﻿Number 
  of 
  Equations 
  

  

  Data 
  were 
  available 
  for 
  developing 
  33 
  volume 
  equa- 
  

   tions, 
  if 
  each 
  species 
  from 
  each 
  area 
  were 
  kept 
  separate. 
  

   Combining 
  some 
  of 
  these 
  data 
  sets 
  was 
  a 
  difficult 
  task 
  

   because 
  few 
  good 
  statistical 
  methods 
  exist 
  for 
  objective 
  

   grouping. 
  My 
  approach 
  used 
  statistical 
  tests 
  between 
  

   groups 
  of 
  regression 
  coefficients 
  and 
  comparative 
  plot- 
  

   ting 
  of 
  regression 
  equations. 
  

  

  Graybill 
  (1976, 
  p. 
  247) 
  presented 
  theory 
  for 
  testing 
  

   whether 
  or 
  not 
  a 
  set 
  of 
  regression 
  coefficients 
  are 
  simi- 
  

   lar. 
  But, 
  for 
  the 
  event 
  of 
  dissimilar 
  coefficients 
  in 
  a 
  set, 
  

   Graybill 
  gave 
  no 
  way 
  to 
  identify 
  which 
  coefficients 
  are 
  

   dissimilar. 
  However, 
  this 
  approach 
  was 
  a 
  good 
  starting 
  

   point. 
  

  

  The 
  data 
  were 
  tested 
  for 
  full 
  and 
  reduced 
  models 
  for 
  

   each 
  species 
  within 
  the 
  two 
  large 
  areas, 
  the 
  Great 
  Basin 
  

  

  States 
  and 
  Colorado 
  Plateau 
  States. 
  A 
  full 
  model 
  had 
  a 
  

   distinct 
  set 
  of 
  regression 
  coefficients 
  for 
  each 
  BLM 
  dis- 
  

   trict 
  or 
  small 
  area 
  within 
  the 
  large 
  area. 
  A 
  reduced 
  

   model 
  had 
  only 
  one 
  set 
  of 
  coefficients 
  for 
  the 
  entire 
  large 
  

   area. 
  

  

  Table 
  1 
  shows 
  no 
  significant 
  difference 
  between 
  the 
  

   full 
  and 
  reduced 
  models 
  for 
  Utah 
  juniper 
  and 
  pinyon 
  in 
  

   the 
  Colorado 
  Plateau 
  States, 
  and 
  for 
  western 
  juniper 
  in 
  

   the 
  Great 
  Basin 
  States. 
  Data 
  for 
  these 
  areas 
  were 
  

   grouped 
  into 
  their 
  respective 
  reduced 
  models. 
  Further 
  

   analysis 
  was 
  done 
  for 
  those 
  areas 
  showing 
  significant 
  

   results 
  in 
  table 
  1. 
  Graphs 
  of 
  the 
  full 
  models 
  were 
  exam- 
  

   ined 
  to 
  distinquish 
  which 
  areas 
  should 
  have 
  separate 
  vol- 
  

   ume 
  equations. 
  

  

  Equations 
  for 
  the 
  Great 
  Basin 
  States 
  are 
  shown 
  in 
  

   figures 
  6 
  and 
  7. 
  The 
  Utah 
  juniper 
  equations 
  for 
  the 
  

   BLM 
  districts 
  of 
  Ely, 
  Elko, 
  and 
  Winnemucca 
  (also 
  

  

  Table 
  1.— 
  Analysis 
  of 
  variance 
  tables 
  comparing 
  full 
  and 
  reduced 
  volume 
  models 
  for 
  

   pinyon 
  and 
  juniper 
  

  

  

  Degrees 
  

  

  Sum 
  of 
  Mean 
  

  

  

  

  Source 
  

  

  of 
  freedom 
  

  

  squares 
  square 
  F-value 
  

  

  ProD 
  > 
  F 
  

  

  

  Utah 
  Juniper 
  In 
  the 
  Great 
  Basin 
  States 
  

  

  

  

  Total 
  

  

  1,339 
  

  

  2,617,659 
  

  

  

  

  Full 
  model 
  

  

  : 
  24 
  

  

  2,573.803 
  

  

  

  

  Reduced 
  model 
  

  

  3 
  

  

  2,568.511 
  

  

  

  

  Gain 
  due 
  to 
  full 
  model 
  21 
  

  

  5.292 
  0.2520 
  

  

  7.56 
  

  

  0.0001 
  * 
  

  

  Error 
  

  

  1,315 
  

  

  43.856 
  .0334 
  

  

  

  

  

  Utah 
  Juniper 
  in 
  the 
  Colorado 
  Plateau 
  States 
  

  

  

  

  Total 
  

  

  397 
  

  

  892.891 
  

  

  

  

  Full 
  model 
  

  

  12 
  

  

  878.180 
  

  

  

  

  Reduced 
  model 
  

  

  3 
  

  

  877.799 
  

  

  

  

  Gain 
  due 
  to 
  full 
  model 
  9 
  - 
  

  

  .381 
  .0423 
  

  

  1.11 
  

  

  .3544"^ 
  

  

  Error 
  

  

  385 
  

  

  14.711 
  .0382 
  

  

  

  

  Rocky 
  Mountain 
  Juniper 
  in 
  the 
  Colorado 
  Plateau 
  States 
  

  

  

  Total 
  

  

  194 
  

  

  359.158 
  

  

  

  

  Full 
  model 
  

  

  

  354.285 
  

  

  

  

  Reduced 
  model 
  

  

  3 
  

  

  353.088 
  

  

  

  

  Gain 
  due 
  to 
  full 
  

  

  model 
  6 
  

  

  1.197 
  .1995 
  

  

  7.59 
  

  

  .0001* 
  

  

  Error 
  

  

  185 
  

  

  4.874 
  .0263 
  

  

  

  

  

  Western 
  Juniper 
  

  

  in 
  the 
  Great 
  Basin 
  States 
  

  

  

  

  Total 
  

  

  177 
  

  

  669.961 
  

  

  

  

  Full 
  model 
  

  

  6 
  

  

  663.288 
  

  

  

  

  Reduced 
  model 
  

  

  3 
  

  

  663.182 
  

  

  

  

  Gain 
  due 
  to 
  full 
  

  

  model 
  3 
  

  

  .106 
  .0353 
  

  

  .91 
  

  

  .4375"^ 
  

  

  Error 
  

  

  171 
  

  

  6.672 
  .0390 
  

  

  

  

  

  Singleleaf 
  Pinyon 
  in 
  the 
  Great 
  Basin 
  States 
  

  

  

  

  Total 
  

  

  1,445 
  

  

  2,931.848 
  

  

  

  

  Full 
  model 
  

  

  20 
  

  

  2,910.769 
  

  

  

  

  Reduced 
  model 
  

  

  3 
  

  

  2,909.535 
  

  

  

  

  Gain 
  due 
  to 
  full 
  model 
  17 
  

  

  1.234 
  .0726 
  

  

  4.91 
  

  

  .0001 
  * 
  

  

  Error 
  

  

  1,425 
  

  

  21.079 
  .0148 
  

  

  

  

  

  Pinyon 
  in 
  the 
  Colorado 
  Plateau 
  States 
  

  

  

  

  Total 
  

  

  350 
  

  

  762.673 
  

  

  

  

  Full 
  model 
  

  

  12 
  

  

  753.441 
  

  

  

  

  Reduced 
  model 
  

  

  3 
  

  

  753.090 
  

  

  

  

  Gain 
  due 
  to 
  full 
  

  

  model 
  9 
  

  

  .351 
  .0390 
  

  

  1.43 
  

  

  .1737"^ 
  

  

  Error 
  

  

  338 
  

  

  9.232 
  .0273 
  

  

  

  

  * 
  This 
  is 
  the 
  probability 
  from 
  an 
  F-distribution 
  (with 
  degrees 
  of 
  freedom 
  from 
  the 
  gain 
  due 
  to 
  the 
  

   full 
  model 
  and 
  from 
  the 
  error) 
  of 
  getting 
  a 
  value 
  larger 
  than 
  the 
  reported 
  F-value. 
  For 
  the 
  o-level 
  set 
  

   at 
  0.05, 
  these 
  are 
  significantly 
  different. 
  

  

  "=For 
  the 
  o-level 
  set 
  at 
  0.05, 
  the 
  full 
  and 
  reduced 
  models 
  are 
  not 
  significantly 
  different. 
  

  

  5 
  

  

  