# Probability of Chance Agreement

This is calculated by ignoring that pe is estimated from the data and treating po as an estimated probability of binomial distribution while using asymptotic normality (i.e. assuming that the number of elements is large and po is not close to 0 or 1). S E κ {displaystyle SE_{kappa }} (and CI in general) can also be estimated using bootstrap methods. Some researchers have been concerned about the tendency of κ to take the frequency of observed categories as data, which may make it unreliable to measure agreement in situations such as the diagnosis of rare diseases. In these situations, κ tends to underestimate the agreement on the rare category.  For this reason, κ is considered too conservative a measure of agreement.  Others[Citation needed] dispute the assertion that kappa „takes into account“ random agreement. To do this, an explicit model of how chance affects evaluators` decisions would be needed. The so-called random adjustment of kappa statistics presupposes that if it is not entirely certain, evaluators simply guess – a very unrealistic scenario. Nevertheless, guidelines on magnitude have appeared in the literature.

Perhaps the first Landis and Koch who characterized the values < 0 as no match and 0–0.20 as weak, 0.21–0.40 as fair, 0.41–0.60 as moderate, 0.61–0.80 as substantial, and 0.81–1 as almost perfect match. However, these guidelines are by no means generally accepted; Landis and Koch provided no evidence of this, but instead relied on personal opinions. It has been found that these guidelines can be harmful rather than useful.  Diligence:218 Equally arbitrary guidelines characterize Kappas above 0.75 as excellent, 0.40 to 0.75 as just to good, and below 0.40 as bad. If statistical significance is not a useful guide, what size of kappa reflects an appropriate match? Guidelines would be helpful, but factors other than matching can affect their size, making interpretation of a certain magnitude problematic. As Sim and Wright noted, two important factors are prevalence (are the codes equipped or do they vary their probabilities) and bias (are the marginal probabilities similar or different for the two observers). When other things are the same, the kappas are higher when the codes are equipped. On the other hand, kappas are higher when codes are distributed asymmetrically by both observers. Unlike variations in probability, the distorting effect is greater when the kappa is small than when it is large. :261-262 Weighted kappa allows for different weighting of disagreements and is particularly useful when ordering codes. :66 Three matrices are involved, the matrix of observed scores, the matrix of expected scores based on random matching and the matrix of weights.

The cells of the weight matrix on the diagonal (top left to bottom right) represent a match and therefore contain zeros. Cells outside the diagonal contain weights that indicate the severity of this disagreement. Often, the cells of one of the diagonals are weighted with 1, these two with 2, etc. To calculate the kappa value, we first need to know the probability of the match (which is why I highlighted the diagonal of the match). This formula is derived by adding up the number of tests that the evaluators agree on and then dividing it by the total number of tests. Using the example in Figure 4, this would mean (A + D) / (A + B + C + D). First, we calculate the relative agreement between the evaluators. It is simply the proportion of overall scores where both evaluators said „yes“ or both „no“. The probability that both will randomly say „yes“ is 0.50 x 0.60 = 0.30 and the probability that both will say „no“ is 0.50 x 0.40 = 0.20.

Thus, the total probability of a random match is \${p_e}\$ = 0.3 + 0.2 = 0.5. To calculate pe (the probability of a random match), we note the following: To interpret your Cohen Kappa results, you can refer to the following guidelines (see Landis, JR & Koch, GG (1977). The measurement of observer harmonization for categorical data. Biometrics, 33, 159-174): Kappa assumes its theoretical maximum value of 1 only if the two observers distribute the codes equally, that is, if the corresponding row and column totals are identical. Everything else is less than a perfect match. Nevertheless, the maximum value that kappa could achieve with uneven distributions helps to interpret the value actually obtained from kappa. The equation for the κ-maximum is as follows: Suppose you are analyzing data for a group of 50 people applying for a grant. Each grant application was read by two readers and each reader said „yes“ or „no“ to the proposal. Suppose the data on the number of disagreements are as follows, where A and B are readers, the data on the main diagonal of the matrix (a and d) count the number of matches, and the data outside the diagonal (b and c) count the number of disagreements: Step 1: Calculate the relative agreement (in) between the evaluators.

Cohen`s kappa is always between 0 and 1, with 0 indicating that there is no match between the two evaluators and 1 indicating a perfect match between the two evaluators. The Cohen-Kappa coefficient (κ) is a statistic used to measure inter-evaluator reliability (and also intra-evaluator reliability) for qualitative (categorical) elements.  It is generally believed that this is a more robust measure than the simple calculation of the percentage of agreement, since κ takes into account the possibility that the agreement may occur randomly. .