13 Statistical Techniques for the Interpretation of Analytical Data 691
13.2.2.1 Applications
Linear calibration is the most important application of this technique in chemical
experimentation (Miller and Miller 2000;Cela 1994; Massart et al. 1990). The fitted
equation can be used to estimate the value ofX(concentration, ˆx 0 ) corresponding to
some measurement ˆy 0 (mean value ofmreplicates) of the responseYon an unknown
sample ( ˆx 0 = (ˆy 0 −b 0 )/b 1 ), and its confidence interval
(
xˆ 0 ±t 1 −α/ 2 ,n− (^2) bs 1
√
1
m+
1
n+
(ˆy 0 − ̄y)^2
b^21
∑
(xi− ̄x)^2
)
(inverse regression). As an example, Tables 13.11 and 13.12
show the results of the linear regression for relative area (Y) vs concentration of the
analyte (X) in the standard solution using two replicates at five points, obtained
with the STATISTICA program (procedureMultiple Regression,intheMultiple
Linear Regressionmodule), that include estimation of the model parameters (Y
= 0.005889 + 0.030635∗X) and the statistics for the fit: correlation coefficient
(r= 0 .9798), determination coefficient (R^2 = 0 .960), and standard error of estima-
tion (s= 0 .0124) and the ANOVA table (there is a statistically significant relation-
ship betweenYandXat the 99% confidence level). Table 13.13 shows the ANOVA
Table 13.11Results of linear regression for relative area (Y) vs concentration of the analyte (X)in
the standard solution. Estimation of the model parameters and statistics for the fit:
Std. error B (regression Std. error
n = 10 Beta of Beta coefficients) of B t-value P-level
Intercept 0.005889 0.007092 0.83033 0.430433
Octanoic 0.979809 0.070687 0.030635 0.002210 13.86117 0.000001∗
R=.97981 R^2 =.96003 Adjusted R^2 =.95503
Std. Error of estimate:.01244
∗Parameterβ 1 =0(P<0.01).
Table 13.12ANOVA table for the regression:
Source of Sums of Degrees of Mean
Variation Squares (SS) freedom (df) Squares (MSS) F-value P-level
Regress. 0.029736 1 0.029736 192.1322 0.000001∗
Residual 0.001238 8 0.000155
Total 0.030974
∗There is a statistically significant relationship between Y and X at the 99% confidence level.
Table 13.13ANOVA table for the regression with the lack of fit test
Source of Sums of Degrees of Mean
Variation Squares (SS) freedom Squares (MSS) F-value P-level
Regress. 0.0297356 1 0.0297356 192.13 0.00000a
Residual 0.00123813 8 0.00015477
Lack-of-Fit 0.00022487 3 0.000074956 0.37 0.779b
Pure Error 0.00101326 5 0.000202653
Total 0.0309737 9
aThere is a statistically significant relationship between Y and X at the 99% confidence level
bthe model appears to be adequate for the observed data (P-value for lack-of-fit>0.10).