The optimal group size K _opt can be used to identify all the unqualified samples by the lowest detection times. However, when the group size is too large, the probability of unqualified sub-samples in a composite sample group that needs to be tested individually will increase. This reduces the detection efficiency, making it essential to select the most appropriate group size based on the estimated sample qualification rate. According to the classical algorithm of the optimal group size proposed by Dorfman in 1943 (Dorfman, 1943), the total number of samples is n, and K represents the number of mixed samples in a group. There are n/K groups in total. Assuming that the qualification rate is q, and the false-positive probability is Z, then according to the probability theory, the probability that all K samples are qualified is q ^K -Z, and one sample group only needs to be tested once to determine that all K sub-samples are qualified. Additionally, the probability of unqualified sub-samples in the K sample group is 1-q ^K + Z. To identify unqualified samples, the sub-samples in the group should be tested individually. Consequently, this group needs to be tested 1 + K times in total.

As PAEs are intentional additives in high concentrations (>1%), the targeted PAEs with detectable concentrations in a batch are rare, meaning that Z is far smaller than q ^K . Therefore, the probability of Z can be ignored. Formula (1) can be derived and used to calculate the expected total average detection times N with different qualification rate q and different group size K. N = K + 1 × 1 − q K × n K + q K × n K = n × 1 − q K + 1 K (1)

When N < n, the saved testing workload (S) is given by Formula (2). S = 1 − N n = 1 − n 1 − q K + 1 K n = q K − 1 K (2)

If the result is not detected or the qualification rate q is fixed, when the saved testing workload S reaches the maximum, the composite group K is the optimal group size in theory.

Table 1 lists the testing time of a single sample obtained from Formula (1), with different qualification rates of samples (q) and group sizes (K), on the premise that the sensitivity of the detection system is sufficient. When q is 70% and K is 3, the testing time is 0.99, then S is 0.01, and each sample is expected to save 1% of detection times by comparing with the routine measurement. When q is 99% and K is 11, the testing time is 0.196, then S is 0.804, which can save 80.4% of detection times. From this table we conclude that as the qualification rate q becomes higher, this quantitative composite testing method can increasingly reduce the detection workload and save the testing cost.

Figure 1 shows the variation trend of expected testing times under different conditions, and the saved testing workload S changes in an opposite tendency. For different qualification rates q, S increases first and then decreases with the increase of the group size K, indicating that too many samples in a group will lead to lower detection efficiency.

To reduce detection times, it is necessary to determine the optimal group size K _opt to obtain the maximum saved testing workload in the quantitative composite testing. As shown in Figure 1A, when q is 69%, the expected testing time is more than 1, which means that S is less than 0. Hence, it can be concluded that if the sample qualification rate q is lower than 70%, composite testing of any group size K will lead to more detection times and thus composite testing is not applicable. With the improvement of the sample qualification rate q, saved testing workload S under the same group size K exponentially increase. Normally, comparing with the traditional individual testing, the higher is the q, the more detection times can be saved by this quantitative composite testing method. For example, Figure 1F shows that when q is 99%, the value of S approximately reaches 0.8, saving 80% of the detection times. Therefore, the quantitative composite testing method is recommended when the sample qualification rate q is higher than 70%.

It is important to consider factors that affect the accuracy of results when determining the composite group size, K, rather than relying solely on a mathematical analysis to maximize the saved testing workload. A comprehensive analysis should be conducted to determine the maximum allowable group size (K _max) that guarantees the accuracy of composite testing results. To prevent the occurrence of undetected mixed samples that contain unqualified sub-samples, the group size should be selected within the range of 2 to K _max.

K _max is based on a comprehensive analysis of factors that affect the precision of the composite testing results. These factors include limits (L), the number of concerned substances in the limit (I), limit of quantification (LOQ), instrument detection limit (IDL), and other factors that vary among laboratories due to differences in testing capabilities and material diversity. Therefore, a safety factor (F) based on experience and historical data is recommended. Taking these factors into account, Formula (3) has been derived to calculate the maximum allowable group size K _max. K max = L × 1 − U r e l   max Q M , max × I × F (3) Q M , max = Q I , max × V m min (4)

In Formula (3), L represents the maximum regulated limit for the target substance. U _{rel max} (%) is the maximum value of relative expanded uncertainty among all U _rel values of the tested substance(s). F denotes the safety factor of regulated limits. L, I and F are determined by the practical application. Q _M,max is the maximum value of LOQ among all LOQs of the tested substance(s) for the method. It is the most critical factor, and can be estimated from Formula (4). Q _I,max is the maximum IDL value among all IDLs of the tested substance(s) for the instrument. V represents the final volume of the composite test solution, and m _min is the minimum mass of test portions in the composite test. I denotes the number of substances corresponding to the limit. For instance, the European REACH Directive (European Agency for Safety and Health at Work, 2006) stipulates that the sum concentration of three test items, DINP, DNOP, and DIDP, must not exceed 0.1%. In this case, I = 3. If information about the limit, LOQ, measurement uncertainty, etc., is available, K _max can be calculated using Formula (3). The final calculated value of K _max should be rounded down.

Before performing quantitative composite testing, LOQ and U _rel in Formula (3) need to be evaluated. LOQ refers to the minimum amount of the analyte in the sample that can be quantitatively determined with defined precision and accuracy under the specified experimental conditions. LOQ is used to measure detection sensitivity and is generally equal to 10 times the standard deviation of multiple parallel testing results. U _rel is the relative expanded uncertainty close to the regulation limit and is used to indicate the possible dispersion of the measured values (Miekisch et al., 2008). To ensure the validity and accuracy of composite testing results, evaluating the relative expanded uncertainty U _rel is necessary. Currently, for detecting PAEs, most laboratories can work out their own data and use them in the composite testing. Figure 2 shows possible sources of measurement uncertainties.

The relative combined standard measurement uncertainty u _rel mainly consists of the relative standard uncertainty of mass u _rel(m) , volume u _rel(V) , standard working solution u _rel(std) , recovery u _rel(rec) , and accuracy u _rel(rsd) . The value of u _rel can be obtained from Formula (5), which refers to ISO/IEC GUIDE 98-3-2008 (ISO, 2008). u r e l = u r e l m 2 + u r e l V 2 + u r e l s t d 2 + u r e l r e c 2 + u r e l r s d 2 (5)

Assuming that the testing result conforms to the normal distribution, then the relative expanded uncertainty U _rel can be calculated using Formula (6), the coverage factor, k, mostly equals 2 when the confidence degree is 95%. U r e l = k × u r e l (6)

K _max is calculated according to Formula (3), Formula (4), Formula (5), and Formula (6).

When the regulation or law requires that the combined concentration of multiple target substances lower than the limit, they need to be simultaneously detected. In this case, K _max is calculated using the maximum values of U _rel and LOQ among the target substances. When K _max ≤ 1, the quantitative composite testing method is not applicable.

The appropriate composite group size K _a can be affected by various factors, such as regulation limits, LOQ, measurement uncertainty, and qualification rate. The laboratory should consider all factors to select a suitable K _a . By comparing K _max and K _opt , K _a should be the smallest value of them. However, considering practical constraints, K _a is limited to no more than 10 (K _a ≤ 10).

The following shows some examples of K _a determination. Scenario A, B, and C are examples related to the America CPSIA and Canada CCPSA regulation, European Union REACH Directive Entry 51and52, and China standards and regulations GB 6675.1: 2014, respectively.

As quantitative composite testing cannot provide the concentration of the target substance in each individual sub-sample, it is assumed that the detected target substance originates entirely from the sub-sample with the minimum mass when interpreting the detection results. To determine the maximum possible concentration, W _max (mg/kg) of the target substance that may exist in a single sub-sample, Formula (7) can be utilized. By comparing W _max with the regulated limit, the testing results can be evaluated, and the presence of unqualified samples in the group can be determined. W max = c × V m min × D (7)

In this formula, c is the concentration of target substance in the solution to be tested following pretreatment of the sample group (mg/L); V is the constant-volume of the extraction liquid (mL); D is the dilution ratio; m _min is the minimum mass of a single sub-sample in the group.

Due to the uncertainty of each step in composite testing, it is necessary to correct the regulated limit (L) according to the measurement uncertainty to ensure the accuracy of the testing results. The calculation of the corrected limit, L _cor, is given by Formula (8). L c o r = L × 1 − U r e l / I (8)

If W _max ≤ L _cor, all the samples in the group are qualified. Otherwise, if W _max > L _cor, individual testing is required.

It is worth noting that in quantitative composite testing, if a composite sample group contains several low-concentration samples and each of these samples is below the corrected limit L _cor , the maximum possible concentration W _max may be higher than L _cor . This can result in a situation where all sub-samples in the group are qualified, but individual testing is still required to confirm whether there are unqualified samples. Therefore, it is not recommended to use quantitative composite testing when a large proportion of samples with low concentrations (50 or 100 mg/kg) are present. This situation should be identified through detection experiments of different detection items and is not included in this model.

Based on the constructed mathematical quantitative composite testing model, a flowchart of the work involved is presented in Figure 3. Firstly, the regulated limit L, LOQ, relative expanded measurement uncertainty U _rel , and sample qualification rate q of the batch of samples need to be determined. With these parameters, K _max is calculated according to the model, and it is used to decide whether the quantitative composite testing can be conducted. If the maximum group size K _max ≤ 1, the method is not recommended. The optimal group size K _opt can be obtained from Table 1. The appropriate group size K _a can be selected as K _a = Min (K _max , K _opt ) to enhance detection efficiency and ensure test accuracy. Next, K _a sub-samples are grouped, weighed, pretreated, and tested. Finally, the sample group’s qualification is determined by comparing the maximum possible concentration W _max with the corrected limit L _cor . If the group may contain unqualified sub-samples, they are further tested individually.

The samples were weighed using a BS124S analytical balance (Germany Sartorius Group). PAEs were extracted from the samples using Elmasonic P type ultrasonic cleaner (Elma, Germany). The analysis of the selected PAEs was performed on a 7890A gas chromatograph hyphenated to a 5975C mass selective detector (Agilent Technologies, Palo Alto, CA).

7 PAEs standard products were purchased from Germany Dr. Ehrenstorfer Company, including Di-iso-butyl phthalate (DIBP, 99.39%), Di-n-butyl phthalate (DBP, 98.78%), Benzyl butyl phthalate (BBP, 99.39%), Di 2-Ethyl Hexyl Phthalate (DEHP, 99.39%), Di-n-octyl phthalate (DNOP, 99.39%), Di-iso-decyl phthalate (DIDP, 99.00%) and Di-iso-nonyl phthalate (DINP, 99.00%). Dichloromethane reagent (Chromatographically pure) was purchased from Fisher Scientific (United States of America).

There are 12 polyvinyl chloride (PVC)-base matrix samples, which include 9 blank ones (#1 ∼ #9) without PAEs and 3 positive samples (#A ∼ #C). #A is national certificated reference material (CRM) GSB 16-3484-2018; #B is business certificated material RMC (reference materials certificate) 010a; #C is quality control (QC) sample from the Technology Center of Guangzhou Customs District. The concentration of the 7 PAEs in the three positive samples are listed in Table 2.

20 mg of the 7 PAEs are respectively transferred into a 100 mL volumetric flask, by adding dichloromethane to the constant-volume line. Then, the PAEs were mixed stock standard solutions with a concentration of 200 mg/L. The standard working solutions are prepared as follows: firstly, transfer 10 mL of the stock standard solution into a 50 mL volumetric flask and add dichloromethane till the constant-volume line, until the solution reaches a concentration of 40 mg/L. Afterwards, transfer 1 mL, 2.5 mL, 12.5 mL, and 25 mL of these solutions into 100 mL volumetric flasks and add dichloromethane until the solution reaches the constant-volume line. Finally, the standard working solutions with concentrations of 0.4 mg/L, 1.0 mg/L, 5.0 mg/L, and 10 mg/L are obtained.

The samples are cut into pieces with a diameter less than 5 mm. For each group, all the sub-samples should be weighed to 0.1 g (with a deviation within 10%) and mixed in a scintillation vial. Dichloromethane should then be added according to the total sample mass (i.e., 25 mL of dichloromethane should be added per 1 g of sample). The mixture should be subjected to ultrasonication in a water bath at 60 °C for 60 min. After the solution has cooled down, filter the supernatant through a 0.45 μm filter membrane.

The experimental conditions for gas chromatography-mass spectrometry (GC-MS) were based on the international standard ISO 8124-6. The GC-MS parameters and total ion flow chromatograms for the seven phthalate esters (PAEs) are presented in Table 3 and Figure 4, respectively. The GC separation of the PAEs was performed using a DB-5MS capillary column (30 m × 0.25 mm inner diameter × 0.25 μm film thickness) from Agilent J&W. Helium (99.999%) was used as the carrier gas and operated at a constant flow rate of 1 mL/min. The injector was operated in splitless mode at a temperature of 280°C, and the injection volume was set to 1 μL. The oven temperature program started at 80°C and ramped linearly to 290°C at a rate of 30°C/min and held for 1 min. The temperature was then increased to 300°C at a rate of 5°C/min and held for 3 min. The MS conditions included an ion source temperature of 280°C, an electron impact ionization source at 70 eV, and full scan mode ranging from m/z 50 to 500 were simultaneously applied for chemical determination using selected ion monitoring (SIM) mode.

Currently, many countries have implemented strict regulations regarding the amount of PAEs allowed in toys and children’s products. Table 4 provides a list of some of the standard regulations and limit requirements of PAEs in consumer products including toys and children’s products issued by some countries and regions. The European REACH Directive is among the most stringent, mandating that the combined content of four PAEs (DEHP, BBP, DBP, and DIBP) in all toys and children’s products must not exceed 0.1%, while the combined content of three other PAEs (DINP, DIDP, and DNOP) in toys and children’s products that can be placed in the mouth must also not exceed 0.1% (European Agency for Safety and Health at Work, 2006; consumerfed, 2008; ASTM F963-2011, 2011; SOR/2016-188, 2017).

According to the constant volume used in Sample Pretreatment (Section 3.3 of this study) and regulatory limit requirements, the concentration range of the standard solution for the 7 PAEs was selected as 0.4–40 mg/L. The mixed standard solutions of 0.4 mg/L, 1.0 mg/L, 5.0 mg/L, 10.0 mg/L, and 40 mg/L, prepared in the Preparation of Solution (Section 3.2 of this study), were quantitatively analyzed according to the instrument conditions specified in GC-MS conditions (Section 3.4 of this study). Table 3 and Figure 4 present the GC parameters and total ion flow chromatograms of the 7 PAEs. The standard working curves of the 7 PAEs showed good linear relationships within the linear concentration range of 0.4–40 mg/L, and the linear correlation coefficients ranged from 0.9997-0.9999.

In composite testing, the mixing of multiple samples can lead to the dilution of the target object, which requires higher detection sensitivity compared to traditional single sample testing. Therefore, it is necessary to evaluate the LOQ in advance to determine whether it meets the requirements of group testing. To determine the LOQ, 10 μg of each of the 7 PAE standard substances was added to 1.0 g of a PVC blank sample (#1), and pretreatment was conducted according to Sample Pretreatment (Section 3.3 of this study). The extracted liquid was then measured by GC-MS 7 times in parallel. The LOQ was determined as the 10 times standard deviation of the testing result of the target substance. The LOQs of the 7 PAEs were found to be 9.1–41.8 mg/kg, which were much lower than the limit requirements of 1000 mg/kg for the summation of 1–4 of PAEs in the China national standards and regulations given in Table 4. Therefore, they easily meet the needs of general detection.

The calculated relevant uncertainty components: u _rel(m), u _rel(V), u _rel(std), u _rel(rec), u _rel(rsd) and the relative expanded uncertainty U _rel are presented in Table 5. Detailed calculation methods are presented in The Maximum Allowable Group Size K _max (Section 2.2 in this study). The relative expanded uncertainties U _rel of the 7 PAEs are 16.6%–23.2%. Then, the maximum allowable group size, K _max, can be calculated based on the regulated limit L, Limit of quantification LOQ, and relative expanded uncertainty U _rel using Formula (3).

Based on the limit requirements of the content summation of 1–4 PAEs in toys and children’s products in Table 4, the total amount of these PAEs should not exceed 0.1% (1000 mg/kg). In the quantitative composite testing method for the 7 PAEs, the maximum LOQ (Q _M,max ) is 41.8 mg/kg and the maximum relative expanded uncertainty (U _rel ) is 23.2%. Using Formula (3), the maximum allowable group size (K _max) is calculated to be 4 or 18 (corresponding to I values of 1 or 4), respectively. Since the detection sensitivity meets the requirements of composite testing (K _max ≥ 2), this method can accurately determine whether composite samples contain any unqualified samples (i.e., samples in which one or more PAEs exceeds the limit).

To verify the accuracy of the quantitative composite testing method for PAEs in toys, testing results were obtained using different group sizes K. A total of 12 PVC samples were used for composite testing, with detailed information given in Instruments, Materials, and Reagents (Section 3.1 in this study). Sample grouping is listed in Table 6, with 6 sample groups of G1∼G6 set up with group sizes K of 3, 6, and 10, respectively. For each group size, there were 2 parallel groups. After pretreatment, GC-MS detection was performed, and the testing results are shown in Table 7.

In Table 7, the reference value of the group was calculated based on the reference PAEs value of the positive sample in Table 2. Using the concentration of each of the 7 PAEs, their corresponding measured value was calculated according to Formula (4). According to Table 7, the quantitative composite testing recovery rates are 91.0–112.3%, and the relative deviations between the measured values and their corresponding reference values are no more than 10%. Additionally, there were no false-positive and false-negative detection results. The variation of the group size had no significant effect on the testing result, indicating the accuracy of the PAEs quantitative composite testing method.

To investigate the false-negative and false-positive cases in the composite testing with a large number of samples, this study analyzed the PAE content in approximately 130,000 toys and children’s products from the baby product lab of the Technology Center of Guangzhou Customs District. Based on the practical statistical results, a simulated database with millions of samples was constructed and the random sample computer simulated group tests according the composite test model were conducted to verify the effectiveness and accuracy of the test results. The results indicate that both the false-positive and false-negative rates compared with the individual tests are very low and are within a controllable range. We have reported it in the ISO/TC 181 (Toy safety technical committee)/WG6 (Toy phthalates working group) meeting in 2021 and we will publish it in another paper. More detailed findings from this study will be reported elsewhere.