The median odds ratio (MOR) quantifies the variability between clusters (e.g., hospitals) using multi-level logistic regression [8, 9]. The main purpose of the MOR is to translate this variability between clusters into the odds ratio scale, which aids in easier interpretation of variability. The MOR’s definition is the median value of the distribution of the odds ratio between the cluster at higher risk and the cluster at lower risk selected randomly from two clusters (e.g., hospitals). For example, we randomly select two persons with same patient-level characteristics (say, present obesity) from two different hospitals and compute the odds ratio for the person with the higher risk of having the outcome versus the person with the lower risk. We then calculate the odds ratio from all pairs of persons to obtain a distribution of such odds ratios. The median OR is the median of this distribution. Alternatively, the MOR might be interpreted as the increased odds of an event when moving from a lower-risk cluster to a higher-risk cluster. The MOR is relatively easy to compute because it depends directly on the variability between clusters. The formula is the following [8, 9]:

$$\text\left(0.675\times \sqrt}\right)$$

where 0.675 is the 75th percentile of the standard normal distribution. Note that the MOR is always greater than or equal to 1, because the formula exponentiates a number greater than or equal to 0. An MOR of 1 indicates no between-cluster variability. It is insightful to compare the magnitude of the MOR, which quantifies the between-cluster variance, with the magnitude of the ORs for the fixed effects to understand the relative contribution of the random intercepts to the outcome’s overall variability, versus the fixed predictors.

To clinically understand what the MOR represents, we return to our example of determining the influence of patient-level, hospital-level (say, ‘variability in ICU admission practice’) and perhaps region-level variables on mortality [11]. An MOR of 2 would indicate that for two pregnant women with the same characteristics, the odds of dying may be twofold higher in one hospital compared to the other, given that we have considered other factors that may confound this association in our multivariable model. There are a number of examples of this concept in other medical and surgical specialties to illustrate the importance of using the MOR to estimate the influence of variability on outcomes [13,14,15]. For example, Wijeysundera and colleagues investigated predictors of pre-operative medical consultation for patients undergoing surgeries across 79 hospitals in Ontario. They concluded that patient-level factors are not meaningfully associated with pre-operative medical consult, yet there is variability of this practice among participated hospitals. Through multi-level logistic regression, they computed an MOR of 3.51, implying for two patients with very similar characteristics from different hospitals, one may be 3.51 times more likely to receive pre-operative medical consultation than another [15].

Both the ICC and MOR are tools to explore variability in multi-level regression. The ICC offers a more intuitive understanding of the variability in multi-level linear regression, while the MOR is more useful for multi-level logistic regression. While the ICC is chiefly a measure of between-cluster variability, it is influenced, by definition, by both between- and within-cluster variability, while the MOR reflects only between-cluster variability.

In frequentist statistics, the odds ratio is reported with a 95% confidence interval (CI), because this parameter is “fixed” and would lie within this interval, say, on average 95 times if the “experiment” was performed 100 times. The MOR, however, is random and has an associated distribution. Thus, the MOR should be reported with the Bayesian analogue of a 95% (CI), termed the 95% credible interval (Crl). When a parameter is “random” with a distribution around the MOR, the parameter lies in an interval of the distribution with a set probability. A 95% Crl of the MOR means that the MOR lies in this interval with 95% probability. Markov chain Monte Carlo methods are required to compute the MOR with a 95% Crl in multi-level logistic regression; however, further details are beyond the scope of this text [9].

The previous paragraph focused on MOR, we now consider the impact of ICU admission practice (a cluster-level variable) on outcomes by incorporating between-cluster variability. The Interval Odds Ratio (IOR) is a measure that estimates the range in which the true odds ratio of a variable at cluster-level lies, considering between-cluster variability. It is another type of interval used for the quantification of the impact of a variable at cluster level (e.g., hospital) on the outcome by incorporating between-cluster variability, in multi-level logistic regression. Note that the IOR does not describe the interval around a MOR, it reflects the size and variation of the OR of a cluster-level variable. An 80% IOR (IOR-80) is defined as the interval centered on the median of the distribution that comprises 80% of the odds ratio’s values. An 80% IOR is often recommended, because it covers a large portion of the odds ratios [8, 16]; however, the percentage is arbitrary, and other percentages could be considered.

Suppose that ICU admission practice (expressed as an annual incidence of ICU admissions, i.e., the number of ICU admissions divided by total number of hospitalized pregnant and postpartum women) is dichotomized as high versus low with the median value being the cutoff. When we compute a common OR of this ICU admission practice, an OR is calculated based on the comparison of mean odds in each group (high versus low). To compute the IOR of ICU admission practice, we might think about all possible pairs of subjects with the same covariates. For example, one woman is admitted to a hospital that more often admits pregnant and postpartum patients to the ICU (high group) and the other woman from a hospital that less frequently admits said patients to the ICU (low group). For each pair, we can calculate the OR between these subjects by accounting for the admission practice and variability between the two hospitals. We obtain the distribution of these ORs from considering all possible pairs. The IOR-80 is centered at the median of this distribution and covers 80% of the distribution of the ORs. Using this, we can calculate the lower and upper bounds of the IOR-80 with the following formulas that are similar to formulas we use for CIs:

$$}_=exp\left(\beta -1.2816\times \sqrt}\right)$$

$$}_=exp\left(\beta +1.2816\times \sqrt}\right)$$

In the above formulas, β is the regression coefficient for the cluster-level variable (e.g., ICU admission practice), Between-Cluster Variability is the cluster-level variance (variability), and − 1.2816 and 1.2816 represent the 10th and 90th percentiles of the standard normal distribution, respectively. Again, the IOR-80 is not a “confidence interval” around the MOR.

How do we interpret the IOR-80 in addition to a MOR? In our example, we explore the association of ICU admission practice for pregnant and postpartum women with mortality and measure variability induced by hospitals (cluster heterogeneity). We assess the MOR with the impact of the ICU admission practice among hospitals on mortality, in which simple odds ratios and 95% CIs and IOR-80 are used. We can employ multi-level logistic regression, to account for patient-level characteristics and the “clustering effect” of having many patients in a certain number of unique hospitals. In our case, we can interpret the different measures as follows:

The Median Odds Ratio reflects the between-hospital variability or heterogeneity with regard to mortality; the 95% CrI reflects an interval where this Median Odds Ratio lies, with 95% probability.

The Odds Ratio reflects the association of ICU admission practice with mortality; the 95% CI reflects the interval where this Odds Ratio lies when this experiment is repeated on multiple samples.

The 80% Interval Odds Ratio (IOR-80) is a value centered at the median of the distribution of the ORs and covers 80% of the distribution, quantifying the association of ICU admission practice with mortality while accounting for “between cluster” (i.e., between hospital) variability.

How can we illustrate the importance of the above? Suppose both results show that women admitted to low ICU admission hospitals, compared with high ICU admission hospitals, have a higher odds of dying (say, OR = 2.34, with a 95% CI 1.77–3.09).

Assume we found the MOR to be 4.4 and the IOR-80 to be 0.14:39.4 (Fig. 4). This MOR is high, implying high variability among clusters. This is also reflected by the wide IOR. Since the interval includes 1, the IOR-80 informs ICU admission practices may not be significantly associated with mortality. This illustration implies substantial unexplained variability exists, which could be patient level or cluster level, or both.

Suppose instead we found a MOR of 1.45 and IOR-80 1.164.73 (Fig. 5). This is a rather small odds ratio, implying little variability between clusters. The IOR-80 seems to be narrower due to little variability between clusters, just as the MOR implied. Moreover, the interval excludes 1, which means the influence of ICU admission practice is likely significantly associated with mortality.