On Some Statistical Aspects of Agreement Among

Measurements

BIKAS K SINHA [ISI, Kolkata]

TampereAugust 28, 2009

Part II : Statistical Assessment of

Agreement

Understanding Agreement among Raters

involving Continuous Measurements….

• Theory & Applications…..

Continuous Measurements

Evaluation of agreement when the data are measured on a continuous

scale…… Pearson correlation coefficient,

regression analysis, paired t-tests, least squares analysis for slope and intercept, within-subject coefficient of variation, and intra-class correlation coefficient…..

General Overview….• 1. Comparison of Gold Standard or

Reference Method and one (or more) New or Test Method(s)

If the two agree fairly well, we can use them interchangeably or the New One which is possibly cheaper or more convenient in place of the Gold Standard !

2. Calibration : Establish mathematical relationship between the two sets of measurements.

Overview…contd.

• 3. Conversion : Compare two approx.

methods, measuring same underlying quantity.

Goal : Interpret results of one in terms of the

other

Temperature recorded in two instruments….

..one in ^oF and the other in ^oC.

Talk focuses on # 1 : Comparison of GS & TM

GS : Gold Standard & TM : Test Method

Two Approaches…..

• Bland & Altman :Limits Of Agreement [LOA] Approach [with over 6000 citations in the Institute for Scientific Information Database]

• Lawrence Lin : Use of Concordance Correlation Coefficient

• Lin & Collaborators…..serious in-depth study with pharmaceutical applications

LOA Approach…..• Subjects Rater 1 Rater 2

• 1 x_1 y_1

• 2 x_2 y_2

• ……………………………..

• n x_n y_n

Model : x_ j = S_ j + Beta_1 + e_ 1j

y_ j = S_ j + Beta_2 + e_2 j

S_ j : True Unobservable Measurement for the j-th subject…randomly distributed as

LOA Approach….Model….• N(Mu, sigma^2_s)

• Beta_1 & Beta_2 : Fixed Raters’ Bias Terms

• e_1 j : iid N(0, sigma^2_e1)

• e_2 j : iid N(0, sigma^2_e2)

• S_ j, e_1j, e_2 j ….all independent

• This is Grubbs’ Model

• sigma^2_s : Between-subject variance

• sigma^2_e = measurement error variance

LOA Approach…..Model….• E(X) = Mu + Beta_1,

V(X) = sigma^2_s + sigma^2_e1 = sigma^2_x

• E(Y) = Mu + Beta_2

V(Y) = sigma^2_s + sigma^2_e2 = sigma^2_y

• Cov(X, Y) = sigma^2_s

• Rho = sigma^2_s / sigma_x . sigma_y

• Rho_x = sigma^2_s / sigma^2_x

• = Reliability Coeff. for Rater 1

•

LOA Approach…..Model…

• Rho_y = sigma^2_s / sigma^2_y

• = Reliability Coeff. for Rater 2

• Rho^2 = Rho^2_x . Rho^2_y

• Notion of Perfect Agreement :

• All paired observations (x_ j, y_ j) lie on the 45^o line through Origin

• Equivalent Conditions : Same means, same variances and Rho = 1

• Leads to Testing Issues……

LOA Approach….Data Analysis

• m = E(X – Y) = (m1–m2)

• 2 = Var(X-Y) = (12 +2

2 - 212)

• Estimates are based on paired data• LOA has 2 components :• (i) 95% LOA, defined by m^ +/- 1.96^• (ii) Plot of mean (x+y)/2 vs D = x – y, with

LOA superimposed....Bland-Altman Plot...SAS JMP produces Plot

•

LOA Approach….If a large proportion of the paired differences

[D’s] are sufficiently close to zero, the two

methods have satisfactory agreement.

Step I : Estimate the set m +/- 1.96Step II : Declare ‘sufficient’ agreement if the

differences within these limits are not clinically

important as determined by the investigator

specified threshold value delta_o depending

on the question of clinical judgement.

Lin Approach….

• Lin et al in a series of papers made in-depth

study of agreement using such notions as concordance correlation coefficient, total deviation index, coverage probability etc

We will now elaborate on these concepts.

Continuous Measurements

• Two raters – n units for measurement

• Data : [{xi, yi}; 1 ≤ i ≤ n]

• Scatter Plot : Visual Checking

• Product Moment Corr. Coeff.:

High +ve : What does it mean ?

• Squared Deviation : D2 = (X-Y)2

MSD:E[D2]=(m1–m2)2 + (12 +2

2 - 212)

Carotid Stenosis Screening StudyEmory Univ.1994-1996

• Gold Standard : Invasive Intra-arterial Angiogram [IA] Method

• Non-invasive Magnetic Resonance Angiography [MRA] Method

Two Measurements under MRA: 2D & 3D Time of Flight Three Technicians : Each on Left &

Right Arteries for 55 Patients by IA & MRA [2d & 3D] :3x3x2 =18 Obs. / patient

Data Structure….

• Between Technicians : No Difference

• Left vs Right : Difference

• 2D vs 3D : Difference

Q. Agreement between IA & 2D ? 3D ?

Barnhart & Williamson (2001, 2002) :

Biometrics papers …..no indication of any strong agreement

Scatter Plot : IA-2D-3D

Right vs Left Arteries [IA]

Descriptive Statistics :Carotid Stenosis Screening Study

Sample Means

Methods 1A, MRA-2D & MRA-3D by Sides

Method N Left Artery Right Artery

----------------------------------------------------------

1A 55 4.99 4.71

MRA-2D 55 5.36 5.73

MRA-3D 55 5.80 5.52

Descriptive Statistics (contd.) Sample Variance – Covariance Matrix 1A MRA-2D MRA-3D

L R L R L R

1A-L 11.86 1.40 8.18 1.18 6.80 1.08

1A-R 10.61 2.67 7.53 1.78 7.17

2D-L 10.98 2.70 8.69 1.74

2d-R 8.95 2.19 7.69

3D-L 11.02 2.65

3D-R 10.24

Data Analysis : Lin Approach

• Recall MSD = E[(X-Y)2] : Normed? No !• Lin (1989):Converted MSD to Corr.Coeff• Concordance Corr. Coeff. [CCC]

• CCC = 1 – [MSD / MSDInd.]

= 212 /[(m1–m2)2 + (12 +2

2) Properties :Perfect Agreement [CCC =

1] Perfect Disagreement [CCC = -1] No Agreement [CCC = 0]

CCC…• CCC = 212 /[(m1–m2)2 + (1

2 +22)]

= . a

= Accuracy Coefficient a = Precision Coeff. [<=1] a = [2 / { + 1/ + 2}] where = 1/2 and 2 = (m1–m2)2 / 12

CCC = 1 iff = 1 & a = 1 a = 1 iff [ m1 = m2 ] & [1 = 2 ] hold simultaneously !!

Study of CCC….• Identity of Marginals:Max.Precision

• High value of : High Accuracy

• Needed BOTH for Agreement

• Simultaneous Inference on

H0 : 0, [m1 = m2] & [ 1 = 2 ]

LRT & Other Tests based on CCC

Pornpis/Montip/Bimal Sinha (2006)

Thermo Pukkila Volume…..

Total Deviation Index

Lin (1991) & Lin et al (JASA, 2002) Assume BVN distribution of (X,Y) = P[ |Y – X| < k] = P[ D2 < k2 ]; D = Y - X

= 2 [k2, 1, mD2 / D

2 ]..non-central 2 TDI = Value of k for given Inference based on TDIChoice of : 90 % or more

Coverage Probability

Lin et al (JASA, 2002) : BVN distributionCP(d) = P[ |Y – X| < d]

= [(d - mD) / D ] - [(- d - mD) / D ] Emphasis is on given d and high CP.CP^ : Plug-in Estimator using sample means, variances & corr. coeff. Var[CP^] : LSA V^[CP^] : Plug-in Estimator

Graphical Display …

Back to Data Analysis…

• Carotid Stenosis Screening StudyEmory Univ.1994-1996

• GS : Method IA

• Competitors : 2D & 3D Methods

• Left & Right Arteries : Different

• Range of readings : 0 – 100 %

• Choice of d : 2%

Doctoral Thesis…

Robieson, W. Z. (1999) : On the Weighted

Kappa and Concordance Correlation Coefficient.

Ph.D. Thesis, University of Illinois at Chicago, USA

Lou, Congrong (2006) : Assessment of Agree-

ment : Multi-Rater Case.

Ph D Thesis, University of Illinois at Chicago, USA

Data Analysis….

• Lou(2006) derived expressions for

CP(d)^, V^(CP^(d)), COV^(…,…)

where CPiJ = P[|Xi – XJ|< d]

Data Analysis : CP12, CP13 & CP23

• Estimated Coverage Probability [CP] & Estimated Var. & Cov. for Screening Study

• Side Pairwise CP^ V^(CP^) COV^• Left CP12(L)^=0.56 0.0019 0.0009• Left CP13(L)^=0.47 0.0015• Right CP12(R)^=0.60 0.0021 0.0010• Right CP13(R)^=0.54 0.0019 • Left CP23(L)^ =0.64 0.0021 • Right CP23(R)^=0.69 0.0021

95% Lower Confidence Limits

• Left Side• CP12(L)^=0.56 95% Lower CL = 0.48• CP13(L)^=0.47 95% Lower CL = 0.40

• Right Side • CP12(R)^=0.60 95% Lower CL = 0.51• CP13(R)^=0.54 95% Lower CL = 0.46

Conclusion : Poor Agreement in all cases

Data Analysis (contd.)

Testing Hyp. Statistic p - value

H0L : CP12(L)= CP13(L) Z-score 0.0366 [against both-sided alternatives ]

H0R : CP12(R)= CP13(R) Z-score 0.1393Conclusions : For “Left Side”, CP for [1A vs2D] & for [1A vs 3D] are likely to be differentwhile for “Right Side” these are likely to beequal.

Testing Multiple Hypotheses

• For “K” alternatives [1, 2, …, K] to the Gold Standard [0], interest lies in

H0L : CP01(L)= CP02(L) = … = CP0K(L)

H0R : CP01(R)= CP02(R) = … = CP0K(R)

This is accomplished by performing

Large Sample Chi-Square Test [Rao (1973)]

Set for “Left Side”

L= (CP01(L)^ CP02(L)^ ….CP0 K(L)^)

Chi-Square Test… Chi-Sq.Test Statistic

L W^-1 L- [LW^-11]2 / [1 W^-1 1]

where

Wtt = Var (CP0t(L)^); t = 1, 2, ..

Wst = Cov (CP0s(L)^, CP0t(L)^); s # t

Asymptotic Chi-Sq. with K-1 df

Slly…for “Right Side” Hypothesis.

Simultaneous Lower Confidence Limits

Pr[ CP01 L1,CP02 L2, …,CP0K Lk] 95%

Set Zt = [CP0t^ – CP0t ] /Var^(CP0t^)

Assume : Zt ‘s Jointly follow Multivariate Normal Dist.

Work out estimated Correlation Matrix as usual.Solve for “z” such that

Pr[Z1 z, Z2 z, Z3 z,…, ZK z] 95%

Then Lt = CP0t^ – z.Var^(CP0t^) t = 1, 2, .., KStat Package : Available with Lou (2006).

Other Approaches…..

• Union-Intersection Principle….

• H_o : [│m│ > d_m] U [

││UU

Excellent Review Paper by Choudhary & Nagaraja :

Journal of Stat Planning & Inference .....

Conclusion

• Useful Statistical Concepts• Sound Technical Tools• Diverse Application Areas• Scope for Further Research on

Combining Evidences from Multi-Location Experiments...Meta Analysis !

Thanks !