1150 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
An Analysis of Variance (ANOVA) table provides statistics
about the overall significance of the developed model. The
sum of squares for the model is determined as
yi
yi
n Model sum of squares
i
n
i
n
1
2
1
2
=-
==
//
where yi is the model prediction for the ith observation and
n is the number of observations. Model Mean square is the
average squared error for the observation data, or the sum-
of-squares of errors divided by the number of observation
and is expressed as
Model mean sum of square Degree of freedom
Model sum of square =
F-test for the model indicates the level of significance of the
model prediction. This tests the full model against a model
with no variables and with the estimate of the dependant
variable being the mean of the values of the dependant vari-
ables. The F-value is the ratio of mean model sum of square
by the mean error sum of squares and is expressed as
F Test Estimate of residual variance
Estimate model variance -=
The ANOVA for all three response models are given in
Table 5. The F-value of grade and recovery are 15.82 and
417.5 respectively at higher than 99.99% confidence level.
The Prob F for both the model are acceptable (less than
0.05) which indicated the developed models were signifi-
cant. Experimental results and the predicted values obtained
using model Eqs. (2) and (3) are tabulated in Table 6. Also
the relationship between the predicted and observed value
of the responses is shown in Figure 4, which shows that the
fit is quite good as the R2 value for grade and recovery of the
concentrate fraction of the MGS are 0.97 and 0.99 respec-
tively. The standard deviations of both the predicted models
are 1.05 and 1.08 for grade and recovery respectively which
are acceptable values. Further, the residual plots for the pre-
dicted values of grade and recovery are plotted in Figure 3.
From the figure, it can be observed that, the residual values
are uniformly distributed. Hence, it can be seen that the
errors for both grade and recovery are well distributed. In
order to check the validity of the proposed equations within
the range of the variables selected, a few random experi-
ments were also carried out following the afore-mentioned
methodology. The comparisons between the actual and
model predicted data at different combinations of variables
are presented in Table 7. It is evident from Table 7, the pro-
posed quadratic equations to predict the actual grade and
recovery (%Fe) of concentrate fraction of MGS is within
average errors of 3.31 and 4.42%, respectively. Therefore, it
may be considered that the proposed quadratic (Eqs. 2 and
Table 5. Analysis of variance for grade and recovery
Statistics Grade (%Fe) Recovery (Wt%)
Sum of squares 156.17 4349.24
Mean sum of squares 17.35 483.25
F Value 15.82 417.50
Probability of F 0.0036 0.0001
R2 0.97 0.99
Standard deviation 1.05 1.08
Table 6. Observed and predicted values of concentrate fraction grade and recovery
Test
No.
Condition Grade (%Fe) Recovery (Wt%)
X
1 (Inclination)
X
2 (Wash water)
X
3 (Rotational Speed
of the drum) Observed Predicted Observed Predicted
1 –1 –1 0 54.60 54.88 47.70 47.90
2 1 –1 0 57.00 58.02 53.60 53.10
3 0 0 0 56.80 56.80 43.44 43.44
4 –1 1 0 62.00 61.00 38.34 38.84
5 1 1 0 57.70 57.42 49.60 49.40
6 –1 0 –1 62.00 62.72 12.00 10.89
7 1 0 –1 60.80 60.80 19.00 18.57
8 0 0 0 56.80 56.80 43.44 43.44
9 –1 0 1 53.60 53.60 54.34 54.75
10 1 0 1 55.80 55.08 61.73 62.83
11 0 –1 –1 60.80 59.78 19.50 20.41
12 0 1 –1 59.60 59.88 13.70 14.31
13 0 –1 1 50.00 49.70 65.37 64.75
14 0 1 1 54.10 55.12 59.00 58.09
15 0 0 0 56.80 56.80 43.44 43.44
An Analysis of Variance (ANOVA) table provides statistics
about the overall significance of the developed model. The
sum of squares for the model is determined as
yi
yi
n Model sum of squares
i
n
i
n
1
2
1
2
=-
==
//
where yi is the model prediction for the ith observation and
n is the number of observations. Model Mean square is the
average squared error for the observation data, or the sum-
of-squares of errors divided by the number of observation
and is expressed as
Model mean sum of square Degree of freedom
Model sum of square =
F-test for the model indicates the level of significance of the
model prediction. This tests the full model against a model
with no variables and with the estimate of the dependant
variable being the mean of the values of the dependant vari-
ables. The F-value is the ratio of mean model sum of square
by the mean error sum of squares and is expressed as
F Test Estimate of residual variance
Estimate model variance -=
The ANOVA for all three response models are given in
Table 5. The F-value of grade and recovery are 15.82 and
417.5 respectively at higher than 99.99% confidence level.
The Prob F for both the model are acceptable (less than
0.05) which indicated the developed models were signifi-
cant. Experimental results and the predicted values obtained
using model Eqs. (2) and (3) are tabulated in Table 6. Also
the relationship between the predicted and observed value
of the responses is shown in Figure 4, which shows that the
fit is quite good as the R2 value for grade and recovery of the
concentrate fraction of the MGS are 0.97 and 0.99 respec-
tively. The standard deviations of both the predicted models
are 1.05 and 1.08 for grade and recovery respectively which
are acceptable values. Further, the residual plots for the pre-
dicted values of grade and recovery are plotted in Figure 3.
From the figure, it can be observed that, the residual values
are uniformly distributed. Hence, it can be seen that the
errors for both grade and recovery are well distributed. In
order to check the validity of the proposed equations within
the range of the variables selected, a few random experi-
ments were also carried out following the afore-mentioned
methodology. The comparisons between the actual and
model predicted data at different combinations of variables
are presented in Table 7. It is evident from Table 7, the pro-
posed quadratic equations to predict the actual grade and
recovery (%Fe) of concentrate fraction of MGS is within
average errors of 3.31 and 4.42%, respectively. Therefore, it
may be considered that the proposed quadratic (Eqs. 2 and
Table 5. Analysis of variance for grade and recovery
Statistics Grade (%Fe) Recovery (Wt%)
Sum of squares 156.17 4349.24
Mean sum of squares 17.35 483.25
F Value 15.82 417.50
Probability of F 0.0036 0.0001
R2 0.97 0.99
Standard deviation 1.05 1.08
Table 6. Observed and predicted values of concentrate fraction grade and recovery
Test
No.
Condition Grade (%Fe) Recovery (Wt%)
X
1 (Inclination)
X
2 (Wash water)
X
3 (Rotational Speed
of the drum) Observed Predicted Observed Predicted
1 –1 –1 0 54.60 54.88 47.70 47.90
2 1 –1 0 57.00 58.02 53.60 53.10
3 0 0 0 56.80 56.80 43.44 43.44
4 –1 1 0 62.00 61.00 38.34 38.84
5 1 1 0 57.70 57.42 49.60 49.40
6 –1 0 –1 62.00 62.72 12.00 10.89
7 1 0 –1 60.80 60.80 19.00 18.57
8 0 0 0 56.80 56.80 43.44 43.44
9 –1 0 1 53.60 53.60 54.34 54.75
10 1 0 1 55.80 55.08 61.73 62.83
11 0 –1 –1 60.80 59.78 19.50 20.41
12 0 1 –1 59.60 59.88 13.70 14.31
13 0 –1 1 50.00 49.70 65.37 64.75
14 0 1 1 54.10 55.12 59.00 58.09
15 0 0 0 56.80 56.80 43.44 43.44