3 Resampling Methods: The Jackknife
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1 3 Resamplig Methods: The Jackkife 3.1 Itroductio I this sectio, much of the cotet is a summary of material from Efro ad Tibshirai (1993) ad Maly (2007). Here are several useful referece texts o resamplig methods. 1. Daviso ad Hikley (1997) Bootstrappig ad its Applicatios, Cambridge Uiversity Press. 2. Efro, B. (1982) The Jackkife, the Bootstrap ad Other Resamplig Plas, SIAM. 3. Efro, B. ad Tibshirai, R.J. (1993) A Itroductio to the Bootstrap, Chapma & Hall. 4. Maly, Brya F.J. (2007) Radomizatio, Bootstrap, ad Mote Carlo Methods i Biology, Chapma & Hall/CRC Press. 5. Shao, J. ad Tu, T. (1995) The Jackkife ad Bootstrap, Spriger-Verlag. I statistical iferece, the goal is ofte the estimatio of parameters or some characteristics of a probability distributio f based o a radom sample x = (x 1, x 2,..., x ) of size take from f. The empirical distributio fuctio f is the discrete distributio that assigs probability 1/ to each value x i (i = 1, 2,..., ) i the sample. Note: this is ot a cumulative distributio fuctio. Note: the x i values do ot have to be uique. That is, x i could equal x j for some i j. If there are repeated data values, we ca express f as a vector of observed frequecies f k for each observed value k. That is, f k = Number of cases whe x i = k Therefore, f assigs to ay set A the empirical probability P({A}) = Number of cases whe x i A Table 3.1 cotais data from a populatio of 82 pairs of average LSAT scores ad average GPA of 82 America law schools who participated i a study of admissio practices. The LSAT is a exam required for admittace ito a US law school. The GPA is the studet grade poit average. A radom sample of 15 pairs of LSAT scores ad GPA was selected from the 82 schools. The sampled schools are i paretheses (). Figure 3.1 cotais scatterplots of the the average LSAT scores versus the average GPA scores for (i) the populatio ad (ii) the sample. The empirical distributio f assigs probability 1/15 to each of the 15 data poits. Cosider the set A = {(y, z) : y < 600, z < 3.00} where y is the LSAT score ad z is the GPA. The the empirical probability P({A}) = 5/15 because 5 of the 15 schools are elemets of the set A. I statistical iferece, we wat to make ifereces about the populatio show i the left-side plot based o the sample show i the right-side plot (Figure 3.1). Specifically, we may wat to make ifereces about the populatio correlatio coefficiet based o the sample correlatio coefficiet. 38
2 Table 3.1: Law School data i Efro ad Tibshirai (1993) school LSAT GPA school LSAT GPA school LSAT GPA school LSAT GPA (45) (4) (47) (6) (50) (70) (31) (52) (53) (13) (35) (15) (36) (79) (82) Sampled schools have bold-faced school umbers. Figure
3 Cosider the followig radom sample of 50 values: The empirical distributio f would assig probability 1/50 to each of the 50 data values. But, for a radom sample, we ca summarize f by ( ( f 1, f 2, f 3, f 12 4 ) = 50, 20 50, 11 50, 7 ) = (.24,.40,.22,.14) 50 where f i = the proportio of x i values = 0, 1, 2, ad 3, respectively for i = 1, 2, 3, 4. Mathematical statistics results: It ca be proved that the vector of observed frequecies f = ( f 1, f 2, f 3,...) is a sufficiet statistic for the true distributio f. That is, all of the iformatio about f that is cotaied i the sample x = (x 1, x 2,..., x ) is also cotaied i f = ( f 1, f 2, f 3,...). Remember that the sufficiecy assumes that the data set has bee geerated by radom samplig. For other data sets (such as time series data) this will ot be true. We ca also write a parameter θ directly as a fuctio of f. That is, θ = t(f ). For example, µ X = the expectatio of a radom variable X ca be writte as the parameter θ = µ X = E f (X) = t(f). A parameter is a fuctio of a probability distributio F. A statistic is a fuctio of a sample x. For the law school data, the parameters µ y = , µ z = 3.135, ad ρ =.761. Based o the sample of size = 15, the correspodig statistics are y = 600.3, z = 3.095, ad Pearso s sample correlatio coefficiet r = The Jackkife Method Let θ be some parameter associated with a distributio f. Suppose We have a radom sample x = (x 1, x 2,..., x ) ad a estimator θ of θ, ad the goal is to estimate the bias ad stadard error of θ. Let x (i) be the sample but with the i th observatio removed: x (i) is the i th jackkife sample. x (i) = (x 1, x 2,..., x i 1, x i+1,..., x ) (1) The jackkife method of estimatio is based o the jackkife samples x (i) (i = 1, 2,..., ). Suppose θ = g(x) for some fuctio g of the data. The i th jackkife replicatio of θ is θ (i) = g(x (i) ) (2) 40
4 For example, cosider the followig example from Maly (2007) summarized i Table 3.2. The goal is to estimate the populatio stadard deviatio. That is, θ = σ. The data set has = 20 values give i the row labeled Data. Maly (2007) calculated the square root of the biased MLE estimator of the variace havig istead of 1 i the deomiator: θ = σ = 1 20 (x i x) 2 = 1.03 (ad ot the sample stadard deviatio s which has ( 1) = 19 df). Rows labeled 1 to 20 have oe value removed, leavig 19 remaiig values. The 19 values i colums 1 to 20 form the 20 jackkife samples. The i th jackkife replicatio is σ i = 1 (x j x) 19 2 which is calculated from the 19 values (without x i ) i the i th jackkife sample. The values are give i the first SD colum. j i Jackkife Bias Estimatio Let θ ( ) = θ (i) /. The jackkife estimate of bias is defied as bias( θ) = ( 1)( θ ( ) θ) (3) The bias-corrected jackkife estimate of θ is θ jack = θ bias( θ) = θ ( 1) θ ( ) Mathematically, it ca be show that bias( θ) is a ubiased estimator of the true bias for may statistics. For other statistics, although θ jack is a biased estimator of the true bias, the bias of θ jack is reduced i compariso to the uadjusted estimate θ. 41
5 42
6 3.2.2 Jackkife Stadard Error Estimatio Cosider what we usually do whe estimatig a mea of a distributio. We use the sample mea X = X i as our estimator. To see the relatioship with jackkife estimatio, we ca write the mea with the i th observatio removed as X (i) as: ( ) j=1 X j X i X (i) = 1 Therefore a idividual X i ca be writte as ( ) X i = X j ( 1)X (i) = X ( 1)X (i). (4) j=1 That is, it is possible to determie a observatio from the sample mea ad from the sample mea with the i th observatio removed. We ow apply this approach to statistics other tha the sample mea. For example, whe estimatig a parameter θ with estimator θ, if we replace the sample meas i (4) with the correspodig estimators of θ, we have PV(x (i) ) is called the i th pseudo-value. PV(x (i) ) = θ ( 1) θ (i) Note that we expect that PV(x (i) ) θ ( 1)θ = θ. Thus, each pseudo-value ca be viewed as a estimate of θ, ad the average of the pseudo-values is PV = 1 ) ( θ ( 1) θ (i) ( ) 1 = θ + ( 1) θ (i) = θ ( 1) θ ( ) which is the bias-corrected jackkife estimate. That is, PV = θ jack. This suggests the atural estimator for the variace of θ jack is s 2 jack / where s2 jack is the sample variace of the pseudo-values: var( θ jack ) = s2 jack = 1 1 ( PV(x(i) ) PV ) 2. (5) 1 ad the jackkife stadard error of θ jack is s.e.( θ jack ) = var( θ jack ). I their textbook, Efro ad Tibshirai (1993) state that the jackkife stadard error is [ ] 1/2 1 [ s.e.( θ jack ) = ( θ (i) θ ( 1) ( ) ) 2 2 1/2 = (s 2 for the θ (i) values)] (6) It ca be show that the stadard errors i (5) ad (6) are computatioally equivalet. 43
7 Look agai at the example i Table 3.2. The goal was to estimate the populatio stadard deviatio σ. So, i this example, θ = σ. The ucorrected estimate θ = σ = (i the SD colum of Data row). The 20 sample the jackkife replicatios ( σ (i) values) appear i the SD colum. The 20 pseudo-values (the PV(x (i) ) values) appear i the PV i colum. The average of the pseudo-values PV = σ jack = Therefore, bias( σ) (19)( ) = (slight roud-off error). Or, you ca estimate the bias usig bias( σ) = θ θ jack (more accurate). Suppose we wat to apply the jackkife method to 3 other situatios: 1. Suppose we wated to estimate θ = µ? What would happe if we use the sample mea ( θ = X)? 2. Suppose we wated to estimate θ = σ? What would happe if we use the sample stadard deviatio ( θ = s)? 3. Suppose we wated to estimate θ = σ 2? What would happe if we use the sample stadard variace ( θ = s 2 )? The results are summarized i the followig table. Parameter Statistic θ( ) θ bias( θ) θjack s.e.( θ jack ) µ x σ s σ σ MLE σ 2 s There are several importat thigs to otice i these results. 1. We kow x ad s 2 are ubiased estimators of µ ad σ 2. Therefore, applyig the jackkife method to a ubiased estimator has o effect. That is, θ jack = θ, ad the estimated bias will the be Both bias-corrected estimates of σ are still biased, but the bias is reduced compared to the origial estimates (statistics). After bias correctio, the two θ jack values for estimatig σ are very close. 3. The stadard errors ca the be used to calculate t-based cofidece itervals: θ jack ± t s.e.( θ jack ) where t is the 1 α/2 critical value from a t-distributio with 1 degrees of freedom. 44
8 Theta(hat) values for 4 differet estimates mea s s^2 s_mle Jackkife Samples ad Jackkife Replicatios (Table 3.2 data) Obs y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y Obs y16 y17 y18 y19 y20 j(mea) j(s) j(s^2) j(s_mle) R code for jackkifig the Maly data library(bootstrap) # The Maly data x <- c(3.56, 0.69, 0.10, 1.84, 3.93, 1.25, 0.18, 1.13, 0.27, 0.50, 0.67, 0.01, 0.61, 0.82, 1.70, 0.39, 0.11, 1.20, 1.21, 0.72) # Sample mea mea(x) # Jackkife the mea jackmea <- jackkife(x,mea) jackmea 45
9 # Bias-corrected jackkife estimate meajack = mea(x) - jackmea$jack.bias meajack # Sample stadard deviatio sd(x) # Jackkife the stadard deviatio jacksd <- jackkife(x,sd) jacksd # Bias-corrected jackkife estimate sdjack = sd(x) - jacksd$jack.bias sdjack # Sample stadard deviatio with i deomiator (MLE) sdmle <- fuctio(x)(sqrt((legth(x)-1)/legth(x))*sd(x)) sdmle(x) # Jackkife the MLE stadard deviatio (deomiator with ) jacksdmle <- jackkife(x,sdmle) jacksdmle # Bias-corrected jackkife estimate sdmlejack = sdmle(x) - jacksdmle$jack.bias sdmlejack # Sample variace var(x) # Jackkife the variace jackvar <- jackkife(x,var) jackvar # Bias-corrected jackkife estimate varjack = var(x) - jackvar$jack.bias varjack R output for Jackkifig the Maly Data # Jackkife the mea # Sample mea jack.se jack.bias 0 jack.values [1] [8] [15] # Bias-corrected jackkife estimate # Jackkife the stadard deviatio # Sample stadard deviatio jack.se jack.bias jack.values [1] [8] [15] # Bias-corrected jackkife estimate
10 # Jackkife the MLE stadard deviatio (deomiator with ) # Sample stadard deviatio with i deomiator (MLE) jack.se jack.bias jack.values [1] [8] [15] # Bias-corrected jackkife estimate # Jackkife the variace # Sample variace jack.se jack.bias 0 jack.values [1] [8] [15] # Bias-corrected jackkife estimate Jackkifig the Correlatio Coefficiet Let us cosider estimatio of a correlatio coefficiet ρ. We will calculate the jackkife estimates from the 15 (y, z) poits from the followig law school data (see Figure 3.1 ad Table 3.1). Recall: LSAT score = y, GPA = z. School LSAT GPA School LSAT GPA School LSAT GPA
11 R code for Jackkifig the law school data # Jackkifig correlatio coefficiets for the law school data library(bootstrap) =15; # the umber of desig poits yzdata <- as.matrix(c(576,3.39,580,3.07,653,3.12, 635,3.30,555,3.00,575,2.74,558,2.81,661,3.43,545,2.76, 578,3.03,651,3.36,572,2.88,666,3.44,605,3.13,594,2.96)) dim(yzdata) <- c(2,) idata <- t(yzdata) # # Jackkifig Pearso s correlatio coefficiet # corr <- fuctio(yz,idata) { cor(idata[yz,1],idata[yz,2]) } # Pearso s correlatio coefficiet r sampcorr <- cor(idata[1:,1],idata[1:,2]) sampcorr # Jackkife Pearso s correlatio coefficiet jacklaw <- jackkife(1:,corr,idata) jacklaw # Bias-corrected jackkife estimate corrjack = sampcorr - jacklaw$jack.bias corrjack R output for Jackkifig the law school data > # Pearso s correlatio coefficiet r [1] > > # Jackkife Pearso s correlatio coefficiet $jack.se [1] $jack.bias [1] $jack.values [1] [8] [15] > # Bias-corrected jackkife estimate [1]
12 3.4 Jackkifig Bioequivalece The followig example was take from Efro ad Tibshirai (1993). estimatio. It ivolves ratio I this study, 8 subjects wore medical patches that release a hormoe ito the blood stream. Each subject had blood levels of the hormoe measured after wearig 3 differet patches: (1) a patch cotaiig o hormoe (placebo), (2) a patch cotaiig hormoes maufactured at the old site (oldpatch), ad (3) a patch cotaiig hormoes maufactured at the ew site (ewpatch). Two differeces were calculated; z = oldpatch placebo y = ewpatch oldpatch The goal of the study was to establish bioequivalece. The patches maufactured at the ew site eeded to be approved by the US Food ad Drug Admiistratio (FDA). This required showig that they were bioequivalet to patches maufactured at the old site. The goal is to establish that patches maufactured at the ew ad old site match (bioequivalet). The followig table summarizes the experimetal data: subject placebo oldpatch ewpatch z y z = y = Statistically, the FDA criterio for bioequivalece is E(ewpatch) E(oldpatch) E(oldpatch) E(placebo).20 That is, the FDA wats the patches from the ew site match the patches from the old site withi 20% of the amout of hormoe added to the blood stream by patches from the old site compared to placebo levels. Let the parameter θ be the ratio The estimate of θ is θ = E(ewpatch) E(oldpatch) E(oldpatch) E(placebo) θ = y/z = 452.3/6342 Now we just use jackkife estimatio to calculate the eight θ (i) replicatios, take their mea to get θ ( ), ad calculate the bias ad stadard errors usig θ, θ (i), ad θ ( ). 49
13 R code for jackkifig the bioequivlece data # Jackkifig bioequivalece library(bootstrap) = 8; # the umber of desig poits xx= c( 9243, 9671, 11792, 13357, 9055, 6290, 12412, 18806) zz= c(17649, 12013, 19979, 21816, 13850, 9806, 17208, 29044) yy= c(16449, 14614, 17274, 23798, 12560, 10157, 16570, 26325) z = zz-xx y = yy-zz idata <- as.matrix(cbid(y,z)) # Sample bioequivalece sampbioeq = mea(y)/mea(z) sampbioeq # Defie the bioequivalece fuctio bioeq <- fuctio(yz,idata) { mea(idata[yz,1])/mea(idata[yz,2]) } # Jackkife the bioequivalece jackbioeq <- jackkife(1:,bioeq,idata) jackbioeq # Bias-corrected jackkife estimate bioeqjack = sampbioeq - jackbioeq$jack.bias bioeqjack R output for jackkifig the bioequivlece data > # Sample bioequivalece [1] > > # Jackkife the bioequivalece $jack.se [1] $jack.bias [1] $jack.values [1] [7] > # Bias-corrected jackkife estimate [1]
14 3.5 Limitatios of the Jackkife The jackkife method of estimatio ca fail if the statistic θ is ot smooth. Smoothess implies that relatively small chages to data values will cause oly a small chage i the statistic. The sample media is a example of a statistic that is ot smooth. For example, look at the Maly (2007) data. The ordered values are Note that there are oly 2 differet jackkife estimate values. 1/2 of the jackkife estimates = 0.72 (based deletig each of the first 10 values). 1/2 of the jackkife estimates = 0.69 (based deletig each of the last 10 values) has a value of Therefore, the jackkife is ot a good estimatio method for estimatig percetiles (such as the media), or whe usig ay other o-smooth estimator. This will ot be the case usig the bootstrap method of estimatio. A alterate the jackkife method of deletig oe observatio at a time is to delete d observatios at a time (d 2). This is kow as the delete-d jackkife. Because we are deletig d, there will be ( ) d) jackkife samples of size d ad therefore jackkife replicatios, ad the estimate of the stadard error is ( d ( d d ( ) ) 1/2 ( θ (i) θ ( ) ) 2 d where the summatio is take over all ( d) subsets of poits. Oe reaso to use the delete-d jackkife is to hadle cases of o-smooth estimators. I practice, if is large ad d is chose such that < d <, the the problems of o-smoothess are removed. 51
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