Dr Sebastian Scholl
email: sebastian.scholl@med.uni-jena.de

Dr. med.; Bärbel Spies-Weißhart

PhD Anne Klink

Lars-Olof Mügge

Hans-Joerg Fricke

Andreas Hochhaus

Baerbel Spies-Weisshart

Lars-Olof Muegge

Dr Sebastian Pd

Scholl

Secondary resistance to sorafenib in two patients with acute myeloid leukemia (AML) harbouring FLT3-ITD mutations

Keywords: AML, sorafenib, FLT3-ITD, resistance AML, sorafenib, FLT3-ITD, resistance AML, sorafenib, FLT3-ITD, resistance

Dear Editor, we report on two patients with acute myeloid leukemia (AML) initially presenting in 2007. Both patients were tested positive for a FLT3-ITD mutation located in the juxtamembrane domain and were therefore treated with the FLT3 tyrosine kinase inhibitor sorafenib (400 mg/ day) after progress of AML following several protocols of conventional chemotherapy. Both patients responded with a significant reduction of peripheral blast counts after 10 to 17 days leading to hematological response for 12 and 14 weeks, respectively. At the time of progression, molecular analysis investigating mutations of both tyrosine kinase domains by sequencing of each individual FLT3-ITD cDNA did not demonstrate any additional mutations. We therefore suggest that the secondary resistance to sorafenib is mediated by other mechanisms than the acquisition of secondary mutations of FLT3 in these patients.

Response to Reviewers: Dear Professor Ganser, thank you and the reviewer(s) very much for reviewing our Letter to the Editor entitled "Secondary resistance to sorafenib in two patients with acute myeloid leukemia (AML) harbouring FLT3-ITD mutations".

We carefully read the reviewer´s recommendations and would like to explain our changes point by point.

1.)

In patient 1, there was a new translocation t(9;18) detectable at early relapse as reported on page 3. After this, there were no further cytogenetic aberrations of AML blasts in patient 1 until progression following sorafenib treatment. In contrast, in patient 2 who initially presented with a partial monosomy 7 we could only detect leukemic blasts without chromosomal aberrations prior to start of sorafenib. These data are now introduced in the paper.

2.)

In the meanwhile, we also cloned and sequenced the genomic region containing the internal tandem duplication. The sequencing data confirm that both patients carried a FLT3-ITD located in the JM domain and not in the TKD1 region.

3.)

Unfortunately, we did not measure plasma concentrations of sorafenib or even intracellular reduction of STAT5 phosphorylation in the blasts of these patients. 4.)

The FLT3mut/ wildtype ratio was determined in patient 1 (ratio 0,729) as well as in patient 2 at diagnosis (ratio 0,694). Indeed, in patient 1 there was a secondary loss of the FLT3 wildtype allele during AML progression. These aspects are now introduced in the manuscript.

5.)

Indeed, the benefit of figure 2C is limited. We agree with the reviewer and introduced a schematic presentation of the receptor while the primer sequences are now indicated in the text. 6.)

The term "relapse" was replaced by the term "progression". 7.)

We changed the spelling between British and American English.

We would like to submit our revised version and would be happy if our manuscript could be considered for publication in Annals of Hematology.

Yours sincerely, Sebastian Scholl on behalf of the authors *Conflict of interest Click here to download Conflict of interest: Conflictofinterest_030310.doc

Dear Editor, we report on two patients with acute myeloid leukaemia (AML) initially presenting in 2007.

Both patients were tested positive for the FLT3-ITD mutation and were therefore treated with the FLT3 tyrosine kinase inhibitor sorafenib (400 mg/ day) after progress of AML following several protocols of conventional chemotherapy. Both patients responded with a significant reduction of peripheral blast counts after 10 to 17 days leading to hematological response for 12 and 14 weeks, respectively. At the time of relapse, molecular analysis investigating mutations of both tyrosine kinase domains by sequencing of each individual FLT3-ITD cDNA did not demonstrate any additional mutations. We therefore suggest that the secondary resistance to sorafenib is mediated by other mechanisms than the acquisition of secondary mutations of FLT3 in these patients.

FLT3-ITD mutations represent the second most frequent molecular aberration in AML leading to a constitutive activity of the class III receptor tyrosine kinase FLT3. FLT3-ITD mutations can be found in 25-30% of all AML patients and are associated with a reduced disease free survival and overall survival [1][2][3].

The biaryl urea sorafenib (BAY 43-9006, Bayer Schering, Leverkusen, Germany) that has been approved for the treatment of several solid tumors, e.g. renal and hepatocellular carcinoma, is able to significantly inhibit FLT3 including activating FLT3 mutations like FLT3-ITD at a nanomolar concentration [4,5]. Furthermore, in vitro analysis of several FLT3 inhibitors revealed a different spectrum of additional mutations that may confer resistance to these inhibitors [6]. Recently, several clinical studies investigating the feasibility and efficacy of sorafenib in a small cohort of patients with relapsed FLT3-ITD positive AML have been reported including a case report demonstrating even a molecular complete remission following sorafenib treatment in a patient with AML who relapsed after allogeneic stem cell transplantation [5,[7][8][9]. So far, there is no report on analyses of FLT3-ITD kinase domain sequences in patients with progress of AML after treatment with sorafenib.

Patient #1 was a 59-year-old man with AML M2 according to the French American British 1). Debulking of leukemia was associated with persistent anemia and thrombocytopenia. After 88 days of sorafenib treatment, a further progress of AML was observed demonstrated by rapidly increasing WBC counts including the reappearance of a high percentage of peripheral blasts. Later on, the patient died of a cerebral hemorrhage.

Patient #2 was a 69-year-old male patient AML (FAB M2) diagnosed in 2007. At diagnosis, AML presented with 24% bone marrow blasts and signs of dyplasia. These morphological aspects and the result of cytogenetic analysis (del7q) prompted the assumption of an antecedent myelodysplastic syndrome. The patient underwent an induction chemotherapy consisting of mitoxantrone and intermediate-dosed cytarabine resulting in a complete remission. After two courses of consolidation therapy (mitoxantrone and intermediate-dosed cytarabine) persistent complete hematological remission was demonstrated. Due to the unfavourable karyotype, preparation for an allogeneic stem cell transplantation including a reduced intensity conditioning was initiated. However, the patient unfortunately relapsed within two months after completion of the consolidation therapy. While there was no severe co-morbidity in this patient, conventional chemotherapy was continued using an age-adapted regimen containing cytarabine and cyclophosphamide but bone marrow blasts persisted. Thus, we did not pursuit a curative concept any longer and palliative cytoreductive chemotherapy was started. After failure of three consecutive cytoreductive regimens treatment with sorafenib (200 mg b.i.d.) was commenced. This patient obtained HR as shown by a continuously decreasing number of peripheral blasts while there was no objective remission in the bone marrow. In detail, WBC declined from 59/nl prior to the start of sorafenib therapy to 1.1/nl at day 16 while a significant reduction of the proportion of peripheral blasts was observed (from 67% to 5%). Importantly, this patient had a persistent transfusion dependent anemia and thrombocytopenia. Treatment with sorafenib resulted in a stable HR for 101 days.

At this time, WBC as well as peripheral blasts increased reflecting a secondary resistance to sorafenib treatment. Consecutive therapy with low dose cytarabine only led to a transient disease control. The patient died of a sepsis due to pneumonia and severe infectious colitis.

In both patients, a FLT3-ITD mutation was detected at diagnosis on the level of genomic DNA. We therefore confirmed the stability of these molecular aberrations before the treatment with sorafenib was started. Furthermore, FLT3-ITD expression was detected on cDNA level as well in both patients. At the time of progression during sorafenib therapy, two cDNA fragments were cloned containing the entire sequences of the tyrosine kinase domain (TKD) 1 and 2 of FLT3 in order to elucidate TKD mutations that might confer resistance to the tyrosine kinase inhibitor (Figure 2C). Since sequence analyses of several clones were not able to demonstrate any additional mutation within one of the FLT3-TKD´s when both patients became refractory to sorafenib, an additional methodological approach was performed. In detail, direct sequencing of cDNA fragments with primers spanning either TKD1 or 2 did also not reveal further mutations of FLT3-ITD in both patients.

The course of these two patients treated with sorafenib shows not only the response to a monotherapy with this tyrosine kinase inhibitor after failure of different regimens with conventional chemotherapy. We can in particular demonstrate that sequence analysis after cloning of each individual FLT3-ITD allele did not reveal additional mutations within the tyrosine kinase domains. In contrast to in vitro studies with inhibitors of FLT3-ITD, we can show that apart from the acquisition of TKD mutations additional mechanisms may be responsible for the secondary resistance to sorafenib in AML patients.

Such target-independent mechanisms of resistance to tyrosine kinase inhibitors have already been demonstrated in the treatment of chronic myelogenous leukemia with imatinib. These mechanisms that are not mediated by mutations of the target gene and include clonal cytogenetic aberrations, activation of growth factor receptors, and drug influx and efflux pumps [10][START_REF] White | OCT-1-mediated influx is a key determinant of the intracellular uptake of imatinib but not nilotinib (AMN107): reduced OCT-1 activity is the cause of low in vitro sensitivity to imatinib[END_REF][START_REF] Wang | Expression of the uptake drug transporter hOCT1 is an important clinical determinant of the response to imatinib in chronic myeloid leukemia[END_REF][START_REF] Rosée | Molecular pathogenesis of tyrosine kinase resistance in chronic myeloid leukemia[END_REF]. Both patients were tested positive for a FLT3-ITD mutation located in the juxtamembrane domain and were therefore treated with the FLT3 tyrosine kinase inhibitor sorafenib (400 mg/ day) after progress of AML following several protocols of conventional chemotherapy. Both patients responded with a significant reduction of peripheral blast counts after 10 to 17 days leading to hematological response for 12 and 14 weeks, respectively. At the time of progression, molecular analysis investigating mutations of both tyrosine kinase domains by sequencing of each individual FLT3-ITD cDNA did not demonstrate any additional mutations. We therefore suggest that the secondary resistance to sorafenib is mediated by other mechanisms than the acquisition of secondary mutations of FLT3 in these patients.

FLT3-ITD mutations represent the second most frequent molecular aberration in AML leading to a constitutive activity of the class III receptor tyrosine kinase FLT3. FLT3-ITD mutations can be found in 25-30% of all AML patients and are associated with a reduced disease free survival and overall survival [1][2][3].

The biaryl urea sorafenib (BAY 43-9006, Bayer Schering, Leverkusen, Germany) that has been approved for the treatment of several solid tumors, e.g. renal and hepatocellular carcinoma, is able to significantly inhibit FLT3 including activating FLT3 mutations like FLT3-ITD at a nanomolar concentration [4,5]. Furthermore, in vitro analysis of several FLT3 inhibitors revealed a different spectrum of additional mutations that may confer resistance to these inhibitors [6]. Recently, several clinical studies investigating the feasibility and efficacy of sorafenib in a small cohort of patients with relapsed FLT3-ITD positive AML have been reported including a case report demonstrating even a molecular complete remission following sorafenib treatment in a patient with AML who relapsed after allogeneic stem cell transplantation [5,[7][8][9]. So far, there is no report on analyses of FLT3-ITD kinase domain sequences in patients with progress of AML after treatment with sorafenib.

Patient #1 was a 59-year-old man with AML M2 according to the French American British Consolidation chemotherapy consisted of three courses with alternating anthracyclines (idarubicine versus mitoxantrone) combined with intermediate-dosed cytarabine due to a lack of availability of a sibling or unrelated donor at this time point. Only three month after consolidation therapy has been completed, disease reoccured not only in the bone marrow but also as meningeosis leukemica. Furthermore, karyotype analysis demonstrated a translocation t(9;18) at early relapse. This patient was subsequently treated with high-dosed cytarabine and cyclophosphamide and received an additional intrathecal therapy. After the first course of this salvage protocol a partial remission could be documented while there was a second progress of AML shortly after hematological recovery following the second course of high-dosed cytarabine and cyclophosphamide. In the absence of an appropriate donor, the patient subsequently received several lines of conventional cytoreductive chemotherapies without any evidence of disease control. In this situation, molecular analysis could still demonstrate the FLT3-ITD mutation while the wildtype allele of FLT3 was lost. Thus, treatment with sorafenib (200 mg b.i.d.) was started. The patient obtained a hematological response (HR) demonstrated by a delayed clearance of peripheral blasts while there was no objective remission in the bone marrow. In detail, WBC declined from 38/ nl prior to the commencement of sorafenib to 2.9/ nl at day 17 and 4.2/ nl at day 26. Contemporaneously, the percentage of peripheral blasts dropped-down from 66% to 14% at day 17 and 5% at day 26, respectively (Figure 1). Debulking of leukemia was associated with persistent anemia and thrombocytopenia. After 88 days of sorafenib treatment, a further progress of AML was observed demonstrated by rapidly increasing WBC counts including the reappearance of a high percentage of peripheral blasts. Later on, the patient died of a cerebral hemorrhage.

Patient #2 was a 69-year-old male patient with FLT3-ITD-positive (allelic ratio FLT3 preparation for an allogeneic stem cell transplantation including a reduced intensity conditioning was initiated. However, the patient unfortunately relapsed within two months after completion of the consolidation therapy. While there was no severe co-morbidity in this patient, conventional chemotherapy was continued using an age-adapted regimen containing cytarabine and cyclophosphamide but bone marrow blasts persisted. Thus, we did not pursuit a curative concept any longer and palliative cytoreductive chemotherapy was started.

Interestingly, leukemic blasts demonstrated a normal karyotype at relapse and in all further cytogenetic analyses. After failure of three consecutive cytoreductive regimens treatment with sorafenib (200 mg b.i.d.) was commenced. This patient obtained HR as shown by a continuously decreasing number of peripheral blasts while there was no objective remission in the bone marrow. In detail, WBC declined from 59/nl prior to the start of sorafenib therapy to 1.1/nl at day 16 while a significant reduction of the proportion of peripheral blasts was observed (from 67% to 5%). Importantly, this patient had a persistent transfusion dependent anemia and thrombocytopenia. Treatment with sorafenib resulted in a stable HR for 101 days.

At this time, WBC as well as peripheral blasts increased reflecting a secondary resistance to sorafenib treatment. Consecutive therapy with low dose cytarabine only led to a transient disease control. The patient died of a sepsis due to pneumonia and severe infectious colitis.

In both patients, a FLT3-ITD mutation was detected at diagnosis on the level of genomic DNA. We therefore confirmed the stability of these molecular aberrations before the treatment with sorafenib was started. Furthermore, FLT3-ITD expression was detected on cDNA level as well in both patients. At the time of progression during sorafenib therapy, two cDNA fragments were cloned containing the entire sequences of the tyrosine kinase domain (TKD) 1 and 2 of FLT3 in order to elucidate TKD mutations that might confer resistance to the tyrosine kinase inhibitor (Figure 2C). Since sequence analyses of several clones were not able to demonstrate any additional mutation within one of the FLT3-TKD´s when both patients became refractory to sorafenib, an additional methodological approach was performed. In detail, direct sequencing of cDNA fragments with primers spanning either TKD1 or 2 did also not reveal further mutations of FLT3-ITD in both patients.

The course of these two patients treated with sorafenib shows not only the response to a monotherapy with this tyrosine kinase inhibitor after failure of different regimens with conventional chemotherapy. We can in particular demonstrate that sequence analysis after cloning of each individual FLT3-ITD allele did not reveal additional mutations within the tyrosine kinase domains. In contrast to in vitro studies with inhibitors of FLT3-ITD, we can show that apart from the acquisition of TKD mutations additional mechanisms may be responsible for the secondary resistance to sorafenib in AML patients.

Such target-independent mechanisms of resistance to tyrosine kinase inhibitors have already been demonstrated in the treatment of chronic myelogenous leukemia with imatinib. These mechanisms that are not mediated by mutations of the target gene and include clonal cytogenetic aberrations, activation of growth factor receptors, and drug influx and efflux pumps [10][START_REF] White | OCT-1-mediated influx is a key determinant of the intracellular uptake of imatinib but not nilotinib (AMN107): reduced OCT-1 activity is the cause of low in vitro sensitivity to imatinib[END_REF][START_REF] Wang | Expression of the uptake drug transporter hOCT1 is an important clinical determinant of the response to imatinib in chronic myeloid leukemia[END_REF][START_REF] Rosée | Molecular pathogenesis of tyrosine kinase resistance in chronic myeloid leukemia[END_REF]. To our knowledge, there are no data about such mechanisms in AML patiens who were treated with a FLT3 inhibitor. Sequence analysis as demonstrated in our patients let us hypothesise that secondary resistance to kinase inhibitors like sorafenib might also be mediated by such target-independent mechanisms in FLT3-ITD positive AML.

Figure 1 Time course of peripheral blood counts during sorafenib treatment in patient 1 and 2 .Figure 2

 122 Figure 1 Time course of peripheral blood counts during sorafenib treatment in patient 1 and 2.WBC (/nl), leukemic blasts (%), platelets (/nl), hemoglobin (mmol/l)

Figure 2 1801

 2 Figure 1

(

  FAB) classification presenting with a hyperleukocytosis at primary diagnosis. Cytogenetic and molecular genetic analyses revealed a normal karyotype and the presence of a FLT3-ITD as well as a NPM1 mutation. At initial diagnosis, the allelic ratio FLT3 mutated/ wildtype was 0.729. The patient received an induction chemotherapy consisting of idarubicine and intermediate-dosed cytarabine resulting in a complete hematologic remission of AML.

  mutated/ wildtype 0.694) AML (FAB M2) diagnosed in 2007. At diagnosis, AML presented with 24% bone marrow blasts and signs of dyplasia. These morphological aspects and the result of cytogenetic analysis (del7q) prompted the assumption of an antecedent myelodysplastic syndrome. The patient underwent an induction chemotherapy consisting of mitoxantrone and intermediate-dosed cytarabine resulting in a complete remission. After two courses of consolidation therapy (mitoxantrone and intermediate-dosed cytarabine) persistent complete hematological remission was demonstrated. Due to the unfavourable karyotype,

Figure 1 Time course of peripheral blood counts during sorafenib treatment in patient 1 and 2 .Figure 2

 122 Figure 1 Time course of peripheral blood counts during sorafenib treatment in patient 1 and 2.WBC (/nl), leukemic blasts (%), platelets (/nl), hemoglobin (mmol/l)

  Figure 1

  Figure 2

  secondary resistance to kinase inhibitors like sorafenib might also be mediated by such target-independent mechanisms in FLT3-ITD positive AML. 11. White DL, Saunders VA, Dang P, Engler J, Zennettino AC, Cambareri AC, et al. (2006)OCT-1-mediated influx is a key determinant of the intracellular uptake of imatinib but not nilotinib (AMN107): reduced OCT-1 activity is the cause of low in vitro sensitivity to

	imatinib. Blood 108:697-704
	12. Wang L, Giannoudis A, Lane S, Williamson P, Pirmohamed M, Clark RE (2008)
	Expression of the uptake drug transporter hOCT1 is an important clinical determinant of
	the response to imatinib in chronic myeloid leukemia. Clin Pharmacol Ther 83:258-264
	13. La Rosée P, Hochhaus A (2010) Molecular pathogenesis of tyrosine kinase resistance in
	chronic myeloid leukemia. Curr Opin Hematol 17:91-96

To our knowledge, there are no data about such mechanisms in AML patiens who were treated with a FLT3 inhibitor. Sequence analysis as demonstrated in our patients let us hypothesise that


Edwin R Van Den Heuvel

A COMPARISON OF ESTIMATION METHODS ON THE COVERAGE PROBABILITY OF SATTERTHWAITE CONFIDENCE INTERVALS FOR ASSAY PRECISION WITH UNBALANCED DATA

Keywords: Henderson III Moment Estimators, (Restricted) Maximum Likelihood Estimators, Variance Components, Unbalanced Data, Satterthwaite Confidence Intervals, Assay Precision Henderson III Moment Estimators, Likelihood Estimators, Measures of Precision, Satterthwaite Confidence Intervals, Unbalanced data, Variance Components. 1 Dutch (Netherlands

Construction of closed-form confidence intervals on linear combinations of variance components were developed generically for balanced data and studied for simple analysis of variance models. Satterthwaite approach is generalized to unbalanced data and modified to increase its coverage probability. They are applied on measures of precision in combination with (restricted) maximum likelihood and Henderson III Type 1 & 3 estimation. Simulation studies do not show superiority of any of the possible combinations of estimation methods and Satterthwaite approaches on three precision measures. However, the modified Satterthwaite approach with Henderson III Type 3 estimation is often preferred above the other combinations.

(bio)chemical assays to simulate the assay variation within and between laboratories for routine use of these assays. Studies like this focus on the measures of precision "repeatability", "intermediate precision" and "reproducibility", see for instance the international guideline ICH (1996). In statistical terms, measures of precision are defined by specific linear combinations of variance components which represent the random effects of (combinations of) assay sources of variation. The variance components are generally defined through the effects of sources of variation incorporated in the mixed or random effects analysis of variance models used to analyze the experimental data. The linear combinations are often the sum of individual variance components.

The choice of estimation method for the individual variance components in unbalanced designs has been debated in the statistical literature for a long time. [START_REF] Henderson | Estimation of variance and covariance components[END_REF] provided three methods which are based on the moments of particular sets of sums of squares. The moment estimators are not unique because there are different ways of partitioning the sums of squares of sources of variation in unbalanced designs. Henderson's third method is most generally applicable, since it can be applied to any random or mixed effects analysis of variance model. But also for this method there exist different sets of sums of squares. The order of including terms for the sources of variation in the analysis of variance model determines the different sets of sums of squares, see Searle, Casella and McCulloch (1992) and [START_REF] Littell | SAS ® for mixed models[END_REF]. All these choices reduce to the same unique set of sums of squares for balanced designs, which then posseses nice properties. Indeed, the sums of squares are mutually independently distributed and they are distributed as multiples of chi-square random variables. In unbalanced designs these properties no longer hold in general.

Restricted maximum likelihood (REML) estimation is often preferred above other methods of estimation for unbalanced data, see [START_REF] Mcculloch | Generalized, Linear, and Mixed Models[END_REF]. REML estimation provides an unique set of non-negative variance component estimates, but it does require the (strong) assumption of normality. The argument for REML estimation is for instance supported by Swallow and Monahan (1984). Their simulation study for the one-way (balanced) random effects analysis of variance model Whatever estimation method is applied, it is good statistical practice to construct appropriate confidence intervals on assay precision. [START_REF] Burdick | Confidence Intervals on Variance Components[END_REF] give a nice overview of several generic approaches, but the closed-form approaches are essentially developed for balanced analysis of variance models because they require the nice properties of the sums of squares in balanced designs. Not much attention has yet been given to confidence intervals for measures of precision for unbalanced data, except though for the one-way random effects analysis of variance model, see [START_REF] Li | Confidence intervals on sum of variance components with unbalanced designs[END_REF] and [START_REF] Burdick | The present status of confidence interval estimation for one-factor random models[END_REF], because then the mathematical complexity is limited. However, the Satterthwaite approach, described by for instance Satterthwaite (1946), Welch (1956) or [START_REF] Burdick | Confidence Intervals on Variance Components[END_REF], can easily be adapted to general unbalanced designs and it can also be applied to any method of variance component estimation. This original approach can then also be modified to increase its coverage probability and make it possibly conservative, because the original Satterthwaite approach is known to lead to too small coverage probabilities in some cases.

The goal of this article is to study coverage probabilities of the original and modified Satterthwaite approach to confidence intervals on measures of assay precision for unbalanced data using maximum likelihood, restricted maximum likelihood, Henderson III Type1 and Henderson III Type 3 variance component estimators. Simulation studies are conducted to investigate three measures of precision for a particular three-way classification random effects analysis of variance model. This model is often applied to data of interlaboratory studies, see [START_REF] Nijhuis | Closed-form confidence intervals on measures of precision for an interlaboratory study[END_REF].

Emphasize is given to experiments with small sample sizes and to situations were individual variance components may be close to zero. A detailed description of the original and modified Satterhwaite approach and a detailed description of the four estimation methods are provided too.

GENERAL THEORY

∑ ∑ = = + + = m j j j n i i i V U X y 1 1 ε γ β
, with y the vector of N observations, X an ( p N × ) matrix of known constants of rank p , β a vector of p unknown fixed parameters, i U an ( i q N × ) matrix of known constants, i γ a random vector of i q independent normally distributed "assay unrelated" variables with mean 0 and variance

2 1 + - + i n m σ , j
V an ( j r N × ) matrix of known constants, j ε a random vector of j r independent normally distributed "assay related" variables with mean 0 and variance

2 1 + -j m σ
, and all random vectors mutually independently distributed.

The variance components for the "assay related" group are now defined by

2 1 σ , 2 2 σ ,….., 2 m σ , with 2 1
σ the "residual" variance component, representing the last term in model ( 1). The n variance components for the "assay unrelated" group are indexed by

2 1 + m σ , 2 2 + m σ ,….., 2 n m+
σ . The order in the indices for the variance components is directly related to the inverse order in the random effects in model ( 1). For all or most assay precision studies, the contribution in the variation of the observations y is firstly determined by "assay unrelated" sources of variation and secondly by "assay related" sources of variation. Indeed, differences between for instance batches will enter the response before it is measured and then it will be (possibly) increased when it is observed with the assay. Within the "assay related" group, a similar reasoning and ordening can often be made, but within the "assay unrelated" group this is less relevant since we are interested in the estimation of the assay precision.

The "assay related" group may contain random effects that represent interaction effects between "assay related" sources of variation and "assay unrelated" sources of variation, since they are considered to represent assay variation. For instance, the interaction effect between batches and laboratories is considered part of the assay variation, because it will be assumed that batches are homogeneous and their mean levels are constant during the time period of the study. (2)

∑ = = 0 0 1 2 2 m q q q m c σ γ , 0 ≥ q c .
The coefficients in (2) are considered non-negative since we are interested in (part of) the total contribution of assay variation. Moreover, the coefficients are often equal to one, but other choices may be applicable in particular when a "reportable value" is some kind of average from replicated results. For repeatability, 0 m is usually one, while for intermediate precision 0 m is selected in such a way that all sources of variation contributing to the within laboratory variation (intermediate precision)

are included in (2). For reproducibility 0 m is equal to m , including all assay related sources of variation. Thus the different measures of assay precision are strictly ordered in size.

HENDERSON III MOMENT ESTIMATORS

For the terms contained in the analysis of variance model a set of "mean squares" may be defined, see [START_REF] Henderson | Estimation of variance and covariance components[END_REF]

( ) ∑ = = = k q q q k k k a M 1 2 , E σ θ .
Here E is the symbol for expectation and 1

1 , = k a , 0 2 , ≥ k a ,….., 0 1 , ≥ - k k a , 0 , > k k
a are known constants related to the matrices for the random effects in model (1). Due to the order or hierarchy of the terms in the model, the variance components Based on the expected mean squares in (3), the variance components can be expressed as linear combinations of the expected mean squares, i.e.

2 1 + k σ , 2 2 + k σ ,…..
(4) σ is obtained by substituting q M for the expected mean square ( )

( ) ∑ = = k q q k,q k M a 1 2 E σ , with 1 , ~k a , 2 , ~k a ,….., k k a , ~ known functions of the constants 1 , 1 a , 1 , 2 a , 2 , 2 a , 1 , 3 a ,…, 3 , 3 a ,…..,
q M E in (4).
Substituting the standard moment estimators for the variance components in (2) and using relation ( 4) we obtain the moment estimator 2 0 ˆm γ for the measure of precision in (2), i.e.

(5)

∑ ∑ = = = ≡ 0 0 0 1 1 2 2 m q q q m q q q m M c c σ γ , where 1 c , 2 c ,….., 0 ~m c are known functions of the constants 1 c , 2 c ,….., 0 m c and 1 , 1 ã , 1 , 2 ã , 2 , 2 ã , 1 , 3 ã ,…, 3 , 3 ã ,….., 1 , 0 ~m a ,…,
( ) ( ) ( ) ∑ ∑ ∑ - = + = = + = - = 1 1 1 2 2 1 2 2 2 2 2 2 ˆ0 0 0 0 0 2 0 , ĉov 2 var ˆmq m q r r q r q m q q q m m c c c E m σ σ σ γ γ σ γ .
which in case the experimental design is balanced reduces to

∑ = ⋅ = 0 2 0 1 2 2 2 ˆ2 m q q q q n c m θ σ γ .
This follows from the Chi-square distribution related to the mean squares It is well known that, except for the residual variance, the standard moment estimates for individual variance components can become negative for both balanced and unbalanced designs, see for instance Searle et al. (1992) and [START_REF] Mcculloch | Generalized, Linear, and Mixed Models[END_REF]. Although the standard moment estimators are unbiased, they have the disadvantage that different and specifically ordered measures of precision may become incorrectly ordered when the standard moment estimators are used, due to the negative variance component estimates for individual variance components. For example, the estimator in (5) may be less than the standard moment estimator 1 M for the residual variance 2 1 σ . [START_REF] Nijhuis | Closed-form confidence intervals on measures of precision for an interlaboratory study[END_REF] proposed to use the adjusted or truncated moment estimators to maintain a proper order in the estimates for the measures of precision.

The adjusted moment estimators are defined by the standard moment estimators truncated to nonnegative variance components estimators. This method of estimation is similar to the (but not exactly the same as) restricted maximum likelihood estimation method, see Subsection 2.2. The estimator for the measure of precision when the adjusted moment estimators are applied becomes x is an element of set A and zero elsewhere. The adjusted moment estimators will be used for the estimation of measures of precision here as well.

(7) [ ) ( ) ∑ = ∞ ⋅ ≡ 0 0 1 2 , 0 2 2 1 m q q q q m c σ σ γ
Henderson III moment estimators are not unique. Different ways of positioning the sums of squares are possible. The positioning has much to do with the order in which the statistical terms enter the statistical model (1), see for instance Searle et al. (1992). Although general rules may not be given, for assay precision studies the order of terms in the model is not at all complicated. Recall that the contribution of variation into the observations is induced by the "assay unrelated" sources of variation first and then followed by the "assay related" sources of variation. More specific, the order of terms in the model is usually related to the order in time in which variation may have entered the response. This Henderson III estimation method is referred to the Type 1 estimation method, see Searle et al.

(1992) and [START_REF] Littell | SAS ® for mixed models[END_REF]. Another way is the Type 3 method, which calculates the additional sum of squares of each term when it would be added to the model as the last term (but remaining the is not relevant at all, see [START_REF] Littell | SAS ® for mixed models[END_REF]. It should be clear that the residual variance component estimator 2 1 σ is the same for both Type 1 and Type 3 estimation method. It is also the only variance component estimator which always follows a Chi-square distribution (when appropriately normalized).

It is expected that the Type 3 estimation method is less sensitive to unbalanced data than the Type 1 method, since it is less susceptive to extreme results, in particular for small sample sizes.

MAXIMUM LIKELIHOOD BASED ESTIMATORS

Maximum likelihood estimation for random and mixed effects analysis of variance models is an estimation method that determines the fixed and random model parameters by maximizing the likelihood function for the selected linear model. A disadvantage of maximum likelihood estimation is that the variance component estimators are biased, see [START_REF] Mcculloch | Generalized, Linear, and Mixed Models[END_REF]. The restricted maximum likelihood (REML) estimation method for estimating variance components in random and mixed effects analysis of variance models is the maximum likelihood estimation method applied to a linear transformation of the observations which do not contain any of the fixed effects anymore (expectation zero), see page 176 of [START_REF] Mcculloch | Generalized, Linear, and Mixed Models[END_REF].

For balanced data sets and non-negative standard moment estimates for variance components, the REML estimates are equal to the standard moment estimates, see page 177 of [START_REF] Mcculloch | Generalized, Linear, and Mixed Models[END_REF]. REML estimates and maximum likelihood estimators though are always non-negative even if (some of) the standard moment estimates are negative. The non-negative estimates are essentially obtained by removing or dropping the appropriate terms from the random or mixed effects analysis of variance model, see Searle et al. (1992), page 235. This implies that the restricted maximum likelihood estimators are indeed similar to the adjusted moment estimators, with the small distinction that the adjusted moment estimation method does not drop any terms from the model.

Dropping terms from the model is in our opinion a (philosophical) disadvantage of the (restricted) maximum likelihood estimation method, in particular if small sample sizes are used. Indeed, for small sample sizes relevant contributions of variation sources may not necessarily be positively estimated since they could be less in size with respect to the contribution of other sources of variation. Eliminating this relevant source of variation on the basis of an observed negative variance component estimate is then actually incorrectly allocating the effect of this source of variation to other sources of variation. Thus a possible consequence of the (restricted) maximum likelihood estimation method is that not only the "truncated" variance component estimators are biased (due to the truncation itself), but also several other variance component estimators which has incorporated the variation of the eliminated sources of variation. This would be the case for both balanced as unbalanced designs. On the other hand, this disadvantage does not imply that the (restricted) maximum likelihood estimation method does not work very well in practice.

Using the likelihood equations (cf.

∑ = = 0 0 1 2 2 m q q q m s c γ& .
The first moment for the (restricted) maximum likelihood estimator in ( 8) is not necessarily an unbiased estimator for the true measure of precision. However, asymptotically the (restricted) maximum likelihood estimator in (8) will be unbiased when sample sizes converge to infinity in an appropriate way. The variance of the estimator in ( 8) is of the same form as (6), i.e.

(9)

( ) ( ) ( ) ∑ ∑ ∑ - = + = = + = - = 1 1 1 2 2 1 2 2 2 2 2 2 0 0 0 0 0 2 0 , cov 2 var m q m q r r q r q m q q q m m s s c c s c E m γ γ σ γ & &
The variances and covariances in (9) are obtained by asymptotics using Fisher's Information matrix or the inverse Hessian matrix, which is defined by the second derivatives of the likelihood function with respect to the model parameters, see Searle et al. (1992) and [START_REF] Littell | SAS ® for mixed models[END_REF].

SATTERTHWAITE APPROACH

The approach of Satterthwaite ( 1946) is quite simple and is based on an approximation of the estimator as a Chi-square variable. Let T τ . This approximation can be applied to any estimator and not just to estimators for the measures of precision, but the approximation seems to make sense when the estimator is somewhat related to a Chi-square distributed variable as for measures of precision.

Confidence intervals are now constructed using the standard theory for Chi-square distributed random variables. A ( α -1

)100% confidence interval for the measure of precision is defined by In the literature, the intuitive interpretation of the number of degrees of freedom has never been connected with Satterthwaite approach, because Satterthwaite approach has only been described as a way of approximating the variance estimator at hand, whatever the outcome of the number of degrees of freedom. But the number of degrees of freedom also indicates the available amount of The literature on Satterthwaite approximation has described its performance and it does not always provide good coverage probabilities for the estimation of linear combinations of variance components, see [START_REF] Burdick | Confidence Intervals on Variance Components[END_REF]. The performance becomes less when one or more of the coefficients 1 c , 2 c ,….., 0 m c are negative. However, for positive coefficients, the performance is diminished as well when sample sizes and variance components vary substantially in size. Some researchers have tried to improve the Satterthwaite approximation. [START_REF] Ames | On estimating approximate degrees of freedom[END_REF] tried to correct the bias in the estimator for the number of degrees of freedom by changing the estimator in (10). They indeed improved the estimator, but their solution was developed for the situation that the estimator in ( 7) contains only two mean squares (

(11) ( ) ( ) [ ] 2 ; 2 1 ˆ2 2 ˆ0 0 α χ α χ - - ⋅ - ⋅ S S f d m S f d m S T f d T f d with ( )
2 0 = m
). A generalization of this work seems not straightforward. Zou and McDermott (1999) extended the Satterthwaite approach by including higher order moments of the Chi-square distribution to obtain the best number of degrees of freedom. Their simulation study shows that the coverage probability becomes conservative, which may be positive for some situations.

However, the complexity to unbalanced experimental designs will become too high for practical considerations.

The main disadvantage of Satterthwaite's approximation is in our view the use of the "estimate" for the number of degrees of freedom. We believe that the method would perform better when the number of degrees of freedom does not have to be estimated. Practically, this is of course impossible, but it indicates that the variation from the estimator may have a negative effect on the performance of the coverage probability. Thus limiting the variation of the estimator, by using the bounds 1 and or even tighter constraints may certainly be helpful in increasing the performance of Satterthwaite's approximation.

Henderson III Moment Estimators

For the construction of confidence intervals for the measure of precision in (2) using Henderson III moment estimators, the estimator 0 m T in ( 10) and ( 11) should first be replaced by the adjusted moment estimator 2 0 ~m γ from (7). The first and second moments of this estimator can be established for balanced designs but they become too complicated for unbalanced designs, due to the loss of nice properties of the mean squares. Therefore, we will approximate the standard error by

(12) ( ) ( ) [ ) ( ) ( ) [ ) ( ) [ ) ( ) ∑ ∑ ∑ - = + = ∞ ∞ = ∞ + = 1 1 1 2 , 0 2 , 0 2 2 1 2 , 0 2 2 2 ~0 0 0 2 0 1 1 , ĉov 2 1 var 1 m q m q r r q r q r q m q q q q c c c m σ σ σ σ σ σ σ γ
, with 2 ˆq σ the standard moment estimator for the variance component 2 q σ . The variances and covariances in ( 12) can be estimated, see Searle et al. (1992). The estimates will be denoted by

r q, ν with ( ) 2 , var q q q σ ν = and ( ) 2 2 , , ĉov r q r q σ σ ν =
, when r q ≠ . The estimator for the standard error of the moment estimator ν in (6). The constraints though were still the same because it is a property of the Satterthwaite approximation for balanced designs. It is expected that the estimated number of degrees of freedom in (13) would be lower than or equal to the choice given by Nijhuis and [START_REF] Nijhuis | Closed-form confidence intervals on measures of precision for an interlaboratory study[END_REF]. Hence, the approach here will result usually in larger or equal coverage probabilities. We consider this as beneficial, because the literature reports that one of the weaknesses of Satterthwaite approach is that the confidence intervals produces coverage probabilities which are too low in some cases, e.g. [START_REF] Burdick | Confidence Intervals on Variance Components[END_REF].

To accommodate this even further, one could argue that the standard error for the moment estimator 2 0 ~m γ in ( 7) is better approximated by only the first sum in (12), because this term usually dominates the standard error and because the covariances in ( 12) are expected to be negative as well. Thus an alternative or modified approximate standard error can be defined by ( 14)

( ) ( ) [ ) ( ) ∑ = ∞ = 0 2 0 1 2 , 0 2 2 2 ~1 var 2 m q q q q c m σ σ σ γ .
Still using the same constraints as in (13), the modified Satterthwaite approach provides the following number of degrees of freedom (15)

( ) { } ( ) { } { } 2 2 , min , , , , min max 2 ˆ2 4 2 1 2 1 0 0 0 0 m m m m M df df df df df df f d γ σ γ ⋅ + + = L K , with ( ) 2 ˆ2 ~20 m γ σ the estimator for ( ) 2 2 ~20 m γ σ in (14), by substituting q q, ν for ( ) 2 var q σ in (14).
It should be noted that for repeatability ( 1 0 = m ), the lower and upper constraint on the number of degrees of freedom become equal to 1 df , the number of degrees of freedom for the residual variance.

Thus the selected Satterthwaite approximation with ( 13) and ( 15) both become exact in case the estimator is a true mean square, such as for the residual variance component. For the Henderson III moment estimators there are now four ways of constructing confidence intervals for the measure of precision in (2). The two estimation methods Type 1 and Type 3 will be combined with both the original and modified Satterthwaite approach for constructing standard errors of the estimator of the measure of precision, see the number of degrees of freedom in ( 13) and (15).

Maximum Likelihood Based Estimators

The variances and covariances in (9) can be estimated based on the inverse of the Hessian of the likelihood function or the Fisher Information matrix, see Searle et al. (1992). If these estimators are (also) denoted by

r q, ν , with ( ) 2 , var q q q s = ν and ( ) 2 2 ,
, cov r q r q s s = ν when r q ≠ , then the estimator for the standard error in (9) for the (restricted) maximum likelihood estimator

2 0 m γ& in (8) is then ( ) 1 2 2 0 m s γ& ,
which is obtained by substituting the estimators r q, ν into (9), i.e. ( 16)

( ) ∑ ∑ ∑ - = + = = + = 1 1 1 , 1 , 2 2 0 0 0 2 0 2 1 m q m q r r q r q m q q q q c c c s m ν ν γ& .
The confidence interval for the measure of precision with the (restricted) maximum likelihood estimator is now obtained by ( 11), with

( ) ( ) { } { } 1 2 , 1 min , 1 max 1 ˆ2 4 0 0 m s N f d m L γ γ & & ⋅ - = .
Similar to the reasoning for the Henderson III moment estimators, the dominating term in ( 16) would be the first sum alone. The confidence interval would then be constructed with the modified number of degrees of freedom

(18) ( ) ( ) { } { } 2 2 , 1 min , 1 max 2 ˆ2 4 2 0 0 m s N f d m L γ γ & & ⋅ - = with ( ) 2 2 2 0 m s γ&
given by the modified estimator ( )

0 0 0 2 0 , 2 2 , 2 2 2 1 , 1 2 1 2 2 m m m c c c s m ν ν ν γ + + + = L & .
For the likelihood based estimators, the constraints 1 and Indeed, due to the possibility of dropping terms from the model for variance components which are estimated at zero, changes the "number of degrees of freedom" for the estimation of other variance components. This would also hold for repeatability, i.e. 1 0 = m . Another reason is that even the residual variance component estimator, under the assumption that the "full" analysis of variance model would represent the data, does not follow a chi-square distribution in general and the expectation of the estimator may include variance components which are unrelated to the residual variance component. Although it may be possible to narrow the constraints in ( 17) and ( 18) with the use of the "number of levels" for each term in the model, we have chosen to use only the boundaries in ( 17) and ( 18), which would always be correct in all situations (as described in the beginning of this subsection).

For the likelihood based estimation method there are also four ways to construct confidence intervals for the measure of precision in (2). Both the maximum likelihood and the restricted maximum likelihood method may be used with the original and modified Satterthwaite approach for the standard error of the estimator of the measure of precision, see the number of degrees of freedom in ( 17) and ( 18).

SIMULATION STUDY

A simulation study was conducted using SAS ® software package to investigate the performance of the estimation methods with the original and modified Satterthwaite approach to construct confidence intervals on the measures of assay precision with unbalanced data. The performance measure for the simulation study is the coverage probability for 95% confidence intervals. In line with [START_REF] Jennings | How Do We Judge Confidence-Interval Adequacy[END_REF] the left tail and right tail probabilities are investigated too, to determine the symmetry or bias of the confidence intervals. Of particular interest is the performance of these methods in small but frequently applied interlaboratory studies.

The simulated statistical model is a three-way random effects analysis of variance model which was Five "practical cases" of parameter choices for the variance components of the statistical model are used in the simulation study and they are presented in Table 1. Table 1 also presents the true measures of precision. The variance component for batch was always kept equal to 2 and the Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 6 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 6 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 6 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 3 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 3 pt residual variance is omitted in the table because it is represented by repeatability. Each case represents a possible situation in practice. Case E represents a method with a dominating repeatability. Case D has a dominating lab-to-lab variation. Case B has a dominating run-to-run variation, which could be caused by analysts. Case C represents a dominating interaction between batch and run, indicating that the assay is sensitive to the batches. This case is sometimes observed for biological assays. Case A has all of these effects without being one dominant.

( ) ( ) ( ) ( ) hijk i hj i j hi i h hijk y ε αδ δ αβ β α µ + + + + + + = , with = h 1,2,…, H , = i 1,2,…, I , = j 1,2,…,

< Table 1 >

The selected sample sizes or experimental designs for the different variation sources used in the simulation study are presented in Table 2. These sample sizes were all combined with the practical cases from Table 1. The last column in Table 2 indicates the total number of test samples in the study to give an impression of the size of the intralaboratory experiment.

< Table 2 >

Unbalanced data in the simulation study is implemented by introducing missing data to the complete statistical model in ( 19) that has been simulated first. The selected mechanism is "completely missing at random" (cf. Little and Rubin (2003)) at individual results. The missing data is generated with a Bernoulli random variable indicating which result is omitted from the full study. The selected probabilities for introducing missing data were set at 0, 0.1 and 0.2. For each selected probability all combinations of sample sizes in Table 2 with all practical cases in Table 1 were considered. The number of simulation runs for a particular set of input parameter was selected at 10000, i.e. the number of times model ( 19) is generated. This would result in a standard error of approximately 0.0022 (or 0.22%) for the targeted coverage probability of the simulated confidence intervals.

The coverage probabilities from the simulation study are summarized in Tables 3 to 9. Instead of presenting huge tables of coverage probabilities, only the ranges of coverage probabilities for the different sample sizes or experimental designs are given. Tables 3 and4 present the results for intermediate precision and reproducibility for balanced data and Tables 5 and6 present the same information for unbalanced data with 20% missing data. Each table presents the four estimation methods applied with the original (OS) and modified (MS) Satterthwaite approach and the five different practical cases (A to E). Repeatability is not presented, because the coverage probabilities were very close to the target of 95%. Tables 7,8, and 9 present the ranges on the left and right tail probabilities when 10% missing data is generated for repeatability, intermediate precision and reproducibility, respectively. Only the modified Satterthwaite approach is presented, but again the four estimation methods and the five practical cases are included.

< Table 3 > < Table 4 > < Table 5 > < Table 6 > < Table 7 > < Table 8 > < Table 9 >

CONCLUSIONS

A few generic rules were observed from the simulation study. The original and modified Satterthwaite approaches are generally reasonable close to the target coverage probability of 95%, despite the small sample sizes and the complexity of the structure of the data. The modified Satterthwaite approach showed higher coverage probabilities than the original Satterthwaite approach and in some cases even substantially higher coverages are observed. The maximum likelihood estimation method seems to produce a little bit lower coverage probabilities than the restricted maximum likelihood method. The Henderson III Type 3 method has somewhat larger coverage probabilities than the Deleted: , Deleted: The modified approach showed higher coverage probabilities than the original Satterthwaite approach, which were in some case even substantially higher. 19 methods were asymmetrical, the right tail would generally give lower probabilities than the left tail, making the upper confidence limit conservative, except for some particular simulation cases.

For balanced data two sets of simulation cases clearly stand out, namely practical case B for intermediate precision and practical case D for reproducibility. The ML and REML estimation methods do not seem to perform very well, not even when the modified Satterthwaite approach is used, although the modified approach does increase the coverage probabilities with respect to the original Satterthwaite approach. The modified Satterthwaite approach for the Henderson III Type 1 & 3 estimation methods do seem to give more appropriate coverage probabilities for these practical cases, although the modified Satterthwaite approach shows too high coverage probabilities for the other simulation cases in general with all estimation methods.

For the intermediate precision in simulation case B and for reproducibility in simulation case D, the likelihood based estimation methods with the modified Satterthwaite approach improve in coverage probability when missing data occurs. Contrary to this, Henderson III Type 3 estimation method reduces in coverage probability for these simulation cases and the coverage probabilities become similar to the coverage probabilities for the REML estimation method when both methods use the modified Satterthwaite approach. For the original Satterthwaite approach though, Henderson III Type 3 estimation method remains better than REML estimation method for these two sets of simulation cases B and D for unbalanced data.

It is quite difficult to make a firm choice between the two Satterthwaite approaches or between any of the estimation methods. However, REML is preferred above ML estimation and Henderson III Type 3 is preferred above Henderson III Type 1 estimation. Thus either REML or Henderson III Type 3 should be selected. We believe that the simulation results give a small preference to Henderson III Type 3 estimation method using the modified Satterthwaite approach. For particular simulation cases it provides similar or better results as any other combination of estimation method with one of the two Satterthwaite approaches and it provides in general higher coverage probabilities then aimed for, which makes it a good but conservative method to construct confidence intervals. Furthermore, it provides somewhat more symmetrical coverage probabilities than the maximum likelihood based Table 4: Ranges of coverage probabilities for reproducibility for interlaboratory studies with balanced designs.

CI-Method

Case A Case B Case C Case D Case E OS 92. Deleted: in

  frequently used to estimate measures of assay precision, which describe the variation induced by the measurement or assay processes. For example, method transfer studies studies) are performed regularly in pharmaceutical industry for biological and

  likelihood based estimators have in general similar or smaller biases and mean squared errors than the moment estimators.

  assay precision is now defined by the linear combination of variance components for the first 0 m random effects in the "assay related" group, i.e.

  but this is not necessarily true for unbalanced situations.The first moment of the estimator in (5) is of course the true measure of precision in (2), since the estimator is based on the first moments of the mean squares. The variance of the estimator in (5

  terms of the full model). For this type of estimation the order of terms in model (1)

Formatted

0 mT

 0 be an arbitrary estimator for the measure of precision in (2) and let 0 m T τ be its corresponding standard error. The Satterthwaite approach approximates the Deleted: Then altering the linear model due toDeleted: e variation to Deleted: s Deleted: with relative more substantial Deleted: from other not related Deleted: based on the X a Chi-square distributed variable, and with S df the number of degrees of freedom defined by number of degrees of freedom makes the first two moments of 2). Since the number of degrees of freedom is unknown it should be estimated before it can be used in practice. The usual estimator is τ is an estimator for the standard error 0 m

  p th percentile of the Chi-square distribution with d degrees of freedom. The approximate confidence interval in (11) works best when the tails of the distributions of

  to estimate a variance component from observations of a normal distribution. Thus for a data set with N observations, the maximum number of degrees of freedom would be 1 -N. Obviously, the minimum number of degrees of freedom to estimate a variance component would be at least equal to one since it takes at least two observations to estimate a variance component. Thus a consequence of the intuitive interpretation is that the number of degrees of freedom in (10) should be bounded from below by (at least) 1 and from above with (at most)

.

  This standard error reduces to the usual form in Satterthwaite (1946) for balanced designs, when all variance component estimates are nonnumber of degrees of freedom" corresponding to the mean square k M . These bounds indeed indicate the minimum and maximum amount of information present in the mean squares. Using these constraints and the Satterthwaite approximation, the estimated number of degrees of freedom S f d ˆ in (11) should be replaced by, noted that Nijhuis and Van den Heuvel (2007) did use a different estimator for the number of degrees of freedom in (13). They replaced

  of "degrees of freedom" which are used for the moment estimators. The likelihood based estimators do not have the Deleted: the moment estimators in unbalanced designs, seeSearle et al. (1992).

  already discussed by[START_REF] Nijhuis | Closed-form confidence intervals on measures of precision for an interlaboratory study[END_REF] for balanced designs. For unbalanced data the model is described by

σ

  αβ : the interaction effect of batch h and laboratory i , analytical run j within laboratory i , effect for batch h with analytical run j within laboratory i , random terms mutually independently distributed. Note that the imbalance of the design is incorporated by the dependency of the repeated number of observations hij K on batch, all belong to the "assay related" group, because it is assumed that the batches are homogeneous. The variance component 2 6 σ belongs to the "assay unrelated" group, because it quantifies the variation between production batches. The measures of precision repeatability, intermediate precsion and reproducibility are now defined by Repeatability:

Formatted:

  Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 6 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 7 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 7 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 7 pt Formatted: Font: (Default) Arial, 10 pt, English (U.S.), Lowered by 7 Deleted: the main random effects for batch h , laboratory i , and Deleted: respectively, ( ) hi αβ and Deleted: random Deleted: s for batch h and laboratory i and Deleted: respectively and with Deleted: . Deleted: It is assumed that h

  Henderson III Type 1 method for intermediate precison and reproducibility (for unbalanced data of course). The coverage probabilities for the Henderson III methods reduce with the amount of missing data while the maximum likelihood methods are less affected. Henderson III Type 1 method is affected most by the imbalance, clearly more than any of the other estimation methods. The likelihood based estimation methods gave left tail probabilities which are mostly underestimated (asymmetry), indicating lower confidence limits that are too low, even for repeatability in some of the simulation cases. The Henderson III estimation methods gave less extreme asymmetrical coverage probabilities than the likelihood based estimation methods. When the coverage probabilities for theHenderson III

Font

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  orSearle et al. (1992). Let

		k M be the mean square for the source of variation
	related to variance component	2 k σ . A characteristic of the set of mean squares for Henderson's third
	method is that the expectation (or expected mean square) is of the form
	(3)

, 2 n m+ σ are not involved in the expectation of k M and are therefore not present in (3), see page 104 of Searle et al. (1992). For balanced designs and normally distributed observations the statistic

	F o r
	P
	e	Deleted: variation
	e r
	R	Deleted: Deleted: III
	e
	v i e
	w
	k n degrees of freedom. The number of degrees of freedom k k k k M n X θ ≡ follows a Chi-square distribution with k n is related to the number of levels k q or k r for the source of variation in model (1) for the "assay unrelated" or "assay related" group, respectively. O n l y	Deleted: , depending on whether k M belongs to Deleted: sources of variation
	6

  Searle et al. (1992)), the likelihood based estimation method would

	provide estimators	2 1 s , 2 2 s ,…, 2 m 0 s , which could represent ML or REML estimators, for the variance
	components	2 1 σ , 2 2 σ ,…, 2 m 0 σ . Let 2 m 0 γ& be the (restricted) maximum likelihood estimator of the
	measure of precision given by
	(8)

  J , and = k 1,2,…, hij K and with

	µ :	the overall mean,
	h α :	the effect of batch h ,	α	h	( ) 2 6 , 0 ~σ N	,
	i β :	the effect of laboratory i ,

ca Communications in Statistics -Simulation and Computation

  Searle, S.R.,Casella, G. and McCulloch, C.E. (1992), Variance Components, Wiley, New York. Swallow, W.H. and Monahan, J.F. (1984), Monte Carlo Comparison of ANOVA, MIVQUE, REML, and ML Estimators of Variance Components, Technometrics, 26 (2), 47-57.

		Page 22 of 24
	estimation methods, which produces asymmetrical confidence intervals even for repeatability. The Satterthwaite, F.E. (1946), An approximate distribution of estimates of variance components,
		Deleted: trol
	REML estimation method with the modified Satterthwaite approach would be an appropriate Biometrics Bulletin, 2, 110-114.	Deleted: .
	alternative in our opinin.
		Deleted: the
		Deleted: situation (no missing data)
	Welch, B.L. (1956), On linear combinations of several variances, Journal of the American Statistical
		Deleted: it
	Association, 52 (273), 132-148.	Deleted: s
	Zou, K.H. and McDermott, M.P. (1999), Higher-moment approaches to approximate interval
		Deleted: es
	estimation for a certain intraclass correlation coefficient, Statistics in Medicine, 18, 2051-2061. F o r F o r F o r	Deleted: reasonable Deleted: . However, Deleted: s Deleted: in general
	P P P	Deleted: surprisingly seem to
	e e e
	e r e r e r	Deleted: almost to the Deleted: This seems to hold true
		also for the other cases when missing
	R R R	data is present. Deleted: i
	e e e
	v i e v i e v i e
	w w w	Deleted: However, w
	O n l O n l O n l	Deleted: s
	y y y
	20 21 22

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

Table 1 :

 1 Choices of model parameter in the simulation study.

	Variance Components	Case A	Case B	Case C	Case D	Case E
	Lab	0.6	0.6	0.2	1.0	0.2
	Batch*Lab	0.3	0.1	0.1	0.1	0.1
	Run(Lab)	0.7	1.2	0.2	0.2	0.2
	Batch*Run(Lab)	0.6	0.1	1.1	0.1	0.1
	Repeatability	0.5	0.5	0.5	0.5	0.5
	Intermediate Precision	1.8	1.8	1.8	0.8	0.8
	Reproducibility	2.7	2.4	2.1	1.9	1.1

Table 2 :

 2 Selected samples sizes for batches, laboratories, runs, and replicates used in the simulation study.

	Number of	Number of	Number of	Number of	Total number of
	batches	laboratories	analytical runs	replicates	test samples
	(H)	(I)	(J)	(K)	(HIJK)

Table 3 :

 3 Ranges of coverage probabilities for intermediate precision for interlaboratory studies with balanced designs. -94.04 82.20 -91.96 90.90 -94.98 93.04 -95.45 93.18 -95.31 REML MS

	CI-Method	Case A	Case B	Case C	Case D	Case E
	OS	88.67

Table 5 :

 5 Ranges of coverage probabilities for intermediate precision for interlaboratory studies with 20% missing data.

	CI-Method	Case A	Case B	Case C	Case D	Case E

Table 6 :

 6 Ranges of coverage probabilities for reproducibility for interlaboratory studies with 20% missing data. .54 91.54 -94.56 95.42 -97.02 80.74 -89.95 93.39 -95.94 REML MS 96.47 -98.77 94.35 -97.64 97.69 -99.28 87.29 -95.14 96.87 -98.68 OS 90.41 -94.28 88.32 -92.68 93.11 -96.27 75.16 -87.95 91.55 -95.41 ML MS 95.25 -98.69 92.90 -97.13 96.55 -99.20 85.21 -95.00 96.26 -98.53 OS 87.58 -96.73 85.51 -94.01 82.41 -96.77 82.23 -94.10 90.14 -97.67 TYPE 1 MS 87.71 -98.13 85.74 -95.61 82.45 -97.67 84.88 -95.96 90.13 -98.91 OS 91.36 -97.33 89.62 -95.41 86.96 -97.17 82.03 -94.15 91.17 -97.61 TYPE 3 MS 91.59 -98.58 90.07 -97.26 86.87 -98.16 84.16 -95.40 91.12 -98.80

	CI-Method	Case A	Case B	Case C	Case D	Case E
	OS	93.30 -95

This research has been sponsored by the Dutch Organization of Scientific Research (NWO) under Casimir project 018.002.029 in collaboration with the Probability and Statistics group of the Eindhoven University of Technology in the Netherlands.

URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.caCommunications in Statistics -Simulation and Computation

Communications in Statistics -Simulation and Computation

ACKNOWLEDGEMENTS

I would like to thank E. Talens for verifying the SAS programs for the simulation study and O. Almalik and H. van Zwam for verifying the SAS output with the article. Furthermore, I would like to thank A. Di Bucchianico for critically reading the article and providing good suggestions.


Pascal Wandner

Derk Schellenberg

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Rainer Paulus

Michael Sahle

Christoph Sternemann

Pascal Link

Derk Wandner

Inga Schellenberg

Rainer Pöttgen

Michael Paulus

Eu x Sr 1-x C 2 (0 ≤ x ≤ 1): A Dicarbide Solid Solution 1 with Perfect Vegard Behavior

Keywords: Dicarbide, Solid solution, Europium, Synchrotron powder diffraction, Mössbauer spectroscopy Dicarbides, Solid Solution, Europium, Valence State, Synchrotron Powder Diffraction, Mössbauer Spectroscopy ____________

A solid solution Eu x Sr 1-x C 2 (0 ≤ x ≤ 1) was synthesized by direct reaction of the elements at 1123 K. The crystal structures of these compounds, investigated by synchrotron powder diffraction, depend upon x. For x > 0.5 the monoclinic ThC 2 type structure (C2/c, Z = 4) is observed and for x ≤ 0.5 the ThC 2 type structure coexists with the tetragonal CaC 2 type structure (I4/mmm, Z = 2). The unit cell volumes per formula unit of all Eu x Sr 1-x C 2 compounds show perfect Vegard behaviour, which is due to the almost identical ionic radii of Eu 2+ and Sr 2+ . Mössbauer spectroscopic investigations indeed reveal that Eu is in the divalent state over the whole composition range. Eu x Sr 1-x C 2 exhibits several temperature dependent phase transitions that were studied by synchrotron powder diffraction and differential thermal analysis. The transition to a cubic high-temperature modification (Fm 3 m, Z = 4) is of special interest, as it contains information about strain effects appearing inside the modifications with ordered C 2 dumbbells (ThC 2 and CaC 2 type structures). The linear temperature dependence of the obtained transition temperatures T Ph shows that no observable strain exists in Eu x Sr 1-x C 2 , which is again due to the almost identical radii of Eu 2+ and Sr 2+ . Eu x Sr 1- x C 2 may therefore be described as a strain free dicarbide solid solution with perfect Vegard behaviour. 1 According to the classical definition Eu x Sr 1-x C 2 is not a solid solution, as SrC 2 (I4/mmm, Z = 2) and EuC 2 (C2/c, Z =4) are not isotypic. In a correct description two solid solutions -Eu dissolved in SrC 2 and Sr dissolved in EuC 2 -are formed. But as a Vegard behavior is found for the complete series Eu x Sr 1-x C 2 the term "solid solution" will be used for it throughout the manuscript. This is in agreement with the common use of this term in large parts of the respective literature.

Introduction

Dicarbides of rare earth (REC 2 , RE = rare earth metal) and alkaline earth metals (AEC 2 , AE = alkaline earth metal) have been known for a long time. Although they exhibit identical crystal structures, their physical properties are completely different, i.e. metallic (REC 2 ) vs. insulating behavior (AEC 2 ). In a simple picture this can be attributed to an additional electron in REC 2 according to RE 3+ (C 2 2-)(e -). This additional electron is delocalized in a conduction band. Only very recently we showed that the dicarbide of Europium (EuC 2 ) shows semiconducting behavior [1]. At ambient conditions it crystallizes -differing to all other REC 2 , which are known to exist in the CaC 2 type structure (I4/mmm, Z = 2) -in the ThC 2 type structure (C2/c, Z = 4). These observations reveal that EuC 2 resembles more the AEC 2 than the REC 2 compounds. Accordingly, measurements of the Mössbauer effect show that Eu is clearly in a divalent state in EuC 2 [1].

Solid solutions of rare earth metal dicarbides RE' x RE'' 1- x C 2 have been known since the seventies [2,3]. However, dicarbides of composition RE x AE 1-x C 2 have been reported only for the system CaC 2 -YC 2 [4]. The formation of solid solutions often leads to dramatic changes in the properties of the respective compounds. Electrical resistivity or magnetism may be changed drastically by mixed occupation of metal sites as well as the stability range of different polymorphs. Solid solutions RE x AE 1-x C 2 seem to be especially interesting in this context, as the binary compounds REC 2 and AEC 2 may differ strongly in their physical properties (s. above). Therefore, we have started to investigate the synthesis of RE x AE 1-x C 2 solid solutions as well as the characterization of their structural and physical properties. In a first attempt we have chosen the system Eu x Sr 1-x C 2 , as Eu 2+ (135 pm [5], CN = 10) and Sr 2+ (136 pm [5], CN = 10) have very similar ionic radii. With these requirements the formation of a strain free solid solution with perfect Vegard behavior was expected.

Results and Discussion

We were able to synthesize a solid solution series Eu x Sr 1- x C 2 with 0 ≤ x ≤ 1. The successful synthesis of 12 compounds with differing x was clearly proven by synchrotron powder diffraction data. All compounds -with the exception of SrC 2 , which is colorless -are black, microcrystalline powders that show a high sensitivity to oxygen and moisture. The gases released during this decomposition have not been analyzed up to now, but our preliminary Mössbauer investigations on EuC 2 with a not completely air-tight sample holder indicated that the final product of this decomposition contains Eu 3+ . Due to strong X-ray absorption of Eu, diffractometers with conventional CuK α1 radiation are not suitable; moreover MoK α1 radiation provides powder diffractograms with only a very poor angular resolution. Therefore, synchrotron radiation using a short wavelength is the appropriate tool to obtain powder data with good resolution and minimized absorption.

The resulting patterns were first indexed and subsequently refined by the Rietveld method to get reliable cell parameters and phase ratios. The known crystal structures of EuC 2 [1] and SrC 2 [6] were used as starting models for the refinement. Figure 1 shows the Rietveld refinement of Eu 0.5 Sr 0.5 C 2 as an example for the quality of the data and the refinement. All samples appeared homogeneous and no reflections for any impurity phases were observed. Neither a splitting of reflections nor unusual broad reflections were observed so that the existence of two phases. i.e. SrC 2 and EuC 2 in varying ratios can be excluded.

In all final refinements the composition x -i.e. the occupancy of the mixed metal sites with Eu and Sr, resp.has not been refined, as it led to physical meaningless results in most cases. Instead the composition x was taken from the ratio of the starting materials Eu and Sr. The inaccuracy of weighing was taken into account by applying an error of ± 0.03 to all compositions (see Figure 2). Neutron diffraction experiments on similar solid solutions with Ytterbium (unpublished) confirm the validity of this assumption. Furthermore, atomic positions and displacement parameters of C were not refined, as the scattering power of C is too low compared with Eu or Sr to obtain reliable results. Instead the atomic positions of C in EuC 2 [1] and SrC 2 [6] were used and U iso was fixed to 400 pm 2 . Details of the refinement of Eu 0.5 Sr 0.5 C 2 , representative for all other compounds, are presented in Table 1. [a] U iso and atomic positions of metal ions have been constrained.

[b] U iso fixed due to small fraction of the phase All compounds show crystal structures that are known for the AEC 2 series (AE = Ca [7], Sr [6], Ba [8]). As can be seen in Figure 2, the structure formed at room temperature (RT) depends on the composition x. For x > 0.5 only the monoclinic ThC 2 type structure (C2/c, Z = 4) is found, which is the stable modification of EuC 2 at RT [1]. For x ≤ 0.5 the ThC 2 type structure coexists with the tetragonal CaC 2 type structure (I4/mmm, Z = 2), the stable modification of SrC 2 at RT [6]. The relative amount of the tetragonal modification decreases with x. All refined volumes per formula unit are in good agreement with Vegard´s law, no strong deviations occur (see Figure 3). These findings already suggest that Eu is divalent over the whole composition range, as a change of the Eu valence to + III should lead to a significant reduction of the unit cell volume. In Figure 3 it can be seen that for 0.1 ≤ x ≤ 0.5 both modifications coexist. For x < 0.33 the CaC 2 type structure shows the higher density, whereas for x > 0.33 the density of the ThC 2 type structure is higher. For x = 0.33 the density of both modifications is almost the same. This is roughly coinciding with the appearance as a major component in this system: below x ≈ 0.2 the CaC 2 type structure dominates, above x ≈ 0.2 the ThC 2 type structure (Figure 2). Thus, this behavior seems to be mainly governed by thermodynamics. But as both phases coexist, there are probably also kinetic effects that control the equilibrium composition as well as its change with increasing temperature. To verify the existence of Eu 2+ in Eu x Sr 1-x C 2 151 Eu Mössbauer spectra of five selected compounds with x = 0.1, 0.25, 0.5, 0.75, and 0.9 were recorded at 77 K (Figure 4). Low temperatures were chosen to obtain spectra of improved quality. Unfortunately, we do not know, whether Eu x Sr 1-x C 2 compounds undergo phase transitions at low temperatures. This is part of our current work. But the very good agreement with the Mössbauer parameters obtained for pure EuC 2 (see below) confirm our assumption that Eu is present as Eu 2+ in Eu x Sr 1-x C 2 .

The corresponding fitting parameters of the Mössbauer measurements on Eu x Sr 1-x C 2 are listed in Table 2. All spectra could be well reproduced with single signals subject to quadrupole splitting due to the non-cubic site symmetry. The isomer shift as well as the quadrupole interaction are comparable to EuC 2 (δ = -11.92(3) mm/s, ∆E Q = 4.3(3) mm/s) [1]. There is no trend in the δ, ∆E Q , and Γ values over the whole x range, indicating that the local electronic situation around the europium atoms is quite similar. As EuC 2 and SrC 2 are known to show temperature dependent phase transformations [1,6], the structural behavior of four selected compounds Eu x Sr 1-x C 2 with x = 0.18, 0.25, 0.5, and 0.75 was studied in the temperature range 300-800 K using synchrotron powder diffraction. All compounds show a continuous transformation from the ThC 2 type structure to the CaC 2 type structure with increasing temperature so that both structures coexist over a broad temperature range. Only for x = 0.18 a complete transformation can be observed: above 580 K the CaC 2 type structure is the only detectable phase. At temperatures between 600 and 700 K all compounds show a sharp transformation to a cubic high temperature modification (Fm 3 m, Z = 4), which is also known for both binary carbides [1,6]. From synchrotron powder diffraction no exact transition temperatures T Ph could be determined due to the large applied temperature steps of 50 K. Representative powder patterns as well as mass fractions of the observable modifications at different temperatures are presented in Figures 5 and6. To determine the transition temperatures T Ph more accurately, the thermal behavior of the compounds was analyzed by differential thermal analysis (DTA). The high temperature phase transition that can be regarded as an order-disorder transformation shows a significant uptake of heat that can be observed as an endothermic peak in the DTA. In contrast, the transition between the ThC 2 and CaC 2 type structures, a displacive transformation in the second coordination sphere, does not show an observable evolution of heat. It is therefore not detectable in the DTA. It is remarkable that the high temperature phase transition peaks of Eu x Sr 1-x C 2 are much broader then the peaks of the pure dicarbides. The halfwidth increases from x = 0 to 0.5 and decreases again from x = 0.5 to 1, which implies that this effect arises from the mixed occupancy of the metal site. As a consequence, the transition temperatures (T Ph ) could not be determined by fitting a Gaussian to the peaks. Instead, the temperatures were taken from the measured curves by inspection with errors estimated from the broadness of the respective peaks. The results can be seen in Table 3 and Figure 7. All temperatures T Ph range from 600 to 700 K, which shows a good agreement between powder diffraction and DTA results. Within the limits of accuracy, all T Ph values lie on a straight line between the pure dicarbides. This finding may be interpreted by using the Lattice Strain Theory of McColm et al. [2,3], who carried out similar studies on RE x RE´1 -x C 2 solid solutions. Within this theory the mixed occupancy of the metal site causes a lattice strain inside the modifications with ordered C 2 dumbbells (ThC 2 and CaC 2 type structures), if the metal ions differ in their ionic radii. The strain should be proportional to the difference in the ionic radii as well as to the amount of mixing, with a theoretical maximum at x = 0.5. This strain energy causes an increase of the enthalpy H of the ordered phase, which results in a decrease of ∆H Ph and T Ph . Therefore the transition temperatures T Ph of RE x RE´1 -x C 2 solid solutions do not show a straight line between x = 0 and x = 1, but exhibit a characteristic minimum, whose value depends on the difference in the cations' radii.

The linear behavior of T Ph observed for Eu x Sr 1-x C 2 can be also explained by using McColm´s strain theory. As the divalent ions Eu 2+ and Sr 2+ do not show a noticeable difference in their ionic radii, no strain should arise in the ordered phases from the mixed occupancy of the metal site. As a consequence, T Ph does not show a local minimum in this solid solution. Thus, the values of T Ph can simply be calculated by the linear relationship T Ph (Eu x Sr 1-x C 2 ) = x•T Ph (EuC 2 ) + (1-x)•T Ph (SrC 2 ), analogous to the respective volumes.

Conclusions

Our investigations show that the solid solution Eu x Sr 1-x C 2 can be prepared without impurity phases by direct reaction of the elements at 1123 K. The system shows miscibility in the whole composition range (0 ≤ x ≤ 1). All samples -with the exception of colourless SrC 2 -are black microcrystalline powders.

The used analytical methods show that the compounds Eu x Sr 1-x C 2 can be described as a solid solution with a perfect Vegard behaviour comprising Eu 2+ ions. Synchrotron powder diffraction experiments reveal that Eu x Sr 1-x C 2 crystallizes in the well-known ThC 2 and CaC 2 type structures at RT, the type of structure depends on x. The unit cell volumes per formula unit of all compounds satisfy Vegard´s law, as they lie on a straight line between the pure dicarbides EuC 2 and SrC 2 . Moreover, measurements of the Mössbauer effect proved Eu 2+ for all Eu x Sr 1-x C 2 compounds investigated, as was also found for pure EuC 2 [1]. All compounds transform to a cubic high temperature modification that is well-known for both REC 2 and AEC 2 . Both temperature dependent synchrotron powder diffraction and DTA measurements revealed that the high temperature phase transition temperatures T Ph of all compounds lie on a straight line between the pure dicarbides. This observation is due to the nearly identical ionic radii of Eu 2+ and Sr 2+ and can be interpreted using the strain theory of McColm et al. [2,3].

The successful synthesis of Eu x Sr 1-x C 2 is not very surprising, as Eu 2+ and Sr 2+ have very similar ionic radii. But in the meantime, we were able to synthesize solid solutions containing metal ions with more differing radii, e.g. Yb x Ca 1-x C 2 . Here, an interesting valence change of Yb from +2 to +3 was observed, depending upon x. This will be part of an upcoming contribution.

Experimental Section

Synthesis. All Eu x Sr 1-x C 2 compounds were prepared by direct reaction of the elements inside a tantalum bomb containing an inert atmosphere. Eu metal (Chempur, 99.99 %) and Sr metal (Sigma Aldrich, 99.99 %) were filed to fine powders and mixed with graphite powder (Sigma Aldrich, 99.9998 %) in the desired molar ratio. Prior to the reaction the graphite powder was heated at 1073 K in a dynamic vacuum for 48 hours to remove water and oxygen inclusions. In a typical reaction 114.0 mg Eu (0.75 mmol), 65.7 mg Sr (0.75 mmol) and 39.6 mg graphite (3.3 mmol) were thoroughly mixed in a glove box (argon atmosphere) and were afterwards transferred to a tantalum ampoule. A small excess of graphite was used to suppress the formation of oxides and consider losses of graphite during the transfer process. The tantalum ampoules were then sealed using an arc furnace with a helium atmosphere of 750 mbar pressure. A current of 10 A was used to melt the tantalum metal. Subsequently the ampoules were encapsulated in silica glass tubes under vacuum to avoid an oxidation of the tantalum at high temperatures. The resulting ampoules were then transferred to a tube furnace and heated to 1123 K within 4 hours. The reaction was maintained at this temperature for 24 hours, afterwards the furnace was cooled down to room temperature within 8 hours. All sample handling was carried out in a glovebox in an argon atmosphere. Sample purity was checked by X-ray powder diffraction (Huber G670, MoK α1 radiation, glass capillary Ø 0.3 mm). Synchrotron powder diffraction. Synchrotron powder patterns were recorded at beamline BL9 [9] of the DELTA synchrotron (University of Dortmund, Germany). All measurements were carried out in glass or silica glass capillaries (Ø 0.3 mm) that were loaded in a glove box (argon atmosphere). For high temperature measurements a furnace was used, which allowed measurements up to 900 K. A photon energy of 19 keV (λ = 65.255 pm) was chosen to minimize sample absorption. For fast data collection a MAR345 area detector was used. The obtained powder rings were subsequently integrated by using the program FIT2D [START_REF] Hammersley | Fit2d user manual[END_REF] to receive one-dimensional powder patterns. All Rietveld refinements were carried out with the GSAS suite of programs [START_REF] Larson | [END_REF]. Mössbauer spectroscopy. The 21.53 keV transition of 151 Eu with an activity of 130 MBq (2 % of the total activity of a 151 Sm:EuF 3 source) was used for the 151 Eu Mössbauer spectroscopic experiments. The measurements were performed in the usual transmission geometry at 77 K. The source was kept at room temperature in all experiments. The samples were placed within thin-walled sealed glass containers at a thickness corresponding to about 10 mg Eu/cm 2 . Fitting of the spectra was performed with the NORMOS-90 program system [START_REF] Brand | Normos Mössbauer fitting Program[END_REF]. Differential thermal analysis. DTA measurements were carried out using a Netzsch STA 409 Cell thermobalance placed in a glove box (nitrogen atmosphere). Approximately 20 mg of the sample were weighed into an aluminum oxide crucible and subsequently heated to 873 K in an argon stream with a heating rate of 10 K/min. An empty aluminum oxide crucible was heated simultaneously as reference material. Afterwards the furnace was cooled down to room temperature with a cooling rate of 10 K/min. The temperature difference between sample and reference was detected by using Pt-Rh/Pt thermocouples.

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Figure 1 .

 1 Figure 1. Rietveld refinement of Eu 0.5 Sr 0.5 C 2 (295 K, λ = 65.255 pm, BL9 / DELTA) showing the observed (+) and calculated patterns (solid line) as well as the difference between them. Vertical bars mark the positions of the reflections of CaC 2 type structure (I4/mmm, Z = 2; upper bars) and ThC 2 type structure (C2/c, Z = 4; lower bars).

Figure 2 .

 2 Figure 2. Mass fraction of the different modifications of Eu x Sr 1-x C 2 as a function of x at RT. Open squares (□) show the ThC 2 type structure, open triangles (∆) the CaC 2 type structure.

Figure 3 .

 3 Figure 3. Unit cell volume per formula unit of the different modifications of Eu x Sr 1-x C 2 as a function of x. Open squares (□) show the ThC 2 type structure, open triangles (∆) the CaC 2 type structure. A behavior according to Vegard´s law is indicated by a dotted straight line. Errors bars for the weighing and the volume errors are given. The latter are in most cases smaller than the symbol used in the figure.

Figure 4 .

 4 Figure 4. Experimental and simulated 151 Eu Mössbauer spectra of the Eu x Sr 1-x C 2 samples with x = 0.1, 0.25, 0.5, 0.75, and 0.9 (77 K data).

Figure 5 .

 5 Figure 5. Synchrotron powder patterns (background corrected) of Eu 0.5 Sr 0.5 C 2 at different temperatures. The pattern obtained at 425 K mainly shows the ThC 2 type structure, whereas at 651 K mainly the CaC 2 type structure and at 727 K the cubic high temperature modification are observed.

Figure 6 .

 6 Figure 6. Mass fraction of the different modifications of Eu 0.5 Sr 0.5 C 2 as a function of temperature. Open squares (□) show the ThC 2 type structure, open triangles (∆) the CaC 2 type structure and open circles (○) the cubic high temperature modification.

Figure 7 .

 7 Figure 7. Transition temperatures T Ph of Eu x Sr 1-x C 2 as a function of x. All values were obtained from DTA measurements. The dotted straight line indicates a hypothetical strain free system (see text).

Table 1 .

 1 Eu 0.5 Sr 0.5 C 2 : Summary of Rietveld refinement (295 K, synchrotron powder diffraction data, λ = 65.255 pm, Beamline BL9 at DELTA / Dortmund).

		ThC 2 type structure	CaC 2 type structure
	Space group, Z	C2/c (no. 15), 4	I4/mmm (no. 139), 2
	a / pm	703.13(4)	411.68(9)
	b / pm	442.74(3)
	c / pm	761.29(4)	671.1(3)
	β / °	107.122(5)
	V / nm 3	0.22649(2)	0.11374(5)
	Eu / Sr [a]	4 e, y = 0.1828(4)	2 a
	U iso / pm 2	100(10)	100 [b]
	C	not refined (s. text)	not refined (s. text)
	No. of reflections 93	22
	R Bragg	0.0360	0.0396
	Mass fraction / % 94.05(4)	5.9(2)
	R P	0.0132
	wR P	0.0195
	No. of refined
	parameters	30

Table 2 .

 2

	Eu x Sr 1-x C 2 0.1	-11.95(7)	2.70*	4.5*
	0.25	-11.86(4)	2.6(1)	4.4(4)
	0.5	-11.91(3)	2.7(1)	3.7(3)
	0.75	-11.87(4)	2.7(1)	4.3(4)
	0.9	-11.91(4)	2.4(1)	4.2(3)

Fitting parameters of 151 Eu Mössbauer spectroscopic measurements (77 K data) of Eu x Sr 1-x C 2 . Numbers in parentheses represent the statistical errors in the last digit. (δ), isomer shift; (Г), experimental line width; (∆E Q ), electric quadrupole splitting parameter. * fixed parameter. Compound x δ / mm . s -1 Γ / mm . s -1 ∆E Q / mm . s -1

Table 3 .

 3 Transition temperatures T Ph of Eu x Sr 1-x C 2 with estimated errors obtained from DTA measurements.

	x	T Ph / K
	0	698 ± 2
	0.1	687 ± 5
	0.18	685 ± 8
	0.25	680 ± 10
	0.33	674 ± 10
	0.5	662 ± 10
	0.66	650 ± 10
	0.75	635 ± 8
	0.9	633 ± 5
	1	631 ± 2

Acknowledgments

This work was financially supported by the Deutsche Forschungsgemeinschaft through SPP 1166 Lanthanoidspezifische Funktionalitäten in Molekül und Material. The authors acknowledge DELTA for providing synchrotron radiation. ____________

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