knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) library(SimplyAgree)
Another feature of this R package is the ability to estimate the reliability of a measurement. This R package allows for the calculation of Intraclass Correlation Coefficients (ICC), various standard errors (SEM, SEE, and SEP), and coefficient of variation. All of the underlying calculations (sans the coefficient of variation) are based on the paper by @weir2005^[The paper by Weir also appears to heavily rely on the work of @shrout1979 and @mcgraw1996]. This is a fairly popular paper within my own field (kinesiology), and hence was the inspiration for creating this function that provides all the calculative approaches included within that manuscript.
For this package, the test-retest reliability statistics can be calculated with the reli_stats
function. This function allow for data to be input in a long (multiple rows of data for each subject) or in wide (one row for each subject but a column for each item/measure).
For the long data form, the column containing the subject identifier (id
), item number (item
), and measurements (measure
) are provided. In this function I refer to items similar to if we were measuring internal consistency for a questionnaire (which is just a special case of test-retest reliability). So, item
could also be refer to time points, which is what is typically seen in human performance settings where test-retest reliability may be evaluated over the course of repeated visits to the same laboratory. If wide
is set to TRUE
then the columns containing the measurements are provided (e.g., c("value1","value2","value3")
).
To demonstrate the function, I will create a data set in the wide format.
# Example from Shrout and Fleiss (1979), pg. 423 dat = data.frame(judge1 = c(9,6,8,7,10,6), judge2 = c(2,1,4,1,5,2), judge3 = c(5,3,6,2,6,4), judge4 = c(8,2,8,6,9,7))
Now, that we have a data set (dat
), I can use it in the reli_stats
function.
test1 = reli_stats( data = dat, wide = TRUE, col.names = c("judge1", "judge2", "judge3", "judge4") )
This function also has generic print and plot functions. The output from print provides the coefficient of variation, standard errors, and a table of various intraclass correlation coefficients. Notice the conclusions about the reliability of the measurement here would vary greatly based on the statistic being reported. What statistic you should report is beyond the current vignette, but is heavily detailed in @weir2005. However, within the table there are columns for model
and measures
which describe the model that is being used and the what these different ICCs are intended to measure, respectively.
print(test1)
Also included in the results is a plot of the measurements across the items (e.g., time points).
plot(test1)
In some cases there are convergence issues for the linear mixed models. For this reason,
I have added a function, reli_aov
, which uses a sums of squares approach
(i.e., analysis of variance) rather than a linear mixed model. As you can see
below the results will often match that of reli_stats
. The only time this
function is necessary to use is when there are convergence issues.
Rows with missing data are dropped when using reli_aov
.
test2 = reli_aov( data = dat, wide = TRUE, col.names = c("judge1", "judge2", "judge3", "judge4") ) test2
The linear mixed model used for the calculations is specified as the following:
$$ Y_{i} \sim\ N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) $$
$$ \alpha_{j} \sim\ N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right) \text{, for id j = 1,} \dots \text{,J} $$
$$ \alpha_{k} \sim\ N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right) \text{, for items k = 1,} \dots \text{,K} $$
Mean Squared Error (MSE)
$$ MSE = \sigma^2 $$
Variance Between Subjects
$$ MSB = n_k \cdot \sigma^2_{\alpha j} + \sigma^2 $$
Variance Between Items/Judges
$$ MSJ = n_{j} \cdot \sigma^2_{\alpha_{k}} + \sigma^2 $$
Variance Within Subjects/Participants
$$ MSW = \sigma^2 + \sigma^2_{\alpha_{k}} $$
$$ ICC_{(1,1)} = \frac{MSB - MSW}{MSB + (n_j-1) \cdot MSW} $$ $$ ICC_{(2,1)} = \frac{MSB - MSE}{MSB+(n_j -1) \cdot MSE + n_j \cdot (MSJ - MSE) \cdot n^{-1}} $$ $$ ICC({3,1)} = \frac{MSB - MSE}{MSB + (n_j -1) \cdot MSE} $$ $$ ICC{(1,k)} = \frac{MSB - MSW}{MSB} $$ $$ ICC_{(2,k)} = \frac{MSB - MSW}{MSB + (MSJ - MSE) \cdot n_j^{-1}} $$ $$ ICC_{(3,k)} = \frac{MSB - MSE}{MSB} $$
$$ F = \frac{MSB}{MSW} $$
$$ df_{n} = n_j - 1 $$
$$ df_{d} = n_j \cdot (n_k - 1) $$
$$ F_{L} = \frac{F}{F_{(1 - \alpha, \space df_{n}, \space df_{d})}} $$
$$ F_{U} = F \cdot F_{(1 - \alpha, \space df_{d}, \space df_{n})} $$
$$ Lower \space CI = \frac{(F_L - 1)}{(F_L + (n_j - 1))} $$
$$ Upper \space CI = \frac{(F_U - 1)}{(F_U + n_j - 1)} $$
$$ F = \frac{MSJ}{MSE} $$
$$ vn = (n_k - 1) \cdot (n_j - 1) \cdot [(nj \cdot ICC_{(2,1)} \cdot F + n_j \cdot (1 + (n_k - 1) \cdot ICC_{(2,1)}) - n_k \cdot ICC_{(2,1)})]^2 $$
$$ vd = (n_j - 1) \cdot n_k^2 \cdot ICC_{(2,1)}^2 \cdot F^2 + (n_j \cdot (1 + (n_k - 1) \cdot ICC_{(2,1)}) - n_k \cdot ICC_{(2,1)})^2 $$ $$ v = \frac{vn}{vd} $$
$$ F_{L} = F_{(1 - \alpha, \space n_j-1, \space v)} $$
$$ F_{U} = F_{(1 - \alpha, \space v, \space n_j - 1)} $$
$$ Lower \space CI = \frac{n_j \cdot (MSB - F_U \cdot MSE)}{(F_U \cdot (n_k \cdot MSJ + (n_k \cdot n_j - n_k - n_j) \cdot MSE) + n_j \cdot MSB)} $$
$$ Upper \space CI = \frac{n_j \cdot (MSB \cdot F_L - MSE)}{(n_k \cdot MSJ + (n_k \cdot n_j - n_k - n_j) \cdot MSE + n_j \cdot F_L \cdot MSB)} $$
$$ F = \frac{MSJ}{MSE} $$ $$ df_{n} = n_j - 1 $$ $$ df_{d} = (n_j-1) \cdot (n_k - 1) $$
$$ F_{L} = \frac{F}{F_{(1 - \alpha, \space df_{n}, \space df_{d})}} $$ $$ F_{U} = F \cdot F_{(1 - \alpha, \space df_{n}, \space df_{d})} $$ $$ F3L <- F31/qf(1 - alpha, df21n, df21d) $$ $$ F3U <- F31 * qf(1 - alpha, df21d, df21n) $$ $$ Lower \space CI = (F3L - 1)/(F3L + n_k - 1) $$
$$ Upper \space CI = (F3U - 1)/(F3U + n_k - 1) $$
$$ F = \frac{MSB - MSW}{MSB} $$ $$ df_{n} = n_j - 1 $$
$$ df_{d} = n_j \cdot (n_k - 1) $$
$$ F_{L} = \frac{F}{F_{(1 - \alpha, \space df_{n}, \space df_{d})}} $$ $$ F_{U} = F \cdot F_{(1 - \alpha, \space df_{d}, \space df_{n})} $$
$$ Lower \space CI = 1-\frac{1}{F_L} $$ $$ Upper \space CI = 1-\frac{1}{F_U} $$
$$ F = \frac{MSB - MSW}{MSB} $$
$$ vn = (n_k - 1) \cdot (n_j - 1) \cdot [(nj \cdot ICC_{(2,k)} \cdot F + n_j \cdot (1 + (n_k - 1) \cdot ICC_{(2,k)}) - n_k \cdot ICC_{(2,k)})]^2 $$
$$ vd = (n_j - 1) \cdot n_k^2 \cdot ICC_{(2,k)}^2 \cdot F^2 + (n_j \cdot (1 + (n_k - 1) \cdot ICC_{(2,k)}) - n_k \cdot ICC_{(2,k)})^2 $$
$$ v = \frac{vn}{vd} $$
$$ F_{L} = F_{(1 - \alpha, \space n_j-1, \space v)} $$ $$ F_{U} = F_{(1 - \alpha, \space v, \space n_j - 1)} $$
$$ L3 = \frac{n_j \cdot (MSB - F_U \cdot MSE)}{(F_U \cdot (n_k \cdot MSJ + (n_k \cdot n_j - n_k - n_j) \cdot MSE) + n_j \cdot MSB)} $$
$$ U3 = \frac{n_j \cdot (MSB \cdot F_L - MSE)}{(n_k \cdot MSJ + (n_k \cdot n_j - n_k - n_j) \cdot MSE + n_j \cdot F_L \cdot MSB)} $$
$$ Lower \space CI = \frac{L3 \cdot n_k}{(1 + L3 \cdot (n_k - 1))} $$
$$ Upper \space CI = \frac{U3 \cdot n_k}{(1 + U3 \cdot (n_k - 1))} $$
$$ F = \frac{MSB}{MSE} $$
$$ df_n = n_j - 1 $$
$$ df_d = (n_j - 1) \cdot (n_k - 1) $$
$$ F_L = \frac{F}{F_{(1 - \alpha, \space df_n, \space df_d})} $$
$$ F_U = F \cdot F_{(1 - \alpha, \space df_d, \space df_n)} $$
$$ Lower \space CI = 1-\frac{1}{F_L} $$
$$ Upper \space CI = 1-\frac{1}{F_U} $$
The standard error of the measurement (SEM), standard error of the estimate (SEE), and standard error of prediction (SEP) are all estimated with the following calculations.^[This section was previously incorrect. The variance calculation used to utilize $n_j$ instead of N.]
The default SEM calculation is the following:
$$
SEM = \sqrt{MSE}
$$
Alternatively, the SEM can be calculated as the following and the ICC is determined through the se_type
argument:
$$
SEM = \sqrt{\frac{SS_{total}}{(N-1)}} \cdot \sqrt{1-ICC}
$$
The other measures default to using the following equations. The default is to use $ICC_{3,1}$, but can be modified with the se_type
argument.
$$ SEE = \sqrt{\frac{SS_{total}}{(N-1)}} \cdot \sqrt{ICC \cdot (1-ICC)} $$
$$ SEP = \sqrt{\frac{SS_{total}}{(N-1)}} \cdot \sqrt{1-ICC^2} $$
The CV is calculated 3 potential ways within reli_stats
. I highly recommend reporting the default version of CV.
$$ CV = \frac{ \sqrt{MSE} }{ \bar y} $$
se_type
is set to "MSE"]$$ CV = \frac{SEM}{ \bar y} $$
$$ CV = \frac{\sqrt{\frac{\Sigma^{N}{i=1}(y_i - \hat y_i)^2}{N{obs}}}}{ \bar y} $$
If the other_ci
argument is set to TRUE then confidence intervals will be calculated for the CV, SEM, SEP, and SEE.
The default method is type = 'chisq
. This method utilizes the chi-square distribution to approximate confidence intervals for these measurements. The accuracy of these estimates is likely variable, and is probably poor for small samples.
For the CV, the calculation is as follows:
$$ Lower \space CI = \frac{CV}{\sqrt{(\frac{\chi^2_{1-\alpha/2}}{df_{error}+1}-1) \cdot CV^2 \cdot \frac{\chi^2_{1-\alpha/2}}{df_{error}}}} $$
$$ Upper \space CI = \frac{CV}{\sqrt{(\frac{\chi^2_{\alpha/2}}{df_{error}+1}-1) \cdot CV^2 \cdot \frac{\chi^2_{\alpha/2}}{df_{error}}}} $$ For the variance based measures (s) the calculation is as follows:
$$ Lower \space CI = s \cdot \sqrt{\frac{df_{error}}{\chi^2_{1-\alpha/2}}} $$
$$ Upper \space CI = s \cdot \sqrt{\frac{df_{error}}{\chi^2_{\alpha/2}}} $$
If type is not set to chisq
then bootstrapping is performed.
reli_aov
utilizes a non-parametric ordinary bootstrap.reli_stats
utilizes a parametric bootstrap.The number of resamples can be set with the replicates argument (default is 1999; increase for greater accuracy or lower for greater speed). The reported confidence intervals are estimated using the percentile method type = 'perc'
, the normal type = 'norm'
, or basic methods type = 'basic'
.
To ensure reproducibility, please use set.seed()
when these confidence intervals are calculated.
In some cases, the reliability of a categorical or ordinal scale may be worth investigating. For example, physicians may want to develop a diagnosis tool and ensure that the diagnosis is reliable (i.e., categorical designation) or severity of the disease (i.e., a Likert-type scale). Coefficients can be calculated to assess the degree of inter-rater reliability. In its simplest form, the percent agreement between all the raters can be calculated. All other coefficients of agreement are essentially trying to "correct" for random guessing of the rater. The function to make these calculations in the SimplyAgree
is agree_coef
, and it produces 4 estimates: percent agreement, Gwet's AC, Fleiss' Kappa, and Krippendorff's Alpha. However, other packages provide much a much greater breadth of calculative approaches [@irrCAC].
In the agree_coef
function, the user can specify weighted = TRUE
. If this argument is set to true than the ratings have quadratic weights applied to them. Essentially, this penalizes values farther away from each other more than those close to each other. For example, a pair of values equal to 3 and 4 would be penalized less than a pair of values to 1 and 4. For more details, on these "agreement coefficients" I refer all users to Gwet's textbook on inter-rater agreement [@gwet].
As a demonstration, we can create a matrix of ratings.
ratermat1 = ("Rater1 Rater2 Rater3 Rater4 1 1 1 NA 1 2 2 2 3 2 3 3 3 3 3 4 3 3 3 3 5 2 2 2 2 6 1 2 3 4 7 4 4 4 4 8 1 1 2 1 9 2 2 2 2 10 NA 5 5 5 11 NA NA 1 1 12 NA NA 3 NA") ratermat2 = as.matrix(read.table(textConnection(ratermat1), header=TRUE, row.names=1))
We can then perform the analysis without the weights.
agree_coef(data = ratermat2, wide = TRUE, weighted = FALSE, col.names = c("Rater1", "Rater2", "Rater3", "Rater4"))
Or, perform it with weighting.
agree_coef(data = ratermat2, wide = TRUE, weighted = TRUE, col.names = c("Rater1", "Rater2", "Rater3", "Rater4"))
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