Use of a replicated Latin square design in a homogeneity test for high purity organic melting point standards

Abstract

The use of a replicated Latin square design for reference material homogeneity assessment is illustrated by application to a homogeneity study of eight high-purity organic materials certified for melting point. The design controlled for both a three-level location effect and a run effect. Variance components were extracted using mixed effects modelling using a restricted maximum likelihood method. An alternative classical ANOVA calculation is also given. The effect of appreciable numerical rounding by the instrument software was investigated and shown to be acceptable for the particular example. Estimation of the scale of location and run effects showed that in this example the location effect was both statistically and practically significant, while the run effect was not statistically significant at the 95% level of confidence. The design allowed unbiased estimates of between-unit variances in the presence of both interfering effects.

Appendix: Manual calculation for a replicated Latin square design with one fixed effect

The statistical model for the replicated Latin square design used here can be written

$$ y_{lijk} = \mu + L_{l} + S_{i} + U_{ij} + R_{ik} + e_{lijk} $$

(4)

where y _lijk is the observation for CRM unit j in run k and at location l in set i, μ denotes the population mean, L _l the effect of location l on the observed melting point where S _i  ~ N(0, σ _S) is a nominal offset for set (~N(θ, σ) denoting ‘distributed as a normal distribution with mean θ and standard deviation σ’), U _ij  ~ N(0, σ _bb) the deviation from μ for unit j in set i, R _ik  ~ N(0, σ _run) the deviation of run k in set i and e _lijk  ~ N(0, σ _w) the usual residual error term. Estimates of the variance components represented by σ _w and σ _bb are desired for homogeneity assessment; the between-run and between-set variance and a test of the between-location difference is also potentially of interest.

The following calculations follow Box, Hunter and Hunter [4] with nomenclature modified for the present paper and with the addition of variance components for the random effects.

Let n _S be the number of sets and let n _L = n _U = n _R = n be the number of locations, the number of units per set and the number of runs per set. Then, the ANOVA table is constructed as follows:

Let $G = \sum\nolimits_{l,i,j,k} {y_{lijk} }$ be the sum of all n _S n ² observations. Further, let

$T_{{{\text{set}},i}} = \sum\nolimits_{l,j,k} {y_{lijk} } ,\quad i = 1, \ldots ,n_{S}$ be the vector of n _S sums of observations for each set.

$T_{{{\text{loc,}}l}} = \sum\nolimits_{i,j,k} {y_{lijk} } ,\quad l = 1, \ldots ,n_{\text{L}}$ be the vector of n _L sums of observations for each location.

$T_{{{\text{run,}}ik}} = \sum\nolimits_{l,j} {y_{lijk} } ,\quad i = 1, \ldots ,n_{\text{S}} ,\quad k = 1, \ldots ,n_{\text{R}}$ be the vector of n _S n _R sums of observations for each run within each set.

$T_{{{\text{unit,}}ij}} = \sum\nolimits_{l,k} {y_{lijk} } ,\quad i = 1, \ldots ,n_{\text{S}} ,\quad j = 1, \ldots ,n_{\text{U}}$ be the vector of n _S n _R sums of observations for each run within each set.

Then, from

$$ S_{\text{A}} = \frac{{G^{2} }}{{n_{\text{S}} n^{2} }}. $$

The analysis of variance table is then constructed as shown in Table 2, where M _set, M _unit, M _loc, M _run and M _w are the ANOVA mean squares corresponding to the set, location, run, unit and residual effects, respectively. Note that the calculation involves subtraction of some potentially large sums of squares, which can result in serious numerical inaccuracy; it is therefore prudent to code the data (that is, to subtract a convenient amount from each value to leave only the digits that show variation) before carrying out the calculation.

Table 2 ANOVA table for a replicated Latin square design

Variance components for the different random effects are extracted in the usual way from the mean squares. The estimate s ²_w of the residual variance σ ²_w estimate is equal to M _w. The estimates s ²_run and s ²_bb of the between-run and between-unit variances are given by

$$ s_{\text{run}}^{ 2} = \frac{{M_{\text{run}} - M_{\text{w}} }}{n},\,s_{\text{bb}}^{ 2} = \frac{{M_{\text{unit}} - M_{\text{w}} }}{n} $$

(5)

while the estimate s ²_S of the between-set variance is given by

$$ s_{\text{S}}^{ 2} = \frac{{M_{\text{set}} - M_{\text{run}} - M_{\text{unit}} + M_{\text{w}} }}{{n^{2} }}. $$

(6)

Note that because location is a fixed effect common to all sets, the mean square for location does not appear in Eq. 6.

Conventionally, variance estimates that fall below zero are set to zero for simplicity.