Title: Dataplot RTF Document
Author: Alan Heckert
Company: Statistical Engineering Division, NIST

Consensus Means Analysis
              (Full Sample Case) 

Data Summary: |
Response Variable: | ALITE
Lab-ID Variable: | LABID
Total Number of Observations: | 46
Grand Mean: | 57.2260857
Grand Standard Deviation: | 1.4274194
Total Number of Labs: | 5
Minimum Lab Mean: | 56.5000000
Maximum Lab Mean: | 61.1999969
Minimum Lab SD: | 0.1414219
Maximum Lab SD: | 1.6800299
Within Lab (pooled) SD: | 0.8369111
Within Lab (pooled) Variance: | 0.7004202

Table 1: Summary Statistics by Lab

Lab ID

n(i)

Mean

Variance

Standard
Deviation

Standard Deviation
of the Mean

1 | 36 | 56.7527771 | 0.5522779 | 0.7431540 | 0.1238590
2 | 4 | 58.4249992 | 2.8225005 | 1.6800299 | 0.8400149
3 | 2 | 56.5000000 | 0.1799991 | 0.4242630 | 0.2999992
4 | 2 | 60.0999985 | 0.0200002 | 0.1414219 | 0.1000004
5 | 2 | 61.1999969 | 0.7200009 | 0.8485287 | 0.6000004

 
1. Method: Mandel-Paule |
Estimate of (unscaled) Consensus Mean: | 58.5663223
Estimate of (scaled) Consensus Mean: | 0.4396437
Between Lab Variance (unscaled): | 4.0465660
Between Lab SD (unscaled): | 2.0116079
Between Lab Variance (scaled): | 0.1831857
Standard Deviation of Consensus Mean: | 0.8317266
Standard Uncertainty (k = 1): | 0.8317266
Expanded Uncertainty (k = 2): | 1.6634532
Expanded Uncertainty (k =  1.9599645): | 1.6301546
Normal PPF of 0.975: | 1.9599645
Lower 95% (normal) Confidence Limit: | 56.9361687
Upper 95% (normal) Confidence Limit: | 60.1964760
Note: Mandel-Paule Best Usage:
6 or More Labs:

 
2. Method: Modified Mandel-Paule |
Estimate of (unscaled) Consensus Mean: | 58.5590630
Estimate of (scaled) Consensus Mean: | 0.4380985
Between Lab Variance (unscaled): | 3.2046051
Between Lab SD (unscaled): | 1.7901411
Between Lab Variance (scaled): | 0.1450706
Standard Deviation of Consensus Mean: | 0.8338748
Standard Uncertainty (k = 1): | 0.8338748
Expanded Uncertainty (k = 2): | 1.6677495
Expanded Uncertainty (k =  1.9599645): | 1.6343650
Normal PPF of 0.975: | 1.9599645
Lower 95% (normal) Confidence Limit: | 56.9246979
Upper 95% (normal) Confidence Limit: | 60.1934280
Note: Modified Mandel-Paule Best Usage:
6 or More Labs:

3. Method: Vangel-Rukhin ML |
Estimate of (unscaled) Consensus Mean: | 58.5534592
Estimate of (scaled) Consensus Mean: | 0.4369068
Between Lab Variance (unscaled): | 3.2312329
Between Lab SD (unscaled): | 1.7975631
Between Lab Variance (scaled): | 0.1462760
Standard Deviation of Consensus Mean: | 0.8306379
Standard Uncertainty (k = 1): | 0.8306379
Expanded Uncertainty (k = 2): | 1.6612757
Expanded Uncertainty (k =  1.9599645): | 1.6280208
Normal PPF of 0.975: | 1.9599645
Lower 95% (normal) Confidence Limit: | 56.9254379
Upper 95% (normal) Confidence Limit: | 60.1814804
Note: Vangel-Rukhin ML Best Usage:
6 or More Labs:

4. Method: BOB (Bound on Bias) |
Estimate of Consensus Mean: | 58.5955544
Within Lab Uncertainty: | 0.2173445
Between Lab Uncertainty: | 1.3567723
Standard Uncertainty (k = 1): | 1.3740704
Expanded Uncertainty (k = 2): | 2.7481408
Lower 95% (normal) Confidence Limit: | 55.8474121
Upper 95% (normal) Confidence Limit: | 61.3436966
Note: BOB Best Usage:
5 or Fewer Labs:

5. Method: Schiller-Eberhardt |
Estimate of  Consensus Mean: | 58.5908279
Estimate of Variance of Mean: | 0.0169179
Bias Allowance: | 2.6091690
Sigmah (heterogeneity): | 0.0000000
Degrees of Freedom for Sigmah: | 1
Standard Uncertainty (k = 1): | 2.7392378
Expanded Uncertainty (k = 2): | 2.8693066
Expanded Uncertainty (k =  2.3645761): | 2.9167264
t Percent Point Value of 0.975: | 2.3645761
Degrees of Freedom: | 7
Lower 95% (normal) Confidence Limit: | 55.6741028
Upper 95% (normal) Confidence Limit: | 61.5075531
Note: Schiller-Eberhardt Best Usage:
5 or Fewer Labs:

6. Method: Mean of Means |
Mean of Lab Means: | 58.5955544
Standard Deviation of Lab Means: | 2.0532134
Standard Uncertainty (sd/sqrt(n)): | 0.9182249
SD of Consensus Mean (sd/sqrt(n)): | 0.9182249
Standard Uncertainty (k = 1): | 0.9182249
Expanded Uncertainty (k = 2): | 1.8364499
Expanded Uncertainty (k =  2.7764461): | 2.5494020
Degrees of Freedom: | 4
t Percent Point Value of 0.975: | 2.7764461
Lower 95% (normal) Confidence Limit: | 56.0461540
Upper 95% (normal) Confidence Limit: | 61.1449547
Note: Mean of Means Best Usage:
Any Number of Labs:

7. Method: Graybill-Deal |
Estimate of Consensus Mean: | 58.6732941
Estimate of Variance (Sinha): | 0.0128360
Estimate of Variance (naive): | 0.0055405
Standard Uncertainty (k = 1): | 0.1132961
Expanded Uncertainty (k = 2): | 0.2265923
Normal PPF of 0.975: | 2.7764461
Lower 95% (Rukhin) Confidence Limit: | 56.2782021
Upper 95% (Rukhin) Confidence Limit: | 61.0683861
Note: Graybill-Deal Best Usage:
Any Number of Labs,
but no Between Lab Variance

8. Method: Grand Mean (No Lab Effect |
Mean of All Data | 57.2260857
Standard Deviation of All Data: | 2.0532134
SD of Consensus Mean (sd/sqrt(n)): | 0.3027298
Standard Uncertainty (k = 1): | 0.3027298
Expanded Uncertainty (k = 2): | 0.6054596
Expanded Uncertainty (k =  2.0141039): | 0.6097292
Degrees of Freedom: | 45
t Percent Point Value of 0.975: | 2.0141039
Lower 95% (t-value) Confidence Limit: | 56.6163559
Upper 95% (t-value) Confidence Limit: | 57.8358154
Note: Grand Mean Best Usage:
Any Number of Labs,
but no Lab-to-Lab Differences

9. Method: Generalized CI |
Estimate of Consensus Mean: | 58.4525528
Standard Uncertainty (k = 1): | 1.2792627
Expanded Uncertainty (k = 2): | 2.5585253
Lower 95% (simulation) Confidence Limit: | 55.9661942
Upper 95% (simulation) Confidence Limit: | 61.0020752
Note: Generalized CI Best Usage:
Any Number of Labs

10. Method: DerSimonian-Laird |
Estimate of Consensus Mean: | 58.5719872
Estimate of Variance of Consensus Mean: | 0.8636000
Estimate of Between-Lab Variance: | 5.0619206
Standard Uncertainty (k = 1): | 0.9293008
Expanded Uncertainty (k = 2): | 1.8586016
Lower 95% (t-value) Confidence Limit: | 55.9918327
Upper 95% (t-value) Confidence Limit: | 61.1521416
Lower 95% (Rukhin) Confidence Limit: | 56.0077209
Upper 95% (Rukhin) Confidence Limit: | 61.1362534
Note: DerSimonian-Laird Best Usage:
Any Number of Labs:

12. Method: BCP (Bayesian Consensus Proced |
Estimate of Consensus Mean: | 58.5955544
Consensus Mean Standard Deviation: | 1.3627561
Standard Uncertainty (k = 1): | 1.3627561
Expanded Uncertainty (k = 2): | 2.7255123
Degrees of Freedom: | 3.8943412
t Percent Point Value: | 2.8064096
Lower 95% (t) Confidence Limit: | 54.7711029
Upper 95% (t) Confidence Limit: | 62.4200058
Note: BCP Best Usage:
6 or Fewer Labs:

Table 2: 95% Confidence Limits

Method

Consensus
Mean

Lower
Limit

Upper
Limit

1. Mandel-Paule | 58.5663223 | 56.9361687 | 60.1964760
2. Modified Mandel-Paule | 58.5590630 | 56.9246979 | 60.1934280
3. Vangel-Rukhin ML | 58.5534592 | 56.9254379 | 60.1814804
4. BOB | 58.5955544 | 55.8474121 | 61.3436966
5. Schiller-Eberhardt | 58.5908279 | 55.6741028 | 61.5075531
6. Mean of Means | 58.5955544 | 56.0461540 | 61.1449547
7. Graybill-Deal | 58.6732941 | 56.2782021 | 61.0683861
8. Grand Mean | 57.2260857 | 56.6163559 | 57.8358154
9. Generalized CI | 58.4525528 | 55.9661942 | 61.0020752
10. DerSimonian-Laird (t) | 58.5719872 | 55.9918327 | 61.1521416
DerSimonian-Laird (Rukhin) | 58.5719872 | 56.0077209 | 61.1362534
12. BCP | 58.5955544 | 54.7711029 | 62.4200058

Table 3: Standard Uncertainties (k = 1)

Method

Consensus
Mean

Standard
Uncertainty
(k=1)

Relative
Standard
Uncertainty (%)

1. Mandel-Paule | 58.5663223 | 0.8317266 | 1.4201448
2. Modified Mandel-Paule | 58.5590630 | 0.8338748 | 1.4239892
3. Vangel-Rukhin ML | 58.5534592 | 0.8306379 | 1.4185975
4. BOB | 58.5955544 | 1.3740704 | 2.3450079
5. Schiller-Eberhardt | 58.5908279 | 2.7392378 | 4.6751986
6. Mean of Means | 58.5955544 | 0.9182249 | 1.5670557
7. Graybill-Deal | 58.6732941 | 0.1132961 | 0.1930966
8. Grand Mean | 57.2260857 | 0.3027298 | 0.5290066
9. Generalized CI | 58.4525528 | 1.2792627 | 2.1885488
10. DerSimonian-Laird | 58.5719872 | 0.9293008 | 1.5865959
12. BCP | 58.5955544 | 1.3627561 | 2.3256989

Table 4: Expanded Uncertainties (k = 2)

Method

Consensus
Mean

Expanded
Expandednty
(k=2)

Relative
Standard
Uncertainty (%)

1. Mandel-Paule | 58.5663223 | 1.6634532 | 2.8402896
2. Modified Mandel-Paule | 58.5590630 | 1.6677495 | 2.8479784
3. Vangel-Rukhin ML | 58.5534592 | 1.6612757 | 2.8371949
4. BOB | 58.5955544 | 2.7481408 | 4.6900158
5. Schiller-Eberhardt | 58.5908279 | 2.8693066 | 4.8971944
6. Mean of Means | 58.5955544 | 1.8364499 | 3.1341114
7. Graybill-Deal | 58.6732941 | 0.2265923 | 0.3861932
8. Grand Mean | 57.2260857 | 0.6054596 | 1.0580132
9. Generalized CI | 58.4525528 | 2.5585253 | 4.3770976
10. DerSimonian-Laird | 58.5719872 | 1.8586016 | 3.1731918
12. BCP | 58.5955544 | 2.7255123 | 4.6513977