CONSENSUS MEAN

Name:

CONSENSUS MEAN Type:

Analysis Command Purpose:

Compute an estimate of a consensus mean, and the associated uncertainty, based on the data from multiple laboratories or multiple methods. Overview:

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.

Grand Mean Model:

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 t-based condidence interval:

where xbar is the overall mean, tppf 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 the estimate of the consensus mean and its uncertainty.

Mean of Means Model:

For this method, we compute the mean for each of the k laboratories. Then we compute xbar and s as the mean and standard deviation of these k means. The estimate of the consensus mean is simply xbar and we compute the following confidence interval for the consensus mean:

xbar +/- tppf(1-alpha/2,nlab-1)*s/sqrt(nlab)

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 equi-weighted mean with no regard to possible differences in within-lab variation or within-lab 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 within-lab variation and sample sizes.

Common Mean Model:

x_ij

n_i

with independent Gaussian errors e(ij) ~ N(0,kappa(i)^2) . All parameters , i = 1, ...., k are unknown and the goal is to estimate , determine its standard error, and to provide a confidence interval for .

Unbiased estimates of the within lab means and variances sigma(i)^2 = kappa(i)^2/n(i) are:

x(i) = xbar(i) = SUM[j=1 to n(i)][x(ij)]/n(i)

s(i)^2 = SUM[j=1 to n(i)][(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 is the weighted means statistics

xtilde = SUM[i=1 to k][(w(i)x(i)]/SUM[i=1 to k][w(i))]

with w(i) = 1/sigma(i)^2 . The formula for the variance is

Var(xtilde) = E(xtilde - mu)^2 = 1/SUM[i=1 to k][w(i)]

In practice, these within lab variances are unknown and so the true w_i are also unknown.

The Graybill-Deal method is based on this model. In the Graybill-Deal model, the estimate of the consensus mean is

xtilde = SUM[i=1 to k][x(i)*(1/s(i)**2)]/SUM[i=1 to k][(1/s(i)**2)]

Dataplot supports four methods for computing the variance of the Graybill-Deal consensus mean.

The naive estimate of the variance is obtained by replacing the with the sample estimates
Although this variance is easy to compute and widely used, it is known to underestimate the true variance (and rather badly for small sample sizes).
Sinha proposed the variance estimator
with
Zhang performed some simulations that indicate that while this estimate of the variance reduces the bias, it still underestimates it.
To reduce the bias further, Zhang proposed the following estimate for the variance
Zhang proposed the following additional estimate for the variance
where

Dataplot currently generates confidence intervals for the Graybill-Deal method using a method proposed by Rukhin (private communication). This method generates conservative intervals.

The Graybill-Deal approach has the following limitations

It does not take into account between lab effects. If the between lab variance is in fact significant, the Graybill-Deal may not be the appropriate approach.
Labs with small variances may recieve unjustifiably large weights and therefore dominate the estimate of the consensus mean.

One Way Random Effects Model:

where there are i = 1, ..., k labs and j = 1, .... n_i observations for each lab. In this model, 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,). That is, sigma(i)**2 are the within lab variances and is the between lab variance.

For convenience, define the following terms:

x_i	=	mean for lab i
	=	variance for lab i
	=	(this was in the common means model)
v_i	=	n_i - 1
	=
	=	(= variance of the mean)

The Mandel-Paule, modified Mandel-Paule, maximum likelihood (ML), DerSimonian-Laird, and generalized confidence interval methods are based on this model. We will discuss each of these in turn.

Mandel-Paule/Modified Mandel-Paule
The Mandel-Paule estimate of the consensus mean is defined as
with w_i denoting the weight function
where y is an estimate of the between lab variance.
The between lab variance is estimated by iteratively solving the following equation:
The modified Mandel-Paule procedure uses k on the right hand side instead of (k-1).
The confidence interval for the consensus mean is computed as (equation 19 in the Ruhkin and Vangel paper).
The Mandel-Paule estimates can be considered an approximation to maximum likelihood estimates, but they are computationally simpler. The Mandel-Paule methods are a reasonable choice when the number of labs is greater than or equal to six. For a smaller number of labs, the uncertainty intervals are generally too small.
Dataplot uses code provided by Mark Vangel to compute the Mandel-Paule estimates.
Rukhin-Vangel Maximum Likelihood (ML)
The Rukhin-Vangel paper gives the likelihood function. From this likelihood function, the ML estimate for the consensus mean is obtained from the equation
The ML estimate of the between lab variance is obtained from the equation
Rukhin and Vangel give a ML estimate of . This estimate is fairly complex and not repeated here. This ML estimate of is solved numerically using an iterative algorithm. The Mandel-Paule estimates are used as starting values for the consensus mean and between lab variance.
The confidence interval for the ML estimate has the same form as the Mandel-Paule confidence interval. However, the are replaced with in the formula and the ML estimate of the between lab variance is used.
The mathematical details of the ML procedure are given in the Rukhin-Vangel paper. They also show why the Mandel-Paule estimates provide a good aproximation to the ML estimates.
This is the recommended method of choice when the number of labs is large (>= 6). As with Mandel-Paule, the uncertainty intervals tend to be too small when the number of labs is small (<= 5).
Dataplot uses code provided by Mark Vangel to compute the maximum likelihood estimates.

DerSimonian-Laird

The DerSimonian-Laird procedure is as follows:

Compute the Graybill-Deal estimate as an initial estimate of the consensus mean (see the above description for Graybill-Deal).

Determine a non-negative estimate of the between lab variance from

Y_DL

where

TERM1	=
TERM2	=
TERM3	=
	=	Graybill-Deal estimate of the consensus mean

Estimate the Dersimonian-Laird weights and use them to compute the Dersimonian-Laird consensus mean
Compute the variance of the Dersimonian-Laird consensus mean estimate
Compute confidence intervals for the DerSimonian Laird consensus mean.
Dataplot computes two confidence limits for the DerSimonian-Laird method. The first is the standard t-based interval:
The second is an interval proposed by Rukhin. The Rukhin interval is conservative and should maintain the nominal coverage even when the sample sizes and variances for the labs vary widely.

Iyer and Wang Generalized Confidence Intervals
Iyer, Wang, and Matthew have applied the generalized confidence interval approach of Weerhandi to the problem of finding confidence limits for the consensus mean.
The description of this method is rather involved and not given here. See the Wang, Iyer, and Matthews article listed in the Reference section below. Dataplot uses code provided by Jack Wang to compute the confidence intervals for this approach.
The primary advantage of this method is that it can be applied to cases where there are a small number of labs. It is also more robust than the Mandel-Paule and maximum likelihood when the normality assumptions are violated.

Some Additional Methods

BOB (Type B on Bias)
This method is discussed in detail in the "An ISO GUM Approach to Combing Results from Multiple Methods" paper (see the Reference section).
This method should only applied if there are between two and five methods. It is based on the type B model of bias (which is where the name BOB comes from).
BOB is based on the model
where is the unknown value of the measurand, is the equally weighted mean of the population means of the methods, and is the possible bias of the as an estimate of . Both and require estimates and uncertainites of the estimates. The estimate of is the sample mean of the set of method (or lab) results.
We assume the best estimate of is 0, but there is uncertainty in this estimate. A probability distribution is placed on the value of that best summarizes the available information. A common choice is to use a uniform distribution (the ISO GUM paper provides the rationale for this choice), and that is the distributional model Dataplot uses. That is, we assume a uniform distribution centered at zero with upper and lower bounds of +a and -a. For this uniform distribution, the standard uncertainty is a/SQRT(3). The choice for a is the difference between the minimum and maximum lab (or method) mean divided by 2. This yields (xmax-xmin)/sqrt(12) as the uncertainty for the bias term.
Dataplot combines these to get the following uncertainty factor:
where
where is the standard deviation of the ith lab mean and
Here, is the within lab variability (where s is the standard deviation of the lab means) and is the between lab variability.
The ISO GUM paper discusses some variations of this basic technique. For example, Dataplot uses a factor of 2 for the expanded uncertainty interval. This can be replaced with a t-value where the degrees of freedom are computed using the Welch-Saitterwaite approximation.
The BOB method is the preferred method for SRM'swhen the number of labs is small (<= 5).
Schiller-Eberhardt
A number of variants of this method have been used. Dataplot implements the method as discussed in the Schiller-Eberhardt paper (see the Reference section below).
The Schiller-Eberhardt estimate of the consensus mean is:
where x_i is the mean of the ith lab and is the weighting function:
where
Here, is the between lab variance and is the variance of the ith lab. The between lab variance is estimated as the smallest non-negative value that satisfies
This is solved iteratively using the Mandel-Paule algorithm. Note that Dataplot uses the between lab variance computed by the Mandel-Paule method described above.
The uncertainty interval for Schiller-Eberhardt is defined as
where is the variance of the consensus mean, is the material variability variance, and bias allowance is defined as
The variance of the Schiller-Eberhardt consensus mean is computed as
where is the variance of the ith lab mean and the omega weight function is defined as
Note that the weight function for omits the between lab variance term that is included in the weight function for the consensus mean.
The variance of the material variability is discussed in the Schiller-Eberhardt paper. This is computed independently of the data given to the CONSENSUS MEAN command. To specify a value for , enter the following commands before entering the CONSENSUS MEAN command.
SIGMAH contains the value of the variance and DFH contains the corresponding degrees of freedom for the materials variance.
The degrees of freedom for the t percent point function in the uncertainty is computed as
This method has been superseeded by the BOB method. As with BOB, the Schiller-Eberhardt method is intended for a small number of labs (<= 5).

Data Analytic Considerations:

What is the definition of the consensus mean. Is this a lab independent number which represents an absolute physical truth or is this a lab-dependent "average" across all participating labs?
How many labs are there? Some methods are more appropriate for a small number of labs while others are based on asymptotic results and are thus more appropriate for a larger number of labs.
Do between-lab differences (biases) exist?
Are there differences in within-lab variation?
Are there differences in within lab sample sizes?
Does a lab with much data have such only because the lab's method is cheaper and thus of potentially poorer quality than other labs?
Are all labs treated equally?
Do "star" labs exist? That is, labs that are known to be either super unbiased or super accurate.
If an engineering equal lab tests out to be a statistically outlying lab, how (a prioori) is that lab to be weighted?

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:

estimation of a consensus mean;
estimation of an uncertainty limits for the consensus mean.

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 self-behaved but biased is a real possibility which can only be solved by engineering judgement.

Description of the Dataplot Input:

Raw Data - there should two columns of data. The first column contains the respone values and the second column contains the corresponding lab-id. The data do not need to be sorted by lab-id.
If your data is the form where each lab is contained in a separate column, you can do something like the following
This example will take the data for six labs stored in Y1, Y2, Y3, Y4, Y5, and Y6 and save it the variables Y and LABID in a format that can be used by the CONSENSUS MEAN command.
Summary Data - there should three columns of data. The first column contains the sample means for the labs, the second column contains the sample standard deviations for the labs, and the third column contains the sample sizes for the labs.

Description of Dataplot Output:

The first section prints summary information about the data. Specifically, it prints the overall mean, the number of observations, and the number of labs. It also prints a table giving the sample size, mean, variance, standard deviation, and standard deviation of the mean for each lab. It then prints the pooled within lab variance (and standard deviation). The pooled within lab variance is computed as: as
with denoting the variance of the ith lab.
The second section prints the detailed output for each method.
The third section prints a summary table containing the 95% confidence limits for each method.
The fourth section prints summary tables containing the uncertainty and the percent relative uncertainty. Separate tables are printed for standard uncertainty (k = 1) and expanded uncertainty (k = 2).

Syntax 1:

This syntax computes the consensus means based on the raw data.

Syntax 2:

This syntax computes the consensus means based on the lab means, standard deviations, and sample sizes.

Examples:

Note:

XGRAND	= the overall mean.
S2POOL	= the pooled within lab variance.
T1STDERR	= the standard error for the t method using all the data.
T2STDERR	= the standard error for the t method using the lab means.
SEMEAN	= the consensus mean using the Schiller-Eberhardt method.
SES2	= the variance of the consensus mean using the Schiller-Eberhardt method.
BIASALLO	= the bias allowance for the Schiller-Eberhardt method.
SEDF	= the degrees of freedom for the Schiller-Eberhardt method.
MPMEAN	= the consensus mean using the Mandel-Paule method.
MPS2	= the between lab variance using the Mandel-Paule method.
SEMP	= the standard error of the Mandel-Paule method.
MMPMEAN	= the consensus mean using the modified Mandel-Paule method.
MMPS2	= the between lab variance using the modified Mandel-Paule method.
SEMMP	= the standard error of the modified Mandel-Paule method.
MLMEAN	= the consensus mean using the Ruhkin-Vangel maximum likelihood method.
MLS2	= the between lab variance using the Ruhkin-Vangel maximum likelihood method.
SEML	= the standard error of the Ruhkin-Vangel maximum likelihood method.
BOBMEAN	= the consensus mean for the BOB method.
BOBS2	= the between lab variance for the BOB method.
BOBS2W	= the within lab variance for the BOB method.
BOBKU	= the uncertainty value for the BOB method.
GDMEAN	= the consensus mean using the Graybill-Deal method.
GDS2	= the variance of the Graybill-Deal consensus mean.
GCIMEAN	= the consensus mean using the generalized confidence interval approach.
DERSMEAN	= the consensus mean using the DerSimonian-Laird approach.
DERSVARI	= the variance of the DerSimonian-Laird consensus mean.
DERSSE	= the standard error of the DerSimonian-Laird consensus mean.

Note:

The following variables are written to the file dpst1f.dat. These are the statistics for the labs.

Lab ID
Number of Observations for Lab
Mean for the Lab
Variance for the Lab
Standard Deviation for the Lab
Standard Deviation of Mean of the Lab

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.

Consensus mean for the method
Lower 95% confidence limit for the method
Upper 95% confidence limit for the method
Method id

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.

Consensus mean for the method
Standard uncertainty (k = 1) for the method
Percentage relative standard uncertainty
Method id

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.

Consensus mean for the method
Expanded uncertainty (k = 2) for the method
Percentage relative expanded uncertainty
Method id

The following variables are written to the file dpst5f.dat.

1. Simulated consensus means from the generalized confidence interval approach

Note:

The Mandel-Paule and maximum likelihood estimates are generated using code provided by Mark Vangel of the NIST Statistical Engineering Division. The BOB estimates were adapted from a Dataplot macro provided by Stefan Leigh of the NIST Statistical Engineering Division. Note:

SET WRITE DECIMALS <value>

If you want to use an exponential format (E15.7), enter

SET WRITE DECIMALS 99

Note:

for details.

Note:

In some cases, it may be convenient to extract the value of a particular consensus mean statistic.

If you have raw data, you can enter one of the following

If you have summary data, you can enter one of the following

Dataplot statistics can be used in a number of other commands. For details, enter

HELP STATISTICS

For the SUMMARY cases, bootstrapping is not currently supported. However, we anticipate adding this capability in a subsequent release.

Default:

None Synonyms:

None Related Commands:

MEAN PLOT	= Generate a mean plot.
SD PLOT	= Generate a standard deviation plot.
YOUDEN PLOT	= Generate a Youden plot.
ANOVA	= Perform an analysis of variance.

Reference:

Controlled Clinical Trials

Graybill and Deal (1959), "Combining Unbiased Estimators", Biometrics, 15, pp. 543-550.

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. 1194-1197.

Robert Paule and John Mandel (1982), "Consensus Values and Weighting Factors", Journal of Research of the National Bureau of Standards, 87, pp. 377-385.

Andrew Ruhkin (2003), "Two Procedures of Meta-analysis in Clinical Trials and Interlaboratory Studies", Tatra Mountains Mathematical Publications, 26, pp. 155-168.

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 Mandel-Paule Algorithm", Journal of Statistical Planning and Inference, 83, pp. 319-330.

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 260-125, NIST, Gaithersburg, MD.

Bimal Kumar Sinha (1985), "Unbiased Estimation of the Variance of the Graybill-Deal Estimator of the Common Mean of Several Normal Populations", The Canadian Journal of Statistics, Vol. 13, No. 3, pp. 243-247.

Mark Vangel and Andrew Ruhkin (1999), "Maximum Likelihood Analysis for Heteroscedastic One-Way Random Effects ANOVA in Interlaboratory Studies", Biometrics 55, 129-136.

Nien-Fan Zhang (2006), "The Uncertainty Associated with The Weighted Mean of Measurement Data", Metrologia, 43, PP. 195-204.

Applications:

Interlaboratory Studies Implementation Date:

Program:

 
SKIP 25
READ STUTZ86.DAT ALITE JUNK2 JUNK3 JUNK4 JUNK5 LABID
.
CONSENSUS MEANS ALITE LABID

                   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
                                                                     Standard
      Lab                                            Standard       Deviation
       ID    n(i)       Mean          Variance      Deviation         of Mean
 ----------------------------------------------------------------------------
        1      36     56.7527771      0.5522779      0.7431540      0.1238590
        2       4     58.4249992      2.8225005      1.6800299      0.8400150
        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 Maximum Likelihood
     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 Maximum Likelihood 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% (k = 2) Confidence Limit:          55.8474121
     Upper 95% (k = 2) 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):                 2.8693065
     Expanded Uncertainty (k =  2.3645761):        2.9167265
     Degrees of Freedom:                            7
     t Percent Point Value (alpha = 0.05):         2.3645761
     Lower 95% Confidence Limit:                  55.6741028
     Upper 95% 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 (alpha = 0.05):         2.7764461
     Lower 95% (t-value) Confidence Limit:        56.0461540
     Upper 95% (t-value) 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 (Sinha) (k = 1):         0.1132961
     Expanded Uncertainty (Sinha) (k = 2):         0.2265923
     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 (alpha = 0.05):         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 Confidence Intervals
     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 Confidence Interval Best Usage:
           Any Number of Labs, but no Between Lab Variance
  
 10. Method: DerSimonian Laird
     Estimate of Consensus Mean:                  58.5719872
     Estimate of Variance of Consensus Mean:       0.8636000
     Estimate of Between-Lab Variance:             5.0619205
     Standard Uncertainty (k = 1):                 0.9293008
     Expanded Uncertainty (k = 2):                 1.8586016
     Degrees of Freedom:                            4
     t Percent Point Value:                        2.7764461
     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
  
  
 Table 2:  95% Confidence Limits
                                Consensus            Lower            Upper
 Method                              Mean            Limit            Limit
 --------------------------------------------------------------------------
  1. Mandel-Paule              58.5663223       56.9361677       60.1964770
  2. Modified Mandel-Paule     58.5590630       56.9246980       60.1934279
  3. Vangel-Rukhin ML          58.5534592       56.9254384       60.1814799
  4. BOB                       58.5955544       55.8474121       61.3436966
  5. Schiller-Eberhardt        58.5908279       55.6741015       61.5075544
  6. Mean of Means             58.5955544       56.0461540       61.1449547
  7. Graybill-Deal             58.6732941       56.2782006       61.0683875
  8. Grand Mean                57.2260857       56.6163559       57.8358154
  9. Generalized CI            58.4525528       55.9661958       61.0020750
 10. DerSimonian-Laird (t)     58.5719872       55.9918327       61.1521416
     (Rukhin)                  58.5719872       56.0077202       61.1362541
 --------------------------------------------------------------------------
  
  
 Table 3:  Standard Uncertainties (k = 1)
  
                                                  Standard        Relative
                                Consensus      Uncertainty        Standard
 Method                              Mean          (k = 1)  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
 --------------------------------------------------------------------------
  
  
 Table 4:  Expanded Uncertainties (k = 2)
  
                                                  Expanded        Relative
                                Consensus      Uncertainty        Expanded
 Method                              Mean          (k = 2)  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
 --------------------------------------------------------------------------

Date created: 6/5/2001
Last updated: 4/24/2006
Please email comments on this WWW page to alan.heckert@nist.gov.