Informative graphing of continuous safety variables relative to normal reference limits

Datasets used for methods development and analyses

Graphical display of untransformed lab datasets

Graphs are first created for the untransformed data using the variable for time (e.g., VISIT in MDLD.JMP) as a continuous variable for the X-axis. The RESULT is a continuous dependent variable on the Y-axis. For the purposes of demonstration, lines for the normal reference limits are added to the Y-axis designating the ULN and LLN for each of the laboratory testing sites. Colored row states are created designating H as blue + signs, N as orange dots, and L as black inverted triangles.

Graphic inspection of untransformed data

Figure 1 is a presentation of RESULTS from the MDLD dataset by visit month with site-specific reference lines for the ULN and LLN. At the lower magnification, one can appreciate that values below the LLN are relatively compressed compared to those above the ULN (Fig. 1a). One can appreciate at higher magnification that the region of the LLN reference limits contains a mixture of normal and abnormally low values (Fig. 1b); this may be seen at the level of the values around the ULN lines as well. These issues may be exacerbated in trials with multiple sites (e.g., if sites use different assays), with pediatric patients, and where both genders are represented (e.g., if there are gender-specific reference ages).

Derivation of formulas to transform data for graphing

The formulae needed to graph the possible scenarios (L, N, H) are found in the following set of equations:

Formulae needed to graph transformed lab datasets in JMP

Figure 2 presents these formulae as they appear in the JMP Formula Editor. The steps necessary to arrive at the formula for the MI₀ in the JMP dataset include:

1. 1.

   Create columns for the RESULT and corresponding LLN

    
2. 2.

   Create a Column MI using the formula LLN ÷ RESULT

    
3. 3.

   Create a Column Max M using the formula Col Maximum(M). This is done using Col Maximum from the Statistical function of the Functions (grouped) dialogue box and designating Column M as the variable of operation.

    
4. 4.

   A column (Row of Max M) with the conditional formula, IF [M] = [MAX M] → [ROW()] specifies the row containing the maximum M for the next step. All of the cells in this column will appear to be missing values, except the rows corresponding to the maximum M value.

    
5. 5.

   The formula for MI₀ is written as one of the scenarios (last line, Fig. 2) for the conditional function that will be graphed on the Y Axis. The MIo formula makes use of Subscript from the Row function of the Functions (grouped) dialogue box to exclusively use the RESULT and LLN of the maximum multiple of the LLN.

    

Graphical display of transformed data

Figure 3 displays the results when graphing continuous variables where values that are both abnormally low and high are presented with a similar scale. Normal values are displayed in a linear distribution between the ULN and LLN. The black arrows pointing to purple inverted triangles indicate the locations where the multiplicative inverse of the LLN is located for RESULTS of zero based on the equation in Eq. 4.3. The black arrow pointing to the light blue inverted triangle indicates the location of the multiplicative inverse, MI_max, described in Eq. 4.1a. This is the point that is used to derive the position of the MI₀, as demonstrated in Eq. 4.

Advantages of graphic presentation of safety data

Graphic presentation of safety variables provides a means to enhance the accuracy of signal detection. Evidence for this has been reviewed in a recent publication by Michard [7]. Businesses, such as aeronautics, have long known this, designing key displays with graphics rather than numeric displays. The advantage of graphical presentation has been systematically studied with respect to vital sign data in the disciplines of cardiology [8, 9], anesthesiology [10], and critical care medicine [11]. Evaluation of clinical data in graphic form or a combination of numeric and graphic data has been shown to produce improved data interpretation speed and accuracy in several clinical scenarios. In one study of home blood pressure monitoring, this difference in monitoring resulted in differences in patient management, as measured by medications used. While definitive evidence was not found that graphic display could have a meaningful effect on mortality, several studies mentioned above have demonstrated significant improvements in the detection of myocardial ischemia or acute coronary syndrome, which can be fatal.

An unmet need in graphical presentation of data

Potential discrepancies and deficiencies in standard numeric reporting of clinical safety data values have been long recognized. In the 1970’s, Duboff reported creatine kinase values for members of a family with malignant hyperthermia that differed based on the testing site [12]. In this same period of time, the College of American Pathologists’ Enzyme Survey demonstrated that results of liver enzyme tests varied by site, presumably because of differences in instruments, reagents, and assay conditions [13, 14]. The findings from these studies in the 1970’s coincided with a revolution in the use of graphical presentation of data, prompted largely by the work of Cleveland [1, 15, 16] and similar biometricians.

Recent attention to the use of graphics in the presentation of safety data has brought refinements [17, 18] including the use of individual patient graphic profiles [19]; however, limitations and misperceptions still exist.

Rodbard discussed potential issues in perception when viewing data graphed for analytes such as glucose, which may have abnormal data that are both above the ULN and below the LLN. Despite being potentially more critical in the acute setting, hypoglycemic values may be overlooked when presented graphically because, in contrast, hyperglycemic values can be so relatively large that the appearance of the low values does not appear meaningfully different than those that are normal when graphed using standard methods [20]. While the concept of compression discussed by Bottger and Balzer [4] relates more to distortions caused by two unequal stretching operations, the phenomenon described by Rodbard et al., seems to describe compression of abnormally low values relative to the scaling of those above the ULN.

Previous proposals for graphing safety data

Several authors have proposed alternative methodologies for graphing. The most commonly used is based on multiples of the ULN. This method has the most utility for monitoring plasma levels for enzyme biomarkers of tissue damage such as AST, ALT, and CPK. One of the original methods proposed “centrinormalized units” that divide the ULN by 100 and scale all results by this factor so the normal range would always be 100 (upper limit) to LLN*100/ULN (lower limit) [21]. More recently, a system based on the ULN has been proposed as the method to evaluate drug-induced liver toxicity based on ‘Hy’s Law’ [22]. A closely related alternative methodology proposes to use both multiples of the ULN and baseline in a statistical outlier technique to define the critical boundaries of toxicity methods [23]. While the ULN-based methods have value in certain analyses, this methodology would not be useful when plasma analytes or vital sign measurements have values below the lower limit of normal. Normal values would also be scaled based on the ULN, which would make these data difficult to interpret.

Methods of graphing based on the standard deviation (SD) of data have also been proposed, where for example, data falling outside of 2 [24] or 3 [25] standard deviations of the mean would be considered abnormal. This method has the advantage of being able to described data above the ULN and LLN in a scale that is equivalent; however, the utility of this method is limited because not all sample populations are equivalent between hospital or lab testing sites and the population that comprises the mean ± 2 SD is not necessarily normal.

Rodbard [20] proposed use of a semilogarithmic plot to triple the percentage of the vertical axis allocated to values below the LLN, as is encountered in cases of hypoglycemia, while at the same time compressing the region of the graph containing hyperglycemic values. This method has value in expanding the detail for values below the LLN, yet it is not optimal for simultaneously visualizing data containing low, normal and high values.

Scaling of values with the proposed method

The primary objective in developing this method was to provide a means to graph data where values below the LLN were scaled the same as those above the ULN. The first step was relatively simple; transforming abnormally high values above the ULN is typically done by dividing the value by the ULN reference limit. The scaling of results below the LLN is not typically found in literature and maintaining a scalar, visually intuitive display is not as simple as the ULN. The multiple of LLN is multiplied by − 1 in the last step before graphing so that it occupies a position equidistant to normal values as those above the ULN; this final value is termed the multiplicative inverse. While the issue of zero in the denominator would not exist if the formula was configured the same as that of the ULN (e.g., RESULT/LLN), this relationship cannot be used because zero values would result in a multiple equal to zero, which would place the point in the middle of the normal values, which are scaled between 1 and − 1.

The calculation of the multiples of the abnormal low value below the LLN will be problematic when the RESULT equals zero, since this would make the multiplicative inverse an undefined number using the equation LLN/RESULT. This situation is not typically found with certain data, such as vital signs or chemistry laboratories, where having a result of zero is not typically observed. However, there are laboratories where zero is encountered (e.g., the absolute neutrophil count in profound neutropenia) and so the issue must be addressed. A RESULT of zero would intuitively have a multiple greater than any non-zero value in the column. An additional quantity would need to be added to the column maximum of the greatest multiple of the LLN equal to a ratio that is based on the LLN and the value remaining after the smallest result is subtracted from the LLN, as is demonstrated in the derivation of Eq. 4.

Limitations and concerns

Transformation of data almost always has potential issues with the audiences’ ability to comprehend the relationship of numbers when graphically displayed. Log transformation is one of the procedures most often used and it results in a display that may be uninterpretable by those outside of the scientific field. The method presented in this study has the advantage that it is visually intuitive; abnormal data of equal magnitude are equidistant from the normal reference limits. Most who routinely work with data are so accustomed to compression of results less than the lower limit of normal displayed in the Cartesian system relative to that above the upper limit of normal that visualization of the data in a more naturally expanded state will require familiarization. The advantage of this technique, particularly, the ability to look at data with multiple reference ranges in the same graph, will make such effort worthwhile. The method to graph points with a value of zero is not likely to be used often. Other techniques such as inclusion of a ‘zero-corrector’ (such as the use of ½ when an incidence is zero in the calculation of relative risk) may be simpler though this introduces error in the visual display.