Conducting and Presenting Concordance Correlation Coefficient Analysis

[Jonathan King]

Hello
I have a several sets of paired measurements that I wish to assess for agreement.
I was wondering if it is advisable to test for normality of the error prior to analysing using the concord function.
I understand that normality is not an assumption for this test but it may affect results. I have posted a histogram of the errors below.

Also with presenting this data I intend on presenting ρ_c, as well as the 95%CI of the correlation.
Elsewhere I have seen location shift presented alongside these results. Would you advise presenting these data as well?
Any advice would be appreciated.
Thanks in advance.

[Nick Cox]

I'd solve this issue by ignoring it completely and bootstrapping concordance correlations. Naturally you should present location shift too. If that's meaningless, concordance correlation makes little or no sense.

[Jonathan King]

Thanks for the reply Nick.
Bootstrapping is a little beyond my skill set.
Here is a sample of my paired data from a population of 228 pairs.
I'd like to use the concord to measure their level of agreement. If I don't know how to bootstrap can you suggest any other ways of achieving this?
Thanks again.

site_	predictedcol
4.93	6.290981
6.03	6.944743
4.88	6.33551
6.54	7.202926
6.05	6.617754
4.92	6.303995
8.27	7.772792
7.14	7.792072
8.04	7.282252
7.42	8.384775
7.85	7.420662
8.91	8.368908
5.66	9.160925
4.81	7.950098
4.46	8.815401
4.94	7.891434
4.95	8.30157
4.98	7.822245
7.22	11.33773
10.16	12.58638
8.83	11.91866
9.36	12.31347
7.09	11.17154
6.87	11.38873
11.3	9.911688
11.83	10.48137
11.24	9.863089
11.15	9.958758
13.83	11.19683
12.66	10.89975

[Nick Cox]

If you don't know how to bootstrap, you find out. My reading is that the concord confidence intervals if anything are too cautious. Also, for this moderate degree of agreement, the correlation has a close to Gaussian sampling distribution. The bigger deal is whether the clusters in the raw data tell you anything.

* Example generated by -dataex-. To install: ssc install dataex
clear
input double(site_ predictedcol)
 4.93 6.290981
 6.03 6.944743
 4.88  6.33551
 6.54 7.202926
 6.05 6.617754
 4.92 6.303995
 8.27 7.772792
 7.14 7.792072
 8.04 7.282252
 7.42 8.384775
 7.85 7.420662
 8.91 8.368908
 5.66 9.160925
 4.81 7.950098
 4.46 8.815401
 4.94 7.891434
 4.95  8.30157
 4.98 7.822245
 7.22 11.33773
10.16 12.58638
 8.83 11.91866
 9.36 12.31347
 7.09 11.17154
 6.87 11.38873
 11.3 9.911688
11.83 10.48137
11.24 9.863089
11.15 9.958758
13.83 11.19683
12.66 10.89975
end

concord site_ predictedcol

Concordance correlation coefficient (Lin, 1989, 2000):

 rho_c   SE(rho_c)   Obs    [   95% CI   ]     P        CI type
---------------------------------------------------------------
 0.518     0.114      30     0.294  0.742    0.000   asymptotic
                             0.262  0.706    0.000  z-transform

Pearson's r =  0.624  Pr(r = 0) = 0.000  C_b = rho_c/r =  0.830
Reduced major axis:   Slope =     1.371   Intercept =    -4.579

Difference = site_ - predictedcol

        Difference                 95% Limits Of Agreement
   Average     Std Dev.             (Bland & Altman, 1986)
---------------------------------------------------------------
    -1.246       2.101                 -5.363      2.872

Correlation between difference and mean = 0.380

Bradley-Blackwood F = 8.310 (P = 0.00147)

. set seed 2803


. bootstrap rho_c=r(rho_c), reps(10000) nodots saving(rho_c)  : concord site_ predictedcol
(running concord on estimation sample)

Bootstrap results                               Number of obs     =         30
                                                Replications      =     10,000

      command:  concord site_ predictedcol
        rho_c:  r(rho_c)

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       rho_c |   .5182414   .0836768     6.19   0.000     .3542379     .682245
------------------------------------------------------------------------------

. estat bootstrap, all

Bootstrap results                               Number of obs     =         30
                                                Replications      =      10000

      command:  concord site_ predictedcol
        rho_c:  r(rho_c)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
       rho_c |   .51824145  -.0097181    .0836768    .3542379    .682245   (N)
             |                                       .3405313   .6660519   (P)
             |                                       .3562137   .6778926  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval

. use rho_c
(bootstrap: concord)

. qnorm rho_c, scheme(s1color) ytitle(concordance correlation)

[Jonathan King]

Wow that was great.
Thanks so much Nick!