Statistical methods for assessment of agreement

Professor Dr. Bikas K Sinha Applied Statistics Division Indian Statistical Institute Kolkata, INDIA

Organized by Department of Statistics, RU

17 April, 2012

Lecture Plan

Agreement for Categorical Data [Part I]

09.00 – 10.15 hrs

Coffee Break: 10.15 – 10.30 hrs

Agreement for Continuous Data [Part II]

10.30 – 11.45 hrs

Discussion.....11.45 hrs – 12.00 hrs

Key References

Cohen, J. (1960). A Coefficient of Agreement for Nominal Scales. Educational & Psychological Measurement, 20(1): 37 – 46.

[Famous for Cohen’s Kappa]

Cohen, J. (1968). Weighted Kappa : Nominal Scale Agreement with Provision for Scaled Disagreement or Partial Credit.

Psychological Bulletin, 70(4): 213-220.

References ….contd.

Banerjee, M., Capozzoli, M., Mcsweeney,L. & Sinha, D. (1999). Beyond Kappa : A Review of Interrater Agreement Measures. Canadian Jour. of Statistics, 27(1) : 3 - 23.

Lin. L. I. (1989). A Concordance Correlation Coefficient to Evaluate Reproducibility. Biometrics, 45 : 255 - 268.

References …contd.Lin. L. I. (2000).Total Deviation Index forMeasuring Individual Agreement : WithApplication in Lab Performance and Bioequi-valence .Statistics in Medicine,19:255 - 270.

Lin, L. I., Hedayat, A. S., Sinha, Bikas &Yang, Min (2002). Statistical Methods inAssessing Agreement: Models, Issues, andTools. Jour. Amer. Statist. Assoc. 97 (457) : 257 - 270.

Measurements : Provided by Experts / Observers / Raters

• Could be two or more systems, assessors, chemists, psychologists, radiologists, clinicians, nurses, rating system or raters, diagnosis or treatments, instruments or methods, processes or techniques or formulae……

Diverse Application Areas…

• Cross checking of data for

agreement, Acceptability of a new or generic drug or of test instruments against standard instruments, or of a new method against gold standard method, statistical process control..

Nature of Agreement Problems…• Assessment & Recording of Responses …• Two Assessors for evaluation and recording…• The Raters examine each “unit” Independently of

one another and report separately : “+” for “Affected” or “-” for “OK” : Discrete Type.

Summary Statistics UNIT Assessment Table Assessor \ Assessor # II # I + - + 40% 3% - 3% 54%Q. What is the extent of agreement of the two

assessors ?

Nature of DataAssessor \ Assessor # II # I + - + 93% 2% - 4% 1%Assessor \ Assessor # II # I + - + 3% 40% - 44% 13%Same Question : Extent of Agreement /

Disagreement ?

Cohen’s Kappa : Nominal Scales

Cohen (1960)

Proposed Kappa statistic for measuring agreement when the responses are nominal

Cohen’s Kappa • Rater I vs Rater II : 2 x 2 Case Categories Yes & No : Prop. (i,j) (Y,Y) & (N,N) : Agreement Prop (Y,N) & (N,Y) :Disagrmnt. Prop

0 = (Y,Y) + (N,N) = P[agreement]

e = (Y,.) (.,Y) + (N,.) (.,N) P [Chancy Agreement]

= [ 0 - e ] / [ 1 - e ] Chance-corrected Agreement Index

Kappa Computation…. II Total Yes No I Yes 0.40 0.03 0.43 No 0.03 0.54 0.57 Total 0.43 0.57 1.00

Observed Agreement 0 = (Y,Y) + (N,N)= 0.40 + 0.54 = 0.94 …94%

Chance Factor towards agreement……e = (Y,.) (.,Y) + (N,.) (.,N) = 0.43x0.43 + 0.57x0.57=0.5098 ……51% = [ 0 - e ] / [ 1 - e ]=.4302/.4902 = 87.76%....Chance-Corrected Agreement

Kappa Computations..

• Raters I vs II • 0.40 0.03• 0.03 0.54 K = 0.8776• 0.93 0.02• 0.04 0.01 K = 0.2208

0.03 0.40• 0.54 0.03 K = - 0.8439• 0.02 0.93• 0.01 0.04 K = - 0.0184

Nature of Categorical Data

Illustrative Example

Study on Diabetic Retinopathy Screening

Problem : Interpretation of

Single-Field Digital Fundus Images

Assessment of Agreement

WITHIN / ACROSS 4 EXPERT GROUPS

Retina Specialists / General Opthalmologists /

Photographers / Nurses : 3 from each Group

Description of Study Material

400 Diabetic Patients Selected randomly from a community hospital

One Good Single-Field Digital Fundus Image Taken from each patient with Signed Consent

Approved by Ethical Committee on Research with Human Subjects

Raters : Allowed to Magnify / Move the Images NOT TO MODIFY Brightness / Contrasts

THREE Major Features

#1. Diabetic Retinopathy Severity [6 options]

No Retinopathy / Mild / Moderate NPDR

Severe NPDR / PDR / Ungradable

#2. Macular Edema [ 3 options]

Presence / Absence / Ungradable

#3. Referrals to Opthalmologists [3 options]

Referrals / Non-Referrals / Uncertain

Retina Specialists’ Ratings [DR]

RS1 \ RS2CODES 0 1 2 3 4 9 Total 0 247 2 2 1 0 0 252 1 12 18 7 1 0 0 38 2 22 10 40 8 0 1 81 3 0 0 3 2 2 0 7 4 0 0 0 1 9 0 10 9 5 0 1 0 0 6 12 Total 286 30 53 13 11 7 400

Retina Specialists’ Ratings [DR]

RS1 \ RS3CODES 0 1 2 3 4 9 Total 0 249 2 0 1 0 0 252 1 23 8 7 0 0 0 38 2 31 4 44 2 0 0 81 3 0 0 7 0 0 0 7 4 0 0 0 0 10 0 10 9 9 1 0 0 0 2 12 Total 312 15 58 3 10 2 400

Retina Specialists’ Ratings [DR]

RS2 \ RS3CODES 0 1 2 3 4 9 Total 0 274 5 6 1 0 0 286 1 16 5 8 1 0 0 30 2 15 2 35 0 0 1 53 3 2 2 7 1 1 0 13 4 0 0 2 0 9 0 11 9 5 1 0 0 0 1 7 Total 312 15 58 3 10 2 400

Retina Specialists’ Consensus Rating [DR]

RS1 \ RSCRCODES 0 1 2 3 4 9 Total 0 252 0 0 0 0 0 252 1 17 19 2 0 0 0 38 2 15 19 43 2 1 1 81 3 0 0 2 4 1 0 7 4 0 0 0 0 10 0 10 9 8 0 0 0 0 4 12 Total 292 38 47 6 12 5 400

Retina Specialists’ Ratings [Macular Edema]

RS1 \ RS2CODES Presence Absence Subtotal Ungradable Total Presence 326 11 337 1 338Absence 18 22 40 3 43

Subtotal 344 33 377 -- --

Ungradable 9 0 -- 10 19

Total 353 33 -- 14 400

Retina Specialists’ Ratings [ME]RS1 \ RS3CODES Presence Absence Subtotal Ungradable Total Presence 322 13 335 3 338Absence 8 32 40 3 43

Subtotal 330 45 375

Ungradable 12 0 -- 7 19

Total 342 45 -- 13 400

Retina Specialists’ Consensus Rating [ME]

RS1 \ RSCRCODES Presence Absence Subtotal Ungradable

Total Presence 335 2 337 1 338Absence 10 33 43 0 43

Subtotal 345 35 380 -- --

Ungradable 10 0 -- 9 19

Total 355 35 -- 10 400

Photographers on Diabetic ME PHOTOGRAPHERS 1 vs 2 1 2 Codes Presence Absence SubTotal Ungradable Total

Presence 209 5 214 51 265Absence 65 41 106 4 110

Subtotal 274 46 320 -- --

Ungradable 2 2 --- 21 25

Total 276 48 --- 76 400

Photographers’ Consensus Rating on Diabetic Macular Edema

PHOTOGRAPHERS’s # 1 Consensus Rating Codes Presence Absence SubTotal Ungradable Total

Presence 257 5 262 3 265Absence 74 30 104 6 110

Subtotal 331 35 366 -- --

Ungradable 24 0 --- 1 25

Total 355 35 --- 10 400

Study of RS’s Agreement [ME]2 x 2 Table : Cohen’s Kappa (K) Coefficient Retina Specialist Retina Specialist 2 1 Presence Absence Subtotal Presence 326 11 337 Absence 18 22 40 Subtotal 344 33 377 IGNORED ’Ungradable’ to work with 2 x 2 table % agreement : (326 + 22) / 377 = 0.9231 = Theta_0% Chancy Agreement : %Yes. %Yes + %No. %No(337/377)(344/377) + (40/377)(33/377) = 0.8250 = Theta_e K = [Theta_0 – Theta_e] / [ 1 – Theta_e] = 56% only ! Nett Agreement Standardized

Study of Photographers’’s Agreement on Macular Edema

2 x 2 Table : Cohen’s Kappa (K) Coefficient Photographer Photographer 2 1 Presence Absence Subtotal Presence 209 5 214 Absence 65 41 106 Subtotal 274 46 320 IGNORED ’Ungradable’ to work with 2 x 2 table % agreement : (209 + 41) / 320 = 0.7813 = Theta_0% Chancy Agreement : %Yes. %Yes + %No. %No(214/320)(274/320) + (106/320)(46/320) = 0.6202 = Theta_e K = [Theta_0 – Theta_e] / [ 1 – Theta_e] = 42% only ! Nett Agreement Standardized

What About Multiple Ratings likeDiabetic Retinopathy [DR] ?

1 Retina Specialists 2CODES 0 1 2 3 4 9 Total 0 247 2 2 1 0 0 252 1 12 18 7 1 0 0 38 2 22 10 40 8 0 1 81 3 0 0 3 2 2 0 7 4 0 0 0 1 9 0 10 9 5 0 1 0 0 6 12 Total 286 30 53 13 11 7 400

K Computation……

% Agreement =(247+18+40+2+9+6)/400 = 322/400 =0.8050 = Theta_0% Chance Agreement = (252/400)(286/400) + ….+(12/400)(7/400) = 0.4860 = Theta_e

K = [Theta_0 – Theta_e] / [ 1 – Theta_e] = 62% ! Note : 100% Credit for ’Hit’ & No Credit for ’Miss’. Criticism : Heavy Penalty for narrowly missed ! Concept of Unweighted Versus Weighted Kappa

Table of Weights for 6x6 Ratings

Ratings Ratings [ 1 to 6 ]

1 2 3 4 5 6

1 1 24/25 21/25 16/25 9/25 0

2 24/25 1 24/25 21/25 16/25 9/25

3 21/25 24/25 1 24/25 21/25 16/25

4 16/25 21/25 24/25 1 24/25 21/25

5 9/25 16/25 21/25 24/25 1 24/25

6 0 9/25 16/25 21/25 24/25 1

Formula w_ij = 1 – [(i – j)^2 / (6-1)^2]

Formula for Weighted Kappa

• Theta_0(w) = sum sum w_ij f_ij / n

• Theta_e(w) = sum sum w_ij (f_i. /n)(f_.j/n)

• These sum sum are over ALL cells

• For unweighted Kappa : we take into account only the cell freq. along the main diagonal with 100% weight

Computations for Weighted Kappa

• Theta_0(w) = • Theta_e(w) = • Theta_0(w) – Theta_e(w) Weighted Kappa = ---------------------------------• 1 – Theta_e(w)• Unweighted Kappa =........K works for pairwise evaluation of Raters’

agreement …….

K-statistics for Pairs of Raters…Categories DR ME Referral

Retina Specialists

1 vs 2 0.63 0.58 0.65

1 vs 3 0.55 0.64 0.65

2 vs 3 0.56 0.51 0.59

1 vs CGroup 0.67 0.65 0.66

2 vs CGroup 0.70 0.65 0.66

3 vs CGroup 0.71 0.73 0.72

Unweighted Kappa......

K-statistics for Pairs of Raters…

Categories DR ME Referral

General Opthalmologists

1 vs 2 0.35 0.17 0.23

1 vs 3 0.44 0.27 0.27

2 vs 3 0.33 0.19 0.27

1 vs CGroup 0.33 0.16 0.18

2 vs CGroup 0.58 0.50 0.51

3 vs CGroup 0.38 0.20 0.24

K-statistics for Pairs of Raters…

Categories DR ME ReferralPhotographers….. 1 vs 2 0.33 0.35 0.231 vs 3 0.49 0.38 0.412 vs 3 0.34 0.45 0.321 vs CGroup 0.33 0.29 0.332 vs CGroup 0.26 0.29 0.203 vs CGroup 0.39 0.49 0.49

K-statistics for Pairs of Raters…

Categories DR ME Referral

Nurses………..

1 vs 2 0.28 0.15 0.20

1 vs 3 0.32 NA NA

2 vs 3 0.23 NA NA

1 vs CGroup 0.29 0.27 0.28

2 vs CGroup 0.19 0.15 0.17

3 vs CGroup 0.50 NA NA

NA : Rater #3 did NOT rate ’ungradable’.

K for Multiple Raters’ Agreement

• Judgement on Simultaneous Agreement of Multiple Raters with Multiple Classification of Attributes…....

# Raters = n

# Subjects = k

# Mutually Exclusive & Exhaustive

Nominal Categories = c

Example....Retina Specialists (n = 3),

Patients (k = 400) & DR (6 codes)

Formula for Kappa

• Set k_ij = # raters to assign

ith subject to jth category

P_ j = sum_i k_ij / nk = Prop. Of all

assignments to jth category

Chance-corrected assignment to category j

[sum k^2_ij – knP_ j{1+(n-1)P_ j}

K_ j = -------------------------------------------

kn(n-1)P_ j (1 – P_ j)

Computation of Kappa

• Chance-corrected measure of over-all agreement

• Sum_ j Numerator of K_ j

• K = -----------------------------------------

• Sum_ j Denominator of K_ j

• Interpretation ….Intraclass correlation

K-statistic for multiple raters…

CATEGORIES DR ME Referral Retina Specialsts 0.58 0.58 0.63Gen. Opthalmo. 0.36 0.19 0.24 Photographers 0.37 0.38 0.30Nurses 0.26 0.20 0.20All Raters 0.34 0.27 0.28

Other than Retina Specialists, Photographersalso have good agreement for DR & ME…

Conclusion based on K-Study

• Of all 400 cases…..• 44 warranted Referral to Opthalmologists due to

Retinopathy Severity• 5 warranted Referral to Opthalmologists due to

uncertainty in diagnosis • Fourth Retina Specialist carried out Dilated

Fundus Exam of these 44 patients and substantial agreement [K = 0.68] was noticed for DR severity……

• Exam confirmed Referral of 38 / 44 cases.

Discussion on the Study

• Retina Specialists : All in active clinical practice : Most reliable for digital image interpretation

• Individual Rater’s background and experience play roles in digital image interpretation

• Unusually high % of ungradable images among nonphysician raters, though only 5 out of 400 were declared as ’ungradable’ by consensus of the Retina Specialists’ Group.

• Lack of Confidence of Nonphysicians, rather than true image ambiguity !

• For this study, other factors [blood pressure, blood sugar, cholesterol etc] not taken into account……

Cohen’s Kappa : Need for Further Theoretical Research

• COHEN’S KAPPA STATISTIC: A CRITICAL APPRAISAL AND SOME MODIFICATIONS

• BIKAS K. SINHA^1, PORNPIS YIMPRAYOON^2, AND MONTIP TIENSUWAN^2

• ^1 : ISI, Kolkata

• ^2 : Mahidol Univ., Bangkok, Thailand

• CSA BULLETIN, 2007

CSA Bulletin (2007) Paper…

• ABSTRACT: In this paper we consider the problem of assessing agreement between two raters while the ratings are given independently in 2-point nominal scale and critically examine some features of Cohen’s Kappa Statistic, widely and extensively used in this context. We point out some undesirable features of K and, in the process, propose three modified Kappa Statistics. Properties and features of these statistics are explained with illustrative examples.

Further Theoretical Aspects of Kappa – Statistics….

• Recent Study on Standardization of Kappa

• Why standardization ?

• K = [Theta_0 – Theta_e] / [ 1 – Theta_e]

Range : -1 <= K <= 1

• K = 1 iff 100% Perfect Rankings

• = 0 iff 100% Chancy Ranking

• = -1 iff 100% Imperfect BUT Split-Half

Why Split Half ?

Example

Presence Absence• Presence ---- 30%• Absence 70% ---

K_C = - 73% [& not -100 %]

************************************

Only Split Half ---- 50% provides

50% ---- K_C = - 100 %

K-Modified….

• [Theta_0 – Theta_e] K_C(M) = ------------------------------------------ P_I[Marginal Y]. P_I[Marginal N] + P_II[Marginal Y]. P_II[Marginal N] Y : ’Presence’ Category & N : ’Absence’ Category ’I’ & ’II’ represent the Raters I & II K_C(M) Satisfies K = 1 iff 100% Perfect Rankings..whatever• = 0 iff 100% Chancy Ranking…whatever• = -1 iff 100% Imperfect Ranking…whatever…

Other Formulae..….

• What if it is known that there is 80% Observed Agreement i.e., Theta_0 = 80% ?

• K_max = 1 ? K_min = -1 ?...NOT Really....• So we need standardization of K_C as• K_C(M2) = [K_C – K_C(min)] OVER

[K_C (max) – K_C(min)]

where Max. & Min. are to be evaluated under the stipulated value of observed agreement

Standardization yields….

K_C + (1-Theta_0) / (1+Theta_0)

K_C(M2) = -----------------------------------------------

Theta_0^2 / [1+(1-Theta_0)^2] +

(1-Theta_0) / (1+Theta_0)

K_C(M3)={[K_C(M) + (1-Theta_0)/(1+Theta_0)}

OVER

{[Theta_0 /(2-Theta_0) + (1-Theta_0)/(1+Theta_0)}

Revisiting Cohen’s Kappa…..

2 x 2 Table : Cohen’s Kappa (K) Coefficient Retina Specialist Retina Specialist 2

1 Presence Absence Subtotal Presence 326 11 337 Absence 18 22 40 Subtotal 344 33 377 K_C = 56% [computed earlier]

Kappa - Modified• K_C(M) = 56 % [same as K_C]

Given Theta_0 = 92.30 %

• 0.5600 + 0.0400

• K_C(M2) = ------------------------- = 61 %

• 0.8469 + 0.0400

• 0.5600 + 0.0400

• K_C(M3) = --------------------------- = 67 %

• 0.8570 + 0.0400

Beyond Kappa …..

• A Review of Inter-rater Agreement Measures

• Banerjee et al : Canadian Journal of Statistics : 1999; 3-23

• Modelling Patterns of Agreement :

• Log Linear Models

• Latent Class Models

That’s it for now……

• Thanks for your attention….

• This is the End of Part I of my talk.

Bikas Sinha

UIC, Chicago

April 29, 2011