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).
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:
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. The
nrowfunction is used to determine the number of rows in the
NLA_2007data frame, and the resulting value is assigned to an object named
nr. Finally, the initial six lines and the final six lines in the
NLA_2007data frame are printed using the
headand
tail` functions, respectively.
Load the survey design and analytical variables data set:
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).
Location of lakes that were sampled in the western United States by the U.S. Environmental Protection Agency during the National Lakes Assessment (NLA) conducted in 2007.
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:
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:
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:
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.