Comparison of measurement methods with a mixed effects procedure accounting for replicated evaluations (COM_{3}PARE): method comparison algorithm implementation for head and neck IGRT positional verification
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Abstract
Purpose
Comparison of imaging measurement devices in the absence of a goldstandard comparator remains a vexing problem; especially in scenarios where multiple, nonpaired, replicated measurements occur, as in imageguided radiotherapy (IGRT). As the number of commercially available IGRT presents a challenge to determine whether different IGRT methods may be used interchangeably, an unmet need conceptually parsimonious and statistically robust method to evaluate the agreement between two methods with replicated observations. Consequently, we sought to determine, using an previously reported head and neck positional verification dataset, the feasibility and utility of a Comparison of Measurement Methods with the Mixed Effects Procedure Accounting for Replicated Evaluations (COM_{3}PARE), a unified conceptual schema and analytic algorithm based upon Roy’s linear mixed effects (LME) model with Kronecker product covariance structure in a doubly multivariate setup, for IGRT method comparison.
Methods
An anonymized dataset consisting of 100 paired coordinate (X/ measurements from a sequential series of head and neck cancer patients imaged nearsimultaneously with cone beam CT (CBCT) and kilovoltage Xray (KVX) imaging was used for model implementation. Softwaresuggested CBCT and KVX shifts for the lateral (X), vertical (Y) and longitudinal (Z) dimensions were evaluated for bias, intermethod (betweensubject variation), intramethod (withinsubject variation), and overall agreement using with a script implementing COM_{3}PARE with the MIXED procedure of the statistical software package SAS (SAS Institute, Cary, NC, USA).
Results
COM_{3}PARE showed statistically significant bias agreement and difference in intermethod between CBCT and KVX was observed in the Zaxis (both p − value<0.01). Intramethod and overall agreement differences were noted as statistically significant for both the X and Zaxes (all p − value<0.01). Using prespecified criteria, based on intramethod agreement, CBCT was deemed preferable for Xaxis positional verification, with KVX preferred for superoinferior alignment.
Conclusions
The COM_{3}PARE methodology was validated as feasible and useful in this pilot head and neck cancer positional verification dataset. COM_{3}PARE represents a flexible and robust standardized analytic methodology for IGRT comparison. The implemented SAS script is included to encourage other groups to implement COM_{3}PARE in other anatomic sites or IGRT platforms.
Keywords
Repeat Statement Random Statement Linear Mixed Effect Compound Symmetry Option TypeBackground

First known application of LMEbased COM_{3}PARE hypothesis testing protocol for method comparison using imaging data.

Demonstration of feasibility and utility of COM_{3}PARE using an established head and neck positional verification dataset, previously presented with standard method comparison approaches.
Methods
Datasets
 1.
No significant bias (i.e., no difference between the means of the two methods under a prespecified threshold nor a statistically significant difference between said means).
 2.
No statistically significant difference in the intersubject (betweensubject) variability of the two methods.
 3.
No statistically significant difference in the intrasubject (withinsubject) variability (i.e., repeatability) of the two methods.
For this study, we prespecified a bias threshold of an absolute value of <0.1 cm, with a statistically significant difference designated by α<0.05. To assess the aforementioned criteria, we implemented the LME methodology proposed by Roy^{48}, referred to as COM_{3}PARE (see Appendix A).
Statistical analysis with COM_{3}PARE
As mentioned in the introduction the number of replicated measurements on each patient or subject may not be equal, and also the number of replications of the two methods on the same subject may not be equal. Let \(p^{KVX}_{i}\) and \(p^{CBCT}_{i}\) be the number of replications on subject i by the established method (KVX), and a new method (CBCT) respectively. Let \(p_{i}= \max \left ({p^{KVX}_{i}, p^{CBCT}_{i}}\right)\), and n_{ i }=2p_{ i }. Therefore, the number of observations on the ith subject is n_{ i }, under the assumption that the ith subject has \(\left p^{KVX}_{i} p^{CBCT}_{i}\right \) missing values.
where b_{1},b_{2},…,b_{ N },ε_{1},ε_{2},…,ε_{ N } are independent, and y_{1},y_{2},…,y_{ N } are also all independent. LME model allows for the explicit analysis of betweensubject (D) and withinsubject (R_{ i }) sources of variation of the two methods. We define the two methods by a vector variable Mvar; Mvar=1 for the KVX method and Mvar=2 for the CBCT method. We choose the intercept and the vector variable Mvar as fixed effects, thus the design matrix X_{ i } has three columns, and consequently β=(β_{ o },β_{1},β_{2})^{′} is a 3dimensional vector containing the fixed effects. We also choose the vector variable Mvar as random effects, i.e., Mvar is random across individual subjects; thus the design matrix Z_{ i } has two columns. Therefore, b_{ i }= (b_{1i},b_{2i})^{′} is a 2dimensional vector containing the random effects.
Thus, the covariance matrix has the same structure for each subject, except that of the dimension. The 2×2 block diagonals Block Ω_{ i } in the overall variancecovariance matrix Ω_{ i } represent the overall variancecovariance matrix between the two methods. Similarly, the 2×2 block diagonals in the overall correlation matrix Ω_{ i }_Correlation represent the overall correlation matrix between the two methods. Thus, the offdiagonal element in this 2×2 overall correlation matrix gives the overall correlation between the two methods. It can be easily seen that the overall variability is the sum of betweensubject variability and withinsubject variability (see Roy^{48} for detail). Thus, we see that if there is a disagreement in overall variabilities, then it may be due to the disagreement in either betweensubject variabilities or withinsubject variabilities or both.
MIXED procedure of SAS
We use MIXED procedure (PROC MIXED) of SAS to get the maximum likelihood estimates (MLEs) of β,D, R_{ i } and Ω_{ i }. METHOD=ML specifies MIXED procedure to calculate the maximum likelihood estimates of the parameters. The COVTEST option requests hypothesis tests for the random effects. CLASS statement specifies the categorical variables. DDFM=KR specifies the KenwardRoger^{51} correction for computing the denominator degrees of freedom for the fixed effects. KenwardRoger correction is suggested whenever one has replicated or repeated measures data; also for missing data. The SOLUTION (S) option in the MODEL statement provides the estimate of the difference between the two mean readings (bias) of the two methods. RANDOM and REPEATED statements specify the structure of the covariance matrices D and R_{ i }. See the sample program in Appendix A that demonstrates the use of RANDOM and REPEATED statements. PROC MIXED calculates the (n_{ i }×n_{ i })dimensional submatrix R_{ i } of the ith subject from the (2p×2p)dimensional matrix (V⊗Σ), and eventually calculates (n_{ i }×n_{ i })dimensional submatrix Ω_{ i }. When the number of replications on each subject by respective methods is unequal, PROC MIXED considers the case as missing value situation. Options V=3 and VCORR=3 in the RANDOM statement give the estimate of the overall variancecovariance matrix Ω_{3} and the corresponding Ω_{3}_Correlation matrix, i.e., for the third subject. The option G in the RANDOM statement gives the estimate of the betweensubject variancecovariance matrix D. Option R in the REPEATED statement gives the estimate of the variancecovariance matrix R_{1} for the first subject. One can get the Ω_{ i } variancecovariance matrix and the corresponding Ω_{ i }_Correlation matrix for all subjects by specifying V= 1 to N, and VCORR=1 to N in the RANDOM statement. When the correlation matrix V on the replicated measurements assumes equicorrelated structure and Σ as unstructured, we use the option TYPE=UN along with SUBJECT=REPLICATE(PATIENT) in the REPEATED statement. This gives the 2× 2 withinsubject variancecovariance matrix Σ. See Appendix A.
Related hypotheses testings to test the disagreement between KVX and CBCT
If there is a disagreement between the two methods, it is important to know whether it is due to the bias, due to the difference in betweensubject variabilities or due to the difference in withinsubject variabilities of the two methods. If it is due to the bias between the two methods, it is easy to correct. The output of PROC MIXED always gives the bias, its t − value and its p − value. Nonetheless, it is not straightforward to check the agreement or disagreement in betweensubject variabilities and in withinsubject variabilities of the two methods. We will accomplish these by the indirect use of PROC MIXED in two steps (described below) by using likelihood ratio tests.
Testing of hypothesis of difference between the means of KVX and CBCT
Output of PROC MIXED (Solution for Fixed Effects) gives the bias and the corresponding t − value and p − value.
Testing of hypothesis of difference in betweensubject variabilities of KVX and CBCT
The log likelihood function under both null hypothesis and alternating hypothesis must be maximized separately. We do this by setting the option METHOD=ML in PROC MIXED statement. The option TYPE=UN in the RANDOM statement, along with the option TYPE=UN in the REPEATED statement, is used to calculate the “2 Log Likelihood" for the covariance structure under H_{ d }. Similarly, the option TYPE=CS in the RANDOM statement, along with the option TYPE=UN in the REPEATED statement, is used to calculate the “2 Log Likelihood" for the covariance structure under K_{ d }.
PROC MIXED calculates “LRT df" under the heading of “Null Model Likelihood Ratio Test", see Appendix B.
Testing of hypothesis of difference in withinsubject variabilities of KVX and CBCT
The option TYPE=UN in the RANDOM statement, along with TYPE=UN in the REPEATED statement, is used to calculate the “2 Log Likelihood" for the covariance structure under H_{ σ }. TYPE=UN in the RANDOM statement, along with TYPE=CS in the REPEATED statement, is used to calculate the “2 Log Likelihood" for the covariance structure under K_{ σ }. The test statistic −2 lnΛ_{ σ } under K_{ σ } follows a chisquare distribution with d.f. ν_{ σ }= LRT df (underH_{ σ })−LRT df (underK_{ σ }).
Testing of hypothesis of difference in overall variabilities of KVX and CBCT
The option TYPE=UN in the RANDOM statement, along with TYPE=UN in the REPEATED statement, is used to calculate the “2 Log Likelihood" for the covariance structure under H_{ ω }. The option TYPE=CS in the RANDOM statement, along with TYPE=CS in the REPEATED statement, is used to calculate the “2 Log Likelihood" for the covariance structure under K_{ ω }. The test statistic −2 lnΛ_{ ω } under K_{ ω } follows a chisquare distribution with d.f. ν_{ ω }= LRT df (underH_{ ω })−LRT df (underK_{ ω }).
Results
The pvalue for testing the withinsubject variabilities of the two methods by using IML procedure of SAS is calculated at the third stage (see Appendix A). The p − value= 5.5112E − 9 (see Appendix B).
Betweenmethod bias
Bias (cm)  p − value  

X  0.0335  0.6077 
Y  0.0428  0.2836 
Z  0.0942  0.0253 
Intermethod agreement
KVX (cm)  CBCT (cm)  p − value  

X  0.0413  0.0670  0.4795 
Y  0.0511  0.0464  0.7518 
Z  0.0273  0.0848  0.0010 
Intramethod agreement
KVX (cm)  CBCT (cm)  p − value  

X  0.3396  0.1047  5.5 ×10^{−9} 
Y  0.1687  0.1693  1.0 
Z  0.0485  0.0825  0.0034 
Overall agreement
KVX (cm)  CBCT (cm)  p − value  

X  0.3809  0.1717  3.7 ×10^{−8} 
Y  0.2198  0.2157  0.9512 
Z  0.0758  0.1673  1.3 ×10^{−5} 
Mixed effects estimated correlation coefficient
Correlation  

coefficient  
X  0.5329 
Y  0.8038 
Z  0.7336 
Using the aforementioned criteria, automated shifts from CBCT and kV Xray, acquired and processed in the manner detailed are interchangeable only for measurements of the Yaxis (anteroposterior), and for example, should not be used on alternating days in facilities with both systems in either X or Zaxis. Additionally, our method suggests that, with lower intramethod variability in the Xaxis (lateral), CBCT is the preferred measurement method, while in the Zaxis (superoinferior) kV Xray measurement is preferable.
Discussion
 1.
The bias and overall agreement must fall within a prespecified range (e.g., bias/agreement of <0.1 cm between IGRT devices).
 2.
There should be no statistically significant, using a prespecified threshold (e.g., <0.05) difference in the intersubject variability of the two methods.
 3.
There should be no statistically significant difference in the intrasubject variability (i.e., repeatability) of the two methods.
 4.
In cases where criteria 2 and 3 are NOT met, the preferred IGRT technique is the one exhibiting the lower intrasubject variability (i.e., greater repeatability).
Conclusion
COM_{3}PARE represents an attempt at a unified conceptual schema and analytic algorithm for method compassion of IGRT platforms. Initial application in a head and neck positional verification dataset shows feasibility and utility.
Appendix A
SAS code
Below we provide the sample SAS code to test withinsubject variabilities by fitting the linear mixed effects model to our KVX and CBCT shifts for the lateral (X). We first fit the linear mixed effects model for the null hypothesis, then we fit the linear mixed effects model for the alternating hypothesis, and then find the p − value for the test. Appropriate changes can be made to test betweensubject variabilities and overall variabilities using the SAS commands as described in Sections Testing of hypothesis of difference in betweensubject variabilities of KVX and CBCT and Testing of hypothesis of difference in overall variabilities of KVX and CBCT. Appropriate changes can be also made for vertical (Y) and longitudinal (Z) dimensions and for any other data sets.
Appendix B
SAS output for covariance structure under the null and the alternating hypotheses
Below we provide the selected portions of the output of the above program.
Notes
Acknowledgements
Special thanks to Joseph Ting, PhD for dataset utilization permission.
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