the print function. The object produced by attrisk.analysis is a data frame containing nineteen columns. The data data frame is identical to the one produced by the relrisk.analysis function except that it doesn’t include the columns named Estimate.num and Estimate.denom. Since attributable risk is not calculated using a ratio estimator, values for numerator and denominator estimates are not relevant.

Print the attributable risk estimates:

print(AttRisk_Estimates)
#>           Type Subpopulation       Response         Stressor NResp    Estimate
#> 1   Western_US    Western_US  Chlorophyll_a   Total_Nitrogen   236  0.35632857
#> 2  Lake_Origin      Man-Made  Chlorophyll_a   Total_Nitrogen   152  0.42134049
#> 3  Lake_Origin       Natural  Chlorophyll_a   Total_Nitrogen    84  0.13260811
#> 4   Western_US    Western_US  Chlorophyll_a Total_Phosphorus   236  0.29454985
#> 5  Lake_Origin      Man-Made  Chlorophyll_a Total_Phosphorus   152  0.37332396
#> 6  Lake_Origin       Natural  Chlorophyll_a Total_Phosphorus    84  0.10127632
#> 7   Western_US    Western_US  Chlorophyll_a        Turbidity   236  0.10732593
#> 8  Lake_Origin      Man-Made  Chlorophyll_a        Turbidity   152  0.01794367
#> 9  Lake_Origin       Natural  Chlorophyll_a        Turbidity    84  0.06302202
#> 10  Western_US    Western_US MacroInvert_OE   Total_Nitrogen   234  0.35425260
#> 11 Lake_Origin      Man-Made MacroInvert_OE   Total_Nitrogen   151  0.19279580
#> 12 Lake_Origin       Natural MacroInvert_OE   Total_Nitrogen    83  0.26351638
#> 13  Western_US    Western_US MacroInvert_OE Total_Phosphorus   234  0.14300096
#> 14 Lake_Origin      Man-Made MacroInvert_OE Total_Phosphorus   151 -0.06045589
#> 15 Lake_Origin       Natural MacroInvert_OE Total_Phosphorus    83  0.43296859
#> 16  Western_US    Western_US MacroInvert_OE        Turbidity   234  0.36315945
#> 17 Lake_Origin      Man-Made MacroInvert_OE        Turbidity   151  0.36830221
#> 18 Lake_Origin       Natural MacroInvert_OE        Turbidity    83  0.05787783
#>    StdError.log     LCB95Pct  UCB95Pct WeightTotal CellCounts.11 CellCounts.12
#> 1    0.25520134 -0.061431594 0.6096659    4890.777            41            33
#> 2    0.23787429  0.077636614 0.6369686    2049.445            29            20
#> 3    0.20270096 -0.290490261 0.4169900    2841.333            12            13
#> 4    0.23113933 -0.109717615 0.5515437    4890.777            39            35
#> 5    0.20990967  0.054375912 0.5846945    2049.445            26            23
#> 6    0.24370434 -0.448997696 0.4425773    2841.333            13            12
#> 7    0.09551704 -0.076458205 0.2597325    4890.777            22            52
#> 8    0.12710745 -0.259883685 0.2345050    2049.445            19            30
#> 9    0.05046779 -0.034397925 0.1512669    2841.333             3            22
#> 10   0.22734361 -0.008272434 0.5864315    4882.983            28            26
#> 11   0.27828602 -0.392710659 0.5321508    2045.360            23            17
#> 12   0.22902173 -0.153736794 0.5298684    2837.623             5             9
#> 13   0.20135161 -0.271660373 0.4224501    4882.983            17            37
#> 14   0.23045317 -0.665922146 0.3249584    2045.360            11            29
#> 15   0.34363056 -0.112003067 0.7108599    2837.623             6             8
#> 16   0.17937230  0.094866404 0.5519270    4882.983            20            34
#> 17   0.23493409 -0.001118809 0.6014039    2045.360            19            21
#> 18   0.06577750 -0.071758993 0.1718342    2837.623             1            13
#>    CellCounts.21 CellCounts.22 CellProportions.11 CellProportions.12
#> 1             22           140        0.188105511         0.14624540
#> 2             17            86        0.365103884         0.14193999
#> 3              5            54        0.060437097         0.14935088
#> 4              8           154        0.175320038         0.15903087
#> 5              5            98        0.319782522         0.18726135
#> 6              3            56        0.071119667         0.13866831
#> 7             16           146        0.077115692         0.25723522
#> 8             15            88        0.160770986         0.34627289
#> 9              1            58        0.016775371         0.19301261
#> 10            33           147        0.096304303         0.07551795
#> 11            22            89        0.193412993         0.12432343
#> 12            11            58        0.026308304         0.04033894
#> 13            30           150        0.072616404         0.09920585
#> 14            20            91        0.119440241         0.19829618
#> 15            10            59        0.038865755         0.02778149
#> 16            16           164        0.077364465         0.09445779
#> 17            14            97        0.177880988         0.13985543
#> 18             2            67        0.004912098         0.06173515
#>    CellProportions.21 CellProportions.22
#> 1         0.132353799          0.5332953
#> 2         0.151129239          0.3418269
#> 3         0.118811130          0.6714009
#> 4         0.150442929          0.5152062
#> 5         0.090886158          0.4020700
#> 6         0.193401056          0.5968110
#> 7         0.061027730          0.6046214
#> 8         0.143826029          0.3491301
#> 9         0.001305557          0.7889065
#> 10        0.223070334          0.6051074
#> 11        0.321854037          0.3604095
#> 12        0.151866985          0.7814858
#> 13        0.253666542          0.5745112
#> 14        0.292048553          0.3902150
#> 15        0.226000766          0.7073520
#> 16        0.059403274          0.7687745
#> 17        0.125327293          0.5569363
#> 18        0.011885203          0.9214676

The write.csv function is used to store the attributable risk estimates as a csv file. Write results as a csv file:

write.csv(AttRisk_Estimates, file="AttRisk_Estimates.csv", row.names=FALSE)`

## References

U.S. Environmental Protection Agency. 2009. “National Lakes Assessment: A Collaborative Survey of the Nation’s Lakes.” U.S. Environmental Protection Agency, Office of Water; Office of Research; Development.

Van Sickle, John, and Steven G. Paulsen. 2008. “Assessing the Attributable Risks, Relative Risks, and Regional Extents of Aquatic Stressors.” Journal of the North American Benthological Society 27: 920–31.

Van Sickle, John, John L. Stoddard, Steven G. Paulsen, and Anthony R. Olsen. 2006. “Using Relative Risk to Compare the Effects of Aquatic Stressors at a Regional Scale.” Environmental Management 38: 1020–30.