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EUROMET Project 732 « Towards new Fixed Points » Workshop. COMPONENTS OF VARIANCE MODEL FOR THE ANALYSIS OF REPEATED MEASUREMENTS Eduarda Filipe St Denis, 23 rd November 2006. Summary. Introduction General principles and concepts The Nested or Hierarchical Design - PowerPoint PPT Presentation
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COMPONENTS OF VARIANCE MODEL FOR THE ANALYSIS OF REPEATED MEASUREMENTS
Eduarda FilipeSt Denis, 23rd November 2006
EUROMET Project 732 « Towards new Fixed Points »Workshop
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Summary
• Introduction• General principles and concepts • The Nested or Hierarchical Design• Application example of a three-stage
nested experiment.
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Introduction
• The realization of the International Temperature Scale of 1990 - ITS90 requires that the Laboratories usually have more than one cell for each fixed point.
• The Laboratory may consider one of the cells as a reference cell and its reference value, performing the other(s) the role of working standard(s), or the Laboratory considers its own reference value as the average of the cells.
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Introduction
In both cases, the cells must be regularly compared and the calculation of the uncertainty of these comparisons performed.
A similar situation exists when the laboratory compares its own reference value with the value of a travelling standard during an inter-laboratory comparison.
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These comparisons experiments are usually performed with two or three thermometers to obtain the differences between the cells.
The repeatability measurements are performed each day at the equilibrium plateau and the experiment is repeated in subsequent days. This complete procedure may also be repeated some time after.
Introduction
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The uncertainty calculation should take into account these time-dependent sources of variability, arising from short-term repeatability, the day-to-day or ”medium term” reproducibility and the long-term random variations in the results;
Introduction
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The Type A method of evaluation by the statistical analysis of the data obtained from the experiment is performed using the Analysis of Variance (ANOVA) for designs, consisting of nested or hierarchical sequences of measurements.
Introduction
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ANOVA definitionISO 3534-3, Statistics – Vocabulary and Symbols
“Technique, which subdivides the total variation of a response variable into meaningful components, associated with specific sources of variation”.
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General principles and concepts
Experimental design is a statistical tool concerned with planning the experiments to obtain the maximum amount of information from the available resources
This tool is used generally for the improvement and optimisation of processes
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General principles and concepts
• it can be used to test the homogeneity of a sample(s)
• to identify the results that can be considered as “outliers”
• to evaluate the components of variance between the “controllable” factors
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General principles and concepts
• tool applied to Metrology for the analysis of large amount of repeated measurements
• Measurements in:– short-term repeatability – day-to-day reproducibility– long-term reproducibility
• permitting “mining” the results and to include this “time-dependent sources of variability” information at the uncertainty calculation.
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General principles and concepts
In the comparison experiment to be described, the factors are the standard thermometers, the subsequent days measurements and the run measurements.
These factors are considered as random samples of the population from which we are interested to draw conclusions.
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The nested or hierarchical design. DefinitionISO 3534-3, Statistics – Vocabulary and Symbols
“The experimental design in which each level of a given factor appears in only a single level of any other factor”.
Purpose of this modelTo deduce the values of component variances that cannot be measured directly.
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The nested or hierarchical design. General model
• The factors are hierarchized like a “tree” and any path from the “trunk” to the “extreme branches” will find the same number of nodes.
The analysis of each factor is done with:“Random Effects One-way ANOVA” or components-of-variance model, nested in the subsequent factor.
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Factor D
Factor T
Factor P
Measurements
Observer
Nested design
1
1
A
1
1
1
A
2
1
1
B
1
1
1
B
2
1 2
M = 2
T = 2
D = 10
P = 2
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The nested or hierarchical design. General model
For one factor with a different levels taken randomly from a large population,
where Mi is the expected (random) value of the group of observations i, the overall mean, i the i th group random effect and ij the random error component.
),...,2,1,...,2,1( njandai
My ijiijiij
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• For the hypothesis testing, the errors and the factor-levels effects are assumed to be normally and independently distributed, respectively ij ~ N (0, 2) and i ~ N (0,
2).
The nested or hierarchical design. General model
222 y• The variance of any
observation y is composed by the sum of the variance components
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The nested or hierarchical design. General model
0:
0:2
1
20
H
H• The test is unilateral and the hypotheses are:
• That is, if the null hypothesis is true, all factor-effects are “equal” and each observation is made up of the overall mean plus the random error ij ~ N (0, 2).
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The total sum of squares (SST), a measure of total variability in the data, may be expressed by:
The nested or hierarchical design. General model
a
i
n
jiiji
a
i
a
i
n
jiiji
a
i
n
jiiji
a
i
n
jij
yyyy
yyyynyyyyyy
1 1
1 1 1
22
1 1
2
1 1
2
))((2
)()()()()(
= 0
SSFactor
SSFactor sum of squares of differences between factor-level averages and the grand average or a measure of the differences between factor-level
SSE
SSE a sum of squares of the differences of observations within a factor-level from the factor-level average, due to the random error
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The nested or hierarchical design. General model
Dividing each sum of squares by the respectively degrees of freedom, we obtain the corresponding mean squares (MS):
21 1
2
1
2
)1(
)(
)(1
na
yy
MS
yya
nMS
a
i
n
jiij
Error
a
iiFactor
MSFactor - mean square between factor-level is:
•an unbiased estimate of the variance 2, if H0 is true
•or a surestimate of 2, if H0 is false.
MSError - mean square within factor (error) is always
• the unbiased estimate of the variance 2.
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The nested or hierarchical design. General model
)1(,1,0 ~ naaError
Factor FMS
MSF
F is the Fisher sampling distribution with aand a x (n -1) degrees of freedom
To test the hypotheses, we use the statistic:
If F0 > F, a-1, a(n-1) , we reject the null hypothesis.
and conclude that the variance 2 is significantly
different of zero.
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22
1
2)(1
)( nyya
nEMSE
a
iiFactor
n
MSE Factor2
2 )(
The nested or hierarchical design. General model
The expected value of the MSFactor is:
and the variance component of the factor is obtained by:
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The nested or hierarchical design. General model
Considering now the three level nested design of the figure the mathematical model is:
yrdtm (rdtm) th observation overall meanr P th random level effectd D th random level effect t T th random level effectrtdm random error component.
rdtmtdprdtmy
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The nested or hierarchical design. General model
The errors and the level effects are assumed to be normally and independently distributed, respectively with mean zero and variance 2 and with mean zero and variances r
2, d2 and
2.
The variance of any observation is composed by the sum of the variance components and the total number of measurements N is obtained by the product of the dimensions N = P × D × T × M.
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The nested or hierarchical design. General model
EDPTPDPT
p d t mpdtpdtm
p d tpdpdt
p dppd
pp
p d t mpdtm
SSSSSSSSSS
or
yy
yyMyyTM
yyDTMyy
2
22
22
)(
).()(
)()(
The total variability of the data can be expressed by
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The nested or hierarchical design. General model
• Dividing by the respective degrees of freedom
P –1 P (D -1) P D (T -1) P D T (M -1)
• Equating the mean squares to their expected values
• Solving the resulting equations
• We obtain the estimates of the components of the variance.
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The nested or hierarchical design. General model
2
22
222
2222
)1()(
)1()(
)1()(
1)(
MPDT
SSEMSE
MTPD
SSEMSE
TMMDP
SSEMSE
DTMTMMP
SSEMSE
EE
T
PDT
PDT
DT
PD
PD
PDTP
P
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested experiment
Two water cells, JA and HS
Two standard platinum resistance thermometers (SPRTs) A and B.
A – water vapour
B – water in the liquid phase
C – Ice mantle
D – thermometer (SPRT) well
t = 0,01 ºC
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Four measurement differences obtained daily with the two SPRTs
This set of measurements repeated during ten consecutive days.
And two weeks after a complete run was repeated Run 2/Plateau 2.
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested
experiment
In this nested experiment, are considered the effects of:
Factor‑P from the Plateaus (P = 2)
Factor-D from the Days (D = 10) for the same Plateau
Factor-T from the Thermometers (T = 2) for the same Day and for the same Run
the variation between Measurements (M = 2) for the same Thermometer, the same Day and for the same Plateau or the residual variation.
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested
experiment1 2
A11 93 103B11 93 93
A12 73 78B12 118 68
A13 91 126B13 130 96
A14 93 88B14 118 78
A15 117 97B15 118 118
A16 108 98B16 80 80
A17 105 100B17 128 108
A18 110 80B18 77 67
A19 97 92B19 104 84
A10 106 96B10 68 63
89
67
Measurements (K)Days SPRTs
23
45
Pla
teau
1
101
A21 107 117B21 74 54
A22 123 48B22 103 83
A23 114 64B23 72 102
A24 74 119B24 70 105
A25 68 58B25 104 104
A26 93 83B26 70 100
A27 77 89B27 104 84
A28 62 112B28 91 51
A29 68 63B29 69 68
A20 95 88B20 113 78
48
910
Pla
teau
2
12
35
67
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested
experimentSchematic representation of the observed temperatures differences
0
50
100
150
200
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
Plateau 1 Plateau 2
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested
experiment
Analysis of variance table
F0 values compared with the critical values F ( = 5%)
F0,05, 1, 18 = 4,4139 for the Plateau/Run effect
F0,05, 18,20 = 2,1515 for the Days effect and
F0,05, 20, 40 =1,8389 for the Thermometers effect.
Source of variation Sum of squaresDegrees of
freedomMean square F0
Plateaus 2187,96 1 2187,96 5,7296 2 + 2 T2 + 4 D
2 + 40 R2
Days 6873,64 18 381,87 1,0032 2 + 2 T2 + 4 D
2
Thermometers 7613,19 20 380,66 1,0314 2 + 2 T2
Measurements 14762,50 40 369,06 2
Total 31437,29 79
Expected value of mean square
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested
experiment
F0 values are inferior to the F distribution for the factor-days and factor-thermometers so the null hypotheses are not rejected.
At the Plateau factor, the null hypothesis is rejected for = 5% so a significant difference exists between the two Plateaus
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Example of the comparison of two thermometric Water triple point cells in a three-stage nested
experiment
• equating the mean squares to their expected values
• we can calculate the variance components
and• include them in the budget of the
components of uncertainty
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Example of the comparison of two thermometric Water triple point cells in a three-level nested
experiment
Uncertainty budget for components of uncertainty evaluated by a Type A method
Components of uncertainty Type A evaluation Variance (K)2 Standard -
deviation (K)
Plateaus 45,15 6,7Days 0,30 0,5Thermometers 5,80 2,4Measurements 369,06 19,2
Total 420,32 20,5
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Example of the comparison of two thermometric Water triple point cells in a three-level nested experiment
• These components of uncertainty, evaluated by Type A method, reflect the random components of variance due to the factors effects
• This model for uncertainty evaluation that takes into account the time-dependent sources of variability it is foreseen by the GUM
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Example of the comparison of two thermometric Water triple point cells in a three-level nested experiment
Comparison of different approaches• Components of uncertainty evaluated by
a Type A method– Nested structure: uA= 20,5 mK– Standard deviation of the mean of 80
measurements uA= 2,2 mK– Standard deviation of the mean 1st Plateau
(40 measurements) uA= 2,8 mK (the 2nd plateau considered as reproducibility and evaluated by a type B method)
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Concluding Remarks
The plateaus values continuously taken where previously analyzed in terms of its normality.
The nested-hierarchical design was described as a tool to identify and evaluate components of uncertainty arising from random effects.
Applied to measurement, it is suitable to calculate the components evaluated by a Type A method standard uncertainty in time-dependent situations.
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Concluding Remarks
An application of the design has been used to illustrate the variance components analysis in a three-factor nested model of a short, medium and long-term comparison of two thermometric fixed points.
The same model can be applied to other staged designs, easily treated in an Excel spreadsheet.
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REFERENCES
1. BIPM et al, Guide to the Expression of Uncertainty in Measurement (GUM), 2nd ed., International Organization for Standardization, Genève, 1995, pp 11, 83-87.
2. ISO 3534-3, Statistics – Vocabulary and Symbols – Part 3: Design of Experiments, 2nd ed., Genève, International Organization for Standardization, 1999, pp. 31 (2.6) and 40-42 (3.4)
3. Milliken, G.A., Johnson D. E., Analysis of Messy Data. Vol. I: Designed Experiments. 1st ed., London, Chapmann & Hall, 1997.
4. Montgomery, D., Introduction to Statistical Quality Control, 3rd ed., New York, John Wiley & Sons, 1996, pp. 496-499.
5. ISO TS 21749 Measurement uncertainty for metrological applications — Simple replication and nested experiments Genève, International Organization for Standardization, 2005.
6. Guimarães R.C., Cabral J.S., Estatística, 1st ed., Amadora: Mc-Graw Hill de Portugal, 1999, pp. 444-480.
7. Murteira, B., Probabilidades e Estatística. Vol. II, 2nd ed., Amadora, Mc-Graw Hill de Portugal, 1990, pp. 361-364.
8. Box, G.E.P., Hunter, W.G., Hunter J.S., Statistics for Experimenters. An Introduction to Design, Data Analysis and Model Building, 1st ed., New York, John Wiley & Sons, 1978, pp. 571-582.
9. Poirier J., “Analyse de la Variance et de la Régression. Plans d’Experience”, Techniques de l’Ingenieur, R1, 1993, pp. R260-1 to R260-23.
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Thank you!