Comparison of 3 different methods of linear regression:
  (1) Least-squares
  (2) ``Theil'' linear regression
  (3) Rank regression
Output of program   RANKREGRESS
Written or modified by  YOURNAME

Starting random-number seed: 123456
(Enter a number on the command line to specify a different seed.)

Prepacked Lubricants and Survival Times for Engines
(Source of data unknown)

    Lubricants  Survival      Lubricants  Survival 
---------------------------------------------------------
     1.   29     23           11.   84    429
     2.   21     36           12.   37    489
     3.   30     67           13.   94    504
     4.   45    104           14.   56    925
     5.   48    147           15.   67    980
     6.   92    164           16.   79   1254
     7.   74    247           17.   70   2961
     8.   61    304           18.   93   5351
     9.   99    355           19.   76   9915
    10.   24    408           20.   72  11301
---------------------------------------------------------

(The last two columns below are the errors
    MADf = (1/n)Sum(i=1,n) |Y_i-beta*X_i-mu|  and
    RMS  = Root( (1/n)Sum(i=1,n)(Y_i-beta*X_i-mu)^2 )

REGRESSIONS for Y(Survival Time) on X(Lubricants):
                                                 MADf      RMS
  LeastSqrs:  Y = 37.9242 X + -573.961        2010.12   3050.92
  Theil:      Y =  7.1050 X +  179.280        1614.72   3354.75
  RankRegr:   Y =  6.8281 X +  197.914        1615.16   3355.82

REGRESSIONS for log(Y)(Survival Time) on X(Lubricants):
                                                 MADf      RMS
  LeastSqrs:  log(Y) =  0.0373 X +    3.826      1.23      1.40
  Theil:      log(Y) =  0.0340 X +    3.984      1.23      1.40
  RankRegr:   log(Y) =  0.0357 X +    3.880      1.23      1.40

CONFIDENCE INTERVALS and P-VALUES for SLOPE:

LEAST-SQUARES REGRESSION (normal-theory) for Y = beta*X + mu
  Confidence interval and P-value for beta
  Beta=37.9242   95% CI (-23.9214, 99.7698)  (18 df)
  Pval(beta!=0): 0.213956  (two-sided, 18 df)

(LEAST SQUARES) BOOTSTRAP for beta in Y = beta*X + mu:
  Medians and 95% confidence intervals, 10,000 replications.
BOOTSTRAP ON OBSERVATIONS (least squares, beta=37.9242):
  Beta=36.1494  95% CI (5.7296, 89.3095)   Bias: 0.5383
Bootstrap Pvalue for H_0:beta=0 vs. H_1:beta!=0:
  One-sided P:  P=53/10,000=0.00530
  Two-sided P:  P=0.01060   95% CI: (0.007754, 0.013446)

BOOTSTRAP ON RESIDUALS (least squares, beta=37.9242):
  Beta=38.5161  95% CI (-18.2171, 90.6865)   Bias: 0.4915
Bootstrap Pvalue for H_0:beta=0 vs. H_1:beta!=0:
  One-sided P:  P=829/10,000=0.08290
  Two-sided P:  P=0.16580   95% CI: (0.154991, 0.176609)

LEAST-SQUARES REGRESSION (normal-theory) for log(Y) = beta*X + mu
  Confidence interval and P-value for beta
  Beta=0.0373138   95% CI (0.0090, 0.0656)  (18 df)
  Pval(beta!=0): 0.0126445  (two-sided, 18 df)


DISTRIBUTION-FREE EXACT 95% CI for Theil's estimator:
  Large-sample approximations,  Y = beta*X + mu:
    Beta=7.1050  95% CI (1.3421, 33.0000)
      (attained at diff.ratio offsets 64,127 in (1,190))
    Large-sample P-value for H_0:beta=0:
      Z=2.33599  P=0.0195 (2-sided)

(THEIL REGRESSION) BOOTSTRAP for beta in Y = beta*X + mu:
  Medians and 95% confidence intervals, 10,000 replications.
BOOTSTRAP ON OBSERVATIONS (Theil, beta=7.10497):
  Beta=6.8281  95% CI (1.2766, 51.1250)   Bias: 0.5144
Bootstrap Pvalue for H_0:beta=0 vs. H_1:beta!=0:
  One-sided P:  P=120/10,000=0.01200
  Two-sided P:  P=0.02400   95% CI: (0.019732, 0.028268)

BOOTSTRAP ON RESIDUALS (Theil, beta=7.10497):
  Beta=7.1050  95% CI (-3.2275, 16.9209)   Bias: 0.5388
Bootstrap Pvalue for H_0:beta=0 vs. H_1:beta!=0:
  One-sided P:  P=527/10,000=0.05270
  Two-sided P:  P=0.10540   95% CI: (0.096641, 0.114159)


(RANK REGRESSION) BOOTSTRAP for beta in Y = beta*X + mu:
  Medians and 95% confidence intervals, 10,000 replications.
BOOTSTRAP ON OBSERVATIONS (rank regression, beta=6.82813):
  Beta=6.8281  95% CI (1.3714, 71.6377)   Bias: 0.5091
Bootstrap Pvalue for H_0:beta=0 vs. H_1:beta!=0:
  One-sided P:  P=51/10,000=0.00510
  Two-sided P:  P=0.01020   95% CI: (0.007408, 0.012992)

BOOTSTRAP ON RESIDUALS (rank regression, beta=6.82813):
  Beta=6.8281  95% CI (-6.5741, 18.2744)   Bias: 0.5968
Bootstrap Pvalue for H_0:beta=0 vs. H_1:beta!=0:
  One-sided P:  P=852/10,000=0.08520
  Two-sided P:  P=0.17040   95% CI: (0.159456, 0.181344)

SUMMARY of estimates, confidence intervals, and P-values:
Least Squares (Y):
Classical      37.92423  (-23.92137,  99.76983)  P=0.21396 (df=18)
BootObs        36.14940  (  5.72961,  89.30950)  P=0.01060
BootResid      38.51612  (-18.21708,  90.68653)  P=0.16580
Least Squares (log(Y)):
Classical       0.03731  (  0.00900,   0.06562)  P=0.01264 (df=18)
Theil Regression (Y):
Approx          7.10497  (  1.34211,  33.00000)  P=0.01949
BootObs         6.82813  (  1.27660,  51.12500)  P=0.02400
BootResid       7.10497  ( -3.22754,  16.92085)  P=0.10540
Rank Regression (Y):
BootObs         6.82813  (  1.37143,  71.63768)  P=0.01020
BootResid       6.82813  ( -6.57414,  18.27441)  P=0.17040

Random numbers used:  1,200,000