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help concord                                                  (SJ7-3: st0015_4)
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Title

    concord -- Concordance correlation coefficient and associated measures,
                 tests, and graphs


Syntax

        concord vary varx [if] [in] [weight] [, by(byvar) summary level(#)
                 ccc[(noref ccc_options)] loa[(noref regline loa_options)]
                 qnormd[(qnormd_options)]]


Description

    concord computes Lin's (1989, 2000) concordance correlation coefficient
    for agreement on a continuous measure obtained by two persons or methods.
    (The measure was introduced earlier by Krippendorff (1970).) 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 (i.e., 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).  The Pearson correlation
    coefficient, r, the bias-correction factor, C_b, and the equation of the
    reduced major axis are reported to show these components.  Note that the
    concordance correlation coefficient, rho_c, can be expressed as the
    product of r, the measure of precision, and C_b, the measure of accuracy.

    concord also provides results for Bland and Altman's limits-of-agreement,
    "loa", procedure (1986). The loa, a data-scale assessment of the degree
    of agreement, is a complementary approach to the relationship-scale
    approach of Lin.

    Finally, two other results are reported:

        1. The correlation between difference and mean. In one interpretation
        this is a test statistic for a null hypothesis of equal variances
        given bivariate normality (Pitman 1939; also see Snedecor and Cochran
        1989, 192-193).  Alternatively, it is an exploratory diagnostic.  A
        value near zero implies concordance.

        2. An F test of equality of means and variances. Note that this too
        assumes bivariate normality (Bradley and Blackwood 1989).  See also
        Hsu (1940) and Reynolds and Gregoire (1991).  Nonsignificance implies
        concordance.

    The user provides the pairs of measurements for a single property as
    observations in variables vary and varx.  Frequency weights may be
    specified and used. Missing values (if any) are deleted in a casewise
    manner.

    Various associated graphs may be obtained through options.  See below for
    explanations of the options ccc, loa, and qnormd.


Options

    by(byvar) produces separate results for groups of observations defined by
        byvar.

    summary requests summary statistics.

    level(#) sets the confidence level % for the CI; default is c(level).

    ccc requests a graphical display of the data and the reduced major axis
        of the data. The reduced major axis or SD line goes through the
        intersection of the means and has slope given by the sign of
        Pearson's r and the ratio of the standard deviations. The SD line
        serves as a summary of the center of the data.

        ccc() suboption noref suppresses the reference line of perfect
        concordance, y=x.

        ccc() may also be specified with other options, which should be
        options of scatter. For example, the scheme may be changed by a call
        such as ccc(scheme(lean1).

    loa requests a graphical display of the loa, the mean difference, and the
        data presented as paired differences plotted against pair-wise means.

        loa() suboption noref suppresses the reference line of perfect
        average agreement, y=0.

        loa() suboption regline adds a regression line to the loa plot
        fitting the paired differences to the pair-wise means.

        loa() may also be specified with other options, which should normally
        be options of scatter.  For example, the reference line of perfect
        average agreement was generated as loa(yline(0, lstyle(refline))
        yscale(range(0)) ylabel(0, add)).

    qnormd requests a normal plot of differences.

        qnormd() may also be specified with options, which should be options
        of scatter. For example, qnormd(title(Normal plot of differences))
        adds a title to the graph.


Comments

    Lin (2000) reported typographical errors in his original 1989 paper that
    affected calculation of the standard error of rho_c. These corrections
    were included in concord in version 2.2.7 (January 2002) when the erratum
    was brought to the attention of the program authors by Dr. Benjamin
    Littenberg.  We thank Dr. Littenberg.

    Kevan Polkinghorne pointed out a problem with loa().  Mark Marshall
    pointed out a problem with by() under Stata 9.

    Dunn (2004) contains a bibliography on related work.  Cox (2004)
    discusses other graphical approaches to this and related problems. Cox
    (2006) discusses concordance correlation and other numerical and
    graphical methods for assessing agreement with various scientific
    examples.


Saved results

    The following items are returned in r(), if the by() option was not used:

        r(N)         number of observations compared
        r(rho_c)     concordance correlation coefficient rho_c
        r(se_rho_c)  standard error of rho_c
        r(asym_ll)   lower CI limit (asymptotic)
        r(asym_ul)   upper CI limit (asymptotic)
        r(z_tr_ll)   lower CI limit (z-transform)
        r(z_tr_ul)   upper CI limit (z-transform)
        r(C_b)       bias-correction factor C_b
        r(diff)      mean difference
        r(sd_diff)   standard deviation of mean difference
        r(LOA_ll)    lower loa CI limit
        r(LOA_ul)    upper loa CI limit
        r(rdm)       correlation between difference and mean
        r(Fdm)       F from Bradley-Blackwood test


Examples

    . concord rater1 rater2

    . concord rater1 rater2 [fw=freq]

    . concord rater1 rater2, summary ccc

    . concord rater1 rater2, summary ccc(noref)

    . concord rater1 rater2, level(90) by(grp)

    . concord rater1 rater2, loa(regline noref)

    . concord rater1 rater2, qnormd(title(Normal plot))


Authors

    Thomas J. Steichen, Winston-Salem, NC, USA, steichen@triad.rr.com

    Nicholas J. Cox, Durham University, UK, n.j.cox@durham.ac.uk


References

    Bland, J. M., and D. G. Altman.  1986. Statistical methods for assessing
        agreement between two methods of clinical measurement. Lancet I:
        307-310.

    Bradley, E. L., and L. G. Blackwood. 1989. Comparing paired data:  a
        simultaneous test for means and variances. American Statistician 43:
        234-235.

    Cox, N. J. 2004. Graphing agreement and disagreement.  Stata Journal 4:
        329-349.

    ------. 2006.  Assessing agreement of measurements and predictions in
        geomorphology.  Geomorphology 76: 332-346.

    Dunn, G. 2004.  Statistical Evaluation of Measurement Errors: Design and
        Analysis of Reliability Studies.  London: Arnold.

    Hsu, C. T. 1940. On samples from a normal bivariate population.  Annals
        of Mathematical Statistics 11: 410-426.

    Krippendorff, K. 1970. Bivariate agreement coefficients for reliability
        of data.  In Borgatta, E.F. and G.W. Bohrnstedt (eds) Sociological
        Methodology. San Francisco: Jossey-Bass, 139-150.  [a.k.a.
        Sociological Methodology 2: 139-150]

    Lin, L. I-K. 1989. A concordance correlation coefficient to evaluate
        reproducibility. Biometrics 45: 255-268.

    ------. 2000. A note on the concordance correlation coefficient.
        Biometrics 56: 324-325.

    Pitman, E. J. G. 1939. A note on normal correlation.  Biometrika 31:
        9-12.

    Reynolds, M., and T. G. Gregoire. 1991. Comment on Bradley and Blackwood.
        American Statistician 45: 163-164.

    Snedecor, G. W., and W. G. Cochran. 1989. Statistical Methods.  Ames, IA:
        Iowa State University Press.


Also see

    STB:  STB-43 sg84; STB-45 sg84.1; STB-54 sg84.2; STB-58 sg84.3

    SJ:   SJ2-2 st0015; SJ4-4 st0015_1; SJ5-3: st0015_2; SJ6-2: st0015_3