*The author of this computation has been verified*

R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Tue, 15 Dec 2009 06:41:47 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/15/t12608845876173q9rt2jixan7.htm/, Retrieved Tue, 15 Dec 2009 14:43:20 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Dec/15/t12608845876173q9rt2jixan7.htm/}, year = {2009}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

2350.44 0 2440.25 0 2408.64 0 2472.81 0 2407.6 0 2454.62 0 2448.05 0 2497.84 0 2645.64 0 2756.76 0 2849.27 0 2921.44 0 2981.85 0 3080.58 0 3106.22 0 3119.31 0 3061.26 0 3097.31 0 3161.69 0 3257.16 0 3277.01 0 3295.32 0 3363.99 0 3494.17 0 3667.03 0 3813.06 0 3917.96 0 3895.51 0 3801.06 0 3570.12 0 3701.61 0 3862.27 0 3970.1 0 4138.52 0 4199.75 0 4290.89 0 4443.91 0 4502.64 0 4356.98 0 4591.27 0 4696.96 0 4621.4 0 4562.84 0 4202.52 0 4296.49 0 4435.23 0 4105.18 0 4116.68 0 3844.49 0 3720.98 0 3674.4 0 3857.62 0 3801.06 0 3504.37 0 3032.6 0 3047.03 0 2962.34 1 2197.82 1 2014.45 1 1862.83 1 1905.41 1 1810.99 1 1670.07 1 1864.44 1 2052.02 1 2029.6 1 2070.83 1 2293.41 1 2443.27 1 2513.17 1 2466.92 1

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 2616.26163480885 -2504.74878571429X[t] + 31.7550804828974t + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) 2616.26163480885 116.540389 22.4494 0 0 X -2504.74878571429 178.15981 -14.059 0 0 t 31.7550804828974 3.548676 8.9484 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.863663571436697 R-squared 0.74591476462679 Adjusted R-squared 0.738441669468754 F-TEST (value) 99.8133636535767 F-TEST (DF numerator) 2 F-TEST (DF denominator) 68 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 433.316484208088 Sum Squared Residuals 12767895.9330791 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2350.44 2648.01671529174 -297.576715291744 2 2440.25 2679.77179577465 -239.521795774646 3 2408.64 2711.52687625755 -302.886876257547 4 2472.81 2743.28195674044 -270.471956740443 5 2407.6 2775.03703722334 -367.437037223341 6 2454.62 2806.79211770624 -352.172117706238 7 2448.05 2838.54719818914 -390.497198189135 8 2497.84 2870.30227867203 -372.462278672032 9 2645.64 2902.05735915493 -256.417359154930 10 2756.76 2933.81243963783 -177.052439637827 11 2849.27 2965.56752012073 -116.297520120725 12 2921.44 2997.32260060362 -75.8826006036221 13 2981.85 3029.07768108652 -47.2276810865197 14 3080.58 3060.83276156942 19.7472384305830 15 3106.22 3092.58784205231 13.6321579476855 16 3119.31 3124.34292253521 -5.03292253521172 17 3061.26 3156.09800301811 -94.8380030181089 18 3097.31 3187.85308350101 -90.5430835010065 19 3161.69 3219.60816398390 -57.9181639839037 20 3257.16 3251.3632444668 5.79675553319869 21 3277.01 3283.1183249497 -6.10832494969829 22 3295.32 3314.8734054326 -19.5534054325957 23 3363.99 3346.62848591549 17.3615140845065 24 3494.17 3378.38356639839 115.786433601609 25 3667.03 3410.13864688129 256.891353118712 26 3813.06 3441.89372736419 371.166272635815 27 3917.96 3473.64880784708 444.311192152917 28 3895.51 3505.40388832998 390.10611167002 29 3801.06 3537.15896881288 263.901031187122 30 3570.12 3568.91404929577 1.20595070422507 31 3701.61 3600.66912977867 100.940870221328 32 3862.27 3632.42421026157 229.845789738430 33 3970.1 3664.17929074447 305.920709255533 34 4138.52 3695.93437122736 442.585628772636 35 4199.75 3727.68945171026 472.060548289738 36 4290.89 3759.44453219316 531.445467806841 37 4443.91 3791.19961267606 652.710387323943 38 4502.64 3822.95469315895 679.685306841047 39 4356.98 3854.70977364185 502.270226358148 40 4591.27 3886.46485412475 704.805145875252 41 4696.96 3918.21993460765 778.740065392354 42 4621.4 3949.97501509054 671.424984909456 43 4562.84 3981.73009557344 581.10990442656 44 4202.52 4013.48517605634 189.034823943663 45 4296.49 4045.24025653924 251.249743460765 46 4435.23 4076.99533702213 358.234662977867 47 4105.18 4108.75041750503 -3.57041750502977 48 4116.68 4140.50549798793 -23.8254979879271 49 3844.49 4172.26057847082 -327.770578470825 50 3720.98 4204.01565895372 -483.035658953722 51 3674.4 4235.77073943662 -561.370739436619 52 3857.62 4267.52581991952 -409.905819919517 53 3801.06 4299.28090040241 -498.220900402414 54 3504.37 4331.03598088531 -826.665980885312 55 3032.6 4362.79106136821 -1330.19106136821 56 3047.03 4394.54614185111 -1347.51614185111 57 2962.34 1921.55243661972 1040.78756338028 58 2197.82 1953.30751710262 244.512482897384 59 2014.45 1985.06259758551 29.3874024144868 60 1862.83 2016.81767806841 -153.987678068411 61 1905.41 2048.57275855131 -143.162758551308 62 1810.99 2080.32783903421 -269.337839034205 63 1670.07 2112.08291951710 -442.012919517103 64 1864.44 2143.838 -279.398 65 2052.02 2175.59308048290 -123.573080482897 66 2029.6 2207.34816096579 -177.748160965795 67 2070.83 2239.10324144869 -168.273241448692 68 2293.41 2270.85832193159 22.5516780684104 69 2443.27 2302.61340241449 140.656597585513 70 2513.17 2334.36848289738 178.801517102616 71 2466.92 2366.12356338028 100.796436619718 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.00151011811247572 0.00302023622495144 0.998489881887524 7 0.000144352144107520 0.000288704288215039 0.999855647855892 8 1.39148607140614e-05 2.78297214281228e-05 0.999986085139286 9 3.01554850427761e-05 6.03109700855523e-05 0.999969844514957 10 4.40188391927983e-05 8.80376783855965e-05 0.999955981160807 11 3.72957694601254e-05 7.45915389202508e-05 0.99996270423054 12 1.99287989931083e-05 3.98575979862166e-05 0.999980071201007 13 7.69786953915654e-06 1.53957390783131e-05 0.99999230213046 14 3.43962596611471e-06 6.87925193222942e-06 0.999996560374034 15 9.48856598999028e-07 1.89771319799806e-06 0.9999990511434 16 2.15901004267837e-07 4.31802008535674e-07 0.999999784098996 17 8.91246123078892e-08 1.78249224615778e-07 0.999999910875388 18 3.75941169940563e-08 7.51882339881126e-08 0.999999962405883 19 1.27608935496053e-08 2.55217870992106e-08 0.999999987239106 20 3.40750335061979e-09 6.81500670123958e-09 0.999999996592497 21 1.06222345456862e-09 2.12444690913724e-09 0.999999998937777 22 4.32428593435386e-10 8.64857186870772e-10 0.999999999567571 23 1.57162291156359e-10 3.14324582312719e-10 0.999999999842838 24 5.70102058885406e-11 1.14020411777081e-10 0.99999999994299 25 7.41265553290512e-11 1.48253110658102e-10 0.999999999925874 26 2.82394947881023e-10 5.64789895762046e-10 0.999999999717605 27 9.27123656911928e-10 1.85424731382386e-09 0.999999999072876 28 5.53041721855833e-10 1.10608344371167e-09 0.999999999446958 29 2.18197471631136e-10 4.36394943262272e-10 0.999999999781803 30 4.76736089789194e-09 9.53472179578388e-09 0.99999999523264 31 1.28047221183891e-08 2.56094442367782e-08 0.999999987195278 32 1.12588038149198e-08 2.25176076298396e-08 0.999999988741196 33 8.04315889580827e-09 1.60863177916165e-08 0.99999999195684 34 5.84355362018205e-09 1.16871072403641e-08 0.999999994156446 35 3.81404565445302e-09 7.62809130890603e-09 0.999999996185954 36 2.48384768388293e-09 4.96769536776587e-09 0.999999997516152 37 2.79194798226586e-09 5.58389596453171e-09 0.999999997208052 38 2.33428080744888e-09 4.66856161489775e-09 0.99999999766572 39 7.81679626986045e-10 1.56335925397209e-09 0.99999999921832 40 4.99657415831162e-10 9.99314831662324e-10 0.999999999500343 41 6.52665580652561e-10 1.30533116130512e-09 0.999999999347334 42 4.34159857745183e-10 8.68319715490367e-10 0.99999999956584 43 4.31659790707522e-10 8.63319581415044e-10 0.99999999956834 44 2.31720793430465e-08 4.6344158686093e-08 0.99999997682792 45 1.99947372606731e-07 3.99894745213461e-07 0.999999800052627 46 1.35358126330797e-06 2.70716252661594e-06 0.999998646418737 47 6.28849725178746e-05 0.000125769945035749 0.999937115027482 48 0.000987625480653735 0.00197525096130747 0.999012374519346 49 0.0143183914360002 0.0286367828720004 0.985681608564 50 0.0717337637668312 0.143467527533662 0.928266236233169 51 0.169885959182516 0.339771918365032 0.830114040817484 52 0.28607553415652 0.57215106831304 0.71392446584348 53 0.464734735420832 0.929469470841665 0.535265264579168 54 0.632926988067686 0.734146023864628 0.367073011932314 55 0.76393576227948 0.472128475441041 0.236064237720520 56 0.810753445462712 0.378493109074576 0.189246554537288 57 0.995902424837987 0.00819515032402563 0.00409757516201282 58 0.99917490553825 0.00165018892349879 0.000825094461749397 59 0.999729184517507 0.000541630964986531 0.000270815482493266 60 0.999695823975455 0.000608352049089402 0.000304176024544701 61 0.999881085253548 0.000237829492904351 0.000118914746452175 62 0.999795944922802 0.000408110154395974 0.000204055077197987 63 0.99936890447052 0.00126219105895859 0.000631095529479294 64 0.996548822757322 0.00690235448535697 0.00345117724267848 65 0.987067183865405 0.0258656322691896 0.0129328161345948 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 51 0.85 NOK 5% type I error level 53 0.883333333333333 NOK 10% type I error level 53 0.883333333333333 NOK

Charts produced by software:

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Parameters (Session):

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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