components of variance model for the analysis of repeated measurements eduarda filipe

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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• 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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• 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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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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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).


222 y• The variance of any

observation y is composed by the sum of the variance components

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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:


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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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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)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 expected value of the MSFactor is:

and the variance component of the factor is obtained by:

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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 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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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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• 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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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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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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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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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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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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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!

components of variance model for the analysis of repeated measurements eduarda filipe

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