The document discusses the analysis of contingency tables, focusing on measures of effect and agreement in health studies, particularly relating to disease prevention and risk assessment. It covers various statistical methods such as attributable risk, relative risk, and reliability measures including intraclass correlation coefficients and kappa statistics. The document also includes examples and calculations related to smoking's impact on colorectal cancer mortality and other health-related metrics.

1. The analysis of contingency tables
(measures of effect and agreement
July 2019
Fatemeh soleimany { soleimany2013@ut.ac.ir}
Amirhossein Aliakbar {amirhossein.aliakbar@gmail.com}

2. ‫توافقی‬ ‫جداول‬ ‫تحلیل‬3(‫توافق‬ ‫و‬ ‫تاثیر‬ ‫معیارهای‬)
‫آذر‬ ‫ایران‬ ‫متابولیسم‬ ‫و‬ ‫دیابت‬ ‫مجله‬-‫دی‬1393‫دوره‬14‫صفحه‬ ‫دوم‬ ‫جلد‬75‫تا‬92
Topics stored from:

3. 3
 Mohammad asghari jafar abadi Associate Professor of Biostatistics, Tabriz
university of medical science
 Seyede Momeneh Mohammadi , Phd candidate of anatomical scienses at Medical
faculty/Tabriz University of Medical Science
 Akbar Soltani, Associate Professor of Endocrinology, Tehran University of Medical Sciences

4. Objectives ( first part)
 To develop approaches for disease prevention
 Attributable risk/fraction
4
Case-control
Attribute Fraction
(AF)
Attribute fraction for
exposed group( AFE)
Attribute fraction for
population( AFP)
Attribute Risk (AR)
Attribute risk
reduction
(ARR)
Relative Risk
Reduction
(RRR)
Number
needed to treat
(NNT)

5. Objectives (second part)
5
reliability
Quantitative data
ICC
Bland’s and
Altman
Lin’s coefficient
corellation
Qualitative data
Kappa and
Kohen’s Kappa
Fleiss Kappa

6. Impact criteria
 AR: Attributable Risk
 AF: Attributable Fraction
 AFE: Attributable Fraction for Exposed Group
 AFP: Attributable Fraction for Population

7. AR/AF To develop approaches for disease prevention
Overall risk
Risk among non-
exposed
AR =
0
20
40
60
80
100
+ -
Excess
Risk
AR = [ (a+c) / n ] – [ c / ( c+d) ]
a b
c d

8. AF (attribute fraction )
AF= AR ÷ total risk
AF = AR ÷ 𝑎 + 𝑐 ÷ 𝑛 = 𝑎 + 𝑐 ÷ 𝑛 − 𝑐 ÷ 𝑐 + 𝑑 ÷ [ 𝑎 + 𝑐 ÷ 𝑛 ]
𝒑𝒆 = [ 𝒂 + 𝒃 ÷ 𝒏] 𝑨𝑭 = 𝒑𝒆 𝑹𝑹 − 𝟏 ÷ [ 𝟏 + 𝒑𝒆 𝑹𝑹 − 𝟏 ]
𝑷𝒄 = [ 𝒂 ÷ 𝒂 + 𝒄 ] 𝑨𝑭 = 𝒑𝒄 𝑹𝑹 − 𝟏 ÷ [ 𝑹𝑹 − 𝟏 ]
a b
c d

9. Relative risk (RR) =
𝑎
𝑎+𝑏
𝑐
𝑐+𝑑
Odds-ratio (OR) =
𝑎𝑑
𝑏𝑐
yes no
yes a b
no c d
event
exposure

10. Death rate Total number
Number of
living people
Number of
death
0.342 284 187 97 Have used
0.27 832 607 225 Non user
0.289 1116 794 322 total
Frequency of mortality in colorectal cancer patients in terms of the variables of smoking status
 AR = %28.9 – %27.0 = %1.9 ( %1.9 of the risk of mortality from colon cancer can be attributed to smoking )
 𝐀𝐅 =
𝐴𝑅
𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑟𝑖𝑠𝑘
=
1.9
28.9
= 0.063 or 6.3% (In comparison with the whole community, almost 6% of death from
colon cancer can be attributed to cigarette smoking.
Overall risk
Probability of
risk in non-
exposed

11. AFE =
𝑅𝐷
𝑡ℎ𝑒 𝑟𝑖𝑠𝑘 𝑖𝑛 𝑒𝑥𝑝𝑜𝑠𝑒𝑑 𝑔𝑟𝑜𝑢𝑝
=
𝑅𝑅−1
𝑅𝑅
=
( 0.342 −0.27 )
0.342
= 0.208 (By eliminating smoking in the exposure group, 21% of
deaths from smoking can be prevented. )
𝑹𝑹 =
97
284
225
832
= 1.263
Confidence Interval for AFE: ( 0.04 , 0.35 ) ( In comparison with smokers, Minimum and maximum
fraction of mortality attributed to cigarette smoking of colon cancer patients is in the range )
Death rate Total number
Number of
living people
Number of
death
0.342 284 187 97 Have used
0.27 832 607 225 Non user
0.289 1116 794 322 total

12. Rate of
smokers
Total number
smoking
Number of non-
smoking ones
Number of
smokers
0.671003367Exposed
0.202559446113Non exposed
0.273659479180Total
0.1520.0690.372The rate of
AR = [(a+c) ÷n]-[c÷(c+d)] = [(67+113) ÷ 659]-[113 ÷(113+446)] = (27.3-20.2) = 7.1 (7.1% of the risk of
being smoker can be attributed to peers’insistence )
AF = AR ÷ total risk = (7.1 ÷ 27.3) = 0.26 (In comparison with whole population, 26% of smoking can be
attributed to peer’s insistence)
OR = (a × d) ÷ (b × c) = (67 × 446) ÷ (33×113) = 8.01
AFE = [OR-1] ÷[OR] = (8.01-1) ÷8.01 = 0.88 (By eliminating the effect of peers’insistence, 8.8 percent of
smoking may be prevented in exposed group)

13. Confidence Interval for AFE: ( 0.80 , 0.92 ) {Therefore, in the statistical
population with 95% confidence, For those with a peer's insistence,
minimum and maximum fraction attributed to peer's insistence is in the
domain above.}
AFP = AFE × percent exposed among cases = 0.875 × (67 ÷180) = 0.326
{By eliminating the effect of peers’insistence, 8.8 percent of smoking may be
prevented in whole population)

14. ARR and RRR in randomized clinical trial
 ARR: Absolute Risk Reduction
 RRR: Relative Risk Reduction
 NNT: Number Needed to Treat
 NNH: number needed to harm
 ARR= NNT if ………

15.  ARR =
𝑐
𝑐+𝑑
−
𝑎
𝑎+𝑏
=
21
21+9
+
11
11+16
= 0.70 − 0.41 = 0.29 (the contribution of omega-3 in treatment of neuropathy
is only 29%.)
NNT =
1
𝐴𝑅𝑅
=
1
0.29
=3.4 ≈3 (it's expected that using this supplement in about 3 patients, will improve one of them. )
RRR ==
(
𝑐
𝑐+𝑑
)−(
𝑎
𝑎+𝑏
)
(
𝑐
𝑐+𝑑
)
=
21
21+9
−(
11
11+16
)
(
11
11+16
)
=
0.29
0.41
= 0.71 (The relative reduction in neuropathy induced by intervention was 71%
compared with the placebo group.)
The risk of
neuropathy
Chance of
recovery
Total
number
neuropathy
Non normalnormal
0.300.7030921Omega 3
0.590.41271611Placebo

16.  NNH= Number needed to harm
Experimental
group ( E)
Control group
(C)
total
Events (E) EE = 75 CE = 100 115
Non-events
(N)
EN = 75 CN = 150 285
Total subjects
(S)
ES = EE + EN
= 150
CS = CE+ CN
= 250
400
Event rate
(ER)
EER = EE/ ES
= 0.5 or 50%
CER = CE/ CS
= 0.4 or 40%
equation variable Abbr. Value
EER-CER Absolute risk
increase
ARI 0.1 or 10%
(EER –CER)
/ CER
Relative risk
increase
RRI 0.25 or 25%
1 / (EER –
CER)
Number
needed to
harm
NNH 10
EER/CER Risk ratio RR 1.25
(EE / EN) /
(CE/CN)
Odds ratio OR 1.5
(EER –CER)
/ EER
Attributable
fraction
among
exposed
AFE 0.2

17. Agreement measures
Golden standard
ICC and SEM
Bland and Altman’s limit of agreement
Lin’s concordance correlation coefficient
Cohen's kappa and Weighted Cohen’s Kappa
ROC

18. 𝐿𝑅+
and 𝐿𝑅−
 PPV: Positive predictive value
 NPV: Negative predictive value
Diagnosed sick Diagnosed healthy
Sick
True positive False negative
healthy
False positive True negative

19. Prior and Posterior probability
 If LR > 1 then Posterior probability > Prior probability disease
 If LR < 1 then Posterior probability < Prior probability

20. Sensitivity =
𝑇𝑟𝑢𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
𝑇𝑟𝑢𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒+𝐹𝑎𝑙𝑠𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
=
23
23+13
= 0.64 → 64%
Specificity =
𝑇𝑟𝑢𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
𝑇𝑟𝑢𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒+𝐹𝑎𝑙𝑠𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
=
51
51+13
= 0.80 → 80%
PPV =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑢𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑢𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒+𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑎𝑠𝑙𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
=
23
23+13
= 0.64 → 64%
NPV =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑢𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑢𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒+𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑎𝑙𝑠𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
=
53
53+13
= 0.80 → 80%
GDM
Total number Health Sick
36 13 23 +
64 51 13 -
100 64 36 Total number

21.  𝑳𝑹+=
𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦
1−𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑡𝑦
=
𝑜𝑑𝑑𝑠 𝑜𝑓 𝑡𝑟𝑢𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
𝑜𝑑𝑑𝑠 𝑜𝑓 𝑓𝑎𝑙𝑠𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
=
23
36
13
64
= 3.15 (The chance to see a real patient to a false
patient in a positive test result is approximately 3 times of the average effect size)
 𝑳𝑹−
=
1−𝑆𝑒𝑛𝑠𝑖𝑣𝑖𝑡𝑦
𝑆𝑝𝑒𝑐𝑖𝑣𝑖𝑡𝑦
=
𝑜𝑑𝑑𝑠 𝑜𝑓 𝑓𝑎𝑙𝑠𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
𝑜𝑑𝑑𝑠 𝑜𝑓 𝑡𝑟𝑢𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
=
13
36
56
64
= 0.45 (The chance to see a healthy, false patient to
actual normal one, in the negative test results is almost half of the effect size.)
GDM
Total number Health Sick
36 13 23 +
64 51 13 -
100 64 36 Total number

22. Index Value
Lower
bound
Upper
bound
prevalence 36% 27% 46.2%
Sensitivity 63.9% 46.2% 79.2%
Specificity 79.7% 67.8% 88.7%
PPV 63.9% 46.2% 79.2%
NPV 79.7% 67.8% 88.7%
𝐿𝑅+ 3.15% 1.83 5.42
𝐿𝑅− 0.45% 0.29 0.71
Note in practice: it’s required to calculate
confidence interval for specificity, Sensivity,
𝐿𝑅+ , 𝐿𝑅− , NPV and PPV.

23. Prior probability of disease= 36 ÷ 100 = 0.36 ( equal for everyone and means prevalence)
prior chance of being actual patient = Pre-test-odds (+)= Pre-test- probability / (1 – Pre-test- probability )
= 0.36÷ 1 − 0.36 = 0.56
(posterior chance of being actual patient) Post test Odds (+)=Pretest odds × LR+ = 0.56 × 3.15 = 1.77
(posterior probability of being actual patient) Post Probability (+)=(Pre test Odds × LR+) / (1 + (Pre test Odds ×
LR+
)) = 1.77÷(1+1.77) = 0.64
(posterior chance of being false patient) Post test odds (-)= Pretest odds × LR−
= 0.56 ×0.45 = 0.26
(posterior probability of being false patient) Post Probability (-)=(Pre test Odds × LR−
) / (1 + (Pre test Odds ×
LR−
)) = 0.26÷(1+0.26) = 0.20

24. ICC (Intra class correlation coefficient (ICC) for one way and two
way models)
Missing data are omitted in a list wise way.
When considering which form of ICC is appropriate for an actual set of data, one has
take several decisions (Shrout & Fleiss, 1979):
 1. Should only the subjects be considered as random effects (’"oneway"’model) or are subjects and
raters randomly chosen from a bigger pool of persons (’"twoway"’ model).
 2. If differences in judges’ mean ratings are of interest, interrater ’"agreement"’ instead of
’"consistency"’ should be computed.
 3. If the unit of analysis is a mean of several ratings, unit should be changed to ’"average"’. In most
cases, however, single values (unit=’"single"’) are regarded.

25. Standard Error of Measurement(SEM)
SEM Reliability
SEM = SD× √(1-ICC)
How can I trust
you?

26. Quantitative measures
The concordance correlation coefficient combines measures of both precision and accuracy to
determine how far the observed data deviate from the line of perfect concordance (that is, the
line at 45 degrees on a square scatter plot). Lin's coefficient increases in value as a function of
the nearness of the data's reduced major axis to the line of perfect concordance (the accuracy of
the data) and of the tightness of the data about its reduced major axis (the precision of the data)
Lin's concordance correlation coefficient
<0.9 poor
0.9 to 0.95 moderate
0.95 to 0.99 substantial
> 0.99
Almost
perfect

27. Lin’s coefficient

28. Bland and Altman's limit of agreement (Tukey mean-difference
plot)
 They advocated the use of a graphical method to plot the difference scores of two
or more than two measurements against the mean for each subject and argued that
if the new method agrees sufficiently well with the old, the old may be replaced.
 BlandAltmanLeh and Blandr

30. ICC (Intraclass Correlation Coefficient)
 is a descriptive statistics that can be used when quantitative measurements are
made on units that are organized into groups. It describes how strongly units in the
same group resemble each other.
90 /
100 9 /
10

32. Cohen’s Kappa and weighted Kappa for two raters
Inter observer variation can be measured in any situation in which two or more independent observers
are evaluating the same thing.
Sometimes, we are more interested in the agreement across major categories in which there is
meaningful difference.
Strongly
disagree
disagreeNo IdeaagreeStrongly agree
Strongly agree
agree
No Idea
disagree
Strongly
disagree
Colour lost means weight lost

34. Commonly used weights
Weighted-Kappa requires specific weight, for instance:
 1-
𝒊 −𝒋
𝒌−𝟏
 𝟏 −
𝒊−𝒋
𝒌−𝟏
𝟐
 arbitrary weights
{ I for rows – J for columns
K for maximum number of instruments}
The nearer
apples,
the bigger
weights

35. totalNo
agreement
Somewhat
agreement
Perfect
agreement
( two
instrument)
204106Perfect
appointment
356209Somewhat
appointment
130504040No appointment
185607055total
Weighted kappa based on first weight
=0.094
And based on second weight = 0.07
Weak
agreement

36. The power of different Kappa coefficients in determining the amount of agreement
among observers or referees
Agreement reliabilityKappa statistic
amount
WeakLess than 0
low0 – 0.2
Less than average0.21 – 0.4
average0.41 – 0.6
good0.6 – 0.8
excellent0.81 - 1
Is there any
confidence interval for
kappa? Is it valuable?

37. references
 Understanding Interobserver Agreement:The Kappa Statistic. Anthony J. Viera, MD; Joanne M. Garrett, PhD. Family
Medicine.
 Bland J, Altman D (1986). Statistical methods for assessing agreement between two methods of clinical measurement. The
Lancet 327: 307 - 310.
 Bland J, Altman D (1999). Measuring agreement in method comparison studies. Statistical Methods in Medical Research 8:
135 - 160.
 P. S. Myles, J. Cui, I. Using the Bland–Altman method to measure agreement with repeated measures, BJA: British Journal
ofAnaesthesia, Volume 99, Issue 3, September 2007, Pages 309–311,
 Bland J, Altman D (2007). Agreement between methods of measurement with multiple observations per individual. Journal
of Biopharmaceutical Statistics 17: 571 - 582. (Corrects the formula quoted in the 1999 paper).
 Bradley E, Blackwood L (1989). Comparing paired data: a simultaneous test for means and variances. American
Statistician 43: 234 - 235.
 Burdick RK, Graybill FA (1992). Confidence Intervals on Variance Components. New York: Dekker.
 Dunn G (2004). Statistical Evaluation of Measurement Errors: Design and Analysis of Reliability Studies. London: Arnold.

38. Thank you