Epidemiologic MethodsSeptember 28 Understanding Measurement: Aspects of Reproducibility & Validity

October 3 Study Design

October 10 Measures of Disease Occurrence I

October 17 Measures of Disease Occurrence IIOctober 24 Measures of Disease Association I

October 31 Measures of Disease Association IIOctober 31 Measures of Attributable Risk

November 7 Bias in Epidemiologic Studies: Selection Bias

November 14 Bias in Epidemiologic Studies: Measurement Bias

November 14 Confounding and Interaction I: General Principles

November 21 Confounding and Interaction II: Assessing InteractionNovember 28 Confounding and Interaction II: Stratified Analysis

December 5 Conceptual Approach to Multivariable Analysis I

December 7 Conceptual Approach to Multivariable Analysis II

December 12 Conceptual Approach to Multivariable Analysis III

Definitions of Epidemiology

• The study of the distribution and determinants (causes) of disease– e.g. cardiovascular epidemiology

• The method used to conduct human subject research– the methodologic foundation of any research

where individual humans or groups of humans are the unit of observation

Understanding Measurement: Aspects of Reproducibility and Validity

• Review Measurement Scales

• Reproducibility

– importance– methods of assessment

• by variable type: interval vs categorical• intra- vs. inter-observer comparison

• Validity– methods of assessment

• gold standards present• no gold standard available

Clinical Research

Sample

MeasureAnalyze

Infer

A study can only be as good as the data . . .

-Martin Bland

Measurement Scales

Scale Example

Interval continuousdiscrete

weightWBC count

Categoricalordinalnominaldichotomous

tumor stageracedeath

Reproducibility vs Validity

• Reproducibility– the degree to which a measurement provides the

same result each time it is performed on a given subject or specimen

• Validity– from the Latin validus - strong– the degree to which a measurement truly

measures (represents) what it purports to measure (represent)

Reproducibility vs Validity

• Reproducibility– aka: reliability, repeatability, precision, variability,

dependability, consistency, stability

• Validity– aka: accuracy

Relationship Between Reproducibility and Validity

Good Reproducibility

Poor Validity

Poor Reproducibility

Good Validity

Relationship Between Reproducibility and Validity

Good Reproducibility

Good Validity

Poor Reproducibility

Poor Validity

Why Care About Reproducibility?

Impact on Validity• Mathematically, the upper limit of a measurement’s

validity is a function of its reproducibility

• Consider a study to measure height in the community:

– if we measure height twice on a given person and get two different values, then one of the two values must be wrong (invalid)

– if study measures everyone only once, errors, despite being random, may not balance out

– final inferences are likely to be wrong (invalid)

Why Care About Reproducibility?

Impact on Statistical Precision

• Classical Measurement Theory:

observed value (O) = true value (T) + measurement error (E)

E is random and ~ N (0, 2E)

Therefore, when measuring a group of subjects, the variability of observed values is a combination of:

the variability in their true values and measurement error

2O = 2

T + 2E

Why Care About Reproducibility?

2O = 2

T + 2E

• More measurement error means more variability in observed measurements

• More variability of observed measurements has profound influences on statistical precision/power:– Descriptive study: less precise estimates of given traits– RCT’s: power to detect a treatment difference is reduced– Observational studies: power to detect an influence of a

particular exposure upon a given outcome is reduced.

Conceptual Definition of Reproducibility

• Reproducibility

• Varies from 0 (poor) to 1 (optimal)

• As 2E approaches 0 (no error), reproducibility

approaches 1

2

E

2

T

2

T

2

O

2

T

Phillips and Smith, J Clin Epi 1993

Sources of Measurement Variability

• Observer

• within-observer (intrarater)

• between-observer (interrater)

• Instrument

• within-instrument

• between-instrument

• Subject

• within-subject

Sources of Measurement Variability

• e.g. plasma HIV viral load– observer: measurement to measurement

differences in tube filling, time before processing

– instrument: run to run differences in reagent concentration, PCR cycle times, enzymatic efficiency

– subject: biologic variation in viral load

Assessing Reproducibility

Depends on measurement scale

• Interval Scale– within-subject standard deviation– coefficient of variation

• Categorical Scale

– Cohen’s Kappa

Reproducibility of an Interval Scale Measurement: Peak Flow

• Assessment requires

>1 measurement per subject

• Peak Flow Rate in 17 adults

(Bland & Altman)

Subject Meas. 1 Meas. 21 494 4902 395 3973 516 5124 434 4015 476 4706 557 6117 413 4158 442 4319 650 638

10 433 42911 417 42012 656 63313 267 27514 478 49215 178 16516 423 37217 427 421

Assessment by Simple CorrelationM

eas.

2

Meas. 1200 400 600 800

200

400

600

800

Pearson Product-Moment Correlation Coefficient

• r (rho) ranges from -1 to +1

• r

• r describes the strength of the association

• r2 = proportion of variance (variability) of one variable accounted for by the other variable

22 )()())((YYXXYYXX

r = -1.0

r = 0.8 r = 0.0

r = 1.0

r = 1.0 r = -1.0

r = 0.8 r = 0.0

Correlation Coefficient for Peak Flow Data

r ( meas.1, meas. 2) = 0.98

Limitations of Simple Correlation for Assessment of Reproducibility

• Depends upon range of data– e.g. Peak Flow

• r (full range of data) = 0.98• r (peak flow <450) = 0.97• r (peak flow >450) = 0.94

Limitations of Simple Correlation for Assessment of Reproducibility

• Depends upon ordering of data

• Measures linear association only

Meas.

2

Meas 1100 300 500 700 900 1100 1300 1500 1700

100

300

500

700

900

1100

1300

1500

1700

Limitations of Simple Correlation for Assessment of Reproducibility

• Gives no meaningful parameter for the issue

Within-Subject Standard Deviation

• Mean within-subject standard deviation (sw)

= 15.3 l/min

subject meas1 meas2 mean s1 494 490 492 2.832 395 397 396 1.413 516 512 514 2.83. . . . .. . . . .. . . . .

15 178 165 172 9.1916 423 372 398 36.0617 427 421 424 4.24

17)24.4...83.2( 222

n

si i

Computationally easier with ANOVA table:

• Mean within-subject standard deviation (sw) :

Analysis of Variance Source SS df MS F Prob > F-----------------------------------------------------------------------Between groups 441598.529 16 27599.9081 117.80 0.0000 Within groups 3983.00 17 234.294118----------------------------------------------------------------------- Total 445581.529 33 13502.4706

squares of sum group- withins2

i

234 squaremean group- within17s2

i

l/min 15.3 squaremean group-within

sw: Further Interpretation

• If assume that replicate results:– normally distributed – mean of replicates estimates true value

– standard deviation estimated by sw

• Then 95% of replicates will be within (1.96)(sw) of the true value

• For Peak Flow data: – 95% of replicates will be within (1.96)(15.3) =

30.0 l/min of the true value

sw: Further Interpretation

• Difference between any 2 replicates for same person = diff = meas1 - meas2

• Because var(diff) = var(meas1) + var(meas2), therefore,

s2diff = sw

2 + sw2 = 2sw

2

sdiff

• If assume the distribution of the differences between pairs is N(0, 2

diff), therefore, – The difference between 2 measurements for the same subject is

expected to be less than (1.96)(sdiff) = (1.96)(1.41)sw = 2.77sw for 95% of all pairs of measurements

w

2

w

2

diffs2s2s

sw: Further Interpretation

• For Peak Flow data:

• The difference between 2 measurements for the same subject is expected to be less than 2.77sw

=(2.77)(15.3) = 42.4 l/min for 95% of all pairs

• Bland-Altman refer to this as the “repeatability” of the measurement

Interpreting sw

• Appropriate only if there is one sw

• if sw does not vary with the true underlying value

Wit

hin

-Su

bje

ct

Std

Devia

tion

Subject Mean Peak Flow

100 300 500 7000

10

20

30

40 Kendall’s correlation coefficient = 0.17, p = 0.36

Another Interval Scale Example

• Salivary cotinine in children (Bland-Altman)

• n = 20 participants measured twicesubject trial 1 trial 2

1 0.1 0.12 0.2 0.13 0.2 0.3. . .. . .. . .

18 4.9 1.419 4.9 3.920 7.0 4.0

Simple Correlation of Two Trialstr

ial

1

trial 20 2 4 6

0

2

4

6

Correlation of Cotinine Replicates

Range of data r

Full range 0.70

< 1.0 0.37

1.0 to 2.7 0.57

> 2.7 -0.01

>2.0

Cotinine: Absolute Difference vs. MeanS

ub

ject

Ab

solu

te D

iffe

ren

ce

Subject Mean Cotinine0 2 4 6

0

1

2

3

4 Kendall’s tau = 0.62, p = 0.001

Logarithmic Transformation

subject trial1 trial2 log trial 1 log trial 21 0.1 0.1 -1 -12 0.2 0.1 -0.69897 -13 0.2 0.3 -0.69897 -0.52288. . . . .. . . . .. . . . .

18 4.9 1.4 0.690196 0.14612819 4.9 3.9 0.690196 0.59106520 7 4 0.845098 0.60206

Log Transformed: Absolute Difference vs. MeanS

ub

ject

ab

s lo

g d

iff

Subject mean log cotinine-1 -.5 0 .5 1

0

.2

.4

.6 Kendall’s tau=0.07, p=0.7

sw for log-transformed cotinine data

• sw

• back-transforming to original units:

• antilog(sw) = antilog(0.175) = 1.49

175.00305.0

Analysis of Variance Source SS df MS F Prob > F------------------------------------------------------------------------Between groups 10.4912641 19 .552171793 18.10 0.0000 Within groups .610149715 20 .030507486------------------------------------------------------------------------ Total 11.1014138 39 .284651636

Coefficient of Variation

• On the natural scale, there is not one common within-subject standard deviation for the cotinine data

• Therefore, there is not one absolute number that can represent the difference any replicate is expected to be from the true value or from another replicate

• Instead,

= coefficient of variation

meansubject -withindeviation standardsubject -within

1- )antilog(sw

Cotinine Data

• Coefficient of variation = 1.49 -1 = 0.49

• At any level of cotinine, the within-subject standard deviation of repeated measures is 49% of the level

Coefficient of Variation for Peak Flow Data

• By definition, when the within-subject standard deviation is not proportional to the mean value, as in the Peak Flow data, then there is not a constant ratio between the within-subject standard deviation and the mean.

• Therefore, there is not one common coefficient of variation

• Estimating the coefficient of variation by taking the common within-subject standard deviation and dividing by the overall mean of the subjects is not very meaningful

Intraclass Correlation Coefficient, rI

• rI

• Averages correlation across all possible ordering of replicates• Varies from 0 (poor) to 1 (optimal)

• As 2E approaches 0 (no error), rI approaches 1

• Advantages: not dependent upon ordering of replicates; does not mistake linear association for agreement; allows >2 replicates

• Disadvantages: still dependent upon range of data in sample, still does not give a meaningful parameter on the actual scale of measurement in question

22

2

2

2

ET

T

O

T

Intraclass Correlation Coefficient, rI

• rI

• where:– m = no. of replicates per person

– SSb = sum of squares between subjects

– SSt = total sum of squares

• rI(peak flow) = 0.98

• rI(cotinine) = 0.69

t

tb

SSmSSmSS

)1(

Reproducibility of a Categorical Measurement: Chest X-Rays

• On 2 different occasions, a radiologist is given the same 100 CXR’s from a group of high-risk smokers to evaluate for masses

• How should reproducibility in reading be assessed?

time2 time1 | No Mass Mass | Total-----------+----------------------+---------- Agreement No Mass | 50 0 | 50 Mass | 0 50 | 50 = (50+50)/100 = 100%-----------+----------------------+---------- Total | 50 50 | 100

time2 time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | 0 50 | 50 Mass | 50 0 | 50 = (0 +0)/100 = 0%-----------+----------------------+---------- Total | 50 50 | 100

time2 time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | 40 10 | 50 Mass | 10 40 | 50 = (40 +40)/100 = 80%-----------+----------------------+---------- Total | 50 50 | 100

time2 time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | 25 25 | 50 Mass | 25 25 | 50 = (25 +25)/100 = 50%-----------+----------------------+---------- Total | 50 50 | 100

Kappa

• Agreement above that expected by chance

• (observed agreement - chance agreement) is the amount of agreement above chance

• If maximum amount of agreement is 1.0, then (1 - chance

agreement) is the maximum amount of agreement above

chance that is possible

• Therefore, kappa is the ratio of “agreement beyond chance” to

“maximal possible agreement beyond chance”

agreement chance -1agreement chance -agreement observed

kappa

Determining agreement expected by chanceObserved

time2 time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | 42 3 | 45 Observed Mass | 18 37 | 55 agreement = 79%-----------+----------------------+---------- Total | 60 40 | 100

Fix marginstime2

time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | | 45 Mass | | 55-----------+----------------------+---------- Total | 60 40 | 100

Fill in expected values for cells under assumption of independencetime2

time1 | No Mass Mass | Total-----------+----------------------+---------- Agreement No Mass | 27 18 | 45 expected by Mass | 33 22 | 55 chance = 49%-----------+----------------------+---------- Total | 60 40 | 100

• Suggested interpretations for kappa

59.00.49 -1

0.49 - 0.79 kappa

kappa Interpretation< 0 No agreement

0 – 0.19 Poor agreement0.20 – 0.39 Fair agreement0.40 – 0.59 Moderate agreement0.60 – 0.79 Substantial agreement0.80 – 1.00 Near perfect agreement

Kappa: problematic at the extremes of prevalenceObserved

time2 time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | 92 3 | 95 Observed Mass | 3 2 | 5 agreement = 94%-----------+----------------------+---------- Total | 95 5 | 100

Fix marginstime2

time1 | No Mass Mass | Total-----------+----------------------+---------- No Mass | | 95 Mass | | 5-----------+----------------------+---------- Total | 95 5 | 100

Fill in expected values for cells under assumption of independencetime2

time1 | No Mass Mass | Total-----------+----------------------+---------- Agreement No Mass | 90.25 4.75 | 95 expected by Mass | 4.75 0.25 | 5 chance = 90.5%-----------+----------------------+---------- Total | 95 5 | 100 Kappa = 0.37

Sources of Measurement Variability: Which to Assess?

• Observer • within-observer (intrarater)• between-observer (interrater)

• Instrument • within-instrument• between-instrument

• Subject• within-subject

• Which to assess depends upon the use of the measurement and how it will be made. – For clinical use: all of the above are needed– For research: depends upon logistics of study (i.e.

intrarater and within-instrument only if just one person/instrument used throughout study)

Improving Reproducibility

• See Hulley text

• Make more than one measurement!– But know where the source of your variation

exists!

Assessing Validity - With Gold Standards

• A new and simpler device to measure peak flow becomes available (Bland-Altman)

subject gold std new1 494 5122 395 4303 516 520. . .. . .. . .

15 178 25916 423 35017 427 451

Plot of Difference vs. Gold StandardD

iffe

ren

ce

Gold standard

0 200 400 600 800 1000

-200

-100

0

100

200

Fre

qu

en

cy

diff-100 -50 0 50 100

0

2

4

6

8

• The mean difference describes any systematic difference between the gold standard and the new device:

• The standard deviation of the differences:

• 95% of differences will lie between -2.3 + (1.96)(38.8), or from -78 to 74 l/min.

• These are the 95% limits of agreement

i

i nd

nd 3.2)]427451(..)494512[(

11

8.381

)( 2

n

dds i i

d

Assessing Validity of Categorical Measures

• Dichotomous

• More than 2 levels– Collapse or– Kappa

Gold StandardPresent Absent

New Present a bMeasurement Absent c d

Sensitivity = a/(a+c)Specificity = d/(b+d)

Assessing Validity - Without Gold Standards

• When gold standards are not present, measures can be assessed for validity in 3 ways:

– Content validity• Face• Sampling

– Construct validity– Empirical validity (aka criterion)

• Concurrent• Predictive

Conclusions• Measurement reproducibility plays a key role in determining validity

and statistical precision in all different study designs– When assessing reproducibility,

• avoid correlation coefficients • use within-subject standard deviation if constant • or coefficient of variation if within-subject sd is proportional to

the magnitude of measurement

• Acceptable reproducibility depends upon desired use

• For validity, plot difference vs mean and determine “limits of agreement” or determine sensitivity/specificity

– Be aware of how your measurements have been validated!