1 / 37

On Some Statistical Aspects of Agreement Among Measurements BIKAS K SINHA [ISI, Kolkata]

Tampere August 28, 2009. On Some Statistical Aspects of Agreement Among Measurements BIKAS K SINHA [ISI, Kolkata] . Part II : Statistical Assessment of Agreement. Understanding Agreement among Raters involving Continuous Measurements…. Theory & Applications…. Continuous Measurements.

benjamin
Télécharger la présentation

On Some Statistical Aspects of Agreement Among Measurements BIKAS K SINHA [ISI, Kolkata]

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Tampere August 28, 2009 On Some Statistical Aspects of Agreement Among MeasurementsBIKAS K SINHA [ISI, Kolkata]

  2. Part II : Statistical Assessment of Agreement Understanding Agreement among Raters involving Continuous Measurements…. • Theory & Applications…..

  3. 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…..

  4. 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.

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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……

  11. LOA Approach….Data Analysis • m = E(X – Y) = (m1–m2) • 2 = Var(X-Y) = (12 +22 - 212) • 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

  12. 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 +/- 1.96  Step 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.

  13. 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.

  14. 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 +22 - 212)

  15. 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

  16. 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

  17. Scatter Plot : IA-2D-3D

  18. Right vs Left Arteries [IA]

  19. 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

  20. 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

  21. 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.] = 212 /[(m1–m2)2 + (12 +22) Properties :Perfect Agreement [CCC = 1] Perfect Disagreement [CCC = -1] No Agreement [CCC = 0]

  22. CCC… • CCC = 212 /[(m1–m2)2 + (12 +22)] = . a  = Accuracy Coefficient a = Precision Coeff. [<=1] a = [2 / { + 1/ + 2}] where  = 1/2 and 2 = (m1–m2)2 / 12 CCC = 1 iff  = 1 & a = 1 a = 1 iff [ m1 = m2 ] & [1 = 2 ] hold simultaneously !!

  23. 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…..

  24. 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 / D2]..non-central 2 TDI = Value of k for given  Inference based on TDI Choice of  : 90 % or more

  25. Coverage Probability Lin et al (JASA, 2002) : BVN distribution CP(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

  26. Graphical Display …

  27. 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%

  28. 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

  29. Data Analysis…. • Lou(2006) derived expressions for CP(d)^, V^(CP^(d)), COV^(…,…)‏ where CPiJ = P[|Xi – XJ|< d]

  30. 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

  31. 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

  32. 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 0.1393 Conclusions : For “Left Side”, CP for [1A vs 2D] & for [1A vs 3D] are likely to be different while for “Right Side” these are likely to be equal.

  33. 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)^)‏

  34. Chi-Square Test… Chi-Sq.Test Statistic LW^-1 L- [LW^-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.

  35. 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, .., K Stat Package : Available with Lou (2006).

  36. Other Approaches….. • Union-Intersection Principle…. • H_o : [│m│ > d_m] U [ > d_s] H_o :│m│>d_m U d_1< _2 / _1<d_2 U r<d_r Excellent Review Paper by Choudhary & Nagaraja : Journal of Stat Planning & Inference .....

  37. Conclusion • Useful Statistical Concepts • Sound Technical Tools • Diverse Application Areas • Scope for Further Research on Combining Evidences from Multi-Location Experiments...Meta Analysis ! Thanks !

More Related