The model formula (1) of the branching-type ANOVA is shown below:
y ijk ¼ μ þ α i þ b j i
ð Þ þ γ k þ u ijk
Here, μ is the grand mean (i.e., the average of all data), αi is the main effect (i.e.,
the difference between the control system and the addition system), b j(i) is the
difference between the flask and the assumption of randomness (i.e., stochasticity),
γ k is the main effect of daily fluctuations after the addition of a chemical substance,
and u ijk represents error. In other words, the grand mean of μ is the average of all of
the data, such that:
(1) i represents the number of groups and is equal to 1 in the control system and 2 in
the addition system.
(2) j represents the difference between the flask and shows the number of repeated
experiment times, and it becomes 1 in the first experiment and 2 in the second
experiment.
(3) k represents the days in progress after the addition of a chemical substance (i.e.,
test material); it becomes 1 for the first day after addition and 2 for the
second day.
(4) It is assumed that certain measurement data, y ijk , is the summed value of μ, α i , b j
(i) , γ k , and u ijk and measurement data, y ijk , for the third experiment, on the fourth
day after the addition of a chemical substance, is expressed as y 234 ¼ μ + α2 + b 3
(2) + γ 4 + u 234 .
(5) According to this, the measurement data y 234 are judged as to whether (1) the test
material addition (α 2 ) has an influence, (2) there is a difference between the
flasks (b 3(2) ), and (3) the changes over time (γ 4 ) are due to error (u 234 ).
(6) A branching-type ANOVA for higher addition concentrations is performed
according to a closed test procedure, and the highest concentration with no
recognized effect on the microcosm N-system is determined as the microcosm
no-effect concentration (m-NOEC). The flow chart of branching-type ANOVA
calculation is shown in Fig. 6.5.
The instructions for the practical calculation procedure are shown below.
– Procedure (1): From all of the data, y ijk , calculate each mean. The following
means (1–5) are the averages of each element:
(1) Mean of each microcosm
y ij: ;i ¼ 1,2,‧‧‧, a;j ¼ 1,2, ‧‧‧, b
(2) Mean of each population
y i:: ;i ¼ 1,2, ‧‧‧, a
(3) Mean of each population in each time
y i:k ;i ¼ 1,2,‧‧‧, a;k ¼ 1,2, ‧‧‧,
c
(4) Mean of each time
y ::k ;k ¼ 1,2, ‧‧‧, c
(5) Grand mean
y ...
(population, a ¼ 2, experiment, b ¼ 3 or more, and time, c ¼ 14, in this
microcosm test method)
– Procedure (2): Calculate each sum of squares from the next expressions (1–8):
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K. Kakazu et al.
y ijk ¼ μ þ α i þ b j i
ð Þ þ γ k þ u ijk
Here, μ is the grand mean (i.e., the average of all data), αi is the main effect (i.e.,
the difference between the control system and the addition system), b j(i) is the
difference between the flask and the assumption of randomness (i.e., stochasticity),
γ k is the main effect of daily fluctuations after the addition of a chemical substance,
and u ijk represents error. In other words, the grand mean of μ is the average of all of
the data, such that:
(1) i represents the number of groups and is equal to 1 in the control system and 2 in
the addition system.
(2) j represents the difference between the flask and shows the number of repeated
experiment times, and it becomes 1 in the first experiment and 2 in the second
experiment.
(3) k represents the days in progress after the addition of a chemical substance (i.e.,
test material); it becomes 1 for the first day after addition and 2 for the
second day.
(4) It is assumed that certain measurement data, y ijk , is the summed value of μ, α i , b j
(i) , γ k , and u ijk and measurement data, y ijk , for the third experiment, on the fourth
day after the addition of a chemical substance, is expressed as y 234 ¼ μ + α2 + b 3
(2) + γ 4 + u 234 .
(5) According to this, the measurement data y 234 are judged as to whether (1) the test
material addition (α 2 ) has an influence, (2) there is a difference between the
flasks (b 3(2) ), and (3) the changes over time (γ 4 ) are due to error (u 234 ).
(6) A branching-type ANOVA for higher addition concentrations is performed
according to a closed test procedure, and the highest concentration with no
recognized effect on the microcosm N-system is determined as the microcosm
no-effect concentration (m-NOEC). The flow chart of branching-type ANOVA
calculation is shown in Fig. 6.5.
The instructions for the practical calculation procedure are shown below.
– Procedure (1): From all of the data, y ijk , calculate each mean. The following
means (1–5) are the averages of each element:
(1) Mean of each microcosm
y ij: ;i ¼ 1,2,‧‧‧, a;j ¼ 1,2, ‧‧‧, b
(2) Mean of each population
y i:: ;i ¼ 1,2, ‧‧‧, a
(3) Mean of each population in each time
y i:k ;i ¼ 1,2,‧‧‧, a;k ¼ 1,2, ‧‧‧,
c
(4) Mean of each time
y ::k ;k ¼ 1,2, ‧‧‧, c
(5) Grand mean
y ...
(population, a ¼ 2, experiment, b ¼ 3 or more, and time, c ¼ 14, in this
microcosm test method)
– Procedure (2): Calculate each sum of squares from the next expressions (1–8):
54
K. Kakazu et al.