Reducing Parameter Variance in a Tree Biomass Model
/*--------------------------------------------------------------
SAS Sample Library
Name: modex22.sas
Description: Example program from SAS/ETS User's Guide,
The MODEL Procedure
Title: Reducing Parameter Variance in a Tree Biomass Model
Product: SAS/ETS Software
Keys: SUR, systems of equations, heteroscedasticity
PROC: MODEL
Notes:
--------------------------------------------------------------*/
data trees;
input tree Dbh Height LCL Age Wood Bark Crown total;
datalines;
1 73.2 29.0 16.2 93 4463.4 572.9 186.4 5222.7
2 30.5 18.3 10.4 40 550.7 83.9 44.9 679.5
3 48.3 22.9 11.9 69 1689.2 225.0 93.4 2007.6
4 69.6 27.4 18.3 74 3441.5 435.5 178.3 4055.3
5 28.7 19.8 11.6 38 482.2 75.3 45.4 602.9
6 53.1 32.0 14.9 78 2281.6 307.5 57.2 2646.3
7 45.7 32.0 17.7 79 1771.3 230.4 24.9 2026.6
8 46.5 30.5 17.4 83 1611.6 228.2 17.2 1857.0
9 33.8 25.9 10.1 68 861.8 122.5 20.0 1004.3
10 55.4 30.5 13.1 70 2952.9 367.4 65.3 3385.6
11 30.5 24.4 9.1 75 679.9 125.2 17.7 822.8
12 70.1 30.5 13.4 81 3867.8 546.1 79.4 4493.3
13 41.9 24.4 12.2 79 1289.1 185.5 36.7 1511.3
14 42.9 25.9 13.7 76 1495.5 201.4 71.7 1768.6
15 66.3 35.1 17.7 81 4091.0 413.2 99.3 4603.5
16 40.6 25.9 11.6 64 1264.2 175.5 12.7 1452.4
17 28.7 21.3 9.8 75 485.8 67.6 19.1 572.5
18 80.5 32.0 17.4 93 5782.0 657.7 186.9 6626.6
19 66.5 33.8 16.8 106 4085.1 524.4 112.0 4721.5
20 61.2 29.0 17.1 71 2621.4 225.4 82.6 2929.4
21 21.8 25.9 11.3 35 292.6 51.7 5.0 349.3
22 29.0 24.4 8.8 35 616.4 94.3 21.8 732.5
23 49.8 27.4 16.2 35 1757.2 220.0 63.5 2040.7
24 34.3 27.4 10.4 37 902.2 144.7 21.8 1068.7
25 43.2 27.4 15.5 41 1251.5 212.3 20.9 1484.7
26 25.1 24.4 8.8 38 437.7 63.5 6.4 507.6
27 29.7 25.9 13.7 40 704.4 89.4 10.0 803.8
28 34.8 25.9 13.1 40 906.7 117.5 23.1 1047.3
29 38.6 27.4 13.4 41 1309.5 148.8 26.8 1485.1
30 49.8 22.9 14.3 40 1497.3 160.1 49.0 1706.4
31 36.1 25.9 16.5 34 794.3 116.6 20.9 931.8
32 37.1 21.3 11.0 67 846.9 120.7 26.8 994.4
33 33.0 21.3 9.8 67 635.5 89.8 27.7 753.0
34 57.7 25.9 18.6 84 2545.1 371.0 70.3 2986.4
35 53.8 25.9 12.2 87 2275.7 359.3 32.7 2667.7
36 57.9 27.4 10.1 89 2822.3 379.2 61.2 3262.7
37 75.4 27.4 13.7 91 3782.1 579.2 61.7 4423.0
38 57.2 25.9 12.2 86 2055.7 362.0 45.4 2463.1
39 69.1 27.4 14.6 87 3618.4 498.1 87.1 4203.6
;
proc model data=trees;
endo wood bark crown total;
vol = dbh**2*height;
cvol = vol*lcl/1000;
wood = b10 + b11*vol;
bark = b20 + b21*vol;
crown = b30 + b31*cvol + b32*height;
total = b40 + b41*vol + b42*cvol + b43*height;
fit / ols outs=stree out=otree outresid;
quit;
proc model data=trees;
ods output ParameterEstimates=ineff;
endo wood bark crown total;
vol = dbh**2*height;
cvol = vol*lcl/1000;
wood = b10 + b11*vol;
bark = b20 + b21*vol;
crown = b30 + b31*cvol + b32*height;
total = b40 + b41*vol + b42*cvol + b43*height;
fit / nools sur sdata=stree;
restrict b10 + b20 + b30 = b40,
b11 + b21 = b41,
b42 = b31,
b43 = b32;
quit;
proc model data=otree;
ods output parameterestimates=vparms;
vol = dbh**2*height;
cvol = vol*lcl/1000;
eq.varwood = log(wood*wood) - (w1 + w2*log(vol));
eq.varbark = log(bark*bark) - (k1 + k2*log(vol));
eq.varcrown = log(crown*crown) - (c1 + c2*log(cvol) - c3*height**2);
eq.vartotal = log(total*total) - (t1 + t2*log(vol));
fit varwood varbark varcrown vartotal;
quit;
data _null_;
set vparms;
call symput(Parameter,Estimate);
run;
proc model data=trees;
ods output ParameterEstimates=eff;
endo wood bark crown total;
vol = dbh**2*height;
cvol = vol*lcl/1000;
wood = b10 + b11*vol;
bark = b20 + b21*vol;
crown = b30 + b31*cvol + b32*height;
total = b40 + b41*vol + b42*cvol + b43*height;
h.wood = exp(&w1)*vol**&w2;
h.bark = exp(&k1)*vol**&k2;
h.crown = exp(&c1)*cvol**&c2 * exp (-&c3*height*height);
h.total = exp(&t1)*vol**&t2;
fit / sur;
restrict b10 + b20 + b30 = b40,
b11 + b21 = b41,
b42 = b31,
b43 = b32;
quit;
data varred;
keep parameter stderrchange;
merge ineff eff(rename=(estimate=effestimate stderr=efferr));
format stderrchange percent.;
label stderrchange = "StdErr Rel. Change";
stderrchange = (efferr - stderr)/stderr;
run;
proc print data=varred(obs=11) noobs label;
run;
