National Survey of Fishing, Hunting, and Wildlife ... - All About Birds

compute in advance the standard error for every estimate one might obtain from those data sets. Moreover, variance estimatesare based on sample data and have variances of their own. Therefore, some methods of stabilizing these estimates of variance,for example, by generalizing or averaging over time, may be used to improve their reliability. Experience has shown thatcertain groups of estimates have similar relationships between their variances and expected values. Modeling or generalizingmay provide more stable variance estimates by taking advantage of these similarities. The generalized variance function isa simple model that expresses the variance as a function of the expected value of the survey estimate. The parameters of thegeneralized variance function are estimated using direct replicate variances. These generalized variance parameters provide arelatively easy method to obtain approximate standard errors for numerous characteristics. Table D-5 provides the generalizedvariance parameters for FHWAR data. Methods for using the parameters to calculate standard errors of various estimates aregiven in the next sections.StandardStandardStandardErrors ofErrorsErrorsEstimatedof Estimatedof EstimatedNumbers.Numbers.Numbers.The approximateThe approximateThe approximatestandardstandardstandarderror, s x,error,error, sx, of an estimated number shown in this report canbe obtained using the following formulas. Formula (1) is used of to an calculate sx estimated, of an estimatednumbernumbershown inshownthis reportin this report cancan be obtainedbe obtainedthe standard errors of levels of sportspersons,anglers, usingusingand thethefollowingfollowingwildlife-watchers. formulas.formulas.FormulaFormula(1) is(1)usedis usedto calculateto calculatethe standardthe standarderrorserrorsof levelsof levelsof sportspersons,of sportspersons,anglers,anglers,and wildlifeand wildlife-watchers.watchers.s 22s xax ax bx bxx Here, x is the size of the estimate and a and b are the parameters in the tables associated with the particular characteristic.Here, xHere,is thexsizeis theofsizetheofestimatethe estimateand a andandbaareandtheb areparametersthe parametersin theintablesthe tablesassociatedassociatedwith thewithparticularthe particularcharacteristic.characteristic.Formula (2) is Formula used for (2) standard is used errors for standard of aggregates, errors of i.e., aggregates, trips, days, i.e., and trips, expenditures.Formula (2) is used for standard errors of aggregates, i.e., trips, days, anddays,expenditures.and expenditures.s 2xax 2s 2ax bx bx cx xyHere, x Here, is again xHere,is the againx size istheagain of the sizethe estimate; ofsizetheofestimate; y the is the estimate; base y is the of y the baseis the estimate; ofbasetheofestimate; and the a, estimate; b, and and a, c andb, are anda, the b,c parameters areandthec areparametersthe in parameters the tables in the associatedwith ated the with particular atedintablesthe tablesassoci-associ-thewithparticularthe characteristic. particularcharacteristic.characteristic.Illustration Illustration of Illustration the Computation of theofComputationthe Computation of the Standard of theofStandardthe Error Standard of Error Estimated Errorof an Estimatedof an Number EstimatedNumberNumberSuppose Table there 1Table were in this1 an inreport estimated this reportshows 37,397,000 showsthat 33,916,000that persons 33,916,000persons age 16 persons16 years years16 old oldyears and and older olderand who eitherolder either eitherfished fished fishedor or hunted hunted or huntedin in the the inUnited United the UnitedStatesStatesininStates in 2006. 2011. 2006.Using Using Usingformula formula(1) (1) with with (1)the the withparameters the parametersa a = = -0.000027 –0.000070 a = -0.000027and and b b =and = 6,125 16,823 b = 6,125from from tablefrom table D-7,table D-5, theD-7, the approximatethe approximate approximatestandard standarderror the of estimate the estimated numberstandarderror oferror ofnumber of 33,916,000of of 33,916,000 37,397,000 sportspersonssportspersons 16 years16 age oldyears 16 and years oldolderand old isolder and older is is2s x 0. 000027 33, 9162s, 000 6, 125 33, 916, 000 420,330x 0. 000027 33, 916, 000 6, 125 33, 916, 000 420,330The 95-percentThe 95-percentconfidenceconfidenceintervalintervalfor theforestimatethe estimatenumbernumberof sportspersonsof sportspersons16 years16oldyearsandoldolderandisolderfromis33,092,000from 33,092,000totoThe 95-percent 34,740,000,34,740,000, confidence ie., 33,916,000ie., interval 33,916,000 for ± 1.96 the ±x estimated 1.96420,330.x 420,330. number Therefore,Therefore, of sportspersons a conclusiona conclusion 16 that years thethataverage old the and averageestimate older is estimate from derived 35,968,000 derivedfrom allfrom to possibleall possible38,826,000, samples i.e., sampleslies 37,397,000 withinlies withina ± range 1.96 a rangecomputed x 728,857. computedin Therefore, thisinwaythiswould a way conclusion wouldbe correctbe that correctfor the roughly average for roughly95 estimate percent95 percent derived of all possibleof from all possible all samples. possible samples.samples lies within a range computed in this way would be correct for roughly 95 percent of all possible samples.Table 1Tableshows1 showsthat 12,510,000that 12,510,000huntershunters16 years16oldyearsandoldolderandengagedolder engagedin 219,925,000in 219,925,000days ofdaysparticipationof participationin 2006.in 2006.UsingUsingSuppose formula there formula were (2) with an (2) estimated thewithparametersthe 13,674,000 parametersa = -0.000235, hunters a = -0.000235, age b = 16 -85,241, years b = -85,241, old and and c =and older 22,698c who = 22,698from engaged tablefrom in D-9,table 281,884,000 theD-9,approximatethe approximate days of standard participationin 219,925,000 2011. 219,925,000 Using estimated formula estimated (2) days with onstandarderror onerror ondaysan the estimatedon parameters an estimatedbase a = of –0.000284, base12,510,000of 12,510,000 b hunters = –127,863, huntersis and is c = 46,699 from table D-5, theapproximate standard error on 281,884,000 estimated days on an estimated base of 13,674,000 hunters is2222, 698 219, 9252, 000s x 0. 000235 219, 9252, 000 85, 241 219, 925,000 22, 698 219, 925,000s 7, 592,000x 0. 000235 219, 925, 000 85, 241 219, 925,000 12, 510,000 7, 592,00012, 510,000The 95-percentThe 95-percentconfidenceconfidenceintervalintervalon theonestimatethe estimateof 219,925,000of 219,925,000days isdaysfromis205,044,000from 205,044,000to 234,806,000,to 234,806,000,ie., 219,925,000ie., 219,925,000± 1.96±x1.967,592,000.x 7,592,000.Again,Again,a conclusiona conclusionthat thethataveragethe averageestimateestimatederivedderivedfrom allfrompossibleall possiblesamplessampleslies withinlies withina rangea rangeThe 95-percent computedcomputed confidence in thisinwaythis interval wouldway on wouldbe the correct estimate be correctfor roughly of for 281,884,000 roughly95 percent95 days percentof is all from possibleof all 253,295,000 possiblesamples.samples. to 310,473,000, i.e.,281,884,000 ± 1.96 x 14,586,000. Again, a conclusion that the average estimate derived from all possible samples lies withina rangeStandardcomputed StandardErrorsin this ErrorsofwayEstimatedwould of Estimated bePercentages.correct Percentages. for roughlyThe reliabilityThe 95 reliability percentof anofestimatedof all an possible estimatedpercentage,samples. percentage,computedcomputedusing sampleusing sampledata fordatabothfor bothnumeratornumeratorand denominator,and denominator,dependsdependson theonsizetheofsizetheofpercentagethe percentageand itsandbase.its base.EstimatedEstimatedpercentagespercentagesare relativelyare relativelymoremoreStandard reliable Errors reliablethan of Estimated thethancorrespondingthe Percentages. correspondingestimates The estimates reliability of theofnumeratorsthe of an numerators estimated of theofpercentages, percentage, the percentages, computed particularlyparticularly using if the sample percentagesif the data percentages for are both 50arepercent50 percentnumerator more. and or denominator, more.WhenWhenthe numerator depends the numerator on and the theand size denominatorthe of the denominator percentage of theofpercentage and the its percentage base. are Estimated inaredifferentin different percentages categories,categories, are use relatively theuseparameterthe more parameterin thein thereliable tables than the tablesindicated corresponding indicatedby thebynumerator. estimates the numerator. of the numerators of the percentages, particularly if the percentages are 50 percentor more. When the numerator and the denominator of the percentage are in different categories, use the parameter in thetables indicated The by approximate the numerator. standard error, s x,p,can be obtained by use of the formulaThe approximate standard error, s x,p,can be obtained by use of the formulasx,p,psx,p,pbp bp( 100 p) ( 100 p)x xHere, xHere,is thextotalis thenumbertotal numberof sportspersons,of sportspersons,hunters,hunters,etc., whichetc., whichis the baseis theofbasetheofpercentage;the percentage;p is theppercentageis the percentage(0 ≤ p(0≤≤ p ≤124 2011 National 100); Survey and b of is Fishing, the parameter Hunting, and in Wildlife-Associated the tables associated Recreation with the characteristic U.S. Fish in and the Wildlife numerator Service of and the U.S. percentage. Census Bureau100); and b is the parameter in the tables associated with the characteristic in the numerator of the percentage.IllustrationIllustrationof theofComputationthe Computationof theofStandardthe StandardErrorErrorof an Estimatedof an EstimatedPercentagePercentagecxy2(1)(2)(2)(3)(1)(2)(3)