Introduction

Forensic entomology is the analysis of insect evidence for forensic purposes [3]. Depending on the level of accessibility and environmental conditions, necrophagous insects will promptly colonise a fresh corpse. By calculating the age of developing insects on a body and by analysing the composition of its insect fauna, the expert may be able to deduce the time when insects first colonised the body which can infer a minimum time since death or post-mortem interval (PMImin). Establishing this period of time is the most important task of a forensic entomologist.

Because blow flies are usually the first group to colonise a body, the focus is often on them when using entomological evidence to estimate the minimum PMI. Analysing the size of the larvae or identifying which stage of immature development they have achieved allows the approximation of age, and such methods are well supported by existing research and are widely described and accepted in the forensic community. However, due to the recent burst in development of the forensic sciences, new court criteria require the evaluation of scientific evidence prior to its submission to the court. The Daubert Standard states that scientific evidence should be testable, have a known error rate, be peer-reviewed and be accepted by the specific scientific community employing the technique [24]. Moreover, a 2009 US Research Council report indicated a need for major improvements in many disciplines of forensic science in order to increase accuracy and meet those standards. Clearly, one of these improvements may be the introduction of new statistical analyses in forensic entomology for the validation of methods in court proceedings [16]. Estimations of insect age are calculated using curvilinear regression, isomegalen diagrams, isomorphen diagrams and thermal summation models [2, 12, 22]. However, estimation of juvenile age in forensic entomology suffers due to the fact that blow flies grow in a non-linear fashion and show, e.g., variable size distributions at different larval stages which unequally affect the estimation of their age [24]. Not surprisingly, scientists have reported different growth rates for the same species of fly, and experts have come to incongruent conclusions about a PMI based on the same entomological evidence depending on which growth data were utilised [24]. This is at least partly due to the wrongful application and misuse of statistical methods. Several authors realised that problem and have attempted to improve the methods used [e.g. 20 , 21] or introduce new statistical tools and formulas in forensic entomology [12, 13, 24, 26]. However, it is our belief that a scientific paper used as a basic principle for a court report should provide detailed descriptions of the experimental settings, an appropriate presentation of the data exploration and finally a sound validation and justification of the applied statistics.

Hence, the present paper exemplary analyses some growth parameters of the blow fly Calliphora vicina, testing various statistical methods in order to reveal differences in the determined regression parameters and to show their impact on the examined characters of the specimens. Besides the validation of the chosen test, we focus especially on the evaluation of the data, which should reveal their quality and the importance of checking it carefully prior to conducting the statistical tests, a fact which is so far often neglected when publishing growth data in forensic entomology [see [29 ] for a general discussion of this problem in an ecological context].

Material and methods

Flies

Two laboratory cultures of Calliphora vicina were reared in wire cages at the Institute of Forensic Medicine in Frankfurt am Main, Germany. Flies were fed with sugar and water ad libitum. Blood was offered as a protein source every two days. Fresh beef liver was provided as an oviposition medium once a week. After oviposition, the eggs were transferred to a LinTek MKKL 600/2 incubator, set at 20 ± 1°C under a lighting cycle of 12:12 h (light/dark). Constant temperature was validated using a DS1922L Temperatur Logger iButton (Maxim/Dallas), measuring every 1 h. The time needed for hatching was not recorded, but larvae were used soon after emergence.

Sampling

Resulting larvae were reared in population densities of 10, 50 and 100 individuals at three constant temperatures (5°C, 20°C and 30°C). To ensure comparability between the measurements at the three temperatures, sampling was not specified in “days after hatching”, but converted to accumulated degree days (ADD). Based on previously published data regarding developmental landmarks in Calliphora vicina (larval stages two and three) [4, 5, 8, 10, 15], sampling times were defined as 40 ADD and 120 ADD after hatching, assuming a minimum developmental threshold of 0°C (Table 1). Each density/temperature/ADD-combination was repeated five times within a single generation (five replicates of each combination) (n = 4800).

Table 1 Summary of the experimental treatments
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Hatched larvae were placed on ground beef in separate plastic cups (400 mL). To prevent intraspecific competition, the amount of meat was increased proportionally (10 individuals received 25g minced meat, 100 individuals received 250g ground beef, respectively).

Each cup was kept in a large plastic container (5000 mL) which was covered with a fine paper towel held in place by a rubber band. The container was filled with 3 cm of sawdust which served as medium for pupariation. The cups were moved daily within the incubators to count out incubator temperature effects and to prevent pseudoreplication. A datalogger (DS1922L) was placed in each incubator to record their temperature stability at 1-h intervals.

For each sample time, all larvae from the meat were removed and killed with boiling water. Impacts on larval development by handling effects could be ruled out because larvae were not disturbed until sampling time. They were then placed in 95% ethanol to avoid post-mortem changes [1, 23]. Length measurements were performed using a geometrical micrometer to 0.1 mm [25], and their weight was estimated using an electronic balance (reading precision: d = 0.001 g).

Regression analysis

Various regression analyses served to introduce different statistical methods in order to identify faulty model validations and to find ways of reproducing the data adequately.

Statistical analyses like linear regression, GAM and GAMM were carried out using the R statistical package [19].

Length was used as response variable. Weight, Density, Temperature and Time were used as possible explanatory variables, where Density consisted of the characteristic values 10, 50 and 100 (coded in R as: 1, 2 and 3, respectively), Temperature consisted of the values 5°C, 20°C and 30°C (coded in R as: 1, 2 and 3, respectively) and Time was defined through the values 40 ADD and 120 ADD (coded in R as: 1 and 2, respectively).

To find the optimal model in terms of statistical criteria, the Akaike Information Criteria (AIC) was consulted, which is a measure of goodness of fit: The preferred model is the one with the lowest AIC value [18 , 27].

Data exploration

For the detection of outliers, bivariate dotplots [7], boxplots and scatterplots were used. Collinearity and the relationships between variables were revealed using a pairplot. Preliminary exploration of the data indicated a strong collinearity between Weight and Time. Additionally, a non-linear relationship between Weight and Length was highlighted. Therefore, either Weight or Time should be used in the fixed part of the model but never both at the same time as this would lead to difficulties during the model selection process and give larger p-values.

Linear regression

The non-linear relationship between Length and Weight caused the elimination of Weight from the analysis. Therefore, the variables used in this model were Length as response and Density, Temperature and Time as nominal explanatory variables. Additionally, two- and three-way interaction terms were included.

Finding the optimal linear model involves verifying the main underlying assumptions: homogeneity, independence and normality [18 , 27 , 28]. To verify homogeneity, scatterplots were used in which the residuals were plotted against the fitted values as well as against each explanatory variable that was either used or not used in the model.

To detect dependent structures, the residuals were plotted against Weight using a scatterplot. Normality can be checked by a normal QQ-plot.

Homogeneity and independence are the most important assumptions, and if one assumption is not confirmed, the model should be rejected.

Generalised additive model

Because of non-linear relationships and heterogeneous and dependent structures, the next step was fitting a generalised additive model (GAM). Instead of a slope parameter, GAMs use a non-linear smoothing function to summarise the relationship between X and Y [24 , 27].

Due to the collinearity between Weight and Time, the variables used in this model were Length as response and Density, Temperature and their interaction terms as nominal explanatory variables. Time was dropped out and replaced by a smoothing function of Weight.

Based on the results for the residual plots of this model, a normal (Gaussian) distribution with an identity link function was suggested. The optimal number of degrees of freedom for the smoother was estimated through cross-validation (a method in which observations are left out and instead are predicted while the smoother is used for the rest of the data.)

The model validation process is approximately the same as in linear regression, which means that the main underlying assumptions have to be verified. Here, the model still showed dependent structures due to the hierarchical, nested data set (five batches within one approach). One possible solution to improve the model without transforming variables, which means a loss of information, is a generalised additive mixed model.

Generalised additive mixed model (GAMM)

The generalised additive mixed model (GAMM) is an extension of generalised additive modelling [28]. Unknown smooth functions of different types of covariates as well as random effects can be added. Therefore, they provide a broad and flexible framework for regression analyses in complex situations.

For this data set, the consecutive number of the batches of all approaches (BatchID) was included as a random structure. To deal with heterogeneity, a variance structure for the residuals was included in which each temperature level was allowed to have a different variance. Because Temperature was a nominal variable, a VarIdent variance structure was used [17, 28].

The other variables used in this model were Length as response and Weight as a smoothing function. The model validation is still the same as in linear regression and in GAM.

Results and discussion

Data exploration

The exploration of the data was done using all variables. As three of the explanatory variables were categorical and as they therefore cannot have any different values than 1, 2 or 3, it was only checked for outliers in the response variable and in the continuous variable Weight. The spread in length was nearly the same in both time regimes, whereas the spread in weight was bigger for 120 ADD than for 40 ADD. The dotplots indicated that Length is the more accurate measureable variable, and therefore, it presented the better response variable.

Both boxplots and dotplots showed some points that are beyond the centre of the data, indicating that these might be points that could influence the analysis incorrectly. Transformation of data might help here, but this process would lead to a loss of information as it would compress the data, and because the points were not that extreme, it was decided not to transform the data.

Pairplots and conditional plots of Length versus each individual explanatory variable showed no linear relationship between Length and Density, a weak linear correlation between Length and Temperature and, as expected, a strong relationship between Length and Time (Fig. 1). Additionally a collinear relationship between Weight and Time could be highlighted which is obvious too as weight increases with time. A scatterplot of Length versus Weight showed linear patterns at the start of the development and non-linear patterns with a wider margin at length values from 1.5 to 2.2 cm, which means that maggots with a weight of 0.1 g, for example, may have length values between approximately 1.5 and 1.9 cm (Fig. 2).

Fig. 1
figure 1

Pairplot of the response variable Length against each explanatory variable

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Fig. 2
figure 2

Scatterplot of Length (cm) against Weight (g)

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Because of the weak correlation between Length and Temperature and Density, the growth rate patterns do not change by temperature and density (Fig. 3). Additionally, it became apparent that individuals who were exposed to the same amount of accumulated degree days, especially 40 ADD, were longer the higher the temperature was. This means that despite the supposed equality of development, the use of the standard ADD method is suspect and should be taken with care when calculating the age of the larvae as this may lead to an incorrect estimation of the PMI [4, 5] (Fig. 4).

Fig. 3
figure 3

Coplot for Length against Weight, conditional on Density and Temperature (Density: 1 = 10 individuals, 2 = 50 individuals, 3 = 100 individuals; Temperature: 1 = 5°C, 2 = 20°C, 3 = 30°C)

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Fig. 4
figure 4

Coplot for Length against Weight, conditional on Temperature and Time (Temperature: 1 = 5°C, 2 = 20°C, 3 = 30°C; Time: 1 = 40 ADD, 2 = 120 ADD)

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After the evaluation of the data, it became clear that there were no extreme outliers and that Weight and Time were collinear. So, two options for further analysis opened up:

  1. 1.

    Drop Weight out, and use Time as the categorical variable for a linear regression.

  2. 2.

    Drop Time out, and use a smoothing function of Weight for a generalised additive model.

Because of its simplicity, linear regression was the first analysis carried out.

Linear regression

The model in the linear regression analysis contained the explanatory variables Temperature, Density, Time and their interaction terms. The model has the form:

$$ \begin{gathered} {\text{Lengt}}{{\text{h}}_{{ij}}}{ = }\alpha + \beta {1} \times {\text{Densit}}{{\text{y}}_i} + \beta {2} \times {\text{Temperatur}}{{\text{e}}_i} + \beta {3} \times {\text{Tim}}{{\text{e}}_i} + \beta {4} \times {\text{Densit}}{{\text{y}}_i} \\ { } \times {\text{Temperatur}}{{\text{e}}_i} + \beta {5} \times {\text{Densit}}{{\text{y}}_i} \times {\text{Tim}}{{\text{e}}_i} + \beta {6} \times {\text{Temperatur}}{{\text{e}}_i} \times {\text{Tim}}{{\text{e}}_i} \\ { } + \beta {7} \times {\text{Densit}}{{\text{y}}_i} \times {\text{Temperatur}}{{\text{e}}_i} \times {\text{Tim}}{{\text{e}}_i} + {\varepsilon_{{ij}}}{. } \\ \end{gathered} $$
(1)

The notation in Eq. 1 means that Length is modelled as a function of Density, Temperature, Time and their two- and three-way interaction terms. The variable Length ij is the length of individual i in batch j. The index j runs from 1 to 5, whereas the number of individuals per batch differs. The term ε ij is the unexplained noise and is assumed to be normally distributed with a mean of 0 and variance of σ².

The R 2 value for the model was 0.859, indicating that about 86% of the variation in length can be explained by this combination of predictors (F 7,3875 = 3389, p < 2.2e − 16).

All interaction terms and single variables were highly significant, which means that the null hypothesis which states that there is no interaction between Density, Temperature and Time can be rejected (Table 2).

Table 2 Numerical output from the linear regression model containing three explanatory variables and their two- and three-way interaction terms
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After the model selection process, the model validation followed, in which the underlying assumptions of linear regression had to be verified. To test for normality, a QQ-plot can be used, in which the points should lay on a straight line. Only in the middle range are the data normally distributed. However, as this assumption is the least important one, some discrepancy can be accepted (Fig. 5). To verify homogeneity, the residuals were plotted against the fitted values (Fig. 6). Because the spread of the residuals is not the same along the gradient, a violation of homogeneity occurred. To detect the source of this heterogeneity, the residuals were plotted against each explanatory variable. The graphs of the residuals versus Density and Temperature showed differences in spread which was the cause of the variation in the residuals. To test for independence, the residuals were plotted against Weight (Fig. 7). Although Weight is not applied in the model, it can be used to check for a possible lack of independence. The graph shows a clear pattern in which all samples with very low and very high weight values had negative residuals, which means that the samples were over-fitted since the residuals are calculated as observed minus fitted values. In contrast, samples with weight values lying in between had positive residuals, indicating that all of these samples were under-fitted.

Fig. 5
figure 5

Normal QQ-plot checking for normality

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Fig. 6
figure 6

Residuals against fitted values. Violation of homogeneity occurred because the spread is increasing with increasing fitted values

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Fig. 7
figure 7

Residuals against Weight. Note the lack of independence due to the pattern of positive and negative residuals

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The overall conclusion is that linear regression is not the right analysis for this set of data due to heterogeneity and a lack of independence. Although it provided significant values and the R² was about 86%, the main underlying assumptions were not verified. This caused the regression parameters to be unreliable, and they therefore should not be used in an expert opinion as the description of the developmental rate as a linear function of the ambient temperature may lead to an incorrect age determination of the specimen and therefore to an erroneous calculation of the minimum post-mortem interval.

The next step for improving the model was fitting a generalised additive model which allows for non-linear relationships.

Generalised additive model

Due to the high collinearity between Weight and Time, the categorical variable Time was replaced by the continuous variable Weight. A smoothing function of Weight has been used to model its effects. Therefore, the model for the generalised additive model contained the variables Density and Temperature (and their interaction term) which were used as factors and a smoothing function of Weight. It had the form:

$$ {\text{Length}}_{{ij}} = \alpha + \beta 1 \times {\text{Density}}_{i} + \beta 2 \times {\text{Temperature}}_{i} + \beta 3 \times {\text{Density}}_{i} \times {\text{Temperature}}_{i} + f{\left( {{\text{Weight}}_{{{\text{ij}}}} } \right)} + \varepsilon _{{ij}} $$
(2)

The aforementioned notation is almost like the linear regression model with the exeption of the smoothing function of Weight. The term ε ij is assumed to be normally distributed with a mean of 0 and variance of σ².

During the model selection process, the Density/Temperature interaction term was the least significant and could be dropped out. After refitting the model without the interaction, Density was not significant and dropped out too. The final model included the nominal variable Temperature and a smoothing function of Weight (Tables 3 and 4).

Table 3 Numerical output from the final generalised additive model containing the numerical variable Temperature
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Table 4 Numerical output for the smoothing function from the final generalised additive model
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The equivalent statistic to the linear R 2 value in GAMs is the explanation for deviance. For the final model, this was 97.6%, indicating a very good model fit (GCV = 0.0059136). The optimal number of degrees of freedom for the smoother was 8.51, affirming the non-linear relationship between Length and Weight (p <2e − 16).

For the final model, the null hypothesis which states that there is no effect for Temperature and Weight can be rejected.

The model validation process is approximately the same as in a linear regression analysis. To check the normality, the distribution of the residuals is illustrated in a QQ-plot (Fig. 8). The plots showed a little discrepancy, but since normality is not the most important assumption to be verified, the suggested normal (Gaussian) distribution could be accepted. The plot of the residuals versus the fitted values showed no bias in residuals based on Length, which means that the assumption of homogeneity could be verified. To test for independence, the residuals were plotted against all explanatory variables. Their distribution for the variable Weight showed no dependent structures (Fig. 9).

Fig. 8
figure 8

Checking for normality with a QQ-plot

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Fig. 9
figure 9

Residuals versus Weight to test for independence

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However, a model misspecification could be highlighted in the graphs for the residuals versus Temperature. They showed heterogeneity and a clear violation of independence. The samples showed more negative residuals than positive ones. Additionally, the imbalance increased with increasing Temperature level, indicating a dependent structure (Fig. 10). The dependent structures in the model were due to multiple observations within a batch, which means that the five length values within one approach were more related to each other than to values from another approach. As a consequence, the standard errors and variances of the estimated parameters were inadequate, and therefore, they should not be used in an expert opinion either.

Fig. 10
figure 10

Residuals versus Temperature. Note that the spread of the residuals is not the same along the gradient

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Because all linear and additive models assume independence within the observations, a generalised additive model is an improvement in terms of the smoother for Weight, although it is still not the right model. To allow for a hierarchical data set, the next step was applying a generalised additive mixed model.

Generalised additive mixed model

The residual spread in the generalised additive model differed with Temperature, which means that the assumption of normally distributed residuals with a mean of 0 and variance of σ² is wrong. One solution for dealing with heterogeneity is including a variance structure in the residuals. Since Temperature is included as a factor in the fixed part of the model, it was possible to use the VarIdent variance structure for Temperature as a variance covariate. The first model had the form:

$$ \begin{array}{*{20}c} {{{\text{Length}}_{{ij}} = \alpha + \beta 1 \times {\text{Temperature}}_{i} + f{\left( {{\text{Weight}}_{{ij}} } \right)} + \varepsilon _{{ij}} }} \\ {{{\text{where}}\,\varepsilon _{{ij}} \sim N{\left( {0,\,\sigma ^{2} _{a} } \right)}.}} \\ \end{array} $$
(3)

To take the dependent structures into account, a second model with the BatchID as the random part was fitted. Both models were compared using the AIC. The second model had the form:

$$ \begin{array}{*{20}c} {{{\text{Length}}_{{ij}} = \alpha + \beta 1 \times {\text{Temperature}}_{i} + f{\left( {{\text{Weight}}_{{ij}} } \right)} + a_{i} + \varepsilon _{{ij}} }} \\ {{{\text{where}}\,a_{i} \sim N{\left( {0,\,\sigma ^{2} _{a} } \right)}\,{\text{and}}\,\varepsilon _{{ij}} \sim N{\left( {0,\,\sigma _{j} ^{2} } \right)}.}} \\ \end{array} $$
(4)

The aforementioned notations are almost the same as the generalised additive model with the exemption of the random intercept a j which allows for variation between batches.

The model, when using a random intercept (4), has the lowest AIC and is therefore selected as the optimal model (model (4): −9518.457 compared to model (3): −8837.617).

The R 2 value for the model was 0.975, indicating that about 98% of the variation in Length can be explained by this combination of predictors (F 3,3791 = 15995.52, p < 0.0001). The optimal number of degrees of freedom for the smoother was 8.75, affirming the non-linear relationship between Length and Weight (p <2e − 16).

All variables were statistically significant, which means that none of these terms can be left out (Tables 5 and 6).

Table 5 Estimated parameters from the final mixed effect model
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Table 6 Numerical output for the smoothing function from the final generalised additive mixed model
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One part of the output from a generalised additive mixed model shows information about the different multiplication factors of σ, which means that each Temperature treatment has different variances. The estimated value for σ is 0.068. The treatment at 30°C has the largest variance, namely, (0.068 × 1.022)², whereas individuals reared at 20°C have the slightest residual spread (0.068 × 0.96)². The residual spread at 5°C lies in between (0.068 × 1²).

The model validation process is the same as in a linear regression analysis. The plot of the residuals versus the fitted values showed no extreme bias in residuals based on Length, which means that the homogeneity assumption could be verified (Fig. 11). To test for independence, the residuals were plotted against all explanatory variables. For the variable Weight, the residuals showed no dependent structures (Fig. 12). In contrast, the graphs for the residuals versus Temperature still showed patterns with more negative than positive residuals, indicating that a violation of independence occurred (Fig. 13). As the residuals in the model validation are not ordinary but standardised residuals, they should not show any pattern. However, because their distribution is not that rigorous and all terms in the model are highly significant, the results are quite robust. According to this, a generalised additive mixed model is by now the best method for that kind of data set because it allows for non-linear relationships and correlation between the batches.

Fig. 11
figure 11

Normalised residuals versus fitted values to test for homogeneity

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Fig. 12
figure 12

Normalised residuals versus Weight to test for independence

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Fig. 13
figure 13

Normalised Residuals of the final model, including a variance structure. Residuals are grouped per Temperature

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Conclusion and outlook

The common methods for the determination of age in immature blow flies include measurements of the length and the principle of the accumulated degree hours or days (ADH, ADD). The basis of these methods for the determination of individual age is a continuous assumed linear relationship between ambient temperature and development. Since the growth rate of blow flies is only linear during a small period of development and within a certain temperature range [3, 14, 24], the applied linear models do not supply a sufficient reliability. In linear models, the variability of a series of individual observations (y i ) is explained by a series of independent variables (x1, x2,…,xn) and a source of error (ε). One of the greatest limitations of linear models is the underlying assumption of independent observations and homoscedasticity [29]. This means that observations within an experimental approach or vessel have to be independent from each other. If the observations show dependent structures, the required replicates are only pseudoreplicates. Therefore, the use of linear models with a single region of variability (ε) becomes inappropriate.

The error values or the variance of the residuals are negatively influenced and finally lead to incorrect inferences [6, 9]. In addition, completely independent approaches are hardly ensured in biological work because individuals that are reared in the same incubator but in separate vessels cannot be classified as independent [11].

The objective of this study was to present various statistical methods applied to a forensic entomological data set. The differences of these methods concerning the validation of the final model were demonstrated, and a method was found that reflects the data set most adequately.

After the exploration of the data, a step-by-step analysis from a linear regression to a generalised additive model was conducted, finalised by a generalised additive mixed model.

Both the linear regression model and the generalised additive model provided incorrect regression parameters due to existing dependence and correlation structures between the observations. Only with a mixed effects model regression parameters can be calculated, which reflect the data adequately.

Since laboratory experiments are used for the establishment of reference data for forensic entomological casework, they are usually carried out with multiple repetitions. These data sets are automatically nested and imply the use of a mixed model. This is the only way to provide valid regression parameters that lead to statistical valid confidence intervals for the determination of age in insects and therefore to an accurate determination of the minimum time since death.

We used here a simple setting for reasons of clarity, but our model is certainly useful for producing relevant development data for forensically relevant fly species. Amendt et al. [2] compiled development data published so far for some forensically important blow fly species. Besides a certain amount of consensus, it was shown that there is discrepancy for one and the same species between different authors. This might be related to possible geographic variability and population specific features, but could be as well the result of wrong methods and erroneous statistics. However, due to a lack of transparency, this is not always verifiable and could lead to difficult situations in court when using such publications as a reference. Hence, we believe that there is not just the need for a detailed description of the experimental settings when publishing development data, but also for an appropriate exploration of the data and the validation of the statistics applied in such a study. In a next step toward an accepted forensic entomological tool, some could allow for making the published data sets available (and therefore checkable) through an online repository. While this reads desirable at a first glance and might contribute to a traceable use of entomological data sets, there is the question how to organise and control such platforms and repositories. This could be the task of organisations like the European Association for Forensic Entomology (EAFE) or the North American Forensic Entomology Association (NAFEA).