
CONSENSUS MEANName:
There are a number of approaches to this problem. The Dataplot CONSENSUS MEANS command computes estimates for a variety of methods and does not specify which is the most appropriate method for a given data set. Consult with a statistician for guidance on which method is most appropriate for your data.
In this case, the consensus mean is simply the grand mean of all the data and a confidence interval for the consensus mean is simply the standard tbased condidence interval:
where \( \bar{X} \) is the overall mean, t is the percent point function for the t distribution, s is the standard deviation of all the points, and n is the total number of points. The assumption of no lab effect is unrealistic in almost all cases. However, we include the grand mean method as a reference point as it gives an indication of how including the lab effects changes the estimate of the consensus mean and its uncertainty.
For this method, we compute the mean for each of the k laboratories. Then we compute \( \bar{X} \) and s as the mean and standard deviation of these k means. The estimate of the consensus mean is simply \( \bar{X} \) and we compute the following confidence interval for the consensus mean:
The limitations of this method are discussed in the "An ISO GUM Approach to Combining Results from Multiple Methods" paper (see the Reference section). For this method, the consensus mean estimate is an equiweighted mean with no regard to possible differences in withinlab variation or withinlab sample sizes. The advantages of this method are that it is robust and simple to compute. The primary disadvantage is that no consideration is given to possible differences in the withinlab variation and sample sizes. If the laboratory means are not normally distributed (e.g., due to the presence of outliers), this can distort the mean of means estimates. Two more robust procudures are available. The median of means estimate takes the median of the laboratory means. The associated uncertainty is
with \( \tilde{x} \), k, and MADe denoting the number of laboratories, the median of the laboratory means, and the scaled median absolute deviation (the scaled median absolute deviation is the median absolute deviation divided by 0.67449) of the laboratory means, respectively. It is recommended that at least five laboratories be available for this uncertainty to be reliable. The Huber mean of means is based on the Huber's H15 robust mean of the laboratory means. The associated uncertainty is
with \( \hat{\sigma_{\tiny H15}} \) denoting the H15 estimate of scale. The e parameter is a tuning constant that is set to 0.95. The details of the H15 location and scale estimators are given elsewhere. These robust estimators are discussed in the CCQM Guidance Note (see References below). These robust estimators are more commonly used in the context of interlaboratory studies rather than for certifying reference materials. For certified reference materials, laboratories and methods are carefully chosen so outliers are less often a problem. Interlaboratory studies typically involve a greater range of laboratories with a wider range of capabilities and outliers are more likely to be an issue.
with independent Gaussian errors \( e_{ij} \sim N(0,\kappa_{i}^{2}) \). All parameters \( \mu \), \( \kappa_{i}^{2} \) i = 1, ...., k are unknown and the goal is to estimate \( \mu \), determine its standard error, and to provide a confidence interval for \( \mu \). Unbiased estimates of the within lab means and variances \( \sigma_{i}^{2} = \kappa_{i}^{2}/n_{i} \)
\( s_{i}^2 = \sum_{j=1}^{n_i}{\frac{(x_{ij}  x_{i})^2}{n_{i}(n_{i}  1)}} \) When the variaces \( \sigma_{i}^2 \) are known, the best, in terms of mean squared error, unbiased estimator of the reference value \( \mu \) is the weighted means statistics
with \( w_{i} = 1/\sigma_{i}^2 \). The formula for the variance is
In practice, these within lab variances are unknown and so the true w_{i} are also unknown. The GraybillDeal method is based on this model. In the GraybillDeal model, the estimate of the consensus mean is
Dataplot supports four methods for computing the variance of the GraybillDeal consensus mean.
Dataplot currently generates confidence intervals for the GraybillDeal method using a method proposed by Rukhin (private communication). This method generates conservative intervals. The GraybillDeal approach has the following limitations
where there are i = 1, ..., k labs and j = 1, .... n_{i} observations for each lab. In this model, \( \mu \) denotes the consensus mean, b_{i} is the lab effect and e_{ij} is the error term. The b_{i} are distributed as N(0,\( \sigma^2 \)) and the e_{ij} are distributed as N(0,\( \sigma_{i}^2 \)). That is, \( \sigma_{i}^2 \) are the within lab variances and \( \sigma^2 \) is the between lab variance. For convenience, define the following terms:
The MandelPaule, modified MandelPaule, maximum likelihood (ML), DerSimonianLaird, and generalized confidence interval methods are based on this model. We will discuss each of these in turn.
Answers to the above questions will determine how to appropriately weight the labs. The consensus mean will be a weighted mean of the lab means. The weighting can be either fixed (i.e., equal weights) or variable where the variable weights can be based on both engineering and statistical considerations. If the engineering decision is made to treat all labs as equal in importance, then from a statistical point of view the analysis consists primarily of the following two steps:
An additional third step is to carry out formal statistical tests to identify potentially outlying labs. A statistically unsolvable question that persists here is that just because a lab appears "different" does not necessarily mean that the lab is wrong (i.e., biased). The spectre that all of the consistent labs being selfbehaved but biased is a real possibility which can only be solved by engineering judgement.
where <y> is a response variable; <tag> is a lab id variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes the consensus means based on the raw data.
<SUBSET/EXCEPT/FOR qualification> where <ymean> is a variable containing the lab means; <ysd> is a variable containing the lab standard deviations; <ni> is a variable containing the lab sample sizes; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes the consensus means based on the lab means, standard deviations, and sample sizes.
<SUBSET/EXCEPT/FOR qualification> where <ymean> is a variable containing the lab means; <ysd> is a variable containing the lab standard deviations; <ni> is a variable containing the lab sample sizes; <labid> is a variable containing the labid (numeric values); and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax computes the consensus means based on the mean, standard deviation, and sample size for each lab. The <labid> is used for identification purposes and is not used in the computations.
CONSENSUS MEANS Y1 GROUP SUBSET GROUP > 2 CONSENSUS MEANS YMEAN YSD NI
The following variables are written to the file dpst1f.dat. These are the statistics for the labs.
The following variables are written to the file dpst2f.dat. This is the information contained in table 2 of the CONSENSUS MEAN output. These variables can be used to make plots of the consensus mean results.
The following variables are written to the file dpst3f.dat. This is the information contained in table 3 of the CONSENSUS MEAN output. These variables can be used to generate plots of the consensus mean results.
The following variables are written to the file dpst4f.dat. This is the information contained in table 4 of the CONSENSUS MEAN output. These variables can be used to generate plots of the consensus mean results.
The following variables are written to the file dpst5f.dat.
If you want to use an exponential format (E15.7), enter
HELP CAPTURE LATEX for details.
Although the BOB procedure is not recommened when there are more than five laboratories, it is not automatically suppressed in this case. The Fairweather method requires that each lab have a minimum of five measurements. If at least one lab has five or fewer measurements, then the Fairweather method is automatically suppressed.
If this situation is encountered, the following methods can still use the data from that lab
For the remaining methods, these labs will be automatically omitted from the consensus means analysis.
If you have summary data, it may not always be available in this form. Specifically, the following types of summary data are sometimes encountered.
To address these cases, the summary data may be entered in the following ways.
Your data may contain a mix of labs where some have the standard deviation and sample size and others where a standard uncertainty is provided. This is allowed, but the following methods will be suppressed if any of the sample sizes has a nonpositive value
If you have raw data, you can enter one of the following
LET A = DERSIMONIAN LAIRD STANDARD ERROR Y X LET A = DERSIMONIAN LAIRD HHD Y X LET A = DERSIMONIAN LAIRD MINMAX Y X LET A = MANDEL PAULE Y X LET A = MANDEL PAULE STANDARD ERROR Y X LET A = MODIFIED MANDEL PAULE Y X LET A = MODIFIED MANDEL PAULE STANDARD ERROR Y X LET A = VANGEL RUKHIN Y X LET A = VANGEL RUKHIN STANDARD ERROR Y X LET A = GENERALIZED CONFIDENCE INTERVAL Y X LET A = GENERALIZED CONFIDENCE INTERVAL STANDARD ERROR Y X LET A = BOB Y X LET A = BOB STANDARD ERROR Y X LET A = BCP Y X LET A = BCP STANDARD ERROR Y X LET A = MEAN OF MEANS Y X LET A = MEAN OF MEANS STANDARD ERROR Y X LET A = FAIRWEATHER Y X LET A = FAIRWEATHER STANDARD ERROR Y X LET A = SCHILLEREBERHARDT Y X LET A = SCHILLEREBERHARDT STANDARD ERROR Y X LET A = GRAYBILL DEAL Y X LET A = GRAYBILL DEAL SINHA STANDARD ERROR Y X LET A = GRAYBILL DEAL NAIVE STANDARD ERROR Y X LET A = GRAYBILL DEAL ZHANG ONE STANDARD ERROR Y X LET A = GRAYBILL DEAL ZHANG TWO STANDARD ERROR Y X LET A = LINEAR POOL Y X LET A = LINEAR POOL STANDARD ERROR Y X If you have summary data, you can enter one of the following
LET A = SUMMARY DERSIMONIAN LAIRD STANDARD ERROR MEAN SD N LET A = SUMMARY DERSIMONIAN LAIRD HHD MEAN SD N LET A = SUMMARY DERSIMONIAN LAIRD MINMAX MEAN SD N LET A = SUMMARY MANDEL PAULE MEAN SD N LET A = SUMMARY MANDEL PAULE STANDARD ERROR ... MEAN SD N LET A = SUMMARY MODIFIED MANDEL PAULE MEAN SD N LET A = SUMMARY MODIFIED MANDEL PAULE STANDARD ERROR MEAN SD N LET A = SUMMARY VANGEL RUKHIN MEAN SD N LET A = SUMMARY VANGEL RUKHIN STANDARD ERROR MEAN SD N LET A = SUMMARY GENERALIZED CONFIDENCE INTERVAL MEAN SD N LET A = SUMMARY GENERALIZED CONFIDENCE INTERVAL ... STANDARD ERROR MEAN SD N LET A = SUMMARY BOB MEAN SD N LET A = SUMMARY BOB STANDARD ERROR MEAN SD N LET A = SUMMARY BCP MEAN SD N LET A = SUMMARY BCP STANDARD ERROR MEAN SD N LET A = SUMMARY MEAN OF MEANS MEAN SD N LET A = SUMMARY MEAN OF MEANS STANDARD ERROR MEAN SD N LET A = SUMMARY FAIRWEATHER MEAN SD N LET A = SUMMARY FAIRWEATHER STANDARD ERROR MEAN SD N LET A = SUMMARY SCHILLEREBERHARDT MEAN SD N LET A = SUMMARY SCHILLEREBERHARDT STANDARD ERROR MEAN SD N LET A = SUMMARY GRAYBILL DEAL MEAN SD N LET A = SUMMARY GRAYBILL DEAL SINHA STANDARD ERROR MEAN SD N LET A = SUMMARY GRAYBILL DEAL NAIVE STANDARD ERROR MEAN SD N LET A = SUMMARY GRAYBILL DEAL ZHANG ONE STANDARD ERROR ... MEAN SD N LET A = SUMMARY GRAYBILL DEAL ZHANG TWO STANDARD ERROR ... MEAN SD N LET A = SUMMARY LINEAR POOL YMEAN YSD N LET A = SUMMARY LINEAR POOL STANDARD ERROR YMEAN YSD N LET A = LINEAR POOL STANDARD ERROR Y X Dataplot statistics can be used in a number of other commands. For details, enter
For the SUMMARY cases, bootstrapping is not currently supported. However, we anticipate adding this capability in a subsequent release.
The following commands are available, but are for methods that are still under development. These commands should not currently be used.
SET LINEAR POOL SAMPLE SIZE <value> The WEIGHTS option lets you specify the name of a variable that contains the weights for the labs. By default, equal weights are used for all labs. Setting <varname> to OFF (or DEFAULT or NONE) will reset to equal weights. The SAMPLE SIZE option lets you specify the number of samples that are drawn. The default value is 50,000 and this value should typically be in the range 10,000 to 100,000. The linear pool sampling can sometimes result in a multimodal distribution. This may indicate that the linear pool method is not appropriate. If the linear pool method is turned on, the sample values are written to the file dpst5f.dat. Note that several methods write to this file. Look for the line "VALUES FROM LINEAR POOL METHOD". So if this string is on line 101 of dpst5f.dat, you can do something like
Graybill and Deal (1959), "Combining Unbiased Estimators", Biometrics, 15, pp. 543550. M. S. Levenson, D. L. Banks, K. R. Eberhardt, L. M. Gill, W. F. Guthrie, H. K. Liu, M. G. Vangel, J. H. Yen, and N. F. Zhang (2000), "An ISO GUM Approach to Combining Results from Multiple Methods", Journal of Research of the National Institute of Standards and Technology, Volume 105, Number 4. John Mandel and Robert Paule (1970), "Interlaboratory Evaluation of a Material with Unequal Number of Replicates", Analytical Chemistry, 42, pp. 11941197. Robert Paule and John Mandel (1982), "Consensus Values and Weighting Factors", Journal of Research of the National Bureau of Standards, 87, pp. 377385. Andrew Rukhin (2009), "Weighted Means Statistics in Interlaboratory Studies", Metrologia, Vol. 46, pp. 323331. Andrew Ruhkin (2003), "Two Procedures of Metaanalysis in Clinical Trials and Interlaboratory Studies", Tatra Mountains Mathematical Publications, 26, pp. 155168. Andrew Ruhkin and Mark Vangel (1998), "Estimation of a Common Mean and Weighted Means Statistics", Journal of the American Statistical Association, Vol. 93, No. 441. Andrew Ruhkin, B. Biggerstaff, and Mark Vangel (2000), "Restricted Maximum Likelihood Estimation of a Common Mean and MandelPaule Algorithm", Journal of Statistical Planning and Inference, 83, pp. 319330. Mark Vangel and Andrew Ruhkin (1999), "Maximum Likelihood Analysis for Heteroscedastic OneWay Random Effects ANOVA in Interlaboratory Studies", Biometrics 55, 129136. Susannah Schiller and Keith Eberhardt (1991), "Combining Data from Independent Analysis Methods", Spectrochimica, ACTA 46 (12). Susannah Schiller (1996), "Standard Reference Materials: Statistical Aspects of the Certification of Chemical SRMs", NIST SP 260125, NIST, Gaithersburg, MD. Bimal Kumar Sinha (1985), "Unbiased Estimation of the Variance of the GraybillDeal Estimator of the Common Mean of Several Normal Populations", The Canadian Journal of Statistics, Vol. 13, No. 3, pp. 243247. NienFan Zhang (2006), "The Uncertainty Associated with The Weighted Mean of Measurement Data", Metrologia, 43, PP. 195204. Hagwood and Guthrie (2006), "Combining Data in Small MultipleMethods Studies", Technometrics, Vol. 48, No. 2. Iyer, Wang, and Matthew (2004), "Models and Confidence Intervals for True Values in Interlaboratory Trials", Journal of the American Statistical Association, Vol. 99, No. 468, pp. 10601071. Fairweather (1972), "A Method for Obtaining an Exact Confidence Interval for the Common Mean of Several Normal Populations", Applied Statistics, Vol. 21, pp. 229233. Cox (2002), "The Evaluation of Key Comparison Data", Metrologia, Vol. 39, pp. 589595. "CCQM Guidance note: Estimation of a Consensus KCRV and associated Degrees of Equivalence", Version 10, 2013. Stone (1961), "The Opinion Pool", Annals of Mathematical Statistics, Vol. 32, pp. 13391342. Duewer (2004), "A Robust Approach for the determination of CCQM Key Comparison Reference Values and Uncertinties", Technical Report Consultultive Committee for Amount of Substance: Metrology in Chemistry (CCQM) International Bureau of Weights and Measures (BIPM), Sevres, France (9th Annual Meeting, Working Document CCQM/0415). Koepke, Lafarge, Possolo and Toman (2017), "Consensus Building for Interlaboratory Studies, Key Comparisons, and MetaAnalysis", Metrologia, Vol. 54, S34S62.
2002/10: Support for Latex and HTML output 2006/3: Reformat output for consistency and clarity Add Tables 3 and 4 to the output Updated the GraybillDeal method Added the DerSimonianLaird method Added the generalized confidence intervals method Added support for Rich Text Format (RTF) output Added support for SET WRITE DECIMALS 2006/04: Added the Fairweather method 2006/06: Added the Bayesian Consensus Procedure method 2010/06: Five methods can use labs with zero standard deviations 2011/11: For summary data, add optional labid variable 2014/10: For summary data, option to input mean and uncertainty (i.e., s/sqrt(n)) instead of s and n. Not all methods supported for this case. 2017/03: Added support for median of means and Huber mean of means methods. 2017/07: Changed the default for SchillerEberhardt, Fairweather, and Bayesian consensus procedure to OFF. 2023/04: Added support for linear pool method SKIP 25 READ STUTZ86.DAT ALITE JUNK2 JUNK3 JUNK4 JUNK5 LABID . FEEDBACK OFF CONSENSUS MEANS ALITE LABIDThe following output is generated: Consensus Means Analysis (Full Sample Case) Data Summary: Response Variable: ALITE LabID Variable: LABID Number of Observations: 46 Grand Mean: 57.22609 Grand Standard Deviation: 1.42742 Total Number of Labs: 5 Minimum Lab Mean: 56.50000 Maximum Lab Mean: 61.20000 Minimum Lab SD: 0.14142 Maximum Lab SD: 1.68003 Mean of Lab Means: 58.59556 SD of Lab Means: 2.05321 SD of Lab Means (wrt to grand mean): 2.56125 Within Lab (pooled) SD: 0.83691 Within Lab (pooled) Variance: 0.70042 Table 1: Summary Statistics by Lab  Standard Lab Standard Deviation ID n(i) Mean Variance Deviation of the Mean  1 36 56.75278 0.55228 0.74315 0.12386 2 4 58.42500 2.82250 1.68003 0.84001 3 2 56.50000 0.18000 0.42426 0.30000 4 2 60.10000 0.02000 0.14142 0.10000 5 2 61.20000 0.72000 0.84853 0.60000 1. Method: MandelPaule Estimate of (unscaled) Consensus Mean: 58.56633 Estimate of (scaled) Consensus Mean: 0.43964 Between Lab Variance (unscaled): 4.04657 Between Lab SD (unscaled): 2.01161 Between Lab Variance (scaled): 0.18319 Standard Deviation of Consensus Mean: 0.83173 Standard Uncertainty (k = 1): 0.83173 Expanded Uncertainty (k = 2): 1.66345 Expanded Uncertainty (k = 1.9599640): 1.63016 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 56.93617 Upper 95% (normal) Confidence Limit: 60.19648 Note: MandelPaule Best Usage: 6 or More Labs: 2. Method: Modified MandelPaule Estimate of (unscaled) Consensus Mean: 58.55906 Estimate of (scaled) Consensus Mean: 0.43810 Between Lab Variance (unscaled): 3.20461 Between Lab SD (unscaled): 1.79014 Between Lab Variance (scaled): 0.14507 Standard Deviation of Consensus Mean: 0.83388 Standard Uncertainty (k = 1): 0.83388 Expanded Uncertainty (k = 2): 1.66775 Expanded Uncertainty (k = 1.9599640): 1.63437 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 56.92470 Upper 95% (normal) Confidence Limit: 60.19343 Note: Modified MandelPaule Best Usage: 6 or More Labs: 3. Method: VangelRukhin Maximum Likelihood Estimate of (unscaled) Consensus Mean: 58.55346 Estimate of (scaled) Consensus Mean: 0.43691 Between Lab Variance (unscaled): 3.23124 Between Lab SD (unscaled): 1.79756 Between Lab Variance (scaled): 0.14628 Standard Deviation of Consensus Mean: 0.83064 Standard Uncertainty (k = 1): 0.83064 Expanded Uncertainty (k = 2): 1.66128 Expanded Uncertainty (k = 1.9599640): 1.62802 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 56.92544 Upper 95% (normal) Confidence Limit: 60.18148 Note: VangelRukhin Maximum Likelihood Best Usage: 6 or More Labs 4a. Method: DerSimonian Laird (original variance) Estimate of Consensus Mean: 58.55450 Estimate of Variance of Consensus Mean: 0.60832 Estimate of Between Lab Variance: 2.82722 Standard Uncertainty (k = 1): 0.77995 Expanded Uncertainty (k = 2): 1.55990 Degrees of Freedom: 4 t Percent Point Value: 2.77645 Lower 95% (tvalue) Confidence Limit: 56.38900 Upper 95% (tvalue) Confidence Limit: 60.71999 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4b. Method: DerSimonian Laird  HornHornDuncan Variance Estimate of Consensus Mean: 58.55450 Estimate of Variance of Consensus Mean: 0.87653 Estimate of Between Lab Variance: 2.82722 Standard Uncertainty (k = 1): 0.93623 Expanded Uncertainty (k = 2): 1.87246 Degrees of Freedom: 4 t Percent Point Value: 2.77645 Lower 95% (tvalue) Confidence Limit: 55.95511 Upper 95% (tvalue) Confidence Limit: 61.15389 Note: DerSimonianLaird Best Usage: Any Number of Labs: 5. Method: GraybillDeal Estimate of Consensus Mean: 58.67330 Estimate of Variance (Naive): 0.00554 Standard Uncertainty (Naive) (k = 1): 0.07443 Expanded Uncertainty (Naive) (k = 2): 0.14887 Lower 95% (Rukhin) Confidence Limit: 54.47558 Upper 95% (Rukhin) Confidence Limit: 62.87103 Note: GraybillDeal Best Usage: Any Number of Labs, but no Between Lab Variance 7. Method: Generalized Confidence Intervals Estimate of Consensus Mean: 58.45256 Standard Uncertainty (k = 1): 1.27926 Expanded Uncertainty (k = 2): 2.55853 Lower 95% (Simulation) Confidence Limit: 55.96620 Upper 95% (Simulation) Confidence Limit: 61.00745 Note: Generalized Confidence Interval Best Usage: Any Number of Labs: 8. Method: Grand Mean (No Lab Effect) Mean of All Data: 57.22609 Standard Deviation of All Data: 2.05321 SD of Consensus Mean (sd/sqrt(n)): 0.30273 Standard Uncertainty (k = 1): 0.30273 Expanded Uncertainty (k = 2): 0.60546 Expanded Uncertainty (k = 2.0141034): 0.60973 Degrees of Freedom: 45 t Percent Point Value (alpha = 0.05) 2.01410 Lower 95% (tvalue) Confidence Limit: 56.61636 Upper 95% (tvalue) Confidence Limit: 57.83582 Note: Grand Mean Best Usage: Any Number of Labs, but no LabtoLab Differences 9. Method: Mean of Means Mean of Lab Means: 58.59556 Standard Deviation of Lab Means: 2.05321 Standard Uncertainty (sd/sqrt(n)): 0.91823 SD of Consensus Mean (sd/sqrt(n)): 0.91823 Standard Uncertainty (k = 1): 0.91823 Expanded Uncertainty (k = 2): 1.83645 Expanded Uncertainty (k = 2.7764451): 2.54940 Degrees of Freedom: 4 t Percent Point Value (alpha = 0.05): 2.77645 Lower 95% (normal) Confidence Limit: 56.04615 Upper 95% (normal) Confidence Limit: 61.14496 Note: Mean of Means Best Usage: Any Number of Labs: 11. Method: BOB (Bound on Bias) Estimate of Consensus Mean: 58.59556 Within Lab Uncertainty: 0.21734 Between Lab Uncertainty: 1.35677 Standard Uncertainty (k = 1): 1.37407 Expanded Uncertainty (k = 2): 2.74814 Lower 95% (k = 2) Confidence Limit: 55.84741 Upper 95% (k = 2) Confidence Limit: 61.34370 Note: BOB Best Usage: 5 or Fewer Labs: 12. Method:SchillerEberhardt Estimate of Consensus Mean: 58.59083 Estimate of Variance of Mean: 0.01692 Bias Allowance: 2.60917 Sigmah (heterogeneity): 0.00000 Degrees of Freedom for Sigmah: 1 Standard Uncertainty (k = 1): 2.73924 Expanded Uncertainty (k = 2): 2.86931 Expanded Uncertainty (k = 2.3645754): 2.91673 Degrees of Freedom: 7 t Percent Point Value (alpha = 0.05): 2.36458 Lower 95% Confidence Limit: 55.67410 Upper 95% Confidence Limit: 61.50756 Note: SchillerEberhardt Best Usage: 5 or Fewer Labs: 13. Method: BCP (Bayesian Consensus Procedure) Estimate of Consensus Mean: 58.59556 Standard Deviation of Consensus Mean: 1.36276 Standard Uncertainty (k = 1): 1.36276 Expanded Uncertainty (k = 2): 2.72551 Degrees of Freedom: 3.89434 t Percent Point Value: 2.80641 Lower 95% (t) Confidence Limit: 54.77110 Upper 95% (t) Confidence Limit: 62.42001 Note: BCP Best Usage: 6 or Fewer Labs: Table 2: 95% Confidence Limits  Consensus Lower Upper Uncertainty Method Mean Limit Limit (k*SE)  1. MandelPaule 58.56633 56.93617 60.19648 1.63016 2. Modified MandelPaule 58.55906 56.92470 60.19343 1.63437 3a. VangelRukhin ML 58.55346 56.92544 60.18148 1.62802 4a. DerSimonianLaird (original) 58.55450 56.38900 60.71999 2.16549 4b. DerSimonianLaird (HHD) 58.55450 55.95511 61.15389 2.59939 5. GraybillDeal 58.67330 54.47558 62.87103 4.19773 7. Generalized CI 58.45256 55.96620 61.00745 2.55490 8. Grand Mean 57.22609 56.61636 57.83582 0.60973 9. Mean of Means 58.59556 56.04615 61.14496 2.54940 11. BOB 58.59556 55.84741 61.34370 2.74814 12. SchillerEberhardt 58.59083 55.67410 61.50756 2.91673 13. BCP 58.59556 54.77110 62.42001 3.82445 Table 3: Standard Uncertainties (k = 1)  Standard Relative Consensus Uncertainty Standard Method Mean (k = 1) Uncertainty (%)  1. MandelPaule 58.56633 0.83173 1.42015 2. Modified MandelPaule 58.55906 0.83388 1.42399 3a. VangelRukhin ML 58.55346 0.83064 1.41860 4a. DerSimonianLaird (original) 58.55450 0.77995 1.33201 4b. DerSimonianLaird (HHD) 58.55450 0.93623 1.59890 5. GraybillDeal 58.67330 0.07443 0.12686 7. Generalized CI 58.45256 1.27926 2.18855 8. Grand Mean 57.22609 0.30273 0.52901 9. Mean of Means 58.59556 0.91823 1.56706 11. BOB 58.59556 1.37407 2.34501 12. SchillerEberhardt 58.59083 2.73924 4.67520 13. BCP 58.59556 1.36276 2.32570 Table 4: Expanded Uncertainties (k = 2)  Expanded Relative Consensus Uncertainty Expanded Method Mean (k = 2) Uncertainty (%)  1. MandelPaule 58.56633 1.66345 2.84029 2. Modified MandelPaule 58.55906 1.66775 2.84798 3a. VangelRukhin ML 58.55346 1.66128 2.83720 4a. DerSimonianLaird (original) 58.55450 1.55990 2.66402 4b. DerSimonianLaird (HHD) 58.55450 1.87246 3.19781 5. GraybillDeal 58.67330 0.14887 0.25372 7. Generalized CI 58.45256 2.55853 4.37710 8. Grand Mean 57.22609 0.60546 1.05801 9. Mean of Means 58.59556 1.83645 3.13411 11. BOB 58.59556 2.74814 4.69002 12. SchillerEberhardt 58.59083 2.86931 4.89719 13. BCP 58.59556 2.72551 4.65140Program 2: read mx sx nx 3.03 0.36 3 3.27 0.33 3 3.44 0.40 12 1.21 0.12 3 1.44 0.21 3 1.18 0.30 8 13.9 0.3 3 13.6 0.04 3 15.0 1.9 8 18.1 0.7 3 18.4 0.5 3 19.7 2.0 8 end of data . let n = number mx let ind = sequence 1 1 n let tag = 1 for i = 1 1 n let tag = 2 for i = 4 1 6 let tag = 3 for i = 7 1 9 let tag = 4 for i = 10 1 12 . bootstrap samples 100000 set write decimals 5 SET DERSIMONIAN LAIRD BOOTSTRAP ON SET SCHILLER EBERHARDT OFF SET MEAN OF MEANS OFF SET GRAND MEAN OFF SET GRAYBILL DEAL OFF SET GENERALIZED CONFIDENCE INTERVAL OFF SET BAYESIAN CONSENSUS PROCEDURE OFF SET FAIRWEATHER OFF . consensus mean mx sx nx subset tag = 1 consensus mean mx sx nx subset tag = 2 consensus mean mx sx nx subset tag = 3 consensus mean mx sx nx subset tag = 4The following output is generated Consensus Means Analysis (Summary Statistics Case) Data Summary: Mean Variable: MX SD Variable: SX Sample Size Variable: NX Total Number of Observations: 18 Grand Mean: 3.34333 Grand Standard Deviation: 0.45800 Total Number of Labs: 3 Minimum Lab Mean: 3.03000 Maximum Lab Mean: 3.44000 Minimum Lab SD: 0.33000 Maximum Lab SD: 0.40000 Within Lab (pooled) SD: 0.38618 Within Lab (pooled) Variance: 0.14913 Mean of Lab Means: 3.24667 SD of Lab Means: 0.20599 Table 1: Summary Statistics by Lab  Standard Lab Standard Deviation ID n(i) Mean Variance Deviation of the Mean  1 3 3.03000 0.12960 0.36000 0.20785 2 3 3.27000 0.10890 0.33000 0.19053 3 12 3.44000 0.16000 0.40000 0.11547 1. Method: MandelPaule Estimate of (unscaled) Consensus Mean: 3.29713 Estimate of (scaled) Consensus Mean: 0.65154 Between Lab Variance (unscaled): 0.01418 Between Lab SD (unscaled): 0.11909 Between Lab Variance (scaled): 0.08436 Standard Deviation of Consensus Mean: 0.09506 Standard Uncertainty (k = 1): 0.09506 Expanded Uncertainty (k = 2): 0.19012 Expanded Uncertainty (k = 1.9599640): 0.18631 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 3.11081 Upper 95% (normal) Confidence Limit: 3.48344 Note: MandelPaule Best Usage: 6 or More Labs: 2. Method: Modified MandelPaule Estimate of (unscaled) Consensus Mean: 3.32472 Estimate of (scaled) Consensus Mean: 0.71884 Between Lab Variance (unscaled): 0.00076 Between Lab SD (unscaled): 0.02751 Between Lab Variance (scaled): 0.00450 Standard Deviation of Consensus Mean: 0.08848 Standard Uncertainty (k = 1): 0.08848 Expanded Uncertainty (k = 2): 0.17697 Expanded Uncertainty (k = 1.9599640): 0.17342 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 3.15130 Upper 95% (normal) Confidence Limit: 3.49814 Note: Modified MandelPaule Best Usage: 6 or More Labs: 3. Method: VangelRukhin Maximum Likelihood Estimate of (unscaled) Consensus Mean: 3.32039 Estimate of (scaled) Consensus Mean: 0.70827 Between Lab Variance (unscaled): 0.00000 Between Lab SD (unscaled): 0.00000 Between Lab Variance (scaled): 0.00000 Standard Deviation of Consensus Mean: 0.08355 Standard Uncertainty (k = 1): 0.08355 Expanded Uncertainty (k = 2): 0.16711 Expanded Uncertainty (k = 1.9599640): 0.16376 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 3.15662 Upper 95% (normal) Confidence Limit: 3.48415 Note: VangelRukhin Maximum Likelihood Best Usage: 6 or More Labs WARNING: ESTIMATED BETWEEN LAB VARIANCE IS LESS THAN 0.00001. THE ESTIMATED STANDARD ERROR OF THE CONSENSUS MEAN MAY BE SUSPECT. 4a. Method: DerSimonian Laird (original variance) Estimate of Consensus Mean: 3.30565 Estimate of Variance of Consensus Mean: 0.01150 Estimate of Between Lab Variance: 0.00866 Standard Uncertainty (k = 1): 0.10722 Expanded Uncertainty (k = 2): 0.21445 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 2.84430 Upper 95% (tvalue) Confidence Limit: 3.76699 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4b. Method: DerSimonian Laird  HornHornDuncan Variance Estimate of Consensus Mean: 3.30565 Estimate of Variance of Consensus Mean: 0.01524 Estimate of Between Lab Variance: 0.00866 Standard Uncertainty (k = 1): 0.12344 Expanded Uncertainty (k = 2): 0.24688 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 2.77453 Upper 95% (tvalue) Confidence Limit: 3.83677 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4d. Method: DerSimonian Laird  Bootstrap Variance Number of Bootstrap Samples 100000 Estimate of Consensus Mean: 3.30565 Estimate of Variance of Consensus Mean: 0.01352 Standard Uncertainty (k = 1): 0.11628 Expanded Uncertainty (k = 2): 0.23256 Lower 95% (percentile bootstrap) Confidence Limit: 3.07915 Upper 95% (percentile bootstrap) Confidence Limit: 3.53499 Lower 95% (symmetric bootstrap) Confidence Limit: 3.07630 Upper 95% (symmetric bootstrap) Confidence Limit: 3.53499 K (symmetric bootstrap) Coverage Factor: 1.97239 Lower 95% (kernel bootstrap) Confidence Limit: 3.07682 Upper 95% (kernel bootstrap) Confidence Limit: 3.53458 K (kernel bootstrap) Coverage Factor: 1.96884 Note: DerSimonianLaird Best Usage: Any Number of Labs: 11. Method: BOB (Bound on Bias) Estimate of Consensus Mean: 3.24667 Within Lab Uncertainty: 0.10156 Between Lab Uncertainty: 0.11836 Standard Uncertainty (k = 1): 0.15596 Expanded Uncertainty (k = 2): 0.31192 Lower 95% (k = 2) Confidence Limit: 2.93475 Upper 95% (k = 2) Confidence Limit: 3.55858 Note: BOB Best Usage: 5 or Fewer Labs: Table 2: 95% Confidence Limits  Consensus Lower Upper Uncertainty Method Mean Limit Limit (k*SE)  1. MandelPaule 3.29713 3.11081 3.48344 0.18631 2. Modified MandelPaule 3.32472 3.15130 3.49814 0.17342 3a. VangelRukhin ML 3.32039 3.15662 3.48415 0.16376 4a. DerSimonianLaird (original) 3.30565 2.84430 3.76699 0.46134 4b. DerSimonianLaird (HHD) 3.30565 2.77453 3.83677 0.53112 4d. DerSimonianLaird (perc. bootstrap) 3.30565 3.07915 3.53499 0.22934 4d. DerSimonianLaird (symm. bootstrap) 3.30565 3.07630 3.53499 0.22934 4d. DerSimonianLaird (kern bootstrap) 3.30565 3.07682 3.53458 0.22893 11. BOB 3.24667 2.93475 3.55858 0.31192 Table 3: Standard Uncertainties (k = 1)  Standard Relative Consensus Uncertainty Standard Method Mean (k = 1) Uncertainty (%)  1. MandelPaule 3.29713 0.09506 2.88311 2. Modified MandelPaule 3.32472 0.08848 2.66135 3a. VangelRukhin ML 3.32039 0.08355 2.51641 4a. DerSimonianLaird (original) 3.30565 0.10722 3.24363 4b. DerSimonianLaird (HHD) 3.30565 0.12344 3.73421 4d. DerSimonianLaird (bootstrap) 3.30565 0.11628 3.51755 11. BOB 3.24667 0.15596 4.80366 Table 4: Expanded Uncertainties (k = 2)  Expanded Relative Consensus Uncertainty Expanded Method Mean (k = 2) Uncertainty (%)  1. MandelPaule 3.29713 0.19012 5.76621 2. Modified MandelPaule 3.32472 0.17697 5.32271 3a. VangelRukhin ML 3.32039 0.16711 5.03282 4a. DerSimonianLaird (original) 3.30565 0.21445 6.48726 4b. DerSimonianLaird (HHD) 3.30565 0.24688 7.46842 4d. DerSimonianLaird (bootstrap) 3.30565 0.23256 7.03510 11. BOB 3.24667 0.31192 9.60732 Consensus Means Analysis (Summary Statistics Case) Data Summary: Mean Variable: MX SD Variable: SX Sample Size Variable: NX Total Number of Observations: 14 Grand Mean: 1.24214 Grand Standard Deviation: 0.30818 Total Number of Labs: 3 Minimum Lab Mean: 1.18000 Maximum Lab Mean: 1.44000 Minimum Lab SD: 0.12000 Maximum Lab SD: 0.30000 Within Lab (pooled) SD: 0.26059 Within Lab (pooled) Variance: 0.06791 Mean of Lab Means: 1.27667 SD of Lab Means: 0.14224 Table 1: Summary Statistics by Lab  Standard Lab Standard Deviation ID n(i) Mean Variance Deviation of the Mean  1 3 1.21000 0.01440 0.12000 0.06928 2 3 1.44000 0.04410 0.21000 0.12124 3 8 1.18000 0.09000 0.30000 0.10607 1. Method: MandelPaule Estimate of (unscaled) Consensus Mean: 1.25879 Estimate of (scaled) Consensus Mean: 0.30308 Between Lab Variance (unscaled): 0.00754 Between Lab SD (unscaled): 0.08682 Between Lab Variance (scaled): 0.11150 Standard Deviation of Consensus Mean: 0.05569 Standard Uncertainty (k = 1): 0.05569 Expanded Uncertainty (k = 2): 0.11137 Expanded Uncertainty (k = 1.9599640): 0.10914 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 1.14965 Upper 95% (normal) Confidence Limit: 1.36793 Note: MandelPaule Best Usage: 6 or More Labs: 2. Method: Modified MandelPaule Estimate of (unscaled) Consensus Mean: 1.24810 Estimate of (scaled) Consensus Mean: 0.26196 Between Lab Variance (unscaled): 0.00089 Between Lab SD (unscaled): 0.02978 Between Lab Variance (scaled): 0.01312 Standard Deviation of Consensus Mean: 0.04683 Standard Uncertainty (k = 1): 0.04683 Expanded Uncertainty (k = 2): 0.09366 Expanded Uncertainty (k = 1.9599640): 0.09179 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 1.15632 Upper 95% (normal) Confidence Limit: 1.33989 Note: Modified MandelPaule Best Usage: 6 or More Labs: 3. Method: VangelRukhin Maximum Likelihood Estimate of (unscaled) Consensus Mean: 1.24541 Estimate of (scaled) Consensus Mean: 0.25160 Between Lab Variance (unscaled): 0.03538 Between Lab SD (unscaled): 0.18808 Between Lab Variance (scaled): 0.52326 Standard Deviation of Consensus Mean: 0.06383 Standard Uncertainty (k = 1): 0.06383 Expanded Uncertainty (k = 2): 0.12767 Expanded Uncertainty (k = 1.9599640): 0.12511 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 1.12030 Upper 95% (normal) Confidence Limit: 1.37052 Note: VangelRukhin Maximum Likelihood Best Usage: 6 or More Labs 4a. Method: DerSimonian Laird (original variance) Estimate of Consensus Mean: 1.25331 Estimate of Variance of Consensus Mean: 0.00405 Estimate of Between Lab Variance: 0.00333 Standard Uncertainty (k = 1): 0.06363 Expanded Uncertainty (k = 2): 0.12726 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 0.97953 Upper 95% (tvalue) Confidence Limit: 1.52709 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4b. Method: DerSimonian Laird  HornHornDuncan Variance Estimate of Consensus Mean: 1.25331 Estimate of Variance of Consensus Mean: 0.00377 Estimate of Between Lab Variance: 0.00333 Standard Uncertainty (k = 1): 0.06136 Expanded Uncertainty (k = 2): 0.12272 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 0.98930 Upper 95% (tvalue) Confidence Limit: 1.51732 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4d. Method: DerSimonian Laird  Bootstrap Variance Number of Bootstrap Samples 100000 Estimate of Consensus Mean: 1.25331 Estimate of Variance of Consensus Mean: 0.00468 Standard Uncertainty (k = 1): 0.06837 Expanded Uncertainty (k = 2): 0.13675 Lower 95% (percentile bootstrap) Confidence Limit: 1.11892 Upper 95% (percentile bootstrap) Confidence Limit: 1.38668 Lower 95% (symmetric bootstrap) Confidence Limit: 1.11892 Upper 95% (symmetric bootstrap) Confidence Limit: 1.38771 K (symmetric bootstrap) Coverage Factor: 1.96557 Lower 95% (kernel bootstrap) Confidence Limit: 1.11858 Upper 95% (kernel bootstrap) Confidence Limit: 1.38814 K (kernel bootstrap) Coverage Factor: 1.97194 Note: DerSimonianLaird Best Usage: Any Number of Labs: 11. Method: BOB (Bound on Bias) Estimate of Consensus Mean: 1.27667 Within Lab Uncertainty: 0.05845 Between Lab Uncertainty: 0.07506 Standard Uncertainty (k = 1): 0.09513 Expanded Uncertainty (k = 2): 0.19026 Lower 95% (k = 2) Confidence Limit: 1.08640 Upper 95% (k = 2) Confidence Limit: 1.46693 Note: BOB Best Usage: 5 or Fewer Labs: Table 2: 95% Confidence Limits  Consensus Lower Upper Uncertainty Method Mean Limit Limit (k*SE)  1. MandelPaule 1.25879 1.14965 1.36793 0.10914 2. Modified MandelPaule 1.24810 1.15632 1.33989 0.09179 3a. VangelRukhin ML 1.24541 1.12030 1.37052 0.12511 4a. DerSimonianLaird (original) 1.25331 0.97953 1.52709 0.27378 4b. DerSimonianLaird (HHD) 1.25331 0.98930 1.51732 0.26401 4d. DerSimonianLaird (perc. bootstrap) 1.25331 1.11892 1.38668 0.13440 4d. DerSimonianLaird (symm. bootstrap) 1.25331 1.11892 1.38771 0.13440 4d. DerSimonianLaird (kern bootstrap) 1.25331 1.11858 1.38814 0.13483 11. BOB 1.27667 1.08640 1.46693 0.19026 Table 3: Standard Uncertainties (k = 1)  Standard Relative Consensus Uncertainty Standard Method Mean (k = 1) Uncertainty (%)  1. MandelPaule 1.25879 0.05569 4.42371 2. Modified MandelPaule 1.24810 0.04683 3.75215 3a. VangelRukhin ML 1.24541 0.06383 5.12544 4a. DerSimonianLaird (original) 1.25331 0.06363 5.07702 4b. DerSimonianLaird (HHD) 1.25331 0.06136 4.89583 4d. DerSimonianLaird (bootstrap) 1.25331 0.06837 5.45553 11. BOB 1.27667 0.09513 7.45155 Table 4: Expanded Uncertainties (k = 2)  Expanded Relative Consensus Uncertainty Expanded Method Mean (k = 2) Uncertainty (%)  1. MandelPaule 1.25879 0.11137 8.84741 2. Modified MandelPaule 1.24810 0.09366 7.50430 3a. VangelRukhin ML 1.24541 0.12767 10.25087 4a. DerSimonianLaird (original) 1.25331 0.12726 10.15403 4b. DerSimonianLaird (HHD) 1.25331 0.12272 9.79165 4d. DerSimonianLaird (bootstrap) 1.25331 0.13675 10.91107 11. BOB 1.27667 0.19026 14.90311 Consensus Means Analysis (Summary Statistics Case) Data Summary: Mean Variable: MX SD Variable: SX Sample Size Variable: NX Total Number of Observations: 14 Grand Mean: 14.46429 Grand Standard Deviation: 1.47455 Total Number of Labs: 3 Minimum Lab Mean: 13.60000 Maximum Lab Mean: 15.00000 Minimum Lab SD: 0.04000 Maximum Lab SD: 1.90000 Within Lab (pooled) SD: 1.52116 Within Lab (pooled) Variance: 2.31393 Mean of Lab Means: 14.16667 SD of Lab Means: 0.73711 Table 1: Summary Statistics by Lab  Standard Lab Standard Deviation ID n(i) Mean Variance Deviation of the Mean  1 3 13.90000 0.09000 0.30000 0.17321 2 3 13.60000 0.00160 0.04000 0.02309 3 8 15.00000 3.61000 1.90000 0.67175 1. Method: MandelPaule Estimate of (unscaled) Consensus Mean: 13.94840 Estimate of (scaled) Consensus Mean: 0.24886 Between Lab Variance (unscaled): 0.26733 Between Lab SD (unscaled): 0.51704 Between Lab Variance (scaled): 0.13639 Standard Deviation of Consensus Mean: 0.23146 Standard Uncertainty (k = 1): 0.23146 Expanded Uncertainty (k = 2): 0.46292 Expanded Uncertainty (k = 1.9599640): 0.45365 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 13.49475 Upper 95% (normal) Confidence Limit: 14.40205 Note: MandelPaule Best Usage: 6 or More Labs: 2. Method: Modified MandelPaule Estimate of (unscaled) Consensus Mean: 13.85264 Estimate of (scaled) Consensus Mean: 0.18047 Between Lab Variance (unscaled): 0.10383 Between Lab SD (unscaled): 0.32222 Between Lab Variance (scaled): 0.05297 Standard Deviation of Consensus Mean: 0.16986 Standard Uncertainty (k = 1): 0.16986 Expanded Uncertainty (k = 2): 0.33972 Expanded Uncertainty (k = 1.9599640): 0.33292 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 13.51973 Upper 95% (normal) Confidence Limit: 14.18556 Note: Modified MandelPaule Best Usage: 6 or More Labs: 3. Method: VangelRukhin Maximum Likelihood Estimate of (unscaled) Consensus Mean: 13.97095 Estimate of (scaled) Consensus Mean: 0.26497 Between Lab Variance (unscaled): 3.62182 Between Lab SD (unscaled): 1.90311 Between Lab Variance (scaled): 1.84784 Standard Deviation of Consensus Mean: 0.09755 Standard Uncertainty (k = 1): 0.09755 Expanded Uncertainty (k = 2): 0.19509 Expanded Uncertainty (k = 1.9599640): 0.19119 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 13.77976 Upper 95% (normal) Confidence Limit: 14.16214 Note: VangelRukhin Maximum Likelihood Best Usage: 6 or More Labs 4a. Method: DerSimonian Laird (original variance) Estimate of Consensus Mean: 13.63630 Estimate of Variance of Consensus Mean: 0.00296 Estimate of Between Lab Variance: 0.00275 Standard Uncertainty (k = 1): 0.05445 Expanded Uncertainty (k = 2): 0.10889 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 13.40203 Upper 95% (tvalue) Confidence Limit: 13.87057 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4b. Method: DerSimonian Laird  HornHornDuncan Variance Estimate of Consensus Mean: 13.63630 Estimate of Variance of Consensus Mean: 0.01177 Estimate of Between Lab Variance: 0.00275 Standard Uncertainty (k = 1): 0.10851 Expanded Uncertainty (k = 2): 0.21702 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 13.16941 Upper 95% (tvalue) Confidence Limit: 14.10319 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4d. Method: DerSimonian Laird  Bootstrap Variance Number of Bootstrap Samples 100000 Estimate of Consensus Mean: 13.63630 Estimate of Variance of Consensus Mean: 0.00855 Standard Uncertainty (k = 1): 0.09248 Expanded Uncertainty (k = 2): 0.18497 Lower 95% (percentile bootstrap) Confidence Limit: 13.44749 Upper 95% (percentile bootstrap) Confidence Limit: 13.82056 Lower 95% (symmetric bootstrap) Confidence Limit: 13.44749 Upper 95% (symmetric bootstrap) Confidence Limit: 13.82510 K (symmetric bootstrap) Coverage Factor: 2.04152 Lower 95% (kernel bootstrap) Confidence Limit: 13.44899 Upper 95% (kernel bootstrap) Confidence Limit: 13.82378 K (kernel bootstrap) Coverage Factor: 2.02727 Note: DerSimonianLaird Best Usage: Any Number of Labs: 11. Method: BOB (Bound on Bias) Estimate of Consensus Mean: 14.16667 Within Lab Uncertainty: 0.23137 Between Lab Uncertainty: 0.40415 Standard Uncertainty (k = 1): 0.46569 Expanded Uncertainty (k = 2): 0.93137 Lower 95% (k = 2) Confidence Limit: 13.23529 Upper 95% (k = 2) Confidence Limit: 15.09804 Note: BOB Best Usage: 5 or Fewer Labs: Table 2: 95% Confidence Limits  Consensus Lower Upper Uncertainty Method Mean Limit Limit (k*SE)  1. MandelPaule 13.94840 13.49475 14.40205 0.45365 2. Modified MandelPaule 13.85264 13.51973 14.18556 0.33292 3a. VangelRukhin ML 13.97095 13.77976 14.16214 0.19119 4a. DerSimonianLaird (original) 13.63630 13.40203 13.87057 0.23427 4b. DerSimonianLaird (HHD) 13.63630 13.16941 14.10319 0.46689 4d. DerSimonianLaird (perc. bootstrap) 13.63630 13.44749 13.82056 0.18880 4d. DerSimonianLaird (symm. bootstrap) 13.63630 13.44749 13.82510 0.18880 4d. DerSimonianLaird (kern bootstrap) 13.63630 13.44899 13.82378 0.18749 11. BOB 14.16667 13.23529 15.09804 0.93137 Table 3: Standard Uncertainties (k = 1)  Standard Relative Consensus Uncertainty Standard Method Mean (k = 1) Uncertainty (%)  1. MandelPaule 13.94840 0.23146 1.65939 2. Modified MandelPaule 13.85264 0.16986 1.22618 3a. VangelRukhin ML 13.97095 0.09755 0.69821 4a. DerSimonianLaird (original) 13.63630 0.05445 0.39928 4b. DerSimonianLaird (HHD) 13.63630 0.10851 0.79576 4d. DerSimonianLaird (bootstrap) 13.63630 0.09248 0.67821 11. BOB 14.16667 0.46569 3.28721 Table 4: Expanded Uncertainties (k = 2)  Expanded Relative Consensus Uncertainty Expanded Method Mean (k = 2) Uncertainty (%)  1. MandelPaule 13.94840 0.46292 3.31878 2. Modified MandelPaule 13.85264 0.33972 2.45237 3a. VangelRukhin ML 13.97095 0.19509 1.39641 4a. DerSimonianLaird (original) 13.63630 0.10889 0.79856 4b. DerSimonianLaird (HHD) 13.63630 0.21702 1.59152 4d. DerSimonianLaird (bootstrap) 13.63630 0.18497 1.35642 11. BOB 14.16667 0.93137 6.57441 Consensus Means Analysis (Summary Statistics Case) Data Summary: Mean Variable: MX SD Variable: SX Sample Size Variable: NX Total Number of Observations: 14 Grand Mean: 19.07857 Grand Standard Deviation: 1.78182 Total Number of Labs: 3 Minimum Lab Mean: 18.10000 Maximum Lab Mean: 19.70000 Minimum Lab SD: 0.50000 Maximum Lab SD: 2.00000 Within Lab (pooled) SD: 1.63707 Within Lab (pooled) Variance: 2.68000 Mean of Lab Means: 18.73333 SD of Lab Means: 0.85049 Table 1: Summary Statistics by Lab  Standard Lab Standard Deviation ID n(i) Mean Variance Deviation of the Mean  1 3 18.10000 0.49000 0.70000 0.40415 2 3 18.40000 0.25000 0.50000 0.28868 3 8 19.70000 4.00000 2.00000 0.70711 1. Method: MandelPaule Estimate of (unscaled) Consensus Mean: 18.57390 Estimate of (scaled) Consensus Mean: 0.29619 Between Lab Variance (unscaled): 0.34970 Between Lab SD (unscaled): 0.59135 Between Lab Variance (scaled): 0.13660 Standard Deviation of Consensus Mean: 0.30625 Standard Uncertainty (k = 1): 0.30625 Expanded Uncertainty (k = 2): 0.61251 Expanded Uncertainty (k = 1.9599640): 0.60025 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 17.97365 Upper 95% (normal) Confidence Limit: 19.17415 Note: MandelPaule Best Usage: 6 or More Labs: 2. Method: Modified MandelPaule Estimate of (unscaled) Consensus Mean: 18.49855 Estimate of (scaled) Consensus Mean: 0.24910 Between Lab Variance (unscaled): 0.10964 Between Lab SD (unscaled): 0.33112 Between Lab Variance (scaled): 0.04283 Standard Deviation of Consensus Mean: 0.23892 Standard Uncertainty (k = 1): 0.23892 Expanded Uncertainty (k = 2): 0.47784 Expanded Uncertainty (k = 1.9599640): 0.46828 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 18.03028 Upper 95% (normal) Confidence Limit: 18.96683 Note: Modified MandelPaule Best Usage: 6 or More Labs: 3. Method: VangelRukhin Maximum Likelihood Estimate of (unscaled) Consensus Mean: 18.73333 Estimate of (scaled) Consensus Mean: 0.39584 Between Lab Variance (unscaled): 0.48222 Between Lab SD (unscaled): 0.69442 Between Lab Variance (scaled): 0.18837 Standard Deviation of Consensus Mean: 0.40092 Standard Uncertainty (k = 1): 0.40092 Expanded Uncertainty (k = 2): 0.80185 Expanded Uncertainty (k = 1.9599640): 0.78580 Normal PPF of 0.975: 1.95996 Lower 95% (normal) Confidence Limit: 17.94754 Upper 95% (normal) Confidence Limit: 19.51913 Note: VangelRukhin Maximum Likelihood Best Usage: 6 or More Labs 4a. Method: DerSimonian Laird (original variance) Estimate of Consensus Mean: 18.49153 Estimate of Variance of Consensus Mean: 0.08945 Estimate of Between Lab Variance: 0.09459 Standard Uncertainty (k = 1): 0.29908 Expanded Uncertainty (k = 2): 0.59817 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 17.20468 Upper 95% (tvalue) Confidence Limit: 19.77838 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4b. Method: DerSimonian Laird  HornHornDuncan Variance Estimate of Consensus Mean: 18.49153 Estimate of Variance of Consensus Mean: 0.07139 Estimate of Between Lab Variance: 0.09459 Standard Uncertainty (k = 1): 0.26719 Expanded Uncertainty (k = 2): 0.53439 Degrees of Freedom: 2 t Percent Point Value: 4.30265 Lower 95% (tvalue) Confidence Limit: 17.34189 Upper 95% (tvalue) Confidence Limit: 19.64117 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4d. Method: DerSimonian Laird  Bootstrap Variance Number of Bootstrap Samples 100000 Estimate of Consensus Mean: 18.49153 Estimate of Variance of Consensus Mean: 0.10234 Standard Uncertainty (k = 1): 0.31991 Expanded Uncertainty (k = 2): 0.63982 Lower 95% (percentile bootstrap) Confidence Limit: 17.86617 Upper 95% (percentile bootstrap) Confidence Limit: 19.11832 Lower 95% (symmetric bootstrap) Confidence Limit: 17.86474 Upper 95% (symmetric bootstrap) Confidence Limit: 19.11832 K (symmetric bootstrap) Coverage Factor: 1.95928 Lower 95% (kernel bootstrap) Confidence Limit: 17.86219 Upper 95% (kernel bootstrap) Confidence Limit: 19.11986 K (kernel bootstrap) Coverage Factor: 1.96411 Note: DerSimonianLaird Best Usage: Any Number of Labs: 11. Method: BOB (Bound on Bias) Estimate of Consensus Mean: 18.73333 Within Lab Uncertainty: 0.28803 Between Lab Uncertainty: 0.46188 Standard Uncertainty (k = 1): 0.54433 Expanded Uncertainty (k = 2): 1.08866 Lower 95% (k = 2) Confidence Limit: 17.64467 Upper 95% (k = 2) Confidence Limit: 19.82200 Note: BOB Best Usage: 5 or Fewer Labs: Table 2: 95% Confidence Limits  Consensus Lower Upper Uncertainty Method Mean Limit Limit (k*SE)  1. MandelPaule 18.57390 17.97365 19.17415 0.60025 2. Modified MandelPaule 18.49855 18.03028 18.96683 0.46828 3a. VangelRukhin ML 18.73333 17.94754 19.51913 0.78580 4a. DerSimonianLaird (original) 18.49153 17.20468 19.77838 1.28685 4b. DerSimonianLaird (HHD) 18.49153 17.34189 19.64117 1.14964 4d. DerSimonianLaird (perc. bootstrap) 18.49153 17.86617 19.11832 0.62679 4d. DerSimonianLaird (symm. bootstrap) 18.49153 17.86474 19.11832 0.62679 4d. DerSimonianLaird (kern bootstrap) 18.49153 17.86219 19.11986 0.62934 11. BOB 18.73333 17.64467 19.82200 1.08866 Table 3: Standard Uncertainties (k = 1)  Standard Relative Consensus Uncertainty Standard Method Mean (k = 1) Uncertainty (%)  1. MandelPaule 18.57390 0.30625 1.64885 2. Modified MandelPaule 18.49855 0.23892 1.29157 3a. VangelRukhin ML 18.73333 0.40092 2.14017 4a. DerSimonianLaird (original) 18.49153 0.29908 1.61741 4b. DerSimonianLaird (HHD) 18.49153 0.26719 1.44495 4d. DerSimonianLaird (bootstrap) 18.49153 0.31991 1.73002 11. BOB 18.73333 0.54433 2.90568 Table 4: Expanded Uncertainties (k = 2)  Expanded Relative Consensus Uncertainty Expanded Method Mean (k = 2) Uncertainty (%)  1. MandelPaule 18.57390 0.61251 3.29769 2. Modified MandelPaule 18.49855 0.47784 2.58313 3a. VangelRukhin ML 18.73333 0.80185 4.28034 4a. DerSimonianLaird (original) 18.49153 0.59817 3.23482 4b. DerSimonianLaird (HHD) 18.49153 0.53439 2.88989 4d. DerSimonianLaird (bootstrap) 18.49153 0.63982 3.46004 11. BOB 18.73333 1.08866 5.81136Program 3: . Consensus means analysis for data in Possolo, . LaFarge and Koepke paper. This example shows . case where the standard deviation is based on . an unknown degrees of freedom . . Step 1: Read the data . read amean asd ni 6.67248 0.00043 0 6.6729 0.0005 0 6.67398 0.00070 0 6.674255 0.000092 0 6.67559 0.00027 0 6.67422 0.00098 0 6.67387 0.00027 0 6.67222 0.00087 0 6.67425 0.00012 0 6.67349 0.00018 0 6.67234 0.00014 0 6.67554 0.00016 0 6.67191 0.00099 0 6.67435 0.00013 0 end of data . . Step 2: Set options . set write decimals 7 set modified mandel paule off set vangel rukhin off set vangel rukhin bootstrap off set dersimonian laird minmax off set dersimonian laird bootstrap off set schiller eberhardt off set mean of means on set grand mean off set graybill deal off set generalized confidence interval off set fairweather off set bayesian consensus procedure off set bob off set linear pool on set random number generator fibonacci congruential seed 46551 . print "Possolo, LaFarge, Koepke Test Data" print " " print " " consensus mean amean asd ni . seed 46551 let consval = summary linear pool amean asd ni seed 46551 let consunc = summary linear pool standard error amean asd ni print consval consuncThe following output is generated Possolo, LaFarge, Koepke Test Data Consensus Means Analysis (Summary Statistics Case) Data Summary: Mean Variable: AMEAN SD Variable: ASD Sample Size Variable: NI Total Number of Labs: 14 Minimum Lab Mean: 0.6671910E+01 Maximum Lab Mean: 0.6675590E+01 Minimum Lab SD: 0.9200000E04 Maximum Lab SD: 0.9900000E03 Table 1: Summary Statistics by Lab  Lab Effective ID Mean Uncertainty Deg of Freedom  1 0.6672480E+01 0.4300000E03 0 2 0.6672900E+01 0.5000000E03 0 3 0.6673980E+01 0.7000000E03 0 4 0.6674255E+01 0.9200000E04 0 5 0.6675590E+01 0.2700000E03 0 6 0.6674220E+01 0.9800000E03 0 7 0.6673870E+01 0.2700000E03 0 8 0.6672220E+01 0.8700000E03 0 9 0.6674250E+01 0.1200000E03 0 10 0.6673490E+01 0.1800000E03 0 11 0.6672340E+01 0.1400000E03 0 12 0.6675540E+01 0.1600000E03 0 13 0.6671910E+01 0.9900000E03 0 14 0.6674350E+01 0.1300000E03 0 1. Method: MandelPaule Estimate of (unscaled) Consensus Mean: 0.6673773E+01 Estimate of (scaled) Consensus Mean: 0.5076361E+00 Between Lab Variance (unscaled): 0.1116924E05 Between Lab SD (unscaled): 0.1056846E02 Between Lab Variance (scaled): 0.8202967E01 Standard Deviation of Consensus Mean: 0.2980634E03 Standard Uncertainty (k = 1): 0.2980634E03 Expanded Uncertainty (k = 2): 0.5961269E03 Expanded Uncertainty (k = 1.9599640): 0.5841936E03 Normal PPF of 0.975: 0.1959964E+01 Lower 95% (normal) Confidence Limit: 0.6673189E+01 Upper 95% (normal) Confidence Limit: 0.6674357E+01 Note: MandelPaule Best Usage: 6 or More Labs: 4a. Method: DerSimonian Laird (original variance) Estimate of Consensus Mean: 0.6673790E+01 Estimate of Variance of Consensus Mean: 0.7793555E07 Estimate of Between Lab Variance: 0.8946160E06 Standard Uncertainty (k = 1): 0.2791694E03 Expanded Uncertainty (k = 2): 0.5583388E03 Degrees of Freedom: 13 t Percent Point Value: 0.2160369E+01 Lower 95% (tvalue) Confidence Limit: 0.6673187E+01 Upper 95% (tvalue) Confidence Limit: 0.6674393E+01 Note: DerSimonianLaird Best Usage: Any Number of Labs: 4b. Method: DerSimonian Laird  HornHornDuncan Variance Estimate of Consensus Mean: 0.6673790E+01 Estimate of Variance of Consensus Mean: 0.9646140E07 Estimate of Between Lab Variance: 0.8946160E06 Standard Uncertainty (k = 1): 0.3105824E03 Expanded Uncertainty (k = 2): 0.6211647E03 Degrees of Freedom: 13 t Percent Point Value: 0.2160369E+01 Lower 95% (tvalue) Confidence Limit: 0.6673119E+01 Upper 95% (tvalue) Confidence Limit: 0.6674461E+01 Note: DerSimonianLaird Best Usage: Any Number of Labs: 9. Method: Mean of Means Mean of Lab Means: 0.6673671E+01 Standard Deviation of Lab Means: 0.1169264E02 Standard Uncertainty (sd/sqrt(n)): 0.3124989E03 SD of Consensus Mean (sd/sqrt(n)): 0.3124989E03 Standard Uncertainty (k = 1): 0.3124989E03 Expanded Uncertainty (k = 2): 0.6249977E03 Expanded Uncertainty (k = 2.1603687): 0.6751128E03 Degrees of Freedom: 13 t Percent Point Value (alpha = 0.05): 0.2160369E+01 Lower 95% (normal) Confidence Limit: 0.6672996E+01 Upper 95% (normal) Confidence Limit: 0.6674346E+01 Note: Mean of Means Best Usage: Any Number of Labs: 16. Method: Linear Pool Linear Pool Consensus Value: 0.6673675E+01 Standard Uncertainty (k = 1): 0.1245152E02 Expanded Uncertainty (k = 2): 0.2490305E02 Lower 95% (normal) Confidence Limit: 0.6671176E+01 Upper 95% (normal) Confidence Limit: 0.6675784E+01 Table 2: 95% Confidence Limits  Consensus Lower Upper Uncertainty Method Mean Limit Limit (k*SE)  1. MandelPaule 0.6673773E+01 0.6673189E+01 0.6674357E+01 0.5841936E03 4a. DerSimonianLaird (original) 0.6673790E+01 0.6673187E+01 0.6674393E+01 0.6031088E03 4b. DerSimonianLaird (HHD) 0.6673790E+01 0.6673119E+01 0.6674461E+01 0.6709724E03 9. Mean of Means 0.6673671E+01 0.6672996E+01 0.6674346E+01 0.6751128E03 16. Linear Pool 0.6673675E+01 0.6671176E+01 0.6675784E+01 0.2498871E02 Table 3: Standard Uncertainties (k = 1)  Standard Relative Consensus Uncertainty Standard Method Mean (k = 1) Uncertainty (%)  1. MandelPaule 0.6673773E+01 0.2980634E03 0.4466191E02 4a. DerSimonianLaird (original) 0.6673790E+01 0.2791694E03 0.4183071E02 4b. DerSimonianLaird (HHD) 0.6673790E+01 0.3105824E03 0.4653763E02 9. Mean of Means 0.6673671E+01 0.3124989E03 0.4682563E02 16. Linear Pool 0.6673675E+01 0.1245152E02 0.1865767E01 Table 4: Expanded Uncertainties (k = 2)  Expanded Relative Consensus Uncertainty Expanded Method Mean (k = 2) Uncertainty (%)  1. MandelPaule 0.6673773E+01 0.5961269E03 0.8932381E02 4a. DerSimonianLaird (original) 0.6673790E+01 0.5583388E03 0.8366143E02 4b. DerSimonianLaird (HHD) 0.6673790E+01 0.6211647E03 0.9307526E02 9. Mean of Means 0.6673671E+01 0.6249977E03 0.9365127E02 16. Linear Pool 0.6673675E+01 0.2490305E02 0.3731534E01 PARAMETERS AND CONSTANTS CONSVAL  0.6673675E+01 CONSUNC  0.1245152E02  
Date created: 06/05/2001 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 