Entropic measures of complexity in a new medical coding system

Background Transitioning from an old medical coding system to a new one can be challenging, especially when the two coding systems are significantly different. The US experienced such a transition in 2015. Objective This research aims to introduce entropic measures to help users prepare for the migration to a new medical coding system by identifying and focusing preparation initiatives on clinical concepts with more likelihood of adoption challenges. Methods Two entropic measures of coding complexity are introduced. The first measure is a function of the variation in the alphabets of new codes. The second measure is based on the possible number of valid representations of an old code. Results A demonstration of how to implement the proposed techniques is carried out using the 2015 mappings between ICD-9-CM and ICD-10-CM/PCS. The significance of the resulting entropic measures is discussed in the context of clinical concepts that were likely to pose challenges regarding documentation, coding errors, and longitudinal data comparisons. Conclusion The proposed entropic techniques are suitable to assess the complexity between any two medical coding systems where mappings or crosswalks exist. The more the entropy, the more likelihood of adoption challenges. Users can utilize the suggested techniques as a guide to prioritize training efforts to improve documentation and increase the chances of accurate coding, code validity, and longitudinal data comparisons.


Introduction
Medical diagnoses and procedures are reported using standardized codes that are updated periodically to keep up with the latest clinical knowledge and practices. Transitioning from an old medical coding system to a new one can be challenging, especially when the two systems are significantly different. One such transition took place in the United States (US) in 2015 when the country switched from the 9th revision of the International Classification of Diseases (ICD) Clinical Modification (ICD-9-CM) to the 10th revision (ICD-10-CM). This newer revision was accompanied by a very different procedure coding system (PCS) (ICD-10-PCS), as compared to the ICD-9-CM procedure coding system (Volume 3, abbreviated here as Vol. 3). For example, each ICD-10-PCS procedure is made of 7 multi-axial characters where each axis encompasses up to 34 alphanumeric values [1]. This arrangement is a significant departure from the procedure code structure in ICD-9-CM Vol. 3, where all codes are numeric and can only be between 2 and 4 characters long. In 2015, ICD-10-PCS had about 72,000 procedure codes as compared to only about 4000 codes in ICD-9-CM Vol. 3. The diagnosis codes between these two revisions of ICD are also quite different. For example, all diagnosis codes in ICD-10-CM are alphanumeric and can be 3 to 7 characters long, whereas ICD-9-CM diagnosis codes are mostly numeric and can only be between 3 and 5 characters long. In 2015, there were about 14,500 diagnosis codes in ICD-9-CM as compared to about 69,800 codes in ICD-10-CM [2]. Given these differences, some analysts had predicted a costly and challenging transition from ICD-9-CM to ICD-10-CM/PCS [3]. Indeed, some of the feared problems did materialize after the changeover, such as the loss in productivity [4,5], the lack of readiness of computer systems, the inability to find some ICD-9-CM concepts in the ICD-10-CM system, and difficulties mapping ICD-10-CM to other coding systems such as SNOMED-CT [6]. Some ICD-10-CM clinical classes were also found to have more coding deficiencies than others, such as the class of external causes of morbidity (V00-Y99) [7]. In one post-ICD-10 implementation audit, it was found that one of the most significant challenges for coders was selecting the correct character in the 3rd position (Root Operation), the 4th position (Body Part), and the 5th position (Approach) of an ICD-10-PCS code [8]. While little evidence exists to suggest that reimbursement was significantly impacted by the transition, in some practices, a statistical increase in the codingrelated denials was noted [9]. A few of the post-transition qualitative studies concluded that training and education were critical in overcoming many of the previously anticipated challenges [6,10]. Besides the US, other countries have also faced challenges while transitioning to new medical coding systems. The issues ranged from coding errors to discrepancy problems when the same condition was coded in both coding systems. For example, in one analysis [11], it was found that the Swiss transition from ICD-9 to ICD-10 resulted in the initial increase of the number of coding errors for co-morbidities, but, over time, the accuracy improved as the learning curve waned. In one Canadian study [12], the authors were interested in assessing the validity of ICD-10 codes after switching from ICD-9. While the authors did not find much difference in the validity of the codes from these two systems, the discrepancy was apparent for some conditions (e.g., HIV/AIDS, hypothyroidism, and dementia). The authors also observed that the quality of data had not yet improved in ICD-10 as originally expected. Now that many countries are preparing to migrate from ICD-10 to ICD-11 [13], one can expect similar transition challenges to occur, as these two coding systems have different code structures [14], and the equivalence is at times lacking [15]. This research aims to introduce entropic measures to help users prepare for the migration to a new medical coding system by identifying and focusing preparation initiatives on clinical concepts with more likelihood of documentation deficiencies, coding errors, and longitudinal data comparison issues.

Related work
Not many studies have considered how to quantify the complexity of codes between two medical coding systems. In some studies, the equivalence in the number and structure of the codes between two coding systems is considered, but without accompanying measures of the dissimilarity in the codes [15]. In a few studies, an attempt is made to address the complexity between two medical coding systems. For example, in Boyd et al. [16,17], the authors proposed using the science of networks to evaluate the difficulties of transitioning from ICD-9-CM to ICD-10-CM in the US. The authors used general equivalence mappings (GEMs) to create graphs where diagnoses were nodes, and the relationships in the GEMs were edges. From their analysis, the authors derived directional motifs and identified convoluted mappings, where multiple medical codes from both coding systems shared complex, entangled, and non-reciprocal mappings. The authors concluded that clinical classes with convoluted mappings were more likely to be challenging to code and costly to implement after the changeover to the new medical coding system. Besides, these authors also anticipated that clinical classes with a high ratio of ICD-10-CM to ICD-9-CM codes were more likely to affect a smooth transition. Another study that considered the complexity of transitioning between two coding systems relates to the work of Chen et al. [18], where the authors leveraged Shannon's entropy to develop a mapping framework between ICD-10 and ICD-11 coding systems. The authors proposed three entropy-based metrics of standardizing rate (SR), uncertainty rate (UR), and information gain (IG) to validate information changes between ICD-10 and ICD-11. The authors obtained the UR measure by M i=1 p i log 1/p i , where M was the number of ICD-11 candidate codes for a single ICD-10 code, and p i was the probability of each ICD-11 code. In a special case of a uniform distribution, the authors suggested utilizing the average probability of 1/M to measure UR, which implied that UR = log M . Among other conclusions, the authors recommended verifying ICD-10 codes with high UR measures as these codes were more likely to hinder a smooth transition to ICD-11.

Contributions
This research complements previous studies highlighted in the Related Work section. For example, as in Chen et al. [18], this research proposes to apply Shannon's entropy to study the complexity of the transition between two medical coding systems. Unlike in this previous study, the entropic measures in this research account for the variation in the alphabets of candidate codes. Besides, Shannon's entropy is also used to create a measure of coding complexity that considers not only the number of candidate codes (as in the UR measure [18]) but also the number of combinations of these codes. As shown later, failure to account for the latter information may underestimate or overestimate the related coding complexity. It should also be mentioned that the proposed methods have an advantage over convoluted measures suggested in Boyd et al. [16,17]. Unlike in the convoluted approach, where a code is classified as either being involved in a convoluted relationship or not, the proposed methods provide non-dichotomous complexity measures of each code.

A motivating problem
It is imagined that a manager of a given medical care facility is preparing to transition from an old medical coding system X to a new medical coding system Y. The forward ( X → Y ) and backward ( X ← Y ) mappings between X and Y are provided. The manager is unsure about employing these mappings to identify clinical concepts that are more likely to be challenging to translate into the new medical coding system. Some of the benefits of knowing this information include being able to formulate targeted training efforts for coding and clinical documentation to foster the validity of the data in the new coding system. Besides, understanding complex translations may help take the necessary steps to ensure longitudinal data comparisons. This research aims to suggest the techniques that the manager could use to solve this dilemma.

Model and assumptions
Given forward mappings ( X → Y ), the old medical coding system X is termed the source system, while the new coding system Y is termed the target system. In the backward mappings ( X ← Y ), the source and target terminologies are reversed. For model development, only forward mappings are considered here since the backward mappings would obey the same logic. From the prescribed forward mappings ( X → Y ), it is assumed that code x ∈ X corresponds to m number of candidate codes y ∈ Y . This relationship, referred to here as a map, is symbolized as x → y 1 , y 2 , . . . , y m or as in the following matrix form: where each code in the map y i , for i : 1, . . . , m , has n fixed number of characters (also called alphabets) a ij , for j : 1, . . . , n . If necessary, padding may be added to a particular code to ensure a constant length of n as this approach may simplify calculations. Each column represents an axis or simply a position of an alphabet in a code. The columns of a map are assumed to be independent. Each row of a map represents a valid code y ∈ Y . A set of more than one code in a map may be necessary to represent code x ∈ X . If m = 0 , code x has no match in Y, which implies data loss in the new coding system. If m = 1 , code x ∈ X has a one-to-one relationship with code y ∈ Y . In this case, the coding complexity is expected to be zero since little surprise exists about what the new code should be. If m > 1 , the coding complexity will be greater than zero as there is more than one candidate code in Y, thus more complexity and chances of coding or translation errors. In this research, a coding error is defined as the selection of a code where at least one alphabet is wrong or the selection of a set of codes where at least one of the codes is incorrect or missing. The expected coding complexity of a given clinical concept in X is characterized in terms of the uncertainty in the rows and columns of a map, which is measured here in bits units of Shannon's entropy [19]. Two major sources of coding complexity are assumed here, namely source A, which captures the variation in the alphabets of a map, and source B, which relates to the combinations of the rows of a map. The entropy for source A, or H(A), is calculated as: where k j ≤ m is the number of unique alphabets in column a j of matrix (1) and p ij is the probability of alphabet i in position j. The more the H(A) measure, the more requisite detailed documentation to express all the alphabets of a map. Likewise, the more the number of code alphabets that must be chosen separately, the more complex and time-consuming the coding.
Regarding source B, the corresponding entropy H(B) is obtained by: Here, s is the total number of possible scenarios and m 0 represents the number of stand-alone codes and, for a given scenario i, m i1 , . . . , m i(m−m 0 ) denote the number of candidate codes in Y that must be combined to represent code x ∈ X . As before, m is the total number of candidate codes in a map. If a map only includes stand-alone codes, where no combinations of codes are required, Eq. 3 becomes H(m), which is comparable to the UR measure introduced in Chen et al. [18]. The more the H(B) measure, the more complex the coding due to the need for more coding memory and time, since more than one candidate code in the target system is going to be required to represent a single code from the source system. See Appendix A for more details on the derivation of Eqs. 2 and 3.

Implementation
It is recommended that both H(A) and H(B) entropic measures be normalized into Z(α) and Z(β) , as exemplified in Appendix A, to allow for the comparison and ranking of complexity from different sources. If H(A) and H(B) measures (or their normalized counterparts) are to be utilized to prepare for the transition (e.g., documentation improvement), they should be weighed using relevant empirical distribution (e.g., historical frequencies of codes in a given medical facility or general practice area). Accordingly, if, say, a particular facility never performs heart transplants, it shouldn't have to spend too much training efforts on the documentation of this clinical concept. Algorithm 4.1 shows the steps that one can take to implement the suggested entropic methods. Step 4: If empirical data, based on historical visits or future forecasts, were available, one would adjust Z(α) and Z(β) measures by multiplying them with the probability of a corresponding clinical concept.
Step 5: Use the adjusted or unadjusted entropic measures to prioritize transition initiatives between two medical coding systems.

Materials
Algorithm 4.1 can be applied to evaluate entropic measures between any two medical coding systems, provided mappings or crosswalks exist. For demonstration, the 2015 US transition from ICD-9-CM to ICD-10-CM/PCS medical coding systems is considered. For a brief background, when the US was preparing to migrate from ICD-9-CM to ICD-10-CM/PCS, forward and backward general equivalence mappings (GEMs) were made available to users [2,20]. A user could determine the number of candidate codes in the target system from these mappings, given a code in the source system. These files also allowed users to apply the given supplemental five digits codes (referred to as flags) to determine valid combinations of candidate codes in a map. For example, a flag code of 00000 or 10000 was used to represent a one-toone relationship. The flag code of 00000 signified the exact equivalence, whereas a flag code of 10000 represented the approximate equivalence. If the relationship were one-to-many, the third character in the flag code would be 1 (instead of 0), and the fourth and fifth characters would specify combinations of candidate codes. The fourth character enumerated the number of scenarios, while the fifth character established the order that combinations were carried out in each scenario. The data used in this paper can be obtained directly from the CMS website at https:// www. cms. gov/ Medic are/ Coding/ ICD10/ Archi ve-ICD-10-CM-ICD-10-PCS-GEMs. The 2015 GEMs, instead of the newer GEMs, are utilized here since they were the most updated mappings available to users to prepare for the transition from ICD-9-CM to ICD-10-CM/PCS in 2015.

Demonstration
Appendix E demonstrates a Python code to implement Algorithm 4.1. Figure 1 exhibits the application of this algorithm on map 0052. This map relates to an ICD-9-CM Vol.3 code of 00.52 for the implantation or replacement of transvenous electrode into left ventricular coronary venous system.

Results
Algorithm 4.1 was applied to both forward and backward GEMs between ICD-9-CM and ICD-10-CM/PCS. Tables 1 and 2 display the corresponding descriptive statistics for H(A), H(B), and UR entropic measures. Codes without match in the target system were excluded from these statistics. For comparison purposes, the normalization of the UR measure [18] is symbolized as Z(UR).
To implement Step 5 of Algorithm 4.1, clinical concepts were ranked by their entropic measures. Figures 2 and 3 show ranked clinical classes from the least to the most sum of Z(α) , Z(β) , and Z(UR) measures. The classes in these figures were also ranked separately using each entropic measure. As expected, the resulting rankings based on Z(α) , Z(β) , and Z(UR) measures were not always consistent. To assess how much the rankings of these entropic measures agreed, the Kendall tau correlation coefficients were assessed, and the results are presented in Table 3. The closer to 1 the Kendall tau value (the greener the color), the more the given entropic measures agreed. An alternative approach to implementing Step 5 of Algorithm 4.1 is performing outlier and pattern analysis and then segregate concepts that should receive more attention during the transition. An example of how such an analysis may be conducted is shown in Fig. 4. To extract thematic descriptions of the outlier maps, network analysis techniques suggested in Niyirora and Aragones [21] were applied after removing stopwords [22] and residual words (e.g., other, unspecified, etc.). Communities of words in Fig. 4c, d (distinguished by different colors) were isolated using the modularity algorithm in Gephi [23]. To gauge the frequency (or Map 0052 Equation 2  [18]) of this map is obtained by log 2 (m) = log 2 (8) = 3 . As compared to H(B), the UR measure slightly underestimates the complexity of map 0052. While no actual probabilities were available for Step 4, it still can be speculated that if the probability of implanting or replacing any electrodes in the ventricular coronary venous system were zero for a given medical facility, both the Z(α) and Z(β) measures would be 0.572*0 = 0.212*0 = 0. The implementation of Step 5 of this algorithm is discussed in "Results" section significance) of words in the outlier maps, a word cloud analysis was undertaken, where the bigger the word meant, the more significant the word (see Fig. 4e, f ).

Discussion
In 2015, ICD-10-PCS had a significantly greater number of procedure codes (n = 71,924), as compared to ICD-9-CM Vol.3 (n = 3,672) (see Table 1). Equally, Table 2 shows more diagnosis codes for ICD-10-CM vis-à -vis ICD-9-CM. This fact alone implies that more specific information was likely to be gained by migrating from ICD-9-CM to ICD-10-CM/PCS, assuming complete clinical documentation and accurate coding. The mean statistics in these tables reveal that all the entropic measures are higher in the forward mappings than the backward mappings. This revelation further certifies that, on   Table 3 Kendall tau correlation among the rankings of clinical classes using the normalized entropic measures ( Z(α) , Z(β) , and Z(UR)) The symbol <> is used to signify mapping between the indicated medical coding systems Forward mapping Backward mapping average, more information was gained in ICD-10-CM/ PCS as compared to ICD-10-CM. The quartile statistics with a value of zero suggest the minimum percentage of one-to-one mapping from the source system (e.g., a 75% quartile of zero indicates that at least 75% of codes in the source system has a one-to-one relationship with the target system). A one-to-one relationship implies that no information is gained since log(1) = 0 . In other words, one-to-one codes may structurally look different, but if they represent the same clinical concept, then no information is gained. To a computer, a one-to-one mapping is a simple translation, but, of course, to a human coder, more complicated code structures may be more challenging to extract and translate. The scale of the information gained (or lost) between ICD-9-CM and ICD-10-CM/PCS can be appreciated by clinical classes depicted in Figs. 2 and 3. For example, Fig. 2a indicates that in the procedural forward mappings, the most information was gained in the class of the Operations on Musculoskeletal System (76-84). The related box plot in Fig. 2b suggests that all three entropic measures relatively agreed on the characterization of class 76-84, given the small interquartile range. For diagnoses, Fig. 3a suggests that the class of the Injury and Poisoning (800-999) carried more forward information in ICD-10-CM followed by the class of Pregnancy and Childbirth (630-679). Remarkably, Fig. 3c implies that an ICD-10-CM class related to Pregnancy and Childbirth (O00-O9A) also resulted in backward information gain in ICD-9-CM. These conflicting results are due to the convoluted nature of the mappings between these two medical coding systems [16].
From Figs. 2 and 3, one notices that some clinical classes have negative entropic measures. This observation implies that little, or no information, would be gained in the target system. For example, Fig. 2a indicates that for the procedure class of the Diagnostic & Therapeutic Procedures (87-99), not much forward information was gained in ICD-10-PCS. Likewise, little, or no backward information was gained in ICD-9-CM Vol. 3 about the procedure class of Medical and Surgical (0) (see Fig. 2c). However, the entropic measures somewhat disagree on the latter suggestion, given a large interquartile range of class (0) in Fig. 2d. Regarding diagnosis codes, Fig. 3a suggests that little, or no information, was gained in ICD-10-CM about the ICD-9-CM class of Supplementary Classification Of External Causes Of Injury And Poisoning (E000-E999). In an apparent contraction, Fig. 3c points to the lack of information gained in the backward mapping about ICD-10-CM classes of Injury, poisoning and certain other consequences of external causes (S00-T88) and External causes of morbidity (V00-Y99). Again, this ambiguity results from the complex relationship between ICD-9-CM and ICD-10-CM/PCS coding systems [16].
It is noteworthy that, despite a greater number of codes in ICD-9-CM/PCS, the backward max statistics in both Tables 1 and 2 are not zero. This finding implies that, for some clinical concepts, ICD-9-CM captured more information vis-à-vis ICD-10-CM/PCS (e.g., class (F) in Fig. 2c and class (O00-O9A) in Fig. 3c). The implication is that some ICD-9-CM information was lost in ICD-10-CM/ PCS, which created issues with longitudinal data comparisons. This dilemma also likely produced problems with verifying ICD-10-CM/PCS codes' validity, especially for classes where the information was gained in both forward and backward mappings (bidirectional) (e.g. in the pregnancy and childbirth clinical class). Additional challenges resulting from the bidirectional information gain include conflicting documentation requirements, primarily if the new coding system collects different clinical information than what is commonly documented. Naturally, coding errors are likely to ensue if clinical documentation is lacking or inconsistent.
To prepare for the transition to a new medical coding system, the user can utilize the proposed entropic measures as a guide to orient training efforts. To this end, clinical classes can be ranked to gauge where most information is likely to be gained or lost. Of course, the user would have more confidence if the rankings of these entropic measures agreed. Regarding the transition from ICD-9-CM to ICD-10-CM/PCS, the proposed methods tend to provide similar rankings. This fact is particularly true in the forward mappings from ICD-9-CM Vol.3 to ICD-10-PCS, where, as highlighted in Table 3, Kendall tau correlation coefficients between the methods are either 1 or very close to 1. However, in some instances, such as in the backward mappings from ICD-10-CM to ICD-9-CM (see Table 3), the methods may disagree. Significant differences between H(A) (the entropy of the alphabets or columns of a map) and H(m) (the entropy of the rows of a map) typically cause this disagreement. The entropic measures will always agree in cases of a single candidate code in the map ( m = 1 ) since the entropy is zero for all measures. As the number of candidate codes   Fig. 2b) and O00-O9A and ST00-T88 (in Fig. 2d) would have been necessary to ascertain any transition challenges and training needs.
Besides ranking maps or clinical classes by their entropic measures, the user may also prioritize transition efforts from outlier and pattern analysis. That is, instead of working with predefined clinical classes, the user would try to assess the impact of the transition using major themes or ontological groups from the descriptions of outlier maps. Many thematic analysis [24] and ontological learning methods [25,26] are applicable here. For demonstration purposes, a simple graph was constructed and patterns were examined using network algorithms [21,27] (see Fig. 4). For example, a close examination of Fig. 4c, d reveal a collection of terms that relate to the vascular, skeletal, integumentary, and cardiac body systems. In terms of the eigenvector centrality, the most central words were tissue, graft, subcutaneous, skin, repair, and incision. Combining these keywords, one may conclude that the procedures for the musculoskeletal, integumentary, and cardio-vascular systems likely involved significant complex coding in ICD-10-PCS, a deduction that is consistent with the results in Fig. 2a.

Conclusion
Transitioning from an old medical coding system to a new one can be challenging, especially when the two coding systems are significantly different. This research aimed to propose methods that could help users prepare for the transition by identifying and focusing preparation initiatives on clinical concepts with more likelihood of transition challenges. To this end, two entropic measures of coding complexity were introduced. The first measure was a function of the variation in the map's alphabets, and the second measure was based on the possible number of valid combinations of candidate codes in a map. The primary assumption was that the more entropy, the more likelihood of coding errors. So, more prudent documentation was recommended for clinical concepts with high rankings of entropic measures, not only to increase the chances of accurate coding but also code validity and longitudinal data comparisons. It was also recommended that the resulting entropic measures be normalized and adjusted by the probability of a given code before isolating clinical concepts of interest. Medical professionals should conduct audits to ascertain transition challenges and training needs, particularly in the instances of diverging entropic measures. The proposed techniques are suitable for establishing coding complexity between any two medical coding systems, provided mappings or crosswalks exist. A demonstration of how to implement the proposed entropic measures was carried out using the 2015 forward and backward mappings between ICD-9-CM and ICD-10-CM/PCS.

Limitations and future research
A central conjecture of this research was that clinical concepts with more entropic measures were more likely to result in a more challenging transition. The justification of this assumption emanated from the fact that more entropic measures meant more variation in the codes, thus necessitating more prudent documentation and coding. This assumption may be violated if documentation is already complete and an experienced coder knows shortcuts to circumvent the new coding complexity. Accordingly, a medical record review may be necessary to ensure that the apparent complexity from the entropic measures actually exists. Besides, the topic of correlation between code validity and related entropic measures was not explored in this research. A medical record review may also be necessary to see if any lack of validity in the codes is explained by reasons other than the entropic measures. Other relevant topics could not be considered in this research without additional data. For example, the question of how a new medical coding system could affect reimbursement was not entrained. Also, the topic of how coding guidelines and conventions may contribute to coding errors in the new system was not discussed. Future research goals include considering these other topics, especially as they relate to the upcoming (or ongoing for some countries) transition from ICD-10 to ICD-11. Future research efforts will also include applying the proposed methods to the mapping between ICD-10 (or ICD-11) and SNOMED CT.  [19].

Using Shannon's entropy to estimate coding complexity
As noted in the main text, two major sources of coding complexity considered here are source A, which relates to the events of the alphabets in each column a j of the map, for j : 1, . . . , n , and source B, which relates to the events of the combinations of the rows (or codes) of the same map. Shannon's entropy of column a j , H (a j ) , is given by H (a j ) = − k j i=1 p ij log 2 p ij where k j ≤ m is the number of unique alphabets in column a j and p ij is the probability of alphabet i in position j. From Eq. 2, it was indicated that H (A) = n j=1 H (a j ) = − n j=1 k j i=1 p ij log 2 p ij . A more refined measure of H(A) is possible to reflect the fact that the position of an alphabet in a code may carry a different weight of a coding error. For example, in ICD-10, a coding error where the first character is incorrect is typically much worse than a coding error where the last character is wrong since the first alphabet tends to serve as a root node for the classification of clinical concepts. Accordingly, a weighing scheme can be devised to account for the relative influence of the position of an alphabet in a code. If w j were the weight of position j in the code, the resulting weighted average entropy of source A, H(Ā) , may look like this: where, as before, k j ≤ m is the number of unique alphabets in column a j and p ij is the probability of alphabet i in position j, for j : 1, . . . , n . The proof of Eq. A.1 follows from a logarithmic rules of log(x.y) = log(x) + log(y) and a. log(x) = log(x) a .
Hence, it . After weighing each column, it is evident that n j=1 w j · log As for source B, it was indicated in Eq. 3 that H(B) = log(v) . The proof of this equation follows from the fact that, under the assumption of a uniform distribution, each valid representation is equally likely, with probability 1/v. Then by Shannon's entropy, . If all m number of candidate codes in a map have a oneto-one relationship with code x, then all these candidate codes are considered stand-alone and don't have to be combined to form a valid representation of code x. In such case, the number of valid representations v equals the number of stand-alone m 0 , which also equals the number of candidate codes m. This means that: Equation A.2 is comparable to a case of the UR measure in Chen et al. [18]. If a map includes some stand-alone codes and other codes that need to be combined under different scenarios to make valid representations of code x, then v is obtained by: where i represents scenario i out of s total number of scenarios. Here, m 0 is again the number of stand-alone codes and m − m 0 is the number of codes that must be combined in sequential order of their index, as a set, to represent the old code x. That is, for a given scenario i, m i1 is the number of codes that must be sequenced first, followed by m i2 , the total number of codes that must be sequenced second followed by m i3 , the total number of codes that must be sequenced third, and so on until  Fig. 1). If a map had more than one scenario, the total number of valid representations of code x would follow Eq. A.3.

Normalizing the entropic measures
Assuming that H(A) ≡ α and H(B) ≡ β , the normalized entropic scores can be obtained this way: Table 4 Top 30 maps in the forward mapping from ICD-9-CM to ICD-10-CM, ranked by their sum of Z(α) , Z(β) , and Z(UR), from most to least total score The map id and description correspond to ICD-9-CM code and description, respectively where ᾱ is the average of α measures from all maps and β is the average of β measures from all maps. The symbol var() signifies the variance function.

Appendix B
Top 30 maps in the forward mapping from ICD-9-CM to ICD-10-CM See Table 4.

Outliers maps of Z(α) where Z(α) > 2.7 (about top 1% of the maps)
See Table 5. See Table 6. Table 6 Outlier maps arranged in the decreasing order of the Z(β) scores As before, m is the number of candidate codes in a map whereas v is the number of valid representations in a map. The UR measure (obtained by log 2 (m) ) can directly be compared to H(B) (obtained by log 2 (v) ). When UR < H(B) , UR has underestimated the expected coding complexity. When UR > H(B) , UR has overestimated the expected coding complexity. Otherwise, the measures are equal. The map id and description correspond to ICD-9-CM code and description, respectively