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  #^ 
  

  

  

  

  

  

  

  .>rR1chfield 
  

   Battle 
  Mountain 
  

   \as 
  Vegas 
  

  

  Winnemucca 
  

  

  IDOO 
  2000 
  3000 
  UOOO 
  

  

  DRSQH 
  (DHC 
  SQURREO 
  TI^ES 
  HEIGKB 
  

  

  Figure 
  6. 
  — 
  Volume 
  equations 
  for 
  multiple- 
  

   stem 
  Utah 
  juniper 
  in 
  the 
  Great 
  Basin 
  States. 
  

   All 
  area 
  labels 
  refer 
  to 
  BLM 
  districts, 
  except 
  

   Idaho, 
  which 
  refers 
  to 
  southern 
  Idaho. 
  

  

  5000 
  

  

  yrBattle 
  Mountain 
  

   parson 
  City 
  

  

  1000 
  2000 
  3000 
  liOOO 
  

  

  DRSQH 
  IDRC 
  SQUARED 
  TIMES 
  HEIGHT) 
  

  

  5000 
  

  

  Figure 
  7. 
  — 
  Volume 
  equations 
  for 
  single-stem 
  

   singleleaf 
  pinyon 
  in 
  the 
  Great 
  Basin 
  States. 
  

   The 
  area 
  labels 
  refer 
  to 
  BLM 
  districts. 
  

  

  includes 
  Susanville 
  BLM) 
  looked 
  different 
  from 
  the 
  rest 
  

   (fig. 
  6). 
  I 
  kept 
  Ely 
  and 
  Winnemucca 
  separate, 
  but 
  com- 
  

   bined 
  Elko 
  with 
  the 
  rest 
  of 
  the 
  Great 
  Basin 
  area. 
  The 
  

   Elko 
  data 
  contained 
  a 
  large 
  percentage 
  of 
  single-stem 
  

   trees, 
  and 
  in 
  a 
  graph 
  of 
  single-stem 
  equations 
  (not 
  

   shown) 
  the 
  Elko 
  data 
  were 
  not 
  different. 
  The 
  Winnemucca 
  

   and 
  Cedar 
  City 
  singleleaf 
  pinyon 
  volume 
  equations 
  

   appeared 
  distinct 
  from 
  the 
  rest 
  in 
  figure 
  7. 
  However, 
  

   these 
  differences 
  were 
  not 
  meaningful 
  because 
  the 
  

   Winnemucca 
  data 
  contained 
  too 
  few 
  trees 
  and 
  the 
  Cedar 
  

   City 
  data 
  contained 
  mostly 
  small 
  trees 
  (DRSQH 
  less 
  

   than 
  2,000). 
  

  

  For 
  the 
  Colorado 
  Plateau 
  States, 
  the 
  table 
  1 
  results 
  

   indicated 
  further 
  analysis 
  for 
  only 
  Rocky 
  Mountain 
  juni- 
  

   per. 
  Graphs 
  of 
  fuU 
  models 
  for 
  Rocky 
  Mountain 
  juniper 
  

   did 
  show 
  differences, 
  but 
  I 
  combined 
  all 
  the 
  data 
  be- 
  

   cause 
  of 
  small 
  sample 
  sizes 
  within 
  groups. 
  

  

  The 
  final 
  number 
  of 
  P-J 
  equations 
  was 
  based 
  on 
  the 
  

   F-tests 
  and 
  on 
  graphical 
  analysis, 
  as 
  described 
  for 
  most 
  

   of 
  the 
  data. 
  In 
  the 
  case 
  of 
  mountain-mahogany, 
  Rocky 
  

   Mountain 
  juniper, 
  the 
  oaks, 
  and 
  hardwoods, 
  a 
  small 
  

   sample 
  size 
  dictated 
  equations 
  by 
  species 
  without 
  con- 
  

   sideration 
  of 
  geographic 
  areas. 
  Thirteen 
  distinct 
  volume 
  

   equations 
  were 
  developed. 
  A 
  volume 
  table 
  for 
  each 
  equa- 
  

   tion 
  is 
  given 
  in 
  appendix 
  B. 
  Table 
  2 
  lists 
  a 
  guide 
  for 
  

   selecting 
  a 
  volume 
  equation 
  for 
  each 
  area 
  and 
  species. 
  

  

  Reliability 
  of 
  Equations 
  

  

  Additional 
  statistical 
  analysis 
  should 
  be 
  done 
  to 
  exam- 
  

   ine 
  rehabiUty 
  of 
  regression 
  equations 
  when 
  coefficients 
  

   are 
  estimated 
  from 
  transformed 
  data, 
  but 
  equation 
  

   predictions 
  are 
  retransformed 
  for 
  use. 
  Such 
  predictions 
  

  

  are 
  subject 
  to 
  transformation 
  bias, 
  and 
  regression 
  statis- 
  

   tics 
  in 
  transformed 
  units 
  also 
  can 
  be 
  misleading. 
  I 
  exam- 
  

   ined 
  the 
  bias 
  of 
  the 
  cube 
  root 
  transformation, 
  recomputed 
  

   the 
  R^ 
  statistic, 
  and 
  tested 
  some 
  of 
  the 
  volume 
  equa- 
  

   tions 
  against 
  another 
  data 
  set. 
  Duan 
  (1983) 
  presented 
  a 
  

   smearing 
  estimator, 
  a 
  nonparametric 
  retransformation 
  

   method, 
  that 
  can 
  be 
  used 
  to 
  approximate 
  the 
  bias 
  of 
  any 
  

   transformation. 
  This 
  was 
  used 
  to 
  compute 
  an 
  approxi- 
  

   mate 
  bias, 
  defined 
  as 
  the 
  difference 
  between 
  the 
  

   predicted 
  value 
  from 
  regression 
  and 
  the 
  smearing 
  esti- 
  

   mator. 
  The 
  smearing 
  estimator 
  was 
  calculated 
  as: 
  

   n 
  

  

  SE 
  = 
  E 
  h(x'^ 
  + 
  w,?;) 
  (3) 
  

  

  n 
  i=l 
  — 
  

  

  where 
  

  

  SE 
  - 
  smearing 
  estimator 
  

   h(») 
  = 
  inverse 
  of 
  the 
  transformation 
  (the 
  cubic 
  

   function) 
  

  

  X 
  = 
  row 
  vector 
  of 
  regression 
  predictor 
  variables 
  

  

  jL 
  = 
  vector 
  of 
  regression 
  coefficients 
  

  

  = 
  residual 
  from 
  regression 
  for 
  the 
  ith 
  tree 
  

   W; 
  = 
  biweight 
  of 
  the 
  ith 
  tree 
  (eq. 
  2) 
  

   n 
  = 
  number 
  of 
  trees. 
  

  

  The 
  transformation 
  bias 
  is 
  Usted 
  in 
  table 
  3 
  as 
  a 
  per- 
  

   centage 
  for 
  several 
  quantiles 
  of 
  the 
  sample 
  data. 
  

   Because 
  this 
  bias 
  is 
  always 
  negative, 
  the 
  volume 
  equa- 
  

   tion 
  will 
  underestimate 
  by 
  the 
  amount 
  of 
  the 
  biases. 
  No 
  

   attempt 
  was 
  made 
  to 
  correct 
  for 
  the 
  transformation 
  

   bias, 
  because 
  the 
  bias 
  was 
  relatively 
  small 
  and 
  a 
  bias 
  

   adjustment 
  that 
  varied 
  according 
  to 
  tree 
  size 
  would 
  be 
  

   complicated 
  to 
  apply. 
  

  

  6 
  

  

  