Statistical Methods
Data Structure
- hugss.MTW - The minitab worksheet used to do all of this.
Each trial run produces a record of data:
| Heuristic | Benchmark | Rank | Trucks | Distance |
- Heuristic - What heuristic (algorithm or treatment) was used to produce this data? The heuristic is always one of ten: GR, Any, SC, HI, RAR, RCP, RP, RPGR, RR, SC, SS.
- Benchmark - What benchmark (starting point) was used as the basepoint for this data? The benchmark is simply a configuration of customers in space along with different time windows (hours of operation) of the customers.
- Rank - What rank of benchmark was used? We started all of the benchmarks from 1 customer to 1 truck and then ran a brute-force optimization to get better starting points. Which one of these better starting points is this?
- Trucks - How many trucks were used in the solution?
- Distance - How far was the aggregrate distance of the trucks' travels?
Statistical Calculations
One-way ANOVA: Trucks versus Heuristic
We first run a one-way ANOVA (ANalysis Of Variance) at 95% confidence. This is just as if Trucks are the only response (measurement) and Heuristic is the only factor. We then do Fisher's least significant difference method of multiple comparison and find that our best heuristic, GR, is significantly different all other heuristics, except for Any and SC (at 95% confidence).
This makes sense, since Any actually calls GR and SC uses a similar approach (think quickly, try, revert if it doesn't work, etc.).
Analysis of Variance for Trucks
Source DF SS MS F P
Heuristi 9 102.29 11.37 5.96 0.000
Error 470 895.69 1.91
Total 479 997.98
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev ------+---------+---------+---------+
any 48 13.208 1.501 (----*----)
gr 48 12.771 1.561 (----*----)
hi 48 14.625 0.570 (----*----)
rar 48 13.521 1.337 (----*----)
rcp 48 13.917 1.302 (----*----)
rp 48 13.396 1.484 (---*----)
rpgr 48 13.479 1.429 (---*----)
rr 48 13.396 1.484 (---*----)
sc 48 13.271 1.455 (----*----)
ss 48 13.479 1.414 (---*----)
------+---------+---------+---------+
Pooled StDev = 1.380 12.80 13.60 14.40 15.20
Fisher's pairwise comparisons
Family error rate = 0.624
Individual error rate = 0.0500
Critical value = 1.965
Intervals for (column level mean) - (row level mean)
any gr hi rar rcp rp
gr -0.116
0.991
hi -1.970 -2.408
-0.863 -1.300
rar -0.866 -1.304 0.550
0.241 -0.196 1.658
rcp -1.262 -1.700 0.155 -0.950
-0.155 -0.592 1.262 0.158
rp -0.741 -1.179 0.675 -0.429 -0.033
0.366 -0.071 1.783 0.679 1.075
rpgr -0.825 -1.262 0.592 -0.512 -0.116 -0.637
0.283 -0.155 1.700 0.595 0.991 0.470
rr -0.741 -1.179 0.675 -0.429 -0.033 -0.554
0.366 -0.071 1.783 0.679 1.075 0.554
sc -0.616 -1.054 0.800 -0.304 0.092 -0.429
0.491 0.054 1.908 0.804 1.200 0.679
ss -0.825 -1.262 0.592 -0.512 -0.116 -0.637
0.283 -0.155 1.700 0.595 0.991 0.470
rpgr rr sc
rr -0.470
0.637
sc -0.345 -0.429
0.762 0.679
ss -0.554 -0.637 -0.762
0.554 0.470 0.345
2 Sample t-Tests
We run a GR vs. SC and a GR vs. Any t-tests (again, only with Trucks) to see exactly how significantly different the heuristics are:
Two-sample T for GR vs Any
N Mean StDev SE Mean
GR 48 12.77 1.56 0.23
Any 48 13.21 1.50 0.22
Difference = mu GR - mu Any
Estimate for difference: -0.438
83.5% CI for difference: (-0.875, -0.000)
T-Test of difference = 0 (vs not =): T-Value = -1.40 P-Value = 0.165 DF = 93
Two-sample T for GR vs SC
N Mean StDev SE Mean
GR 48 12.77 1.56 0.23
SC 48 13.27 1.45 0.21
Difference = mu GR - mu SC
Estimate for difference: -0.500
89.2% CI for difference: (-1.000, -0.000)
T-Test of difference = 0 (vs not =): T-Value = -1.62 P-Value = 0.108 DF = 93
From these data, we have found that GR is significantly different than Any with a 83.5% confidence level, and is significantly different from SC with a 89.2% confidence level.
ANOVA: Trucks, Distance versus Heuristic, Benchmark, Rank
We now proceed to run a multiple ANOVA, comparing everything all at once.
Factor Type Levels Values
Heuristi fixed 10 any gr hi rar rcp rp rpgr
rr sc ss
Benchmar fixed 8 RC101 RC102 RC103 RC104 RC105 RC106 RC107
RC108
Rank fixed 3 0 10 20
Analysis of Variance for Trucks
Source DF SS MS F P
Heuristi 9 102.294 11.366 28.88 0.000
Benchmar 7 700.831 100.119 254.36 0.000
Rank 2 13.400 6.700 17.02 0.000
Error 461 181.456 0.394
Total 479 997.981
Analysis of Variance for Distance
Source DF SS MS F P
Heuristi 9 1316869 146319 94.59 0.000
Benchmar 7 11757396 1679628 1085.78 0.000
Rank 2 18144 9072 5.86 0.003
Error 461 713134 1547
Total 479 13805544
MANOVA for Heuristi s = 2 m = 3.0 n = 229.0
Criterion Test Statistic F DF P
Wilk's 0.30313 41.722 ( 18, 920) 0.000
Lawley-Hotelling 2.01355 51.345 ( 18, 918) 0.000
Pillai's 0.78339 32.983 ( 18, 922) 0.000
Roy's 1.86011
SSCP Matrix for Heuristi
Trucks Distance
Trucks 102.3 10469
Distance 10469.1 1316869
SSCP Matrix for Error
Trucks Distance
Trucks 181.5 6438
Distance 6437.6 713134
Partial Correlations for the Error SSCP Matrix
Trucks Distance
Trucks 1.00000 0.56591
Distance 0.56591 1.00000
EIGEN Analysis for Heuristi
Eigenvalue 1.8601 0.15344
Proportion 0.9238 0.07621
Cumulative 0.9238 1.00000
Eigenvector 1 2
Trucks -8.0E-03 0.089684
Distance 0.001252 -7.0E-04
MANOVA for Benchmar s = 2 m = 2.0 n = 229.0
Criterion Test Statistic F DF P
Wilk's 0.04482 244.675 ( 14, 920) 0.000
Lawley-Hotelling 16.96169 556.101 ( 14, 918) 0.000
Pillai's 1.15007 89.114 ( 14, 922) 0.000
Roy's 16.70135
SSCP Matrix for Benchmar
Trucks Distance
Trucks 700.8 88643
Distance 88642.5 11757396
EIGEN Analysis for Benchmar
Eigenvalue 16.7013 0.2603
Proportion 0.9847 0.0153
Cumulative 0.9847 1.0000
Eigenvector 1 2
Trucks -0.01028 0.08945
Distance 0.00127 -0.00067
MANOVA for Rank s = 2 m = -0.5 n = 229.0
Criterion Test Statistic F DF P
Wilk's 0.87959 15.237 ( 4, 920) 0.000
Lawley-Hotelling 0.13420 15.399 ( 4, 918) 0.000
Pillai's 0.12277 15.075 ( 4, 922) 0.000
Roy's 0.10968
SSCP Matrix for Rank
Trucks Distance
Trucks 13.40 81.1
Distance 81.11 18144.5
EIGEN Analysis for Rank
Eigenvalue 0.1097 0.02452
Proportion 0.8173 0.18272
Cumulative 0.8173 1.00000
Eigenvector 1 2
Trucks 0.089552 0.009373
Distance -9.3E-04 0.001093
Some things to note: The correlation between Trucks and Distance is only 0.56, essentially meaning that there isn't much of a correlation.
General Linear Model: Trucks, Distance versus Heuristic, Benchmark, Rank
I also ran a General Linear Model, which provides multiple comparisons in addition to an ANOVA test.
Factor Type Levels Values
Heuristi fixed 10 any gr hi rar rcp rp rpgr rr sc ss
Benchmar fixed 8 RC101 RC102 RC103 RC104 RC105 RC106 RC107 RC108
Rank fixed 3 0 10 20
Analysis of Variance for Trucks, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Heuristi 9 102.294 102.294 11.366 28.88 0.000
Benchmar 7 700.831 700.831 100.119 254.36 0.000
Rank 2 13.400 13.400 6.700 17.02 0.000
Error 461 181.456 181.456 0.394
Total 479 997.981
Unusual Observations for Trucks
Obs Trucks Fit SE Fit Residual St Resid
90 11.0000 12.5229 0.1248 -1.5229 -2.48R
97 15.0000 16.3854 0.1248 -1.3854 -2.25R
98 15.0000 16.3854 0.1248 -1.3854 -2.25R
99 15.0000 16.2604 0.1248 -1.2604 -2.05R
100 15.0000 16.2604 0.1248 -1.2604 -2.05R
105 14.0000 15.6437 0.1248 -1.6437 -2.67R
106 14.0000 15.6437 0.1248 -1.6437 -2.67R
115 14.0000 12.7021 0.1248 1.2979 2.11R
116 14.0000 12.7021 0.1248 1.2979 2.11R
117 14.0000 12.5771 0.1248 1.4229 2.31R
118 14.0000 12.5771 0.1248 1.4229 2.31R
121 14.0000 15.3854 0.1248 -1.3854 -2.25R
122 14.0000 15.3854 0.1248 -1.3854 -2.25R
123 14.0000 15.2604 0.1248 -1.2604 -2.05R
124 14.0000 15.2604 0.1248 -1.2604 -2.05R
139 15.0000 13.0521 0.1248 1.9479 3.17R
140 15.0000 13.0521 0.1248 1.9479 3.17R
143 15.0000 13.3271 0.1248 1.6729 2.72R
144 15.0000 13.3271 0.1248 1.6729 2.72R
163 13.0000 11.5979 0.1248 1.4021 2.28R
215 11.0000 12.2687 0.1248 -1.2687 -2.06R
234 15.0000 13.6688 0.1248 1.3312 2.17R
235 14.0000 12.3438 0.1248 1.6562 2.69R
239 15.0000 12.6188 0.1248 2.3812 3.87R
279 14.0000 12.7479 0.1248 1.2521 2.04R
424 14.0000 12.6229 0.1248 1.3771 2.24R
466 15.0000 13.6979 0.1248 1.3021 2.12R
R denotes an observation with a large standardized residual.
Analysis of Variance for Distance, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Heuristi 9 1316869 1316869 146319 94.59 0.000
Benchmar 7 11757396 11757396 1679628 1085.78 0.000
Rank 2 18144 18144 9072 5.86 0.003
Error 461 713134 713134 1547
Total 479 13805544
Unusual Observations for Distance
Obs Distance Fit SE Fit Residual St Resid
97 1660.62 1820.19 7.83 -159.57 -4.14R
98 1660.62 1820.19 7.83 -159.57 -4.14R
99 1724.21 1833.58 7.83 -109.37 -2.84R
100 1724.21 1833.58 7.83 -109.37 -2.84R
107 1611.28 1690.22 7.83 -78.94 -2.05R
108 1611.28 1690.22 7.83 -78.94 -2.05R
115 1477.48 1368.66 7.83 108.82 2.82R
116 1477.48 1368.66 7.83 108.82 2.82R
117 1472.31 1382.05 7.83 90.26 2.34R
118 1472.31 1382.05 7.83 90.26 2.34R
119 1492.56 1381.33 7.83 111.23 2.89R
120 1492.56 1381.33 7.83 111.23 2.89R
121 1556.90 1715.92 7.83 -159.02 -4.13R
122 1556.90 1715.92 7.83 -159.02 -4.13R
125 1584.77 1728.58 7.83 -143.81 -3.73R
126 1584.77 1728.58 7.83 -143.81 -3.73R
133 1542.40 1460.28 7.83 82.12 2.13R
134 1542.40 1460.28 7.83 82.12 2.13R
139 1468.41 1360.55 7.83 107.86 2.80R
140 1468.41 1360.55 7.83 107.86 2.80R
143 1498.64 1373.22 7.83 125.42 3.25R
144 1505.39 1373.22 7.83 132.17 3.43R
171 1656.24 1572.97 7.83 83.27 2.16R
239 1406.79 1255.80 7.83 150.99 3.92R
326 1254.15 1334.73 7.83 -80.58 -2.09R
327 1440.05 1348.12 7.83 91.93 2.38R
364 1711.85 1611.56 7.83 100.29 2.60R
R denotes an observation with a large standardized residual.
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Trucks
All Pairwise Comparisons among Levels of Heuristi
Heuristi = any subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
gr -0.8423 -0.4375 -0.03272 (--*---)
hi 1.0119 1.4167 1.82145 (---*--)
rar -0.0923 0.3125 0.71728 (---*--)
rcp 0.3036 0.7083 1.11312 (--*--)
rp -0.2173 0.1875 0.59228 (---*--)
rpgr -0.1339 0.2708 0.67562 (--*---)
rr -0.2173 0.1875 0.59228 (---*--)
sc -0.3423 0.0625 0.46728 (---*--)
ss -0.1339 0.2708 0.67562 (--*---)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = gr subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
hi 1.44938 1.8542 2.2589 (--*---)
rar 0.34522 0.7500 1.1548 (--*---)
rcp 0.74105 1.1458 1.5506 (---*--)
rp 0.22022 0.6250 1.0298 (--*---)
rpgr 0.30355 0.7083 1.1131 (--*--)
rr 0.22022 0.6250 1.0298 (--*---)
sc 0.09522 0.5000 0.9048 (--*---)
ss 0.30355 0.7083 1.1131 (--*--)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = hi subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
rar -1.509 -1.104 -0.6994 (---*--)
rcp -1.113 -0.708 -0.3036 (--*--)
rp -1.634 -1.229 -0.8244 (---*--)
rpgr -1.551 -1.146 -0.7411 (--*---)
rr -1.634 -1.229 -0.8244 (---*--)
sc -1.759 -1.354 -0.9494 (---*--)
ss -1.551 -1.146 -0.7411 (--*---)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = rar subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
rcp -0.0089 0.3958 0.8006 (--*---)
rp -0.5298 -0.1250 0.2798 (--*--)
rpgr -0.4464 -0.0417 0.3631 (---*--)
rr -0.5298 -0.1250 0.2798 (--*--)
sc -0.6548 -0.2500 0.1548 (--*--)
ss -0.4464 -0.0417 0.3631 (---*--)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = rcp subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
rp -0.926 -0.5208 -0.1161 (---*--)
rpgr -0.842 -0.4375 -0.0327 (--*---)
rr -0.926 -0.5208 -0.1161 (---*--)
sc -1.051 -0.6458 -0.2411 (---*--)
ss -0.842 -0.4375 -0.0327 (--*---)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = rp subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
rpgr -0.3214 0.0833 0.4881 (---*--)
rr -0.4048 -0.0000 0.4048 (--*--)
sc -0.5298 -0.1250 0.2798 (--*--)
ss -0.3214 0.0833 0.4881 (---*--)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = rpgr subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
rr -0.4881 -0.0833 0.3214 (--*---)
sc -0.6131 -0.2083 0.1964 (--*---)
ss -0.4048 -0.0000 0.4048 (--*--)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = rr subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
sc -0.5298 -0.1250 0.2798 (--*--)
ss -0.3214 0.0833 0.4881 (---*--)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Heuristi = sc subtracted from:
Heuristi Lower Center Upper -----+---------+---------+---------+-
ss -0.1964 0.2083 0.6131 (---*--)
-----+---------+---------+---------+-
-1.2 0.0 1.2 2.4
Tukey Simultaneous Tests
Response Variable Trucks
All Pairwise Comparisons among Levels of Heuristi
Heuristi = any subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
gr -0.4375 0.1281 -3.416 0.0224
hi 1.4167 0.1281 11.062 0.0000
rar 0.3125 0.1281 2.440 0.3018
rcp 0.7083 0.1281 5.531 0.0000
rp 0.1875 0.1281 1.464 0.9063
rpgr 0.2708 0.1281 2.115 0.5169
rr 0.1875 0.1281 1.464 0.9063
sc 0.0625 0.1281 0.488 1.0000
ss 0.2708 0.1281 2.115 0.5169
Heuristi = gr subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
hi 1.8542 0.1281 14.478 0.0000
rar 0.7500 0.1281 5.856 0.0000
rcp 1.1458 0.1281 8.947 0.0000
rp 0.6250 0.1281 4.880 0.0001
rpgr 0.7083 0.1281 5.531 0.0000
rr 0.6250 0.1281 4.880 0.0001
sc 0.5000 0.1281 3.904 0.0038
ss 0.7083 0.1281 5.531 0.0000
Heuristi = hi subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rar -1.104 0.1281 -8.62 0.0000
rcp -0.708 0.1281 -5.53 0.0000
rp -1.229 0.1281 -9.60 0.0000
rpgr -1.146 0.1281 -8.95 0.0000
rr -1.229 0.1281 -9.60 0.0000
sc -1.354 0.1281 -10.57 0.0000
ss -1.146 0.1281 -8.95 0.0000
Heuristi = rar subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rcp 0.3958 0.1281 3.091 0.0621
rp -0.1250 0.1281 -0.976 0.9936
rpgr -0.0417 0.1281 -0.325 1.0000
rr -0.1250 0.1281 -0.976 0.9936
sc -0.2500 0.1281 -1.952 0.6329
ss -0.0417 0.1281 -0.325 1.0000
Heuristi = rcp subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rp -0.5208 0.1281 -4.067 0.0020
rpgr -0.4375 0.1281 -3.416 0.0224
rr -0.5208 0.1281 -4.067 0.0020
sc -0.6458 0.1281 -5.043 0.0000
ss -0.4375 0.1281 -3.416 0.0224
Heuristi = rp subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rpgr 0.0833 0.1281 0.6507 0.9997
rr -0.0000 0.1281 -0.0000 1.0000
sc -0.1250 0.1281 -0.9761 0.9936
ss 0.0833 0.1281 0.6507 0.9997
Heuristi = rpgr subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rr -0.0833 0.1281 -0.651 0.9997
sc -0.2083 0.1281 -1.627 0.8353
ss -0.0000 0.1281 -0.000 1.0000
Heuristi = rr subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
sc -0.1250 0.1281 -0.9761 0.9936
ss 0.0833 0.1281 0.6507 0.9997
Heuristi = sc subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
ss 0.2083 0.1281 1.627 0.8353
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Distance
All Pairwise Comparisons among Levels of Heuristi
Heuristi = any subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
gr -30.59 -5.216 20.16 (--*-)
hi 152.45 177.824 203.20 (-*-)
rar -3.88 21.494 46.87 (-*-)
rcp 35.03 60.411 85.79 (-*-)
rp -19.27 6.108 31.48 (--*-)
rpgr 26.91 52.282 77.66 (-*-)
rr 34.71 60.081 85.46 (-*-)
sc -21.81 3.565 28.94 (-*-)
ss -17.46 7.915 33.29 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = gr subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
hi 157.66 183.040 208.42 (-*-)
rar 1.33 26.710 52.09 (-*-)
rcp 40.25 65.627 91.00 (-*--)
rp -14.05 11.324 36.70 (-*-)
rpgr 32.12 57.498 82.87 (-*-)
rr 39.92 65.297 90.67 (-*--)
sc -16.60 8.781 34.16 (-*-)
ss -12.24 13.131 38.51 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = hi subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
rar -181.7 -156.3 -131.0 (-*-)
rcp -142.8 -117.4 -92.0 (-*-)
rp -197.1 -171.7 -146.3 (-*-)
rpgr -150.9 -125.5 -100.2 (--*-)
rr -143.1 -117.7 -92.4 (-*-)
sc -199.6 -174.3 -148.9 (-*--)
ss -195.3 -169.9 -144.5 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = rar subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
rcp 13.54 38.92 64.293 (-*-)
rp -40.76 -15.39 9.990 (-*-)
rpgr 5.41 30.79 56.165 (--*-)
rr 13.21 38.59 63.964 (-*-)
sc -43.30 -17.93 7.447 (--*-)
ss -38.95 -13.58 11.797 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = rcp subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
rp -79.68 -54.30 -28.93 (-*--)
rpgr -33.50 -8.13 17.25 (-*-)
rr -25.71 -0.33 25.05 (-*-)
sc -82.22 -56.85 -31.47 (-*-)
ss -77.87 -52.50 -27.12 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = rp subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
rpgr 20.80 46.175 71.55 (-*-)
rr 28.60 53.974 79.35 (-*--)
sc -27.92 -2.543 22.83 (-*-)
ss -23.57 1.807 27.18 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = rpgr subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
rr -17.58 7.80 33.18 (-*-)
sc -74.09 -48.72 -23.34 (-*-)
ss -69.74 -44.37 -18.99 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = rr subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
sc -81.89 -56.52 -31.14 (-*-)
ss -77.54 -52.17 -26.79 (-*-)
-------+---------+---------+---------
-120 0 120
Heuristi = sc subtracted from:
Heuristi Lower Center Upper -------+---------+---------+---------
ss -21.03 4.350 29.73 (-*-)
-------+---------+---------+---------
-120 0 120
Tukey Simultaneous Tests
Response Variable Distance
All Pairwise Comparisons among Levels of Heuristi
Heuristi = any subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
gr -5.216 8.028 -0.6497 0.9997
hi 177.824 8.028 22.1494 0.0000
rar 21.494 8.028 2.6772 0.1829
rcp 60.411 8.028 7.5246 0.0000
rp 6.108 8.028 0.7608 0.9991
rpgr 52.282 8.028 6.5122 0.0000
rr 60.081 8.028 7.4836 0.0000
sc 3.565 8.028 0.4440 1.0000
ss 7.915 8.028 0.9859 0.9931
Heuristi = gr subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
hi 183.040 8.028 22.799 0.0000
rar 26.710 8.028 3.327 0.0301
rcp 65.627 8.028 8.174 0.0000
rp 11.324 8.028 1.410 0.9244
rpgr 57.498 8.028 7.162 0.0000
rr 65.297 8.028 8.133 0.0000
sc 8.781 8.028 1.094 0.9854
ss 13.131 8.028 1.636 0.8308
Heuristi = hi subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rar -156.3 8.028 -19.47 0.0000
rcp -117.4 8.028 -14.62 0.0000
rp -171.7 8.028 -21.39 0.0000
rpgr -125.5 8.028 -15.64 0.0000
rr -117.7 8.028 -14.67 0.0000
sc -174.3 8.028 -21.71 0.0000
ss -169.9 8.028 -21.16 0.0000
Heuristi = rar subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rcp 38.92 8.028 4.847 0.0001
rp -15.39 8.028 -1.916 0.6579
rpgr 30.79 8.028 3.835 0.0049
rr 38.59 8.028 4.806 0.0001
sc -17.93 8.028 -2.233 0.4337
ss -13.58 8.028 -1.691 0.8008
Heuristi = rcp subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rp -54.30 8.028 -6.764 0.0000
rpgr -8.13 8.028 -1.012 0.9916
rr -0.33 8.028 -0.041 1.0000
sc -56.85 8.028 -7.081 0.0000
ss -52.50 8.028 -6.539 0.0000
Heuristi = rp subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rpgr 46.175 8.028 5.7514 0.0000
rr 53.974 8.028 6.7228 0.0000
sc -2.543 8.028 -0.3167 1.0000
ss 1.807 8.028 0.2251 1.0000
Heuristi = rpgr subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
rr 7.80 8.028 0.971 0.9938
sc -48.72 8.028 -6.068 0.0000
ss -44.37 8.028 -5.526 0.0000
Heuristi = rr subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
sc -56.52 8.028 -7.040 0.0000
ss -52.17 8.028 -6.498 0.0000
Heuristi = sc subtracted from:
Level Difference SE of Adjusted
Heuristi of Means Difference T-Value P-Value
ss 4.350 8.028 0.5419 1.000
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