A statistical method for calibrating flame emission spectrometry which takes account of errors in the calibration standards

Abstract Spiegelman, C.H., Watters, R.L. and Hungwu, L., 1991. A statistical method for calibrating flame emission spectrometry which takes account of errors in the calibration standards. Chemometrics and Intelligent Laboratory Systems , 11: 121–130. The determination of potassium in sample solutions using flame emission spectrometry (FES) requires that the calibration function be estimated. Calibration standard solutions of potassium are made in the laboratory and nebulized into the FES instrument. A total of 240 data points were collected from chemical analyses. However the data come from only eight different values of the standards and are highly correlated within each standard. In the final analysis a sample size of eight was chosen to estimate the calibration curve. The main goal of this paper is to estimate the calibration function based on these data and then measure the amount of potassium in samples using this calibration function. The secondary goal is to show some of the important exploratory data analysis that should be done in any calibration. Since both the underlying theory of emission spectrometry and the scatter plot of data points suggest a linear relationship between the emission intensity and potassium concentration, a linear regression model is applied to fit these data and the residuals are examined based on regression assumptions. Data transformation is then attempted to stabilize the nonconstant variance of the residuals due to the fact that residuals fail to meet the assumptions. However, because the suggested transformation of taking the logarithm or 1/4 power of both x and y is hard to interpret, and because the log transformation would require an addition of an arbitrary constant to the standard values, we proceeded with the untransformed data. Outlier detection was used to find possible outliers. Ten consecutive observations (obs. 201–210) in the data set are potential outliers for they have absolute studentized residuals bigger than 2.7. However, influential observation techniques indicate that their effect on the estimation of the calibration curve is not great. In order to help compensate for the error in the calibration standards, we expand the calibration interval estimates. This compensation is important and helps to avoid the rather ad-hoc deletion of unusually influential data from the analysis. We think that a plausible explanation for the outliers is error in the calibration standards. In recognition of the heterogeneity of variance indicated by our data, we perform a weighted least squares type of confidence interval estimation for our calibration curve. The coefficient and standard error estimates are quite close for all weighted cases; in contrast, the unweighted case yields different values for the standard errors. If heteroscedasticity (nonconstant variance of the observations) is ignored, confidence intervals will be too wide at the low end and too narrow at the high end of the calibration curve. At both ends of the calibration curve, each resulting multiple-use calibration confidence interval is somewhat wider than the corresponding single-use calibration confidence interval. Finally, since the measurement errors that have a known finite bound in working standards have been taken into account, the increase in confidence intervals relative to presumed exact standards is about 0.1%.