Homogeneity Testing

to measurement uncertainty, other than the statistical noise of x-ray counting, is insignifi cant. Homogeneity index . A traditional approach to test for homogeneity is the recently refi ned homogeneity index ( H ) [ 3 , 7 ]:

(Equation 1)

T e homogeneity index compares the variance expected from Poisson counting statistics ( s2

variance actually observed ( s2c ). The combined variance s2c is simply the variance (standard deviation squared) of the

Pois ) to the combined

mass fraction values obtained from a number of analysis points (designated N ) across the sample and includes the variance due to Poisson noise and any compositional variance. For each of the N analysis points an individual variance is calculated based on counting statistics and these are then combined into s2

as the average of the N individual values. Equations for this are given in [ 7 ].

Pois

If the observed variance is larger than the expected variance from counting statistics ( H > 1), compositional heterogeneity may be present. T is signifi cance of this heterogeneity can be tested based on F or chi-squared statistics because H 2 is a ratio of variances. T e answer is strictly valid only for the specifi c set of measurement conditions that were used for the data acquisition. For the null hypothesis that no heterogeneity can be detected, H = 1, the combined variance observed is equal to the variance due to counting statistical noise. T e homogeneity index on its own is not suitable for deciding whether heterogeneity is detected or not, because H itself has an inherent uncertainty. T is is because the variances used to calculate H are statistical estimates and will scatter around true but unknown values. T is scatter will decrease by increasing N . Hence, H > 1 may indicate signifi cant heteroge- neity, or it may just be larger than unity because of statistical scatter, even when there is no heterogeneity present. In order to decide which case applies, a critical homogeneity index ( H crit ) can be calculated based on the chi-squared distribution [ 7 , 9 ].

(Equation 2)

If H > H crit , the null hypothesis has to be rejected and signif- icant heterogeneity was detected. However, because statistical testing cannot provide a defi nitive answer, a level of signifi cance α has to be chosen before H crit is computed; this level is usually 0.05 (5%). At this level, a false rejection of the null hypothesis may occur in 5% of all tests when there is actually no detectable heterogeneity present, a type I error (false detection of hetero- geneity) shown in Figure 1 . Note that χ 2

Excel as function CHISQ.INV where the signifi cance level is stated as 1 - α , that is, 0.95 for a 5% level.

( α,N-1 ) is available in

Because the uncertainty of the homogeneity index depends on the number of measurements N , the critical homogeneity index H crit depends on N . At N = 30 measurements and α = 0.05, the critical homogeneity index is 1.21. At N = 300, it is reduced to 1.07 ( Figure 1 ). To be considered homogeneous, the determined

32

Figure 3 : Electron microprobe x-ray distribution maps by wavelength-dispersive spectrometry obtained on a polished apatite crystal investigated for its suitability as reference material. (a) Distribution of silicon (Si Kα ) shows chemical zoning within the crystal. The material is obviously heterogeneous and, in its present form, not suitable as a reference material. (b) Distribution of cerium (Ce Lα ) shows barely visible heterogeneity. In this case, statistical testing based on quantitative analyses has to be conducted.

homogeneity index must be below this critical value. T erefore, increasing the number of measurements increases the power to detect heterogeneity. Increasing N also decreases the probability of erroneously accepting the null hypothesis, the type II error (false non-detection of heterogeneity) illustrated in Figure 2 . However, this probability cannot be quantifi ed because the true heterogeneity present in the sample is not known—this is what the test is trying to infer statistically. Moreover, increasing N has practical limitations because it requires time and increases the risk that instrumental driſt may become a signifi cant contri- bution to the variations observed. T us, choosing a reasonable N is an important task. Uncertainty budget . A useful concept in homogeneity testing and helpful in choosing a reasonable N is the “uncertainty budget.” In its simplest form this relates the combined variance ( s2c )

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to measurement uncertainty, other than the statistical noise of x-ray counting, is insignifi cant. Homogeneity index . A traditional approach to test for homogeneity is the recently refi ned homogeneity index ( H ) [ 3 , 7 ]:

(Equation 1)

T e homogeneity index compares the variance expected from Poisson counting statistics ( s2

variance actually observed ( s2c ). The combined variance s2c is simply the variance (standard deviation squared) of the

Pois ) to the combined

mass fraction values obtained from a number of analysis points (designated N ) across the sample and includes the variance due to Poisson noise and any compositional variance. For each of the N analysis points an individual variance is calculated based on counting statistics and these are then combined into s2

as the average of the N individual values. Equations for this are given in [ 7 ].

Pois

If the observed variance is larger than the expected variance from counting statistics ( H > 1), compositional heterogeneity may be present. T is signifi cance of this heterogeneity can be tested based on F or chi-squared statistics because H 2 is a ratio of variances. T e answer is strictly valid only for the specifi c set of measurement conditions that were used for the data acquisition. For the null hypothesis that no heterogeneity can be detected, H = 1, the combined variance observed is equal to the variance due to counting statistical noise. T e homogeneity index on its own is not suitable for deciding whether heterogeneity is detected or not, because H itself has an inherent uncertainty. T is is because the variances used to calculate H are statistical estimates and will scatter around true but unknown values. T is scatter will decrease by increasing N . Hence, H > 1 may indicate signifi cant heteroge- neity, or it may just be larger than unity because of statistical scatter, even when there is no heterogeneity present. In order to decide which case applies, a critical homogeneity index ( H crit ) can be calculated based on the chi-squared distribution [ 7 , 9 ].

(Equation 2)

If H > H crit , the null hypothesis has to be rejected and signif- icant heterogeneity was detected. However, because statistical testing cannot provide a defi nitive answer, a level of signifi cance α has to be chosen before H crit is computed; this level is usually 0.05 (5%). At this level, a false rejection of the null hypothesis may occur in 5% of all tests when there is actually no detectable heterogeneity present, a type I error (false detection of hetero- geneity) shown in Figure 1 . Note that χ 2

Excel as function CHISQ.INV where the signifi cance level is stated as 1 - α , that is, 0.95 for a 5% level.

( α,N-1 ) is available in

Because the uncertainty of the homogeneity index depends on the number of measurements N , the critical homogeneity index H crit depends on N . At N = 30 measurements and α = 0.05, the critical homogeneity index is 1.21. At N = 300, it is reduced to 1.07 ( Figure 1 ). To be considered homogeneous, the determined

32

Figure 3 : Electron microprobe x-ray distribution maps by wavelength-dispersive spectrometry obtained on a polished apatite crystal investigated for its suitability as reference material. (a) Distribution of silicon (Si Kα ) shows chemical zoning within the crystal. The material is obviously heterogeneous and, in its present form, not suitable as a reference material. (b) Distribution of cerium (Ce Lα ) shows barely visible heterogeneity. In this case, statistical testing based on quantitative analyses has to be conducted.

homogeneity index must be below this critical value. T erefore, increasing the number of measurements increases the power to detect heterogeneity. Increasing N also decreases the probability of erroneously accepting the null hypothesis, the type II error (false non-detection of heterogeneity) illustrated in Figure 2 . However, this probability cannot be quantifi ed because the true heterogeneity present in the sample is not known—this is what the test is trying to infer statistically. Moreover, increasing N has practical limitations because it requires time and increases the risk that instrumental driſt may become a signifi cant contri- bution to the variations observed. T us, choosing a reasonable N is an important task. Uncertainty budget . A useful concept in homogeneity testing and helpful in choosing a reasonable N is the “uncertainty budget.” In its simplest form this relates the combined variance ( s2c )

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