The Family of Reliability Coefficients Gregory G Brown

The Family of Reliability Coefficients Gregory G. Brown VASDHS/UCSD All Hands Meeting 2005

Reliability Coefficients: The problems § Reliability coefficients were often developed in varying literatures without regard to a cohesive theory § Cohesive theories of reliability available in the literature are not widely known § Reliability terms are used inconsistently § Different terms in the literature at times used to represent the same reliability concept

Levels of the Family Tree Level 1. Study Aim Level 2. # Study Factors Level 3. # Levels within Study Factors Level 4. Score Standardization Level 5. Nesting Level 6. Level of Measurement

The Progenitor Coefficient Correlation Ratio ( 2) Winer et al. , 1991

Correlation ratios § Vary between 0. 0 and 1. 0 § Typically measure the amount of variance accounted for by a factor in the analysis of variance design § Index the strength of association between levels of a study factor and the dependent variable, regardless of whether the functional relationship between study factors and the dependent measure is linear or nonlinear.

The two meanings of error § Definition 1: models: The error term in analysis of variance § Definition 2: All relevant sources of variance in an analysis of variance design besides the source of interest § The two definitions of error are associated with different reliability models and with different reliability coefficients

Levels of the Family Tree Level 1. Study Aim Correlation Ratio Determine Reliability Measures Establish Validity Effect Size Measures

Correlation Ratio and Reliability Measures Correlation ratios based on variance component estimates derived from random effects models are generally consistent measures of reliability (Olkin & Pratt, 1958).

The Correlation Ratio and Effect Size Measures Effect Size: Cohen’s Parameter related to power: Winer et al. , 1991

Cohen’s f is the variance of the means across the various levels of an study factor scaled by the common within group variance.

Caveat: There are Two Definitions of the Correlation Ratio § OLD Definition: The correlation ratio is a ratio of sums of squares (Kerlinger, 1964, pp. 200 -206, Cohen, 1965). § Current Definition: The correlation ratio is a ratio of variance component estimates and their fixed effects analogues (eg. Winer et al. , 1991). This is the definition of the correlation ratio used in this talk.

Correspondence Among Effect Size Measures Effect Size+ Cohen’s f 2 Power* n=20 Small . 10 0. 01 . 10 Medium . 25 0. 06 . 37 Large . 40 0. 14 . 78 + Cohen (1988); *Winer et al. , 1991, pp. 120 -133: F(2, 57, f), p=. 05

Shrout and Fleiss (1979) Example Raters Subjects 1 1 9 2 2 3 5 4 8 2 6 1 3 2 3 8 4 6 8 4 7 1 2 6 5 10 5 6 9 6 6 2 4 7 Entries are ratings on a scale of 1 to 10.

Correlation Ratios for Shrout and Fleiss Example: Random Effects Model For both Validity and Reliability Analyses Model: Xij= + i+ j+ ij i : effect of subject, N(0, ), i assumed to be independent of j and ij j : effect of raters, N(0, ), i assumed to be Independent of i and ij Both i and j are random effects.

Results of Shrout and Fleiss Random Effects Analysis * 2 +Power Raters Mean F-value Square 32. 486 31. 866 <. 001 . 59 ~1. 00 Subjects 11. 242 11. 027 <. 001 . 29 . 64 Error 1. 019 Effect *Based on variance components estimates using total variance for denominator of correlation ratio. on variance components definition of 2 and previously described relationship between 2 and Cohen’s f. +Based

Claim 2 § The 2 for subjects equals the ICC(2, 1) for these data (See Shrout and Fleiss, 1979). § Reliability and validity can both be investigated within an analysis of variance framework.

Levels of the Family Tree Level 1. Study Aim Level 2. # Study Factors

Levels of the Family Tree Level 2. Number of Study Factors Reliability Measures Single Factor Designs Intraclass Correlations Multifactorial Designs Generalizability Theory Coefficients

Examples § A single factor reliability design is one where there is only one only source of variance besides subjects (eg. , Raters judging all subjects). § A multi-factor reliability design is one there are several sources of variance besides subjects (eg. Raters judging all subjects on 2 days).

Intraclass correlations for single facet reliability studies § Just reviewed by Lee Friedman

Generalizability Theory § Measurement always involves some conditions (eg. raters, items, ambient sound) that could be varied without changing the acceptability of the observations. § The experimental design defines a universe of acceptable observations of which a particular measurement is member. § The question of reliability resolves into the question of how accurately the observer can generalize back to the universe of observations.

Generalizability Theory (continued) § A reliable measure is one where the observed value closely estimates the expected score over all acceptable observations, i. e. , the universe score § Generalizability coefficient: Cronback, Gleser, Nanda, & Rajaratnam, 1972

Basic Components of the Generalizability Coefficient § Universe score variance: the estimated variance across the objects of measurement (eg. , people) in the sample at hand: § Relative error: For the Shrout & Fleiss example it is the sum of variance components related to people averaged over raters.

Generalizability Theory (continued) § Generalizability coefficient: Brennan, 2001

The Generalizability Coefficient § Generalizability Coefficient: § A large generalizability coefficient means that person variance can be estimated without large effects from other sources of variance that might effect the expected between-subject variation within raters.

Generalizability Theory and Measurement Precision § Generalizability Theory provides a measurement standard: True variation among objects of measurement, eg. people § Generalizability Theory uses the concept of person variance to provide a clear and simple relationship between reliability coefficients, C, and measurement precision: Standarderror = ((1 -C)/C) 2 person.

Innovative Aspects of Generalizability Theory § Generalizability Theory asserts there exist multiple sources of error rather than the single error term of classical reliability theory. § Analysis of variance can be used to hunt these sources of error. § New definitions: • A reliability measure is one that is stable over unwanted sources of variance • A valid measure is one that varies over wanted sources of variance

Generalizability Coefficient for Shrout & Fleiss (1979) data

ICC(3, k) and the Generalizability Coefficient (continue) § The generalizability coefficient is equivalent to ICC(3, k) and both are measures of rater consistency § ICC(3, 1) can be calculated directly from variance components estimates and is equal to the traditional use of the Correlation Ratio as a measure of amount of variance accounted for.

The Dependability Coefficient § Absolute error = sum of variance components each averaged over their respective numbers of observations § Depedendability coefficient =

Dependability Coefficient for Shrout & Fleiss (1979) data

The Dependability Coefficient and ICC(2, k) § The dependability coefficient of Generalizability Theory is equivalent to ICC(2, k) and both are measures of absolute agreement § ICC(2, 1) can be calculated directly from variance components estimates and is equal to the traditional use of the Correlation Ratio as a measure of amount of variance accounted for.

Summary of Results of ICC and Generalizability Theory Comparisons Intraclass Coefficients* Consistency Absolute Agreement K=1 K>1 ICC(3, 1)= ICC(3, k)= Var. components = Generalizability = . 71 . 91 . 7148 . 9093 ICC(2, 1)= ICC(2, k)= Var. components = Dependability = . 29 . 62 . 2897 . 6200 *Values taken from Shrout & Fleiss, 1979 +Values Generalizability Theory Coefficients+ K=1 K>1 calculated from GENOVA output

Intraclass and Generalizability Coefficients § Intraclass Correlation Coefficients are special cases of the one-facet generalizability study (Shrout & Fleiss, 1979) § The ICC(2, 1), ICC(2, k), ICC(3, 1), and ICC(3, k) intraclass correlations discussed by Shrout and Fleiss can be calculated from generalizability software (eg. , Genova).

Levels of the Family Tree Level 1. Study Aim Level 2. # Study Factors Level 3. # Levels within Study Factors

Levels of the Family Tree Level 3. Number of Levels within Study Factors Intraclass Coefficients Two Level Designs Co-dependency Correlations Multilevel Designs Generalizability Theory Coefficients Two Level Designs Multilevel ICCs Historically no distinction made

Levels of the Family Tree Level 1. Study Aim Level 2. # Study Factors Level 3. # Levels within Study Factors Level 4. Score Standardization

Levels of the Family Tree Level 4. Score Standardization Co-dependency Measures Standardized Scores Pearson Product Moment Correlation Raw or Partially Standardized Scores Intraclass Correlations

Standardized Correlation Ratios Pearson Correlation = =

The Correlation Ratio and Pearson Produce Moment Correlation § When subject scores are standardized within rater, the Pearson Product Moment Correlation is equal to the Correlation ratio, when 2 is defined in terms of total variance Correlation Ratio ( 2) § A generalized Product Moment Correlation can be defined across all raters simultaneously using the variance components calculated on standard scores

Product Moment Correlations Rater 1 2 3 1 2. 745 3. 724. 894 4. 750. 730. 719 Variance components estimate ( 2) of rater 1 vs rater 3 reliability based on Z-scores =. 7448

Multi-level Product Moment Correlation Calculated by standardizing scores within judges then calculating 2 using total variance components definition. For Shrout & Fleiss data this value =. 7602 and represents global standardized consistency rating.

Conclusions § The concept of a correlation ratio relates effect size measures to reliability measures § ICCs are Generalizability Theory coefficients for single facet designs § ICC(3, 1), ICC(3, k), and the Generalizability Coefficient are all measures of consistency § ICC(2, 1), ICC(3, k), and the Dependability Coefficient are all measures of absolute agreement

Conclusions § The Pearson Product Moment Correlation is a singlefacet, 2 -level Correlation Ratio for standard scores and is, thus, a measure of consistency. § A multilevel Product Moment Correlation is a singlefacet, k-level Correlaiton Ratio for standard scores and is a measure of standardized consistency across all raters.

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