Estimation Using the Microcosm N-System

Abstract

In this chapter, the methods for measuring biomass and calculating the amount of production and consumption and an endpoint for a standard ecosystem impact assessment are described.

Appendix for Statistical Analyses

• 1. Statistical Testing

Statistical testing of hypotheses is performed using specimens , with which the verification of various hypotheses regarding a population is performed. A concept of statistical testing is shown in Fig. 6.10).

Fig. 6.10

Concept of statistical testing

◎ Statistical testing: Comparison of the differences among multiple specimens is performed, and it is estimated whether it may be said that there is a difference from the result to the population corresponding to each specimen.【Example】The mean of specimen B is compared with the mean of specimen A, and it can be said that there is a difference between the mean of population A and the mean of population B from the result.

◎ Significant probability , confidence interval: The probability is the possibility that the result for a real specimen will not differ between populations (i.e., when the null hypothesis is supported). It can be said that there is a significant difference when this probability is less than 5% (i.e., the null hypothesis of no difference between populations is rejected).

◎ Why must verification be performed? When data for the entire population are available for all cases, verification is not necessary. In many cases, the data available from specimens represent only small portions of the population being investigated. However, the question being investigated here is not “is there a difference between specimens?” but rather “is there a difference between populations?”, and it must be answered based on conclusions derived from a limited number of specimens. Therefore, when the difference between specimens is too small, it can be concluded that there is no significant difference between the populations, and when the difference between specimens is large, to some extent, it can be said that there is a significant difference between the populations. In that case, the significant difference will be expressed as a significant difference in the probability (●%), and this probability (level of significance ) indicates the possibility of incorrectly arriving at the conclusion that there is a significant difference.

◎ Rearranging of the main term for the statistical test:

• ①Specimen : Sample size of the ※ specimens from some population of the target group to be investigated.

• ②Population: The large (entire) group.

• ③Random sampling : Method of choosing a random specimen from within a population.

• ※Probability sample, random sample: The specimens that were randomly selected.

• ④Parameters: Generic names, such as the mean (population mean) of the population, dispersion (population dispersion), and the standard deviation (population standard deviation).

• ⑤Statistics : Generic names, such as the mean (sample mean) of the specimen, dispersion (sample dispersion), and the standard deviation (sample standard deviation).

• ⑥Regular population : Population distribution of the regular model.

• ※Normal distribution : The majority of experimental values have a central tendency, with values increasing and decreasing away from a central value, presenting a symmetrical bell curve. If samples are normally distributed, statistical analysis is possible. However, if the mean is skewed, some amount of error is possible.

• ⑦Null hypothesis : A hypotheses of no difference between the populations being compared. To determine whether or not there are differences between populations, a null hypothesis of “no difference” is used. An alternative hypothesis (the hypothesis that there is a difference in) is adopted if the null hypothesis is rejected.

• ⑧Confidence interval significant probability : The probability (possibility) that the null hypothesis is supported from the results produced from a real specimen . It is said that there is a significant difference when this probability is less than 5% (i.e., the null hypothesis is rejected as almost impossible). It is denoted as “α” or “p” and expressed as either a probability (p = 0.01) or a percentage (p = 1%).

• 2. Difference Between Test Types and Their Uses

Tests are divided into analyses of variance (ANOVA) for judging the difference among groups, the Student’s t-test for judging the difference between two groups, and multivariate techniques. The t-test is used to examine whether there is a difference between the population means between two independent groups, such as A and B. For three groups such as A, B, and C, for which μ represents the mean, it would be a serious mistake to simply repeat the t-test as below. In this case, multivariate techniques must be used.

• Between A and B, is there a difference in the value? (μ_A = μ_B)

• Between A and C, is there a difference in the value? (μ_A = μ_C)

• Between B and C, is there a difference in the value? (μ_B = μ_C)

• Why must the t-test not simply be repeated for multiple groups?

In the case of an examination with 5% confidence intervals, the probability (of a type I error) to dismiss the null hypothesis , H₀, by mistake is 0.05, assuming that the H₀ of μA = μB is true. When separate data analyses are performed for various problems associated with a certain sample, three H₀ will be considered the truth when analyses are performed three times, and the probability of type I errors occurring somewhere in the article will become unexpectedly high, at 1 − (0.95 × 0.95 × 0.95) = 0.14. When the same analyses are performed 10 times, the probability of a type I error becomes 0.40. Furthermore, a more serious problem occurs when different tests are based on the same data. For example, if a t-test of A is performed among the three groups, A, B, and C, comparisons are performed between A vs. B, B vs. C, and C vs. A. Unfortunately, a sample mean of A will then be higher than the true average; in the case of A vs. B and A vs. C, H₀ becomes easy to reject. Therefore, when three H₀ are considered to be the truth, the probability of a type I error is greater than 1–0.953 = 0.14, and it will be unpredictable.

In this way, when the t-test is simply repeated:

• The probability of a type I error throughout the analysis increases.

• When data are not independent, the probability of the type I error becomes unknown.

In this case, it is necessary to use multivariate techniques. However, in the daily operation of the microcosm and after having examined groups A, B, and C on the first day by multiple comparisons, on the subsequent days, the test will be repeated. Therefore, it may be said that this is statistically a mistake because the probability of rejecting the null hypothesis, H₀ (μA = μB = μC), for groups A, B, and C increases with the repetition of the t-test. A notice matter of the multiplex nature by the repetition of the multiple comparison is shown in Fig. 6.11.

Fig. 6.11

Notice matter of the multiplex nature by the repetition of the multiple comparison

The branching-type ANOVA can examine the control system and the addition system (1 mg/L) for all measured days together. Here, all groups could be examined at the same time, but it was decided that a closed testing order would be used because it is not known whether there is any influence from the addition of 1 mg/L or 2 mg/L. A concept of branching-type ANOVA is shown in Fig. 6.12.

Fig. 6.12

Concept of branching-type ANOVA

A method of using a closed testing order is recommended in the dose-response related examination to resolve the problems associated with the multiplex nature when examining every 1 mg/L and 2 mg/L of addition concentration. The importance for each hypothesis is set, and a closed testing order is a technique that allows for sequential examination by importance. When an insignificant result is provided, verification ends there. In the closed testing order, the entire examination and an equal level of significance can be used for verification, or preset levels of importance may be substituted beforehand. In other words, in the microcosm system, it is thought that higher addition concentrations result in greater effects, so the addition concentration is examined in descending order. Verification is terminated at the concentration with no significant difference, and this is assumed to be the maximum NOEC. The Way of thinking of the closed testing order is shown in Fig. 6.13.

Fig. 6.13

Way of thinking of the closed testing order