W M Reichmann

‡, , , J.F. Maillefert §, ∥, ¶ D J Hunter#

J N Katz

P G Conaghan∥∥

E Losina

Responsiveness to change and reliability of measurement of radiographic joint space width in osteoarthritis of the knee: a systematic review

Keywords: Knee osteoarthritis, X-ray, Radiograph, Responsiveness, Reliability, Standardized response mean

The goal of this systematic review was to report the responsiveness to change and reliability of conventional radiographic joint space width (JSW) measurement.

Method

We searched the PubMed and Embase databases using the following search criteria: [osteoarthritis (OA) (MeSH)] AND (knee) AND (X-ray OR radiography OR diagnostic imaging OR radiology OR disease progression) AND (joint space OR JSW or disease progression). We assessed responsiveness by calculating the standardized response mean (SRM). We assessed reliability using intra-and interreader intra-class correlation (ICC) and coefficient of variation (CV). Random-effects models were used to pool results from multiple studies. Results were stratified by study duration, design, techniques of obtaining radiographs, and measurement method.

Results

We identified 998 articles using the search terms. Of these, 32 articles (43 estimates) reported data on responsiveness of JSW measurement and 24 (50 estimates) articles reported data on measures of reliability. The overall pooled SRM was 0.33 [95% confidence interval (CI): 0.26, 0.41]. Responsiveness of change in JSW measurement was improved substantially in studies of greater than 2 years duration (0.57). Further stratifying this result in studies of greater than 2 years duration, radiographs obtained with the knee in a flexed position yielded an SRM of 0.71. Pooled intra-reader ICC was estimated at 0.97 (95% CI: 0.92, 1.00) and the intra-reader CV estimated at 3.0 (95% CI: 2.0, 4.0). Pooled inter-reader ICC was estimated at 0.93 (95% CI: 0.86, 0.99) and the inter-reader CV estimated at 3.4% (95% CI: 1.3%, 5.5%).

Conclusions

Measurement of JSW obtained from radiographs in persons with knee is reliable. These data will be useful to clinicians who are planning RCTs where the change in minimum JSW is the outcome of interest.

Introduction

Knee osteoarthritis (OA) is a painful and disabling disease for many with 12% of adults 60 years of age or older having symptomatic knee OA1. As the population ages, the prevalence of knee OA continues to rise. Currently, available pharmacologic regimens for knee OA focus on alleviating pain, but do not slow the structural progression of disease2. Disease modifying OA drugs (DMOADS) are in the early developmental stages, and thus it is important to quantify the expected rate of structural progression to facilitate trial planning.

Minimum joint space width (JSW) is commonly used to assess knee OA progression3. It has been shown to be sensitive to change [4] and [5] and change in the minimum JSW has been the primary outcome for previous DMOAD trials [4], [5], [6] and [7]. An analytic literature synthesis by Emrani et al. in 2008 showed an interaction between study design and radiographic technique was associated with annual change in minimum JSW. The greatest annual change was seen in observational studies that used a semi-flexed technique without fluoroscopy, while the smallest annual change was see in randomized controlled trials (RCTs) with the same technique5.

The objective of this paper was to update results of Emrani et al. by adding the most recent studies and report responsiveness of JSW in terms of standardized response mean (SRM). The SRM is defined as the mean change divided by the standard deviation (SD) of change and can be interpreted as the number of SDs of change, which will be useful for planning future DMOAD trials. We also report pooled estimates of reliability, which include inter-and intra-reader intra-class correlations (ICCs) and coefficients of variation (CVs).

Method Eligibility criteria

Studies were eligible for our analyses if they satisfied all four requirements of the PICO (Patients Interventions Controls Outcomes). To be included in the review, the study population had to include patients with knee OA followed over time with radiograph-based measures of JSW. We included studies that reported responsiveness (mean change/SD of change or SRM) or reliability measures (inter-or intra-reader ICC or CV). If the study was a RCT then we used data from the control group. This was done to ensure quantification of the natural history of responsiveness of radiographs in those with knee OA. Studies were not limited by publication date (latest search: April 2009) and we included studies that were published in English, French, Spanish, and German.

Information sources and search

We searched the PubMed and Embase databases using the following search criteria: (osteoarthritis [MeSH]) AND (knee) AND (X-ray OR radiography OR diagnostic imaging OR radiology OR disease progression) AND (joint space OR JSW or disease progression).

Study selection

All abstracts were read by one reviewer. The reviewer obtained full-length articles of all abstracts that were considered as probably relevant or of unknown relevance. These articles were subsequently reviewed and data extracted into a data abstraction form. Abstracts of all potentially relevant references in the full-text review were obtained if probably relevant or of unknown relevance.

Studies were excluded if they did not report change in minimum JSW in the knee or if they did not provide a measure of reliability in measuring minimum JSW.

Data items

We abstracted the following study characteristics from each article: study design, radiographic technique, use of fluoroscopy, method of measurement, follow-up time, whether readers were blinded to the order of the radiographic studies, and sample size. Study design was classified as RCT or observational and radiographic technique was categorized as extended view or flexed (includes semi-flexed). Method of measuring minimum JSW was performed manually or using a computer. Follow-up time was categorized as 1-year or less, 1-2 years, or greater than 2 years.

Summary measures

The principal summary measure for our review is the SRM. In articles that reported the SRM directly, we abstracted the reported value. In articles that only reported mean change and SD of change, we calculated the SRM from the two reported measures. Inter-and intra-reader reliability measures (ICC, CV) were also abstracted from the articles.

Synthesis of results

Random-effects models were built to obtain pooled estimates for the SRM and reliability measures across studies adjusting for variability across the studies. Heterogeneity in the estimates was assessed using I-squared, which assesses the percentage of variation across studies that was due to between study variation. Analyses were performed for all studies that reported these measures and by study characteristics, including study design, radiographic approach, radiographic technique, use of fluoroscopy, method of measurement, and follow-up time. Ninety-five percent CIs were derived for all estimates.

Results

Study selection

We identified 866 articles using our electronic search and another 132 were identified manually for a total of 998 articles. Two-hundred eighty-five articles met the initial abstract screening inclusion criteria and the full-text article was obtained and read for further screening. Of these, 32 articles reported responsiveness results (43 estimates) and 24 articles reported reliability results. Of the 24 articles reporting reliability results, the inter-reader ICC was reported eight times, the intra-reader ICC 17 times, the inter-reader CV six times, and the intra-reader CV 19 times (Fig. 1). Of the 43 estimates on responsiveness, 21 (49%) estimates were obtained from studies with followup of 1 year or less, 10 (23%) estimates were derived from studies with follow-up of 1-2 years, and 12 (28%) came from studies with greater than 2 years of follow-up. The mean sample size was 100 (SD = 86). Sixteen estimates (37%) were obtained from studies that used a radiographic approach with the knee fully extended and 27 (63%) from studies that had the knee in flexion. Fluoroscopy was used for 23 (53%) of the estimates and computerized methods of measuring the minimum JSW was used for 24 of the estimates (56%). Nineteen (44%) of the estimates came from RCTs. Of the 43 estimates, only 21 (49%) disclosed whether the readers were blinded to the sequence of the radiographs. Of these 21 estimates, 19 (90%) came from studies that used blinded readers. Study characteristics for all 32 studies are shown in Table I

Full-size table

Of the eight estimates evaluating the inter-reader ICC, four (50%) used a fully extended radiographic approach, four (50%) used fluoroscopy, and seven (88%) measured the joint space manually. The mean sample size in these studies was 110 (SD = 110).

Of the 17 estimates evaluating the intra-reader ICC, six (35%) used a fully extended radiographic approach, eight (47%) used fluoroscopy, and nine (53%) measured the joint space manually. The mean sample size in these studies was 80 (SD = 88).

Of the six estimates evaluating the inter-reader CV, three (50%) used a fully extended radiographic approach, six (100%) used fluoroscopy, and six (100%) measured the joint space manually. The mean sample size in these studies was 120 (SD = 99).

Of the 19 estimates evaluating the intra-reader CV, six (32%) used a fully extended radiographic approach, 14 (74%) used fluoroscopy, and 11 (58%) measured the joint space manually. The mean sample size was 43 (SD = 38). Study characteristics for all 24 studies are shown in Table II.

Synthesis of responsiveness results

The I-squared value for the 43 estimates was 0.82 [95% confidence interval (CI): 0.76, 0.86] indicating substantial between study variation. The I-squared values are shown in Table III.

Full-size table

The random-effects analysis yielded an overall pooled SRM for the 43 estimates of 0.33 (95% CI: 0.26, 0.41). The pooled SRM was similar when the analysis was stratified by radiographic approach, the use of fluoroscopy, measurement method, and study type. Follow-up time was related to the magnitude of the SRM. Estimates derived from studies with 1 year or less and 1-2 years of follow-up had similar responsiveness (0.24 and 0.25 respectively), while estimates coming from studies with greater than 2 years follow-up had an SRM of 0.57 (95% CI: 0.39, 0.75). Similar effects of follow-up time are shown when use of fluoroscopy, measurement method, and study type were stratified by follow-up time. However, when radiographic approach was stratified by follow-up time, estimates derived from studies that used a flexion-based radiographic approach and had greater than 2 years of follow-up time had a higher SRM of 0.71 (95% CI: 0.44, 0.98).

Synthesis of reliability results

Results of random-effects pooling of the reliability estimates showed good inter-and intra-reader reliability for measuring the minimum JSW. The eight estimates of inter-reader ICC produced an estimate of 0.93 (95% CI: 0.86, 0.99), while the 17 estimates of intra-reader ICC produced an estimate of 0.97 (95% CI: 0.92, 1.00). Additional results stratified by study characteristics are shown in Table IV. Six estimates for the inter-reader CV produced an estimate of 3.4% (95% CI: 1.3%, 5.5%) and 19 estimates for the intra-reader CV produced an estimate of 3.0% (95% CI: 2.0%, 4.0%).

Additional results stratified by study characteristics are shown in Table V.

Discussion

We performed an analytic systematic review of the responsiveness and reliability of knee radiographs when measuring the minimum JSW. We analyzed responsiveness using the SRM. This measure can be interpreted as the number of SDs of change. The overall SRM was 0.33 (95% CI: 0.26, 0.41). Follow-up time was the main study characteristic that was related to responsiveness. Studies with follow-up times greater than 2 years showed greater responsiveness (SRM = 0.57; 95% CI: 0.39, 0.75). It is critical to note that studies with a follow-up of 1 year or shorter showed a responsiveness of 0.24. This limitation of the radiographic technique means that to adequately power a study to demonstrate change over this short interval will require much larger sample sizes. Studies that used a flexed view and had greater than 2 years of follow-up showed the greatest responsiveness (SRM = 0.71; 95% CI: 0.44, 0.98). Based upon this literature there does appear to be some advantage to standardized positioning and fluoroscopy with slight improvements in responsiveness. Despite what one may have expected there does not appear to be any advantage in computerized measurement of JSW over manual measures. In studies with greater than 2 years of follow-up, the responsiveness was higher for those that used computerized methods (0.68) compared to those that used manual methods (0.51). However, the 95% CIs substantially overlap due to substantial variability in these estimates (Table III).

The reliability of measuring minimum JSW provided to be excellent with pooled ICCs ranging from 0.91 to 0.99 and pooled CVs ranging from 1.5 to 5.8. Radiographic method, use of fluoroscopy, and measurement method did not affect reliability albeit the majority of the estimates come from different studies with no direct study comparison.

Our findings complement the work of Emrani et al. who published a systematic review in 2008 on the change in minimum JSW. While they found effects of radiographic approach and study type, they also analyzed the crude change in minimum JSW rather than the SRM. They also found that increased follow-up time was inversely associated with change in minimum JSW, while we found that increasing the follow-up time increased the responsiveness of radiographs to change. This difference may be due to differences in definition of primary outcomes and additional assumption of linearity of change that Emrani et al. used in their analysis5.

A major strength of this study is that it is the first literature synthesis to summarize responsiveness in terms of the SRM. These data will be useful to clinicians who are planning studies where the change in the minimum JSW is the outcome of interest. The results of this analysis suggest that studies using JSW as primary outcome measure based on radiographs should plan to have a follow-up period that is greater than 2 years and have the knee in a flexed position when performing the radiographs to ensure the greatest possible responsiveness. While the pooled SRM was higher for studies that did not blind the reader to the sequence of the radiographs (0.55), it is unlikely that blinding of the readers of the radiographs substantially influenced our results since only two estimates came from studies that did not blind their readers. Also, the pooled SRM for estimates coming from studies that did blind the readers was similar to those that did not report this information (0.30 vs 0.35 respectively).

Also, this is the first known literature synthesis that pools reliability data on measuring minimum JSW. In general, these measurements can be considered to be reliable as the intra-and inter-reader ICCs were large and the CVs were low.

A major limitation of our review is that we did not report our results by risk factors for knee OA progression (body mass index, knee alignment, age, concurrent OA in other joints, synovitis, etc.) since they were not uniformly reported. The fact that we were not able to account for these factors may have contributed to the heterogeneity in the SRMs. It is important for future studies that report results on quantitative changes of knee OA progression to report these risk factors. Also, we did not collect data on the number of readers and the time interval between reads for our reliability data. It would be interesting to examine how these factors affected our estimates of reliability.

We found that radiographs provide moderate responsiveness and good reliability measures for measuring the minimum JSW in persons with knee OA. These data will be useful to clinicians who wish to plan future RCTs in which change in minimum JSW is their primary outcome.

Declaration of funding and role of funding source

The Osteoarthritis Research Society International (OARSI) Food and Drug Administration (FDA) OA Initiative received financial support from the following professional organization: American College of Rheumatology.

Fig. 1 .

 1 Fig. 1. Flow chart of the screening process for articles included in the systematic review. View thumbnail images View high quality image (139K) Study characteristics

Table I . Study Characteristics of the manuscripts reviewed for responsiveness

 I .

	Author, year Author, year Author, year	Study Study Study	Sample Sample Sample	Follow-up Follow-up Follow-up	Radiographic Radiographic Radiographic	Method of Method of Method of	Delta Delta Delta
	(Ref.) (Ref.) (Ref.)	type type type	size size size	months months months	approach approach approach	measurement measurement measurement	(SD) (SD) (SD)
	20004 200731				fluoroscopy fluoroscopy fluoroscopy		(0.94) (0.32) (0.66)
	Mazzuca et al. 20016 Pham et al. 200420 Sharif et al. 200732	Cohort 402 RCT 277 Cohort 115	31.60 12 60	Extension without fluoroscopy Extension without fluoroscopy Extension without fluoroscopy	Manual Manual Manual	0.37 (1.25) 0.09 (0.55) 0.18 (0.93)
	Reginster et al. 20017 Gandy et al. 200211 Uebelhart et al. 200421 Brandt et al. 200522 Le Graverand et al. 200833	RCT Cohort 11 106 RCT 76 RCT 180 62 Cohort 62	36 37 12 30 12 12	Extension with fluoroscopy Extension without fluoroscopy Extension without fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy	Computerized Manual Computerized Manual Computerized Computerized	0.40 (0.92) 0.21 (0.37) 0.32 (1.11) 0.45 (0.70) 0.22 (0.41) -0.01 (0.46)
	Miyazaki et al. 200212 Conrozier et al. 200523	Cohort 74 Cohort 96 27	72 12 12	Flexion without fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy	Manual Computerized Computerized	1.40 (1.20) 0.19 (0.48) 0.25 (0.54)
	Boegard et al. 200313 Michel et al. 200524 Spector et al. 200525 Mazzuca et al. 200834 Mazzuca et al. 200314 Bingham et al. 200626 Gensburger et al. 200935 Pessis et al. 200315 Kahan et al. 200936	Cohort 50 52 RCT 150 RCT 98 27 Cohort 47 Cohort 52 20 269 47 RCT 280 Cohort 81 Cohort 20 112 RCT 313	25 14 24 12 12 12 14 12 24 12 24 48 12 9.84 12	Flexion with fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Extension with fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy	Manual Computerized Computerized Computerized Computerized Computerized Manual Manual Computerized Computerized Computerized Computerized Manual Computerized Computerized	0.06 (0.45) 0.09 (0.31) 0.07 (0.56) 0.12 (0.42) 0.02 (0.40) 0.16 (0.37) -0.09 (0.66) 0.00 (0.60) 0.13 (1.08) -0.01 (0.51) 0.09 (1.31) 0.32 (0.76) 0.10 (0.90) 0.00 (0.53) 0.31 (0.71)
	Sugiyama et al. 200316 Cline et al. 200627	Cohort 110 RCT 85	48 11.76	Flexion with fluoroscopy Flexion with fluoroscopy	Computerized Computerized	0.53 (0.43) 0.12 (0.42)
	Author, year (Ref.) Ayral et al. 19968 Cohort 41 Study type Sample size 58 99 Vignon et al. Cohort 200317 58 Mikesky et al. 200628 RCT 60	Follow-up months 12 24 8.16 24 30	Radiographic approach Extension without fluoroscopy Extension with fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy	Method of measurement Manual Computerized Computerized Computerized Manual	Delta (SD) 0.40 (1.00) 0.17 (0.75) -0.07 (0.63) 0.24 (0.50) 0.54 (0.70)
	Ravaud et al. 19969 Pavelka et al. 200418 Botha-Scheepers et al. 200729	Cohort 55 RCT 89 Cohort 122	12 24 24	Extension without fluoroscopy Extension with fluoroscopy Flexion without fluoroscopy	Manual Manual Computerized	0.42 (1.11) 0.40 (0.79) 0.21 (0.52)
	Listrat et al. 199710 Pham et al. 200419 Krzeski et al. 200730	RCT RCT RCT	17 79 71	12 12 12	Extension without fluoroscopy Extension without fluoroscopy Extension with fluoroscopy	Manual Manual N/A	0.70 (1.20) 0.21 (0.59) 0.14 (0.53)
	Pavelka et al. Nevitt et al.	RCT Cohort 53 139 69	60 12 37	Extension with Extension without Manual Manual Flexion without Computerized	0.42 0.12 0.43

Delta: change in minimum JSW from baseline to follow-up (measured in millimeters).

Table II .

 II Study characteristics of the manuscripts reviewed for reliability

	Author, year (Ref.) Author, year (Ref.) Author, year (Ref.)	Sample size Sample size Sample size	Radiographic approach Radiographic approach Radiographic approach	Method of measurement Method of measurement Method of measurement	Reliability estimator Reliability estimator Reliability estimator	Observer Value Observer Value Observer Value
	Mazzuca et al. 200834	202 292 39	Extension with fluoroscopy Extension without fluoroscopy Flexion with fluoroscopy	Manual Manual Computer	CV ICC CV	Inter Inter Intra	2.6% 0.912 0.80
	Author, year (Ref.) Sharif et al. 200440	Sample size 51 20 42	Radiographic approach Flexion with fluoroscopy Extension without fluoroscopy Flexion with fluoroscopy	Method of measurement Manual Manual Manual	Reliability estimator CV CV ICC	Observer Value Intra 2.3% Intra 11.3% Intra 0.89
	Boegard et al. Cicuttini et al. 200541 Gensburger et al.	5 51 123 42	Flexion with fluoroscopy Flexion with fluoroscopy Extension without fluoroscopy Flexion with fluoroscopy	Computer Manual Computer Manual	CV CV CV CV	Intra Intra Intra Intra	3.8% 1.0% 4.8% 2.9%
	Buckland-Wright 200313 200935 Conrozier et al. et al. 199537 200523	5 51 106 44 7 51 106 44	Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy Flexion with fluoroscopy	Computer Manual Computer Manual Manual Manual Computer Manual	CV CV CV ICC CV CV ICC CV	Intra Inter Intra Inter Intra Inter Intra Inter	1.2% 2.7% 1.15% 0.80 3.6% 1.1% 0.99 0.8%
	Mazzuca et al. 200314 Michel et al. 200524 Kahan et al.	7 71 284 100	Flexion with fluoroscopy Flexion without fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy	Manual Manual Computer Computer	CV CV ICC CV	Intra Intra Intra Intra	0.6% 5.8% 0.98 1.2%
	Ravaud et al. Sugiyama et al. 200316 200936 Szebenyi et al. 19969 Vignon et al. 200642 Full-size table	55 10 60 100 55 20 60	Extension without fluoroscopy Flexion with fluoroscopy Extension without fluoroscopy Flexion with fluoroscopy Extension without fluoroscopy Extension with fluoroscopy fluoroscopy Extension without	Manual Computer Manual Computer Manual Computer Manual	ICC CV ICC ICC ICC ICC ICC	Intra Intra Intra Intra Inter Intra Inter	0.95 1.5% 0.895 0.99 0.85 0.98 0.868
	200317	10 36 80	Extension with fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy	Manual Computer Manual	CV ICC ICC	Intra Intra Intra	2.0% 0.98 0.90
	Pavelka et al. 20004 Mazzuca et al. Nevitt et al. 200439 200731	10 280 30 80 30 25	Extension with fluoroscopy Extension with fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy Flexion with fluoroscopy Flexion without fluoroscopy	Manual Manual Manual Manual Manual Computer	ICC CV ICC ICC ICC ICC	Intra Inter Intra Inter Inter Intra	0.99 6.6% 0.996 0.98 0.956 0.96
	Pavelka et al.	280 89 25	Extension with fluoroscopy Extension with fluoroscopy Flexion without fluoroscopy	Manual Manual Computer	ICC CV CV	Inter Intra Intra	0.97 3.6% 2.9%
	Mazzuca et al. 20016 200418	20 89 36	Extension without fluoroscopy Extension with fluoroscopy Flexion with fluoroscopy	Manual Manual Computer	CV CV ICC	Intra Inter Intra	4.4% 6.5% 0.99
	Myazaki et al. 200212 Pham et al. 200419 Le Graverand et al. 200833	10 156 36	Flexion without fluoroscopy Extension without fluoroscopy Flexion with fluoroscopy	Manual Manual Computer	ICC ICC ICC	Intra Intra Intra	0.92 0.993 0.99
	Pavelka et al. 200238 Pham et al. 200420	40 292 18	Extension with fluoroscopy Extension without fluoroscopy Flexion without fluoroscopy	Manual Manual Computer	CV ICC ICC	Intra Intra Intra	1.9% 0.996 0.99

Table III .

 III Results of random-effects pooling for studies that reported estimates of responsiveness by different study characteristics

		Number of estimates Number of estimates	I-squared (95% CI) I-squared (95% CI)	SRM (95% CI) SRM (95% CI)
	No fluoro Fluoro/1-2 years	20 7	0.79 (0.69 0.86) 0.28 (0.17, 0.39) 0.81 (0.62, 0.91) 0.29 (0.14, 0.44)
	Measurement method Fluoro/greater than 2 years	7	0.87 (0.75, 0.93) 0.58 (0.36, 0.80)
	Manual No fluoro/1-year or less	18 12	0.80 (0.70, 0.87) 0.38 (0.26, 0.50) 0.61 (0.28, 0.79) 0.21 (0.10, 0.32)
	Computerized Study type No fluoro/1-2 years	24 3	0.84 (0.77, 0.89) 0.31 (0.20, 0.41) 0.15 (-0.13, 0.82 (0.45, 0.94) 0.42)
	RCT No fluoro/greater than 2 years	19 5	0.82 (0.73, 0.88) 0.30 (0.20, 0.40) 0.89 (0.78, 0.95) 0.56 (0.24, 0.87)
	Cohort Measurement method by follow-up time 24	0.82 (0.74, 0.87) 0.36 (0.24, 0.49)
	Follow-up time Manual/1-year or less	8	0.20 (0.00, 0.63) 0.28 (0.17, 0.38)
	1-year or less Manual/1-2 years	21 2	0.56 (0.27, 0.73) 0.24 (0.15, 0.32) 0.19 (-0.44, 0.92 (0.73, 0.98) 0.82)
	1-2 years Manual/greater than 2 years	10 8	0.80 (0.63, 0.89) 0.25 (0.13, 0.37) 0.87 (0.77, 0.93) 0.51 (0.31, 0.71)
	Greater than 2 years Computerized/1-year or less	12 12	0.88 (0.81, 0.93) 0.57 (0.39, 0.75) 0.68 (0.42, 0.83) 0.21 (0.08, 0.33)
	Reader blinded to order of radiographs Computerized/1-2 years 8	0.78 (0.56, 0.89) 0.26 (0.13, 0.38)
	Yes No Computerized/greater than 2 years	19 2 4	0.76 (0.63, 0.85) 0.30 (0.19, 0.40) 0.59 (0.00, 0.90) 0.55 (0.33, 0.76) 0.90 (0.77, 0.96) 0.68 (0.31, 1.06)
	Unknown Study type by follow-up time	22	0.85 (0.78, 0.89) 0.35 (0.23, 0.46)
	Knee flexion by follow-up time RCT/1-year or less	10	0.60 (0.19, 0.80) 0.21 (0.11, 0.32)
	Extended/1-year or less RCT/1-2 years Extended/1-2 years RCT/greater than 2 years	Number of estimates 9 5 2 4	I-squared (95% 0.87 (0.72, 0.94) 0.24 (0.07, 0.41) 0.00 (0.00, 0.63) 0.26 (0.19, 0.34) SRM (95% CI) CI) 0.61 (0.00, 0.91) 0.38 (0.10, 0.65) 0.51 (0.00, 0.84) 0.56 (0.41, 0.70)
	Overall Extended/greater than 2 years Cohort/1-year or less	43 5 11	0.82 (0.76, 0.86) 0.33 (0.26, 0.41) 0.32 (0.00, 0.74) 0.34 (0.24, 0.44) 0.51 (0.03, 0.75) 0.26 (0.13, 0.40)
	Knee flexion Flexed/1-year or less Cohort/1-2 years	12 5	0.68 (0.42, 0.83) 0.19 (0.06, 0.32) 0.69 (0.20, 0.88) 0.26 (0.06, 0.46)
	Extended Flexed/1-2 years Cohort/greater than 2 years	16 8 8	0.19 (0.00, 0.55) 0.32 (0.26, 0.37) 0.82 (0.65, 0.90) 0.22 (0.08, 0.36) 0.92 (0.86, 0.95) 0.57 (0.30, 0.85)
	Flexed	27	0.88 (0.84, 0.91) 0.34 (0.22, 0.45)
	Flexed/greater than 2 years	7	0.88 (0.78, 0.94) 0.71 (0.44, 0.98)
	Fluoroscopy
	Fluoroscopy by follow-up time
	Fluoro	23	0.83 (0.76,0.88) 0.38 (0.27, 0.48)
	Fluoro/1-year or less	9	0.33 (0.00, 0.69) 0.29 (0.18, 0.39)

Table IV .

 IV Results of random-effects pooling for studies that reported estimates of intra-ICC by different study characteristics

		Number of	Inter-reader CV	Number of	Intra-reader CV
		estimates	(95% CI)	estimates	(95% CI)
	Overall	6	3.4% (1.3, 5.5)	19	3.0% (2.0, 4.0)
	Knee flexion
	Extended	3	5.2% (2.5, 8.0)	6	4.7% (2.7, 6.7)
	Flexed	3	1.5% (0.3, 2.7)	13	2.2% (1.3, 3.2)
	Fluoroscopy
	Fluoro	6	3.4% (1.3, 5.5)	14	2.0% (1.4, 2.5)
	No fluoro	0	N/A	5	5.8% (3.8, 7.9)
	Measurement method
	Manual	6	3.4% (1.3, 5.5)	11	3.6% (2.1, 5.1)
	Number of Computerized 0	Inter-reader ICC N/A	Number of 8	Intra-reader ICC 2.2% (0.8, 3.5)
	Full-size table	estimates	(95% CI)	estimates	(95% CI)
	Overall	8	0.93 (0.86, 0.99)	17	0.97 (0.92, 1.00)
	Knee flexion
	Extended	4	0.93 (0.85, 1.00)	6	0.98 (0.90, 1.00)
	Flexed	4	0.94 (0.79, 1.00)	11	0.97 (0.90, 1.00)
	Fluoroscopy
	Fluoro	4	0.95 (0.85, 1.00)	8	0.98 (0.88, 1.00)
	No fluoro	4	0.91 (0.82, 1.00)	9	0.97 (0.91, 1.00)
	Measurement method
	Manual	7	0.93 (0.86, 0.99)	9	0.97 (0.89, 1.00)
	Computerized 1	0.99 (N/A)	8	0.98 (0.90, 1.00)
	Full-size table
	Table V. Results of random-effects pooling for studies that reported estimates of CV by different
	study characteristics

Additionally the OARSI FDA OA Initiative received financial support from the following companies: Amgen, ArthroLab, AstraZeneca, Bayer Healthcare, Chondrometrics, CombinatoRx, Cypress BioScience, DePuy Mitek, Expanscience, 4QImaging, Genevrier/IBSA, Genzyme, King (Alpharma), Merck, Merck Serono, NicOx, Pfizer, Rottapharm, Smith & Nephew, Wyeth.

While individuals from pharmaceutical, biotechnology and device companies actively participated in on-going working group discussions, due to the conflict of interest policy enacted by OARSI, these individuals were not allowed to vote on the final recommendations made by OARSI to the FDA. This work was also supported in part by the National Institutes of Health (K24 AR057827 and T32 AR055885) and they had no role in the content being published.

Author contributions

• Conception and design (WMR, JFM, EL).

•

Analysis and interpretation of the data (WMR, JFM, DJH, PGC, JNK, EL).

•

Drafting of the article (WMR, JFM, DJH, PGC, JNK, EL).

•

Critical revision of the article for important intellectual content (WMR, JFM, DJH, PGC, JNK, EL).

•

Final approval of the article (WMR, JFM, DJH, PGC, JNK, EL).

•

Statistical expertise (WMR, EL).

• Collection and assembly of data (WMR, JFM).

The corresponding author had full access to all the data in the study and had final responsibility for the decision to submit for publication.


Florent Autin
email: autin@cmi.univ-mrs.fr

Testing means from sampling populations with undefined labels

Keywords: Asymptotic distribution, hypothesis testing, missing labels, mixturemodels AMS Subject Classification: 62D05, 62F03, 62F05

We consider the problem of testing means from samples of two populations for which the labels are not defined with certainty. We show that this problem is connected to another one that is testing expected values of components of mixture-models from two data samples. The underlying mixture-model is associated with known varying mixing-weights. We provide a testing procedure that performs well. Then we point out the loss of performance of our method due to the mixing-effect by comparing its numerical performances to the Welch's t-test on means which would have been done if true labels were available.

Introduction

In many cases, researchers can be interested in gathering information about two populations in order to compare them. In that setting, tests of significance are useful statistical tools for detecting a difference between two population parameters. Related application fields are numerous. Some examples are genetics, neuronal data analysis, medicine, biology, physics, chemistry, social sciences, among other fields.

Consider the Gaussian setting, for which each data of the two populations under study is assumed to follow a normal distribution. Let us recall that this assumption can be tested beforehand using a normality test, such as the well-known Shapiro-Wilk or Kolmogorov-Smirnov test, or it can be assessed graphically using a normal quantile plot. Comparisons between the means of the populations are usually carried out by using t-statistic and lead to the well known Student's t-test or Welch's t-test (see Welch [START_REF] Welch | The generalization of "Student's" problem when several different population variances are involved[END_REF]). These t-tests are popular because of their ease of use and their good performances. Moreover they are robust in the sense that they still perform well when the components are not really Gaussian, provided that the samples size are large enough. Nevertheless, these testing methods require to know the label of each data, that is the population each data is associated with. Unfortunately, researchers do sometimes not get this information. Indeed, one can imagine some cases where the labels of data are erroneous or uncertain, i.e. some data of each population do not deal with the population we want to compare. To give an example of such a situation with lack of information, we consider two populations -New York and California people -reduced to people that take bus/trolley bus or walk to go working. Focusing on working people that take bus/trolley bus, suppose you are interested to know whether travel time of people from New York is significantly different to the one from California from a sample of people where the place they live -New York or California -is available but the way of travel (the label) associated with each data in hand is not. This kind of situation is the one we are interested in. Indeed, we want to address the problem of testing means of (sub)populations when the labels of data are uncertain. More precisely, we first propose to show that this testing problem can be reformulated as testing the expected values of components from two samples of independent mixture variables. In our study of real data, we shall assume that the mixing-weights are known. It means that proportions of people walking or using bus/trolley bus for each population (New York and California) are known, with respect to an auxiliary variable (age for instance). Then, we provide a testing procedure that takes into account this information on populations -and we discuss about its performances. The testing procedure we propose is directly inspired from ideas in Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF]. In this previous work, a nonparametric procedure has been proposed to test whether the densities of two independent samples of independent random variables result from the same mixture of components or not. The value of the test statistic requires to invert in some sense the mixing-weights operators of samples (see Definition 1) as a preliminary step to be calculated. This testing procedure was proved to be powerful since it is minimax over Besov spaces (more details are given in paragraph 3.1 in Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF]). More focusing on practical purposes, we show that providing a testing procedure that incorporates combinatory ideas -provided that the mixing-weights are knownis quite relevant compared to a procedure usually used in classification.

Paper is organized as follows. In Section 2 we present the mixture-model we are interested in. Connection between the testing problem for which the labels of data are not certain and the problem of testing the expected value of the components involved in the mixture-model is provided. In Section 3, we present three testing procedures. The first one is the Oracle Procedure that uses Welch's t-test on data associated with the label of the components we want to test.

Of course this procedure is not tractable for the testing problem with lack of information about labels but it will be used as a benchmark to assess the loss of performances of the other tractable testing procedures. The second testing procedure we present is the Expert Procedure that uses Welch's t-test on data that are supposed to have, with probability larger than or equal to one half, the label associated with the components we focus on. The third and last procedure, namely the Mixing Procedure, uses combinatory properties leading to a new performing test. Section 4 deals with numerical experiments to point out the good performances of the Mixing Procedure -the one we suggestcompared to the Expert Procedure and to assess the loss of performances due to the mixing-effect compared to the Oracle Procedure. An application to real data is also presented whereas a brief conclusion is postponed in Section 5. Finally, the technical lemmas and the proposition we used to prove our main theoretical result (see Theorem 1) together with their proofs can be found in the appendix.

Model description and hypothesis testing problem 2.1 Mixture-models with varying mixing-weights

Let X 1 , . . . , X n be independent random variables such that, for any 1 ≤ i ≤ n, the density of X i on R, denoted by f X i , is a mixture density with components p 1 and p 2 and mixing-weights ω 1 (i) and ω 2 (i), i.e.

f X i = ω 1 (i)p 1 + ω 2 (i)p 2 .
We also introduce labels attached to X 1 , . . . , X n , denoted by u 1 , . . . , u n . This point of view is one interpretation of mixture-models among others (see Section 1.4 in McLachlan and Peel [START_REF] Mclachlan | Finite mixture-models[END_REF]). The main difference lies in considering varying mixing-weights in our model. This point is very important (see Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF]). Therefore our model cannot be described as a mixture-model in the usual sense.

Similarly to the sample X 1 , . . . , X n , we consider a sample of independent random variables Y 1 , . . . , Y n ′ such that, for any 1 ≤ i ≤ n the density of Y i on R, denoted by f Y i , is a mixture density with components p ′ 1 and p ′ 2 and mixingweights

ω ′ 1 (i) and ω ′ 2 (i), i.e. f Y i = ω ′ 1 (i)p ′ 1 + ω ′ 2 (i)p ′ 2 .
We also introduce labels attached to Y 1 , . . . , Y n , denoted by v 1 , . . . , v n and we assume that this second sample is independent from the first one.

If t . denotes the transpose operator, the two mixture-models we have just introduced can be rewritten in a simpler way as follows: - The reader can legitimatelly wonder why the age is assumed to be known and not the ways of transportation to go working. One can think about at least two good reasons. The first one is an a priori reason. The survey would be rather lengthy if all interesting variables were included. Therefore the survey is restricted to a small set of informative variables strongly linked with the interesting variables. Moreover these informative variables can be chosen as objective as possible and easily recordable. This can be called planned missing values (see Graham [START_REF] Graham | Missing data analysis: making it work in the real world[END_REF]). The other reason is an a posteriori one. During the data analysis of a survey, researchers are often confronted with new hypotheses to test. In many situations, the relevant variables have not been recorded and researchers have to plan a new survey which includes these new variables in order to check these hypotheses. This leads to a waste of time and money. Our testing problem on means from data with undefined labels can be associated with the testing problem [START_REF] Butucea | Nonparametric homogeneity tests[END_REF] in the mixture-model [START_REF] Autin | Test on components of densities mixture[END_REF]. Indeed, it corresponds to a testing problem on means for which labels of data are unavailable: the only information on the X i 's label (resp. Y i 's label) is the probability ω l (i) (resp. ω ′ l (i)) that it corresponds to l, for any l ∈ {1, 2}. In other words, the added information on subpopulations we get is the knowledge of the mixing-weights operators.

f X = Ω X p and f Y = Ω Y p ′ , (1)
f X = t (f X 1 , . . . , f Xn ), f Y = t (f Y 1 , . . . , f Yn ), -p = t (p 1 , p 2 ), p ′ = t (p ′ 1 , p ′ 2 ), -Ω X = (ω l (i)) i,l , Ω Y = (ω ′ l (i)) i,l .

Hypothesis testing problem

We recall that two data samples X = t (X 1 , . . . , X n ) and Y = t (Y 1 , . . . , Y n ) are considered. For a chosen label l ∈ {1, 2}, we are interested in testing whether components p l and p ′ l have the same expected value or not. We want to address this problem in a general context that is: the parameters of variance σ 2 k and σ ′ 2 k of the components p k and p ′ k are unknown whatever k ∈ {1, 2}.

For a fixed l ∈ {1, 2}, when respectively denoting by m l and m ′ l the expected value of the components we focus on, the testing problem we consider is lying on the two following hypotheses:

the null hypothesis H 0 : m l = m ′ l , (2)
the alternative hypothesis

H 1 : m l = m ′ l .
We recall that providing a procedure to solve the testing problem (2) means giving a decision rule (or test) ∆ ∈ {0, 1} that relies on the value of a measurable function T (test statistic) of X 1 , . . . , X n and Y 1 , . . . , Y n .

As usual, ∆ = 1 will mean deciding H 1 whereas ∆ = 0 will mean deciding H 0 .

Description of testing procedures

In this section we introduce the testing procedures we are interested in.

Oracle test: ∆ o

The first testing procedure we present is called the Oracle Procedure. This is a two steps procedure. First step consists in recovering the true labels of data. Second step lies on using the Welch's t-test on data with label l in order to know whether m l and m ′ l can be judged as different. This test cannot be used in our context where the true labels are unknown but it will be used as a benchmark when comparing the performances of the other testing procedures. It corresponds to the procedure proposed by the oracle: any statistician having information on labels.

Here we describe into details the Oracle Procedure. Let us denote by

-n l = n i=1 1{u i = l} and n ′ l = n i=1 1{v i = l}, -X(l) = 1 n l n i=1 X i 1{u i = l} and Ȳ (l) = 1 n ′ l n i=1 Y i 1{v i = l}, -σ2 l = 1 n l n i=1 (X i -X(l) ) 2 1{u i = l} and σ′ 2 l = 1 n ′ l n i=1 (Y i -Ȳ (l) ) 2 1{v i = l}.
The Oracle test ∆ o lies on the test statistic T o defined as follows

T o := | X(l) -Ȳ (l) | σ2 l n l + σ′ 2 l n ′ l .
Under the null hypothesis, the asymptotic law of T o is known to be the Standard Gaussian one, namely N (0, 1). Hence, ∆ o = 1{T o > q r } is a test with asymptotically type I error equal to r (0 < r < 1), where q r is the quantile of order 1 -r 2 of the Standard Gaussian law.

Expert test: ∆ e

The testing procedure we describe now is lying on a method used in classification. It is a two steps procedure. The first step consists in allocating label l to any data X i such that ω l (i) ≥ 1 2 and to any data Y j such that ω ′ l (j) ≥ 1 2 . The second step consists in using the Welch's t-test on the two subsamples of data that have been assigned to label l to know whether m l and m ′ l can be judged as different. Notice that it means that the Welch's t-test is done on data having possible wrong labels.

Put

-n l,e = n i=1 1{ω l (i) ≥ 1 2 } and n ′ l,e = n i=1 1{ω ′ l (i) ≥ 1 2 }, - X(l) e = 1 n l,e n i=1 X i .1 ω l (i) ≥ 1 2 and Ȳ (l) e = 1 n ′ l,e n i=1 Y i .1 ω ′ l (i) ≥ 1 2 , -σ2 l,e = 1 n l,e n i=1 (X i -X(l) e ) 2 1 ω l (i) ≥ 1 2 , - σ′ 2 l,e = 1 n ′ l,e n i=1 (Y i -Ȳ (l) e ) 2 1 ω ′ l (i) ≥ 1 2 .
The Expert test ∆ e relies on the test statistic T e defined as follows

T e := | X(l) e - Ȳ (l) e | σ2 l,e n l,e + σ′ 2 l,e n ′ l,e
.

Then, the decision rule is done by putting ∆ e = 1{T e > q r }.

Mixing test: ∆ m

The last testing procedure we propose is inspired from some ideas provided in Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF]. Using combinatory methods, it proposes to invert in some sense the mixing-weights operators so as to provide a new test that will be proved to perform well. Let us describe this new testing procedure into details.

Let us denote by A X and A Y the matrices with n lines and 2 columns satisfying

t Ω X A X = t Ω Y A Y = n 0 0 n . (3)
Notations: For any (i, l) ∈ {1, . . . , n} × {1, 2}, we denote respectively by a l (i) (resp. a ′ l (i)) the entries of A X (resp. A Y ) associated with line i and column l.

Following Maiboroda [START_REF] Maiboroda | An asymptotically effective estimate for a distribution from a sample with a varying mixture[END_REF] or Pokhyl'ko [START_REF] Pokhyl'ko | Wavelet estimators of a density constructed from observations of a mixture[END_REF], solutions of equations ( 3) are given by

a l (i) = n det( t Ω X Ω X ) 2 k=1 (-1) l+k γ (X) lk ω k (i), a ′ l (i) = n det( t Ω Y Ω Y ) 2 k=1 (-1) l+k γ (Y ) lk ω ′ k (i),
where γ (X) lk and γ (Y ) lk are respectively the minor (l, k) of the matrix t Ω X Ω X and of the matrix t Ω Y Ω Y .

For any l ∈ {1, 2}, (m l , m ′ l ) can be estimated by the method of moments when using estimators ( ml , m′ l ) defined as follows:

( ml , m′ l ) = A (l) X , X n , A (l) Y , Y n := 1 n n i=1 a l (i)X i , 1 n n i=1 a ′ l (i)Y i , provided that A (l) X and A (l)
Y respectively denote the l-th column-vector of matrices A X and A Y .

The Mixing test ∆ m lies on the test statistic T m defined by:

T m := | ml -m′ l | V(l) n , (4)
where V(l) n is the estimated variance of ml -m′ l , that is

V(l) n = 1 n 2 n i=1 a 2 l (i) (X i -ω 1 (i) ml -ω 2 (i) ml ) 2 + a ′ l 2 (i) (Y i -ω ′ 1 (i) m′ l -ω ′ 2 (i) m′ l ) 2 .
Remark 1 As discussed in Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF], for any l ∈ {1, 2} the random variable ml (resp. m′ l ) is a good estimator for m l (resp. m ′ l ). Hence if the distance between ml and m′ l is judged too large, the rejection of the null hypothesis H 0 looks better. This idea motivates the choice of the test statistic T m we defined above.

Under the null hypothesis H 0 , the asymptotic law of T m is known, according to the following Theorem.

Theorem 1 Let l ∈ {1, 2}. Assume that • the components within the mixture-model (1) have moments with order 4,

• the mixing-weights of the mixture-model (1) are such that

lim n→+∞ sup i=1,...,n a 2 l (i) n i=1 a 2 l (i) = lim n→+∞ sup i=1,...,n a ′ 2 l (i) n i=1 a ′ 2 l (i) = 0. ( 5
)
Then, under the null hypothesis H 0 , the law of T m is asymptotically the Standard Gaussian one, i.e.

T m L → N (0, 1). (6)
Hence, ∆ m = 1{T m > q r } is a test with asymptotically type I error equal to r (0 < r < 1).

Remark 2 A wide range of mixing-weights of the mixture-model (1) satisfy the condition [START_REF] Maiboroda | An asymptotically effective estimate for a distribution from a sample with a varying mixture[END_REF]. Examples of such mixing-weights are given in (12) of Section 4.

Proof:

To prove Theorem 1, notice that it suffices to prove that for any l ∈ {1, 2}

1.

1

n n i=1 a l (i)X i -m l 1 n 2 n i=1 a 2 l (i) (X i -ω 1 (i) m1 -ω 2 (i) m2 ) 2 L → N (0, 1), 2.
1

n n j=1 a ′ l (i)Y j -m ′ l 1 n 2 n j=1 a ′ l 2 (i) (Y j -ω ′ 1 (j) m′ 1 -ω ′ 2 (i) m′ 2 ) 2 L → N (0, 1).
Because of the independence between the two samples and the fact that under the null hypothesis H 0 , m l = m ′ l . Since these two results of convergence can be proved by an analogous way, we only focus on proving the first one that can be rewritten as follows for any l ∈ {1, 2}:

n i=1 a l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) n i=1 a 2 l (i) (X i -ω 1 (i) m1 -ω 2 (i) m2 ) 2 L
→ N (0, 1).

Denote, for any n ∈ N * , any l ∈ {1, 2} and any 1

≤ i ≤ n B (l) n = n i=1 a 2 l (i)E (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 , (7)
B(l

) n = n i=1 a 2 l (i) (X i -ω 1 (i) m1 -ω 2 (i) m2 ) 2 . ( 8
)
From Proposition 1,

n i=1 a l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) B (l) n L → N (0, 1).
In the sequel, we aim at proving that a same kind of result holds when replacing parameter

B (l)
n by the estimator B(l) n . In other words, the result we want to prove is the following:

n i=1 a l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) B(l) n L → N (0, 1). ( 9
)
From Slutsky theorem, it suffices to prove that estimator B(l)

n of B (l)
n is consistent. We propose to divide the proof of this consistency into two steps. First we prove

n i=1 a 2 l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 B (l) n P roba → 1. ( 10
)
The second step consists in replacing m 1 and m 2 by their consistent estimators m1 and m2 and in checking that the convergence in probability still holds.

From this point we need more assumptions, that is to say the existence of the fourth order moment for p 1 and p 2 .

Let us prove the first step. We apply Bienayme-Chebyshev inequality, for any ǫ > 0:

P n i=1 a 2 l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 -B (l) n > B (l) n ε ≤ n i=1 a 4 l (i)Var (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 (B (l) n ε) 2 ≤ n i=1 a 4 l (i)E (X i -E(X i )) 4 (B (l) n ε) 2 ≤ sup j=1,...,n a 2 l (j) B (l) n n i=1 a 2 l (i)C(m 1 , m 2 , p 1 , p 2 ) B (l) n ε 2 ≤ sup j=1,...,n a 2 l (j) B (l) n min(σ 2 1 , σ 2 2 ) -1 C(m 1 , m 2 , p 1 , p 2 ) ε 2 .
Last inequalities are obtained by using Lemma 4 and Lemma 2. The right part of the last inequality is the product of two terms. The left one tends to 0 when n goes to infinity because of assumption [START_REF] Maiboroda | An asymptotically effective estimate for a distribution from a sample with a varying mixture[END_REF]. The right one is a constant that only depends on ǫ and the parameters of p 1 and p 2 . When considering the limit in infinity with respect to n, we conclude that property (10) holds.

We end by proving the second step. We have

n i=1 a 2 l (i) (X i -ω 1 (i) m1 -ω 2 (i) m2 ) 2 = n i=1 a 2 l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 +2 n i=1 a 2 l (i)(X i -ω 1 (i)m 1 -ω 2 (i)m 2 )(ω 1 (i)(m 1 -m1 ) + ω 2 (i)(m 2 -m2 )) + n i=1 a 2 l (i) (ω 1 (i)(m 1 -m1 ) + ω 2 (i)(m 2 -m2 )) 2 .
The first term is exactly the one appearing in the first step and also converges to 1 in probability when divided by B (l)

n . We turn to the second term. Cauchy-Schwarz inequality entails that

n i=1 a 2 l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) (ω 1 (i)(m 1 -m1 ) + ω 2 (i)(m 2 -m2 )) ≤ n i=1 a 2 l (i) (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 × 2(m 1 -m1 ) 2 n i=1 a 2 l (i)ω 2 1 (i) + 2(m 2 -m2 ) 2 n i=1 a 2 l (i)ω 2 2 (i).
When divided by B (l)

n , the first term of the righthand-side converges to 1 in probability: it is the result of the first step. By using Lemma 2, one gets

max n i=1 a 2 l (i)ω 2 1 (i), n i=1 a 2 l (i)ω 2 2 (i) ≤ B (l) n min(σ 2 1 , σ 2 2 ) -1 .
Hence the second term of the right-hand side of the inequality converges to 0 in probability when divided by

B (l)
n because of the consistency of estimators ml (see Lemma 5). Hence second the term we are interested in converges to 0 in probability when divided by B (l) n . We can proceed in the same way in order to prove that the third term converges to 0 in probability when divided by B (11)

We conclude that the exact variance B (l)

n can be replaced by the consistent estimator B(l) n for the result of convergence. In other words, the property (9) holds.

4 Numerical experiments

Numerical performances of the Mixing-test

In this section we provide numerical experiments and we discuss about the performances of our testing procedure. What we often expect is a gain of performance of the test ∆ m -that is to say a smaller type II error when the type I error is chosen to be r = 0.05 -comparatively to the test ∆ e . Without loss of generality, we suppose that n is even. We consider the Gaussian setting and we assume in this section that the mixingweights operators Ω X and Ω Y have the following form:

Ω X =         α 1 -α . . . . . . α 1 -α 1 -β β . . . . . . 1 -β β         and Ω Y =         α ′ 1 -α ′ . . . . . . α ′ 1 -α ′ 1 -β ′ β ′ . . . . . . 1 -β ′ β ′         , ( 12
)
where n 2 data from X (resp. Y) deal with the couple of mixing-weights (α, 1-α) (resp. (α ′ , 1α ′ )) and the other n 2 data from X (resp. Y) deal with the couple of mixing-weights (1β, β) (resp. (1β ′ , β ′ )). Suppose now that our testing problem is dealing with the first component, i.e. l = 1 and that Ω X and Ω Y are full rank matrices, i.e. α + β = 1 and α ′ + β ′ = 1.

Mixing-test versus Expert-test

In this paragraph we provide a motivation for the use our testing procedure ∆ m . For the sake of simplicity we suppose that α = β and that α ′ = β ′ . For any value of (α, α ′ ) ∈] 1 2 , 1[ 2 , there are many situations where the performance of the Expert test is quite bad even if the numbers of observations n is large.

• Dealing with two components with equal expected value, ∆ e can most of time detect a difference between these components (wrong decision) whereas our test doesn't. For instance, suppose that

m 1 = m ′ 1 and that m ′ 2 is large away from m 2 as α = α ′ . Since E X(1) e = E Ȳ (1)
e , using ∆ e to detect equality between components m 1 and m ′ 1 would be a very bad choice in that context. For n large enough, it would imply that T e > t r with high probability. Hence, the wrong decision H 1 may often be done.

An example of such a situation is given here in the case where α = α ′ = 0.9. Consider the testing problem (2) and suppose

σ 1 = σ ′ 1 = σ 2 = σ ′ 2 = 1 and
= m ′ 1 = 0, m 2 = 1 and m ′ 2 = m 2 + δ.
For varying values of n, δ and 40 000 repetitions of ∆ e with r = 0.05, we give the percentage of wrong decisions H 1 in Table 2.

Notice that the percentage of wrong decisions by ∆ e turns up as n grows up and can be quite important if m ′ 2 is sufficiently far away from m 2 . Most of time, the expert detects a difference between the components m 1 and m ′ 1 but there is not in that context. Comparatively speaking, the percentage of wrong decisions by ∆ m is around 0.05.

• Most of time ∆ e fails to detect a difference between two components with different expected value whereas our test doesn't. For instance, suppose that m 1 = m ′ 1 and that

m 2 ≈ (1 -α) -1 (α ′ m ′ 1 + (1 -α ′ )m ′ 2 -αm 1 ) . Since E(X i ) ≈ E(Y i ), for any 1 ≤ i ≤ n 2 ,
using ∆ e to detect the difference between m 1 and m ′ 1 would be a very bad choice in that context. Indeed, according to the law of large numbers, with high probability -that increases as n goes up -X(1)

e and Ȳ (1)
e would be very close to each other. It would imply that T e ≤ 1.96 with high probability. True decision H 1 would be taken only in 5 % of cases.

An example of such a situation is given here in the case α = α ′ = 0.9. Consider the testing problem (2) and suppose that

σ 1 = σ ′ 1 = σ 2 = σ ′ 2 = 1 and that m 1 = 0, m ′ 1 = 0.1, m 2 = 1 and m ′ 2 = 2.
For varying values of n and 40 000 repetitions of ∆ m with r = 0.05, we give the percentage of correct decisions from ∆ m in Table 3.

As expected, the percentage of correct decisions by ∆ m goes up as n grows up. But it is not the case for the percentage of correct decisions by ∆ e which are always around 0.05. Most of time, the expert is unable to detect the difference between the components in that context.

Finally we conclude that it is better to choose ∆ m for the problem we are interested in.

Mixing-test versus Oracle-test

In this paragraph we compare the empirical powers of ∆ m to the Oracle test ∆ o ones, when taking r = 0.05 and the same parameters as the last example. We recall that the empirical power of any test ∆ corresponds to the numerical evaluation of the probability to correctly decide H 1 , according to ∆.

According to Table 4 we remark that the bigger n the better the powers of ∆ o and of ∆ m . Moreover we note that the empirical power of the Mixing test is not bad when comparing to the Oracle test.

In Table 5 we give the empirical power P m of ∆ m measured in samplings of size n = 1000 in the case where

m 1 = 0, m ′ 1 = δ, m 2 = 1 and m ′ 2 = 0.
As expected, looking at Table 5, • quantity P m depends on the intrinsic difficulty of the problem. Indeed the larger the absolute value of quantity δ := m ′ 1m 1 , the easier the problem of detection and so the more powerful the test,

• the larger the degree of certainty α the better the power of ∆ m . This is due to the fact that the expectation of the number of wrong labels considered by the Expert Procedure grows up as α goes down. As previously discussed, we expect that the better the degree of certainties of the expert, the better the performance of the test ∆ m . This statement is highlighted here when considering the same parameters as before, δ = 0.5 and many choices of couple (α, α ′ ). For each choice of (α, α ′ ) done, we provide in Table 6 the empirical power P m of our test ∆ m that is the percentage of correct detection of a difference between m 1 and m ′ 1 . Interpretation of the results presented in Table 6 goes in the same way as Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF]: the bigger the smallest eigenvalue of both operators t Ω X Ω X and

t Ω Y Ω Y that is λ min = 1 2 (1 -2 min(α, α ′ )(1 -min(α, α ′ ))
) the better the power of our test ∆ m . Note that, the larger the minimum value between α ∈] 1 2 , 1[ and α ′ ∈] 1 2 , 1[ the bigger λ min .

Brief conclusion

Let us summarize the main facts. First, in some cases experts can be completely wrong because of the overall design, that is to say the link between the means of the components and the mixing-weights. This is a serious issue for the Expert test. The results become worse and worse as the sample size increases. The test adapted to the varying mixing-weights that we propose does not suffer from this drawback. The second fact is the good behavior of our test compared to the Oracle test. Although it is behind, the power is quite acceptable. The last important fact which has already been stressed by Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF] is the effect of the mixing-weights. It is known a priori thanks to the smallest eigenvalue of the operators t Ω X Ω X and t Ω Y Ω Y . This point is important as the statistician can act in order to counter to this effect, e.g. he can improve the accuracy of the expert system giving the mixing-weights or increase the sample sizes.

Application to real data

In this section we apply our methodology to real data and we discuss about the results.

Description of the data

We have selected data from U.S. Census Bureau website, more precisely PUMS 2006 (see [START_REF]American Community Survey. Profile: California, Illinois and New York[END_REF]). We are interested in comparing travel time of people living either in the state of New York (abbreviated in NY) or either in the state of California (abbreviated in CA). Two ways of transportation have been kept: Bus/trolley bus and Walk. We have also kept a variable linked to age as it will be useful for the mixture-model with varying mixing weights. This variable records the fact that a person is over 21 years old or under 20 years old.

Here are few facts to roughly describe the PUMS sample. Table 7 gives one-level information.

In Table 8 we compute the mean and the standard deviation (in parentheses) of the travel time according to the categorical variable means of transportation to go working. As it can be seen in Table 8 there might be no difference between New York and California. Nevertheless if the means of transportation is unavailable, it will be perilous to decide when considering the whole sample without any other information. Indeed as shown in Table 7, the difference between New York and California is decreased because of the structure of the population (less people under 20 years old in New York).

Methodology

We assume in the sequel that the information about the way of transport (labels) are unavailable at the microdata level. We are going to apply the test ∆ m adapted to the varying mixing-weights mixture-model. The age variable is the only auxiliary information available at the microdata level that permits to get the mixing-weights to our mixture-model (1).

We consider the following sample: 500 persons over 21 and 500 persons under 20 were randomly sampled in each state (n = 1000).

We applied three testing procedures:

1. Oracle test, 2. Expert test, 3. Mixing test.

First we test the equality of the averages when the ways of transportation to work is Bus/trolley bus (label 1) in Table 9. In this case, the other means of transportation to work is considered as a nuisance parameter. Next we reverse the set-up. We test the equality of the averages when the means of transportation to work is Walk (label 2) in Table 10. Bus/trolley bus is now a nuisance parameter.

A tough situation

Here we are also interested in comparing travel time of people living either in the state of New York or either in the state of Illinois (abbreviated in IL). Data come from U.S. Census Bureau [START_REF]American Community Survey. Profile: California, Illinois and New York[END_REF]. Two ways of transportation to work have been kept: Bus/trolley bus or Railroad. We have also kept the gender variable as it will be useful for the varying mixing-weights mixture-model. As it will be seen, the situation is much more involved compared to the one in the previous section. Here are few facts to roughly describe the PUMS sample. Table 11 gives onelevel information.

The mixing-weights depend on the gender as illustrated in Table 12.

According to the notations we introduced in (12) and Table 12, (α, β) = (0.558, 0.247) (α ′ , β ′ ) = (0.508, 0.346).

In Table 13 we compute the mean and standard deviation (in parentheses) of the travel time according to the categorical variable way of transportation to work.

Once again the difference in travel time is decreased if we consider the entire population. This is due to its structure. As there are more men and women who use railroad in Illinois, the general average of travel time is increased. This is reverse in New York.

In Table 14 we test the equality of the averages when the ways of transportation to work is Bus/trolley bus.

In Table 15 we reverse the set-up and we test the equality of the averages when the way of transportation to work is Walk.

Conclusion

From our point of view, one of the most interesting point is the usefulness of the varying mixing-weights model. It is a versatile model that can be used in many situations with missing microdata but aggregated information. The application treated exemplifies the modeling.

The second take-away message is the excellent performances of the Mixing test we propose. They can be guessed a priori thanks to the smallest eigenvalue of operators involved within the mixture-model. These nice performances were showed both theoretically and numerically.

To conclude let us precise that this work can be easily extended to mixturemodels with more than two components and can be done in a nonparametric setting when using the testing procedure proposed by Butucea and Tribouley [START_REF] Butucea | Nonparametric homogeneity tests[END_REF] as the Oracle test and the one given by Autin and Pouet [START_REF] Autin | Test on components of densities mixture[END_REF] as the Mixing test.

An interesting extension that should really be considered is the case of mixingweigths with errors. This arises when mixing-weigths are computed from a model with estimated parameters or from experts' evaluation. In this case the solution of ( 3) is no longer exact as the matrices Ω X and Ω Y are random. Preliminary simulation results tend to prove that moderate errors have a small effect.

Appendix

In this section we provide the technical lemmas and the proposition required to prove the asymptotic normality under interest. For the sake of simplicity, we present them with respect to X 1 , . . . , X n whereas an analogous version of them does exist for Y 1 , . . . , Y n . We recall that we assume that, for any l ∈ {1, 2} the mixing-weights of the model satisfy [START_REF] Maiboroda | An asymptotically effective estimate for a distribution from a sample with a varying mixture[END_REF].

n i=1 E W (l) ni 2 1 |W (l) ni | ≥ ǫ = n i=1 |y|≥ǫ y 2 dF W (l) ni (x) ≤ n i=1 a 2 l (i) B (l) n x>κn,x<κ ′ n (x -ω 1 (i)m 1 -ω 2 (i)m 2 ) 2 dF X i (x) ≤ 2 n i=1 a 2 l (i) B (l) n   l∈{1,2} ω l (i) 3 x>κn,x<κ ′ n (x -m l ) 2 p l (x) dx + ω 2 2 (i)ω 1 (i) x>κn,x<κ ′ n (x -m 2 ) 2 p 1 (x) dx + ω 2 1 (i)ω 2 (i) x>κn,x<κ ′ n (x -m 1 ) 2 p 2 (x) dx ≤ 2 n i=1 a 2 l (i) B (l) n sup n∈N *   (k,l)∈{1,2} 2 x>κn,x<κ ′ n (x -m k ) 2 p l (x) dx   . Using Lemma 2, for any n ∈ N * , n i=1 a 2 l (i) B (l) n ≤ min(σ 2 1 , σ 2 2 ) -1 .
Then, since variances under p 1 and p 2 are finite, the supremum over n tends to 0 in the integrals above, according to Lebesgue dominated convergence theorem.

Lemma 4 For any

1 ≤ i ≤ n, E (X i -E(X i )) 4 ≤ C(m 1 , m 2 , p 1 , p 2 ),
where C(m 1 , m 2 , p 1 , p 2 ) := 32 max

(k,l)∈{1,2} 2 (x -m k ) 4 p l (x)dx.
Proof:

We have Proof:

E (X i -E(X i )) 4 = E (X i -ω 1 (i)m 1 -ω 2 (i)m 2 ) 4 ≤ 8 ω 1 (i)
Let ǫ > 0 and l ∈ {1, 2}. We have, using Bienayme-Chebyshev inequality and Lemma 1

P (| ml -m l | > ε) ≤ 1 n 2 ǫ 2 n i=1 a 2 l (i)Var(X i ) = 1 n 2 ε 2 n i=1 a 2 l (i)   l∈{1,2}
ω l (i)σ 2 l + ω l (i)(1ω l (i))

(m 1 + m 2 ) 2 2   ≤ 2 nε 2 K   l∈{1,2} σ 2 l + (m 1 + m 2 ) 2 4   .
Last inequality is obtained by using assumption on the smallest eigenvalue of Ω ′ Ω, that is larger than Kn (with K > 0) and the fact that the supremum value for x ∈ [0, 1] of x → x(1x) is equal to 1 4 . The right-hand side clearly tends to 0 when n goes to infinity. We conclude that ml is consistent.

Proof:

We apply Theorem 4.2 in Petrov [START_REF] Petrov | Limit theorems of probability theory[END_REF]. It is the general setup for the Central Limit Theorem, for the triangular array of series (W ni ) i,n of independent random variables X i (that are not identically distributed).

If the three conditions are satisfied for any ǫ > 0 and any τ > 0 1. lim Let us prove that the three conditions are satisfied. Let ǫ > 0. Using Bienayme-Chebyshev inequality,

n i=1 P |W (l) ni | ≥ ε ≤ ǫ -2 n i=1 E W (l) ni 2 1 |W (l) ni | ≥ ǫ .
Hence, condition 1 is clearly satisfied by using Lemma 3.

Let us move to condition 2. We use the same trick as above. For any τ > 0 Condition 2 is clearly satisfied by using Lemma 3.

We end the proof with condition 3. There are two parts (because of two summands) in this condition. For the first part we proceed exactly as in condition The first summand is exactly equal to 1 and the second one tends to 0 as n goes to infinity, according to Lemma 3. Therefore it remains to prove that the second part tends to 0 when n goes to infinity. Because the variables W Still using Lemma 3, we conclude that the second part we are interested in tends to 0 when n goes to infinity, as expected.

Proposition 1

 1 For any l ∈ {1, 2}, any n ∈ N * and any1 ≤ i ≤ n consider W (l)ni , defined as in (15).

  The first summand is equal to 0 as the variables W

Table

  Definition 1The matrices Ω X and Ω Y involved in the model (1) are called the mixing-weights operators.

	Definition 2 Any mixture-model (1) such that Ω X and Ω Y are full rank ma-
	trices is called mixture-model with varying mixing-weights.
	2.2 Example of modeling with mixture-models
	Let us illustrate this theoretical set-up with the example cited in the introduc-
	tion. The random variables X 1 , . . . , X n correspond to the travel times of people
	in the state of New York and the random variables Y 1 , . . . , Y n to travel times in
	the state of California. The labels are the ways of transportation to go working
	and can be either Bus/trolley bus (label 1) or Walk (label 2) . The last step to
	complete the mixture model is to describe the mixing-weights for each observa-
	tion. In each state the mixing-weights strongly depend on the age (over 21 or
	under 20 years old (y.o.)). Table 1 illustrates this fact.
	This table leads to the following mixing-weights:
	ω 1 (i) = 0.5193, ω 2 (i) = 0.4807
	if the person i is over 21 y.o. and lives in New York,
	ω 1 (i) = 0.3465, ω 2 (i) = 0.6535
	if the person i is under 20 y.o. and lives in New York,
	ω ′ 1 (i) = 0.574, ω ′ 2 (i) = 0.426
	if the person i is over 21 y.o. and lives in California,
	ω ′ 1 (i) = 0.4277, ω ′ 2 (i) = 0.5723
	if the person i is over 21 y.o. and lives in California.

Table 2 :

 2 Percentage of wrong decisions by ∆ e

		δ / n	100	200	500	1000 2000
		0.5	0.057 0.064 0.086 0.121 0.191
		1	0.074 0.098 0.172 0.302 0.521
		2	0.126 0.210 0.462 0.749 0.963
		3	0.188 0.350 0.722 0.950 0.999
		Table 3: Percentage of correct decisions by ∆ m
	n	500	1000 2000 3000 4000 5000 6000
	∆ m 0.146 0.242 0.438 0.595 0.718 0.810 0.879
	that m 1

Table 4 :

 4 Empirical Power of ∆ o and ∆ m

	Test / n	500	1000 2000 3000 4000 5000 6000
	∆ o	0.200 0.349 0.609 0.783 0.886 0.942 0.973
	∆ m	0.149 0.245 0.427 0.585 0.704 0.798 0.868

Table 5 :

 5 Empirical Power P m of ∆ m for varying α = α ′

	δ / α	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	1
	0.1	0.071 0.092 0.125 0.161 0.196 0.239 0.275 0.317 0.353
	0.2	0.130 0.226 0.353 0.484 0.603 0.700 0.780 0.837 0.883
	0.3	0.238 0.446 0.659 0.816 0.912 0.960 0.983 0.999	1
	0.4	0.374 0.671 0.882 0.938 0.993	1	1	1	1

Table 6 :

 6 Percentage of correct decisions by ∆ m

	(α, α ′ ), / n	100	200	500
	(0.90, 0.60) 0.153 0.254 0.533
	(0.80, 0.70) 0.311 0.536 0.896
	(0.75, 0.75) 0.332 0.564 0.918
	4.1.3 Comparisons on performances of ∆ m for varying values of
	(α, α ′ )

Table 7 :

 7 Description

		of the population
		NY	CA
	Total	9189	8935
	Over 21	90.39% (8306) 87.04% (7803)
	Under 20	9.61% (883)	12.96% (1162)
	Walk	49.73% (4570) 44.5% (3989)
	Bus/trolley bus 50.27% (4619) 55.5% (4979)

Table 8 :

 8 One-way analysis of travel time (in minutes)

	Walk	Bus/trolley bus Walk and Bus/trolley bus
	NY 12.25 (12.18)	47.26 (28.79)	29.85 (28.23)
	CA 11.23 (12.23)	45.12 (28.84)	30.04 (28.49)

Table 9

 9

	: Bus/trolleybus decisions
		Decision n = 1000 p-value
	Oracle test	not rejected	0.24
	Expert test	not rejected	0.42
	Mixing test	not rejected	0.11
	Table 10: Walk decisions Decision n = 1000 p-value
	Oracle test	not rejected	0.23
	Expert test	rejected	0.04
	Mixing test	not rejected	0.48
	For comparison purpose we have also applied the so-called Expert test. The
	type I error is chosen to be 0.1.
	According to the notations we introduced in (12) and to Table 1,
	(α, β) = (0.5193, 0.6535) (α ′ , β ′ ) = (0.574, 0.5723).

Table 11

 11

	: Description of the population
		NY	IL
	Total	6974	2899
	Men	46.5% (3247) 48.2% (1398)
	Women	53.5% (3727) 51.8% (1501)
	Bus/trolley bus 66.2 % (4619) 58.4% (1692)
	Railroad	33.8% (2355) 41.6% (1207)
	Table 12: Mixing-weights
		Bus/trolley bus	Railroad
	NY men	55.8 % (1813)	44.2% (1434)
	NY women	75.3 % (2806)	24.7% (921)
	IL men	50.8 % (710)	49.2% (688)
	IL women	65.4 % (982)	34.6% (519)

Table 13

 13

		: One-way analysis of travel time
		Bus/trolley bus	Railroad	Bus/trolley bus and Railroad
	New York	47.3 (28.8)	71 (30)	55.3 (31.3)
	Illinois	41.8 (26.4)	63.1 (25.7)	50.7 (28.2)

  4 E (X im 1 )4 + 8 ω 2 (i) 4 E (X im 2 ) 4 = 8 ω 1 (i) 5 (xm 1 ) 4 p 1 (x) dx + ω 1 (i) 4 ω 2 (i) (xm 1 ) 4 p 2 (x) dx + ω 2 (i) 4 ω 1 (i) (xm 2 ) 4 p 1 (x) dx + ω 2 (i) 5 (xm 2 ) 4 p 2 (x) dx (xm k ) 4 p l (x)dx.Lemma 5 For any 1 ≤ l ≤ 2, the estimator ml = 1 n

	≤ 32 max (k,l)∈{1,2} 2
		n
		a l (i)X i of m l is
		i=1
	consistent, that is
	ml	P roba → m l .

Denote, for any n ∈ N * , any l ∈ {1, 2} and any 1

Lemma 1 For any

Proof:

For any 1 ≤ i ≤ n,

From Lemma 1, we immediately derive:

n be defined as in [START_REF] Pokhyl'ko | Wavelet estimators of a density constructed from observations of a mixture[END_REF].

Proof: For any l ∈ {1, 2}, by using Lemma 1 and the fact that, for any 1


Mehdi Benkhadra
email: m.benkhadra@wanadoo.fr

André Bouchot

Julien Gérard

Denis Genelot

Pierre Trouilloud

Laurent Martin

Claude Girard

Alain Danino

Friedrich Anderhuber

Georg Feigl

Flexibility of Thiel's embalmed cadavers: the explanation is probably in the muscles

Keywords:

Introduction

The flexibility of cadavers conserved using Thiel's embalming method remains unexplained. We aimed to perform microscopic comparison of muscle and tendon fibres from fresh cadavers (FC), formaldehyde-preserved cadavers (FPC) and cadavers conserved by Thiel's method (TC). Methods Muscle and tendon biopsies performed on FC, FPC and TC were conditioned and stained by Masson's trichrome, Sirius red and Ramon y Cajal, then studied under optical microscope. Alignment and integrity of the muscle and tendon fibres were studied.

Results

We observed a modification of the muscle fibres in all specimens from TC, regardless of the type of staining used. The muscle fibres taken from FC and FPC were relatively well conserved, both in terms of alignment and integrity. We did not observe any modification of collagen in either muscle or tendon fibres.

Conclusions

The considerable fragmentation of the muscle proteins, probably caused by certain corrosive chemicals, (e.g. boric acid) present in Thiel's embalming solution, could explain the suppleness of the TC. However, we cannot exclude the possibility of alterations in tendon or muscle collagen, since the experimental methods we used, did not allow for the study of collagen ultrastructure.

Introduction

Cadavers conserved using the method described by Thiel [7][8][9] are known to present a high degree of suppleness, allowing almost natural mobilisation of various body parts. Several applications in surgery and anesthesiology have been described, and all underline the very life-like characteristics of cadavers preserved with Thiel's method, affording working conditions almost as realistic as in living patients [1][2][3][4][5][6]11]. No explanation for this pronounced suppleness has been put forward to date. We aimed to perform histological comparison of muscle and tendon fibres taken from fresh cadavers (FC), formaldehydepreserved cadavers (FPC) and Thiel's embalmed cadavers, with a view to identifying modifications at tissue level that might explain this mobility.

Methods

This study was conducted in the Institute of Anatomy, Medical University Graz, Austria, and the Laboratory of Anatomy and Imagery Platform of IFR 100, Dijon, France.

Cadavers

The cadavers investigated were from persons who had donated their body to scientific research, according to the donation programmes of the two institutes of anatomy involved in this study. Six cadavers were used in our study: two preserved according to Thiel's embalming method, two embalmed with formalin and two unembalmed FC. The FC were obtained less than 24 h after death.

Materials

For each cadaver, biopsies were taken from the biceps brachii muscle and brachioradialis tendon. The biopsy was performed with a no. 23 scalpel, longitudinal to the fibre orientation.

Experimental protocol

The samples fixed by Thiel's method and formalin were dehydrated through progressive increasing concentrations of ethanol, then embedded in paraffin. Conversely, the unfixed fresh samples were embedded directly at -20°C in Tissue Tek Optimal Cutting Temperature (OCT) compound. Samples were sectioned using a microtome for paraffin-embedded samples and a cryostat for the OCT embedded samples.

Staining

The sections were stained using Masson's trichrome, sirius red and according to the Ramon y Cajal procedure. Masson's trichrome makes it possible to visualize type I collagen in green and keratin in red; Sirius red stains collagen types I and III, while Ramon y Cajal staining makes it possible to visualize tissue morphology. With this latter procedure, collagen shows up blue.

Comparison criteria

The sample sections were observed under the optical microscope at different magnifications to evaluate structure and ultrastructure. The parameters retained for comparison were integrity of the collagen fibrils in tendon and muscle tissue, as well as integrity and alignment of muscle fibres.

Results

As shown in Fig. 1d andg, we observed major modification of the integrity and the alignment of muscle fibres in all samples taken from Thiel's embalmed cadavers. These changes were visible regardless of the staining protocol used. The muscle fibres had a cut-up, -minced‖ appearance, but remained aligned. The conjunctive collagen fibrils were also undisturbed. No modification of tendon fibres was observed (Fig. 2).

Discussion

The modifications that we observed in the muscle samples from Thiel's embalmed cadavers, could conceivably explain the exceptional suppleness of these cadavers. It appeared that the collagen structure forming the basic muscle structure is preserved. The muscle fibres appeared as if they had been cut up or ground, and no longer had any hold, but were contained in a collagen sheath that remained intact, thus preserving the overall form of the muscle. Acids are well-known to have very corrosive effects on proteins and in our case muscle proteins. The only acid which is present in Thiel's mixture is boric acid, thus we suspected it to be the reason for the observed damage. To our knowledge, the other chemicals of the Thiel's embalming solution could not be involved in this very singular destruction of the muscles [7][8][9]. We also cannot exclude the possibility of alterations in collagen ultrastructure. By light microscope examination, we did not observe any such changes, but the collagen structure would need to be studied more closely by electron microscopy. It would be interesting to analyse other tissue types, which also maintain a near natural degree of mobility and plasticity. For example, blood vessels that can be dilated, when pulse is simulated using a balloon [4], peripheral nerves, which are stretched in extension-flexion of the limbs, and certain organs such as the lungs.

The muscular changes induced by Thiel's technique require precautions in the possible applications of this method. Appropriate applications, should be those requiring muscle flexibility, as realistic as possible, such as training in laparoscopic [11], or plastic [1, 10] surgery, airway training (tracheal intubation) [3] and ultrasound regional anesthesia [2]. Applications concerning the precise biomechanical properties of muscle fibres must be considered with caution. A study comparing biomechanical properties of muscles and tendons preserved by Thiel's method, formalin or unfixed, would be very interesting, and also refine the model and precise appropriate and unappropriate uses of Thiel's embalmed cadavers.

Conclusion

The alterations in muscle fibres that we observed in Thiel's embalmed cadavers could explain the unique suppleness and flexibility of cadavers embalmed by this method. However, alterations in collagen ultrastructure, undetected by our methodology of light microscope examination, could be an alternative explanation.

Fig. 1

 1 Fig. 1 Muscular fibre structure according to fixative product. Masson's trichrome staining of human muscle tissue observed at different magnifications: ×2.5 (a, b, c), ×10 (d, e, f), ×40 (g, h, i). Samples

Fig. 2

 2 Fig. 2 Tendon structure according to fixative product. Masson's trichrome staining of human tendon fibres observed at different magnifications: ×2.5 (a, b, c), ×10 (d, e, f), ×40 (g, h, i). Samples were fixed with Thiel's solution (a, d, g), or formalin (b, e, h) and embedded in paraffin, or unfixed (c, f, i) and embedded in OCT at -20°C

Acknowledgments

We would like to thank Mr Romain Toussaint and Mrs Amandine Chlemaire, preparers of the Imagery Platform of IFR100, for their helpful technical support. We thank Mrs Fiona Ecarnot for translation and editorial assistance. This study was financed by the fund of the laboratory of anatomy, Medical University of Dijon, France.