Statistička odstupanja u analizama svojstava asfaltnih mješavina

Glavni je cilj ovog istraživanja pronaći metodu kojom se određuju statistička odstupanja u analizama svojstava asfaltnih mješavina AC 22, koja ne nastaju slučajno, već su uzrokovana vanjskim faktorima poput promjene normi, laboratorijske opreme ili osoblja. Proveden je proračun kako bi se odredila odstupanja koeficijenata korelacije u svojstvima asfaltnih mješavina raspoređenih u dvije skupine. Skupine se odnose na dva vremenska razmaka, a proračun je ponovljen nasumičnim odabirom podataka unutar dviju skupina, kako bi se odredila odstupanja unutar skupina.


Introduction
Asphalt mixtures in road construction must be resistant to permanent deformations and cracking caused by weather conditions and fatigue.They are composed of aggregate, binder, and air voids.The preliminary selection of appropriate materials, and determination of their proportions in asphalt mixture, are needed to obtain relevant properties that influence the behaviour of asphalt.Besides the production of asphalt mixtures, it is also important to know the way in which the mixture is built in.Quality control of asphalt mixtures and asphalt layers is implemented on the basis of European standards EN.Among other things, the behaviour of asphalt layers also depends on the quality of aggregate that is exposed to various mechanical influences and weather conditions.The basic purpose of stone grains is to transfer load from the wearing layer to the base layer.In asphalt mixtures, bituminous binder acts as a binding material and exerts a great influence on the behaviour of asphalt mixtures.Bituminous binder is composed of a large number of similar organic compounds.Many asphalt characteristics are dependent on it: resistance to permanent deformation, compaction of asphalt layer, bulk and maximum density of mixture, etc. Main characteristics of bituminous binder that have to be investigated are the softening point (PK), and the penetration and density of binder.Bituminous binders are usually divided on the basis of different properties, most important being the softening point and penetration.Besides the conventional bituminous binder, the polymer-modified bituminous binder is also widely used in road construction.That kind of bituminous binder is modified with elastomers and plastomers to improve its properties (to increase the softening point, reduce the Fraass breaking point, increase resistance to permanent deformation, etc.) [1].In recent years, the evolution in computer technology and automation in measurements have enabled collection of large databases and their analysis.Some of them are based on statistical methods such as the simulation and use of different analyses for the correlation, means, variances, and probability functions.Chou et al. [2] propose an approach that uses simulation models that can be employed to calculate probabilistic costs of highway bridge replacement projects.They use cumulative distribution functions as a user-friendly chart for decision makers, which can be used to assess project risks in the pre-design stage.Franz et al. [3] use different virtual simulations and correlation analyses in the field of architecture to study quantitative relations between the experience of architectural spaces and physical properties.Different authors [4,5] have compared confidence intervals for the difference between two means when the distributions are nonnormal and their variances are unknown.The Monte Carlo method [6,7] is widely used in the field of simulation for repeated random sampling to obtain numerical results.Ozgan [8] has modelled the Marshall stability of asphalt mixtures by changing the temperature and the time of exposure to a given temperature.It turned out that a good correlation exists between experimental results, fuzzy logical model, and statistical model.This means that both the statistical model and fuzzy logical model can be used to model stability of asphalt mixtures when changing the temperature, and the time of exposure to that temperature.Likewise, Tušar and Novič [9] have analysed correlations between different asphalt properties and Marshall stability.In [10], the authors define the dependence of the variable asphalt mix grading on the realisation of physical and mechanical properties of samples, such as the stability, stiffness, density, voids content, and percent voids filled with asphalt.Tušar and Kalman [11] statistically analyse different asphalt and bituminous binder tests in order to establish the relationship between different physical asphalt properties and binder properties, and mechanical properties of asphalt mixtures.The statistical method that can be used to establish measurement differences, and to define outliers if they exist, are described in this paper.A mathematical software is used for statistical evaluations.The primary focus is placed on the identification of changes that occur in the relationship between mechanical and physical properties of asphalt mixtures, i.e. between the Marshall stability and other properties of asphalt mixtures, and between the flow and other properties of asphalt mixtures.The statistics, such as the mean value and variance, describe the basic characteristics of the population.Changes of these statistics usually indicate a change in the population.Correlation coefficients describe the linear relationship between individual parameters of the population.Changes in the correlation coefficients do not indicate changes in the basic properties of the population but point to changes in the connections between parameters.Reasons for these changes often lie in measurement method modifications.So, if we find that correlation coefficients between two time periods change more than it would be expected due to random variations, we can conclude that changes to measurements have occurred.We were interested in two different situations between tests differences that can be expected: -If we divide the asphalt mixture data into two time periods, from 1998 to 2005, and from 2006 to 2009, we can find out how the change of standards in 2005 affects test results, or if the differences are due to the random variations.
-If we divide the same data randomly into two groups, like in the preceding example, we can find out which data deviate significantly and, on this basis, we can establish why the deviation occurred.

Data used in statistical analysis
The collected data are results obtained by tests made on samples of the asphalt mixture AC 22 in the time period from 1998 to 2009 at the Building Materials Institute (IGMAT) [12].Asphalt mixtures were mixed with the carbonate aggregate and bituminous binder B50/70, and so all the mixtures were nominally the same.The following data were obtained by testing: softening point of bituminous binder (ring and ball test -PK), penetration (PEN), index of penetration (IP), viscosity, shares of aggregate fractions, binder content, percentage of voids in mineral aggregate filled with binder, voids content in aggregate, maximum density of aggregate, maximum density of asphalt mixture, stability, flow, Marshall quotient, bulk density of asphalt mixture, and voids content in Statistical deviations in the analysis of asphalt mix properties asphalt mixture.Average values and standard deviations for all data used in the study are presented in Table 1.We considered the data that were obtained by experiments made at the institute for building materials Igmat Ljubljana between 1998 and 2005, and between 2006 and 2009.The reason for this division is the implementation of new standards for some experiments in 2006.

Statistical analysis
Statistical analysis is based on comparison of mean values, variances and correlation coefficients between measured quantities.The data are divided into two groups.The division can be performed according to two time periods in which measurements were made, but also according to two groups of laboratory assistants, two laboratories, etc.In order to determine what are the differences between correlation coefficients and what are the reasons for these differences, the correlation coefficients are calculated for two different cases: data are divided into two groups according to the period, data are randomly divided into two groups like in the first example.
The first example is basic, i.e. the one that is being checked.In the second example, we check if the data contain outliers.These outliers cause different correlation coefficients in both groups.

Correlations and simulations
The covariance S XY is a statistical measure of the linear dependence between two variables.The dimensionless coefficient that describes linear dependence is the correlation coefficient r XY , which is calculated using the formula: (1) where S XY is the covariance, while S X and S Y are standard deviations of variables X and Y observed on the given sample.Nataša Zavrtanik, Aleksander Ljubič, Goran Turk Since we are interested in the impact of other properties on the stability and flow, the focus is placed on the correlation that represents the impact of other properties on the stability and flow.
In order to obtain critical values for the differences of correlation coefficients, we have repeatedly and randomly divided the data into two groups representing division into two periods (1777 data in the first group and 416 data in the second group), and the correlation matrix was calculated in every simulation [13].

Results
We have 2193 measurements for the time period between 1998 and 2009 [12] for the asphalt mixture AC  2 that the data from the first time period changed compared to the second time period, because p-values for averages and variances are small (< 0.05) for almost all properties, except in the case of material passing through the sieve 11.2 mm where values for the average and variance are not significantly different.Here, the p-value is defined as the probability of rejecting the null hypothesis when that hypothesis is true.
Correlation coefficients for the stability, flow and other properties of asphalt mixture are shown in Table 3 for both groups of measurements and their differences.Critical values for differences in correlation coefficients between stability and other properties of asphalt mixture are shown in Table 4.The differences that deviate little from critical values are assumed to be the result of chance.Significantly larger differences mean that changes in measurements have occurred during the time periods.The differences of correlation coefficients that exceeded critical values given in Table 4 are specially marked in Table 3.This means that the difference of correlations is statistically significant with the significance level considerably lower than 0.05.Major changes Nataša Zavrtanik, Aleksander Ljubič, Goran Turk groups (1777 data in the first group and 416 data in the second group).The difference in correlations were computed.The simulations were repeated 10000 times and the critical values were determined using the ranking procedure.Critical values of differences for the significance level of 0.05 are presented in Table 4. Figure 1 shows the frequency diagram of differences of correlation coefficients between the proportion of bitumen and stability in 10000 random simulations.It can be seen that the probability of exceeding the critical value of difference of 0.142 is approximately 0.05, which corresponds to the significance level.Since the actual difference for this parameter was 0.456, which is in excess of the limit value, we had to reject the null hypothesis and conclude that the difference of correlation is statistically significant with the significance level considerably below 0.05.
Division according to the randomized measurements: 2193 measurements were randomly divided into two groups, just like in previous division in two periods (1777 data in the first group and 416 data in the second group).We first examined how the two groups changed the statistical properties such as the mean value and variance for all characteristics.
At this point, it should be noted that Tables 5 and 6  Statistical deviations in the analysis of asphalt mix properties the procedure was repeated and the values in Table 5 were consistently very small, this was also an indicator of incorrect data in the database.These values were excluded from the analysis.
In this case, it can be seen that most of the values in Table 5 are greater than 0.05, which corresponds to the significance level.This means that, in general terms, the averages and variances are not statistically significantly different, which was also expected.In few cases when the p-value was smaller than 0.05, this can be attributed to chance.When the random division was repeated, the p-value was small for some other parameters.
Table 6 shows the correlation coefficients between stability, flow and other properties of asphalt mixture for both types of measurements and their differences.It is evident that all deviations of correlation coefficients do not exceed critical values given in Table 4.This means that correlation differences are not statistically significant, with the significance level considerably below 0.05.

Conclusion
The comparison of differences between correlation coefficients of simulated measurements and actual measurements, randomly divided into two groups for mixture AC 22, has shown that no measurement deviates much from other measurements.
This means that where deviations occurred, they are small and are due to chance.The comparison of differences between correlation coefficients of simulated measurements and actual measurements, which are divided into two time periods (first period runs from 1998 to 2005, and the second from 2006 and 2009), has revealed greater deviations.This means that the resulting differences are not only due to chance but also to changes in standards or any other factors that affect the measurements.The method described in the paper enabled us to determine the effect of changes on measurements, which can not be attributed to chance but to some other external factors, such as modification of standards.We can also compare two laboratories or measurements in one laboratory at different time periods where we could estimate the impact of the replacement of machinery, staff and the like.This method can therefore be used for different kinds of problems, where the effect of various changes on different properties in asphalt pavement design is investigated.The statistical method is simple enough for wide use as it utilises only basic statistical calculations that are supported by all computer programs designed for the management, collection, and analysis of data.

Figure 1 .
Figure 1.Frequency histogram of differences in correlation coefficients for binder content and stability

Table 3 . Correlation coefficients for two periods and their differences Properties Period 1998/2005 Period 2006/2009 Differences in r xy between periods r xy
*The deviations are too large, that they could be coincidental

Properties 1 st group of measurements 2 nd group of measurements Differences in r xy between the 1 st and 2 nd group of measurements r xy
are only one example of random division of measurements.P-values for the average and variance, and values for differences in correlation coefficients, changed in every simulation.The values in both tables are only an illustration of what could happen.When