Page 196 - AJWEP-v22i3

Vasylkovska, et al.

results in the u-the experiment, Where:
(ii) m is the number of repeated measurements (i) Δb is the variance of adequacy,
a
0
(repeatability), (ii) SS is the reproducibility dispersion,
2
(iii) i is the number of repetition, y
(iv) Y is the output parameter at the i-th repetition. (iii) F(0.05;f ;f ) is the critical value of Fisher’s test at
y
ad
ui
5% significance level,
The reproducibility of the experiments was (iv) f is the degrees of freedom for adequacy
ad = n – k–1
determined using Cochran’s test (G-test), where the variance,
condition G < G must be satisfied. If G > G , this (v) k is the factor’s number,
P
P
T
T
indicates a deviation from the hypothesis, suggesting (vi) f is the degrees of freedom of the playback variance,
y
that the effectiveness of adjustments to the object where fy = n (m – 1).
under study and the measuring instruments needs to be 0
2
verified. Variance of adequacy ( S ) was calculated using the
ad
The variance of reproducibility (error of the formula:
experiment) was determined by the formula: 2 1 n 2
1 n S ad = nk−− 1 ⋅ ∑ (Y − u Y u ) (VIII)
2
2
S = n ⋅ ∑ 1 S u (III) Where Y is the calculated value of the response in
1
u=
y
u=
u
Where: the i-th experiment.
(i) u is the observation index, The significance of the regression coefficients was
(ii) n is the number of experiments, determined using Student’s t-test. A coefficient b is
a
(iii) S is the variance representing the dispersion of considered significant if the following condition is met:
2
u
results in the u-the experiment. s
The coefficients of the mathematical model were b ≥∆ b = t (0.05; f y ). y n (IX)
a
a
determined by the following formulas:
n
Where:
b = 1 ⋅ ∑ 1 y u , (IV) (i) b is the regression coefficients b b b and b ,
0
n
u=
0, 1, 2,
a
1 n (ii) Δb is the confidence boundary (threshold), 1,2
a
b = ⋅ ∑ x ⋅ y , (V) (iii) t(0.05;f ) is the critical value of the Student’s t-test
i
n u= 1 iu u at a 5% significance level with degrees of freedom
y
1 n of reproducibility variance f .
b = ⋅ ∑ x ⋅ x ⋅ y u (VI) y
ij
n u= 1 iu ju The significance of the regression coefficients was
Where: assessed by setting the coefficients of the quadratic
(iv) x is the value of the i-th code factor in the matrix regression terms to zero (null hypothesis). To evaluate
iu
row in the u-th trial, this, additional experiments were conducted at the
(v) x is the value of the j-th code factor in the matrix center point of the experimental design. The average
ju
row in the u-th trial, value of the optimization criterion at the center was
Y
(vi) b is the free member of a regression equation, determined, and the deviation from the predicted value
0
0
(vii) b is the experimental coefficient, which indicates the Y ) was calculated. This deviation characterizes
i
degree of influence of the corresponding factor on Y, ( b − 0
0
the curvature of the response surface. The null
(viii) b is the experimental coefficient, which reflects
ij
the degree of influence of factors on Y, hypothesis is accepted if the observed deviation does
(ix) n is the number of experiments, not exceed the experimental error. The significance of
(x) u is the observation index, this deviation ( b − Y ) was evaluated using Student’s
0
0
(xi) y is the mean value of the dependent variable Y. t-test:
u
The adequacy of the equation was assessed using ( 0 Y 0 ) N
b −
Fisher’s test (F-test). Adequacy is confirmed if: t = (X)
d
S y 2
S 2
)
F = ad < F (0.05; f ; f y (VII) (xii) b is the free member of a regression equation,
S 2 y ad (xiii) N is the number of degrees of freedom,
0
Volume 22 Issue 3 (2025) 190 doi: 10.36922/AJWEP025170132