Commentary on: Megyesi MS, Nawrocki SP, Haskell NH. Using accumulated degree‐days to estimate the postmortem interval from decomposed human remains. J Forensic Sci. 2005;50(3):618–26. doi: 10.1520/JFS2004017; and Moffatt C, Simmons T, Lynch‐Aird J. An improved equation for TBS and ADD: Establishing a reliable postmortem interval framework for casework and experimental studies. J Forensic Sci. 2016;61(Suppl 1):S201–S207. doi: 10.1111/1556‐4029.12931

See Original Megyesi et al Article here

See Original Moffatt et al Article here

Editor,

In the article previously published titled, Using Accumulated Degree‐Days to Estimate the Postmortem Interval from Decomposed Human Remains [1], there is an error in the published equation for calculating accumulated degree‐days (ADD) from the total body score (TBS). In a more recent study, Moffatt et al. [2] pointed out the error among several other issues. Moffatt et al. [2] discussed several flaws in the original Megyesi et al. [1] study; however, their method of addressing these issues resulted in reducing the original sample size from 68 to 15 individuals and unnecessarily inverting the original model, making the calculation more cumbersome. The goal of this letter is to correct the published equation [1] while leaving all the other assumptions of the original study in place and to demonstrate the implications for previous research of the error miscalculation. According to Google Scholar, Megyesi et al. [1] have been cited over 550 times since 2005 and over 315 times since the errors in the calculation were published [2]. Even when researchers mention the technical flaws in Megyesi et al. [1], the original equation is frequently used in analyses and published in textbooks [3, 4, 5]. Given that the study is cited so often, a correction to the published equation is warranted. We identify and correct the primary technical flaw in Megyesi et al. [1]. We demonstrate that the corrected model more accurately captures the uncertainty of ADD predictions from TBS. We also describe how Moffatt et al. [2] introduced additional assumptions that are technically correct but lead to results that are less useful in practical applications.

The major technical flaw in Megyesi et al. [1] is the inappropriate interpretation of model uncertainty. The expression for ADD on page 6 of the original report represents an additive uncertainty in ADD predictions, when, in fact, the error must be multiplicative given their modeling assumptions. The original report presents ordinary least‐squares regression (OLS) of the form

$${\log }_{10}\left(\mathrm{ADD}\right)=A·\mathrm{TB}{\mathrm{S}}^2+B+\mathit{\text{error}}.$$

Though not specified in Megyesi et al. [1], it is typically assumed that the error term is normally distributed with mean zero and constant variance ${\sigma }^2$. The value of $\mathrm{\sigma }$ can be readily computed from the residuals of the fit data. This value can then be used to estimate uncertainty in predictions of ${\log }_{10}\left(\mathrm{ADD}\right)$. For instance, the 95% prediction interval (PI) bounds are

$${\log }_{10}\left(\mathrm{ADD}\right)95\% \mathrm{PI}=\left[A·\mathrm{TB}{\mathrm{S}}^2+B-2.00·\mathrm{\sigma },\mathrm{A}·\mathrm{TB}{\mathrm{S}}^2+B+2.00·\mathrm{\sigma }\right]$$

This can be interpreted as saying that 95% of future identically distributed observations at a specified TBS are expected to have values of ${\log }_{10}\left(\mathrm{ADD}\right)$ that fall within these bounds. Importantly, the interpretation of the prediction interval assumes that future observations are generated under the same distribution of circumstances as the samples used to infer the values of A, B, and σ. The value 2.00 multiplying the standard deviation is the 97.5% quantile of the student‐t distribution with 66 degrees of freedom. Treating this factor as 1 instead of 2 (as was done in [1]) results in a 68% PI. The 95% PI shown above neglects the uncertainty due to variance in the estimate of the mean. However, in analysis not shown, we found that this correction is less than 5% for TBS values less than 32 and can therefore be neglected in many circumstances. By exponentiating both boundaries, we obtain the PI for ADD itself:

$$\mathrm{ADD} 95\% \mathrm{PI}={10}^{A·\mathrm{TB}{\mathrm{S}}^2+B\pm 2.00·\mathrm{\sigma }}={10}^{A·\mathrm{TB}{\mathrm{S}}^2+B}·{10}^{\pm 2.00·\mathrm{\sigma }}$$

It is important to note that the error term, which was additive in the expression for ${\log }_{10}\left(\mathrm{ADD}\right)$ becomes multiplicative in the expression for $\mathrm{ADD}$ due to the exponentiation. This contrasts with the expression by [1] where the uncertainty in ADD was additive. Stated in another way, the scale of the uncertainty in ADD is proportional to the scale of the point estimate for ADD. This prevents the prediction interval from including negative values and allows the uncertainty of estimates to increase with TBS as might be expected.

Performing OLS regression as described above on the full set of 68 cases from the original report yields A = 0.00155, B = 1.81, and $\mathrm{\sigma }$ = 0.201 with coefficient of variation $r^2=0.816$ . The regression coefficients are very similar to those reported [1]. The r² is slightly smaller than the 0.84 value reported in [1] since we included two samples that were excluded as outliers in the original report. Inserting these values into the ADD prediction interval expression, we obtain

$$\mathrm{ADD} 95\% \mathrm{PI}={10}^{0.00155\mathrm{TB}{\mathrm{S}}^2+1.81}·{10}^{\pm 2.00·0.201}$$

The first factor in this expression computes a point estimate of ADD given TBS. The second factor evaluates to ${10}^{+2.00·0.201}=2.52$ or ${10}^{-2.00·0.201}=1/2.52$. Thus, the result of model uncertainty is to multiply or divide the point estimate by a constant factor of 2.52. For instance, a TBS of 20 yields a point estimate of 269 and a prediction interval of $\left[269/2.52,269·2.52\right]=\left[107,678\right]$ (Note that these values differ slightly from those in Table 1 due to neglicting the uncertainty arising from estimation of the mean.). In contrast, the additive error approach in the original report results in the interval $\left[269-2.00·388,269+2.00·388\right]=\left[-507,1045\right].$ Apart from the obvious issue that the additive interval includes negative values, it also overestimates the uncertainty. As shown in Figure 1, additive errors result in over‐estimation of uncertainty for small values of TBS and underestimation of errors for large TBS. In contrast, the multiplicative error scales with TBS yielded reasonable 95% intervals throughout the full range of observations. The calculations for each TBS and associated 95% interval, including the uncertainty from variance in the estimate of the mean, are shown in Table 1.

FIGURE 1.

Comparison of additive and multiplicative ADD 95% prediction intervals as a function of TBS. Markers show the observed data. [Color figure can be viewed at wileyonlinelibrary.com]

TABLE 1.

TBS and predicted ADD with 95% prediction Interval using the modified equation. The prediction intervals include the uncertainty resulting from variance in the estimation of the mean.

TBS	ADD (°C)	ADD 95% Prediction Interval (°C)
3	66.7	26–171
4	68.4	26.7–175
5	70.6	27.6–181
6	73.4	28.7–188
7	76.9	30.1–197
8	81.1	31.8–207
9	86.2	33.8–220
10	92.3	36.2–235
11	99.4	39–253
12	108	42.4–275
13	118	46.4–300
14	130	51.1–330
15	144	56.8–366
16	161	63.4–409
17	181	71.4–459
18	205	80.9–520
19	234	92.4–594
20	269	106–683
21	312	123–791
22	363	143–923
23	427	168–1080
24	504	198–1290
25	601	235–1530
26	721	281–1850
27	871	339–2240
28	1060	411–2740
29	1300	501–3370
30	1600	615–4180
31	1990	759–5230
32	2500	944–6600
33	3150	1180–8390
34	4000	1490–10800
35	5110	1880–13900

In their follow‐up critique, Moffatt et al. [2] correctly identify the key technical flaws in the original report; however, their subsequent analysis introduces a new set of assumptions which unnecessarily limit the usability and scope of validity of the results. Firstly, Moffat et al. [2] only include 15 of the 68 original cases—omitting all cases with entomologically derived PMIs, indoor cases, and cases with ADD values greater than 3000. Secondly, pointing to a conceptual error in Megyesi et al. [1], Moffatt et al. [2] treat ADD as the explanatory rather than response variable. As a result, inference about ADD requires inversion of the regression equation, which can only be done through numerical simulation or a somewhat complicated analytic expression. This makes the application of results less straightforward for practitioners. Though there may be a causal argument for treating ADD as an explanatory variable (reasonably, TBS has no causal effect on ADD), from a predictive point of view, treating TBS as the explanatory variable is acceptable. We demonstrate that there is no predictive advantage to inverting the regression equation.

The predictive performance of our model is similar to the model presented by Moffatt et al. [2]. They report ${\mathrm{r}}^2=0.91$ on 15 samples. Using our linear regression estimator reported above, we compute ${\mathrm{r}}^2$ on the same 15 data points obtaining an identical ${\mathrm{r}}^2$ value. This is surprising given that our model has the harder task of fitting all 68 samples. On the contrary, the estimator presented by Moffatt et al. [2] performs slightly worse (${\mathrm{r}}^2=0.801$) than ours (${\mathrm{r}}^2=0.816$) when evaluated on the full 68 data points. As shown in Figure 2, the two models also perform similarly well in their ability to capture prediction uncertainty. Both prediction bands capture the natural variation evident in the data across the range of observations. We conclude that, given the similar predictive characteristics of the two models, practitioners are justified in making use of the simpler linear regression results presented in this letter.

FIGURE 2.

Comparison of our 95% PIs with those from Moffatt et al. The red circles mark the 15 samples used in the latter analysis. The red interval was computed using the formulae provided in [2]. [Color figure can be viewed at wileyonlinelibrary.com]

Both Megyesi et al. [1] and Moffatt et al. [2] are useful, but there are inherent limitations of any regression results based on a small, unrepresentative sample. Our results allow for the straightforward computation of a 95% prediction interval (PI) for the accumulated degree‐days (ADD) that a corpse has been subject to, given the observed Total Body Score (TBS). At face value, this interval can be interpreted as saying that 95% of future observations are expected to fall within the computed range. However, this interpretation assumes that the distribution of circumstances that produce future observations will match the distribution that led to the 68 case used to fit the model. Due to the limited variation in circumstances within the Megyesi et al. [1] sample, this assumption is likely to be violated in important ways. If this assumption is violated, the computed PI is no longer valid and can no longer be thought to constitute a true probabilistic measure. In practice, however, the computed interval may provide reasonable ranges of uncertainty even when the distributional assumption is violated if the changes in distribution do not dramatically alter the underlying relationship between TBS and ADD. Correspondingly, if the environmental circumstances are sufficiently different from those represented in the sample of 68 cases, the computed PI could be entirely unreliable. Investigators should therefore use the computed PI with great care, considering all additional sources of information on a case‐by‐case basis.

In conclusion, we have presented a simple analysis of the relationship between ADD and TBS based on the data from the influential study by Megyesi et al. [1], correcting an important error in the estimation of uncertainty. We demonstrate that our approach models the evident variation in the data more realistically than the original report. We also demonstrate that our approach performs at least as well as the more complicated approach presented by Moffatt et al. [2] while being simpler to apply. Finally, we discussed the intrinsic limitations in the application of any method (including ours) for inference about future observations based on a small, unrepresentative dataset. This limitation points to the need for the collection of more representative decomposition datasets across a wide range of environmental conditions.

FUNDING INFORMATION

This project was not supported by any external funding.