Radar Anti‐Jamming Performance Evaluation: A Novel Model Based on Measurement Error and RCS Distributions

A new and simple method of interpreting the of the backscatter radar cross section (RCS) from ocean waves of individual wavelength is presented. Using the Kirchhoff scattering (Physical Optics) theory, the RCS from the ambient waveheight spectrum is first computed as a function of the wavenumber. Differentiating this cumulative RCS yields the distribution function of the RCS from ocean waves of different wavenumbers. The present study shows, as a general feature, that the maximum RCS contribution tends to come from the ocean waves of wavenumbers closer to the Bragg wavenumber as the radar incidence angle increases. Under low wind speeds (/spl sim/2 m/s) and at incidence angles between 200 and 45/spl deg/, the dominant RCS contribution at both C-band and L-band comes from ocean waves of wavenumbers close to the Bragg wavenumber. Under intermediate to high wind speeds (/spl sim/10-15 m/s) and at the small incidence angle of 200, the radar backscatter is dominated by long ocean waves with little contribution from the Bragg waves at both bands. At the intermediate incidence angles (/spl sim/35/spl deg/ to 45/spl deg/), the RCS distribution is centered at wavenumbers near the Bragg wavenumber at all wind speeds from 2 m/s to 15 m/s. These features are more pronounced at C-band than at L-band. The present theory is based on the Kirchhoff scattering model, and as such, its validity may also be limited to a range of small to intermediate incidence angles where the polarization effect is not very significant. The RCS distribution is evaluated for the ambient sea surface and the surface perturbed by varying currents induced by the interaction between the current and bottom topography.

The problem of statistical pattern recognition with noisy or imprecise feature measurements is considered. An exact analytical expression is found for the probability of misclassification under this condition, for multiclass multivariate systems. The probability of error exceeds that of the ideal case for the special case of two classes, the a priori conditional probability density functions are assumed to be normal, along with the two cases of feature measurement error, namely normal and uniform probability density functions. Monotonicity of the misclassification probability with measurement error variance is shown. Numerical results are presented for both cases over a workable range of parameters. The study is useful in practical pattern recognition problems.

The detection of aircraft wingtip vortices has always been a key issue for aviation safety and aircraft stealth. Previous work mostly used density variation and water vapor variation to calculate dielectric parameters and the distribution of radar cross section(RCS). Here, the atmospheric pressure variation around the aircraft are calculated and simulated using Ansys Fluent software and numerical models such as fluid mechanics N-S equation. To investigate the change of its scattering cross section, we calculated the fluctuations of the dielectric parameters around aircraft for the given data of atmospheric pressure fluctuations. Finally, we show the relation between RCS and radar receiving horizontal angle under three incident wave angles, and the change rule of RCS and receiving angle both in the horizontal polarization and vertical polarization cases. This conclusion has a positive effect on the study of the scattering characteristics of wingtip vortices.

A measurement method and system based on conventional radars has been proposed to address the problem of radar cross-section (RCS) measurement of ship targets in open seas. Calibration errors for RCS measurement on ground plane range (GPR) are analysed, and the error elimination method and accurate RCS calculation formula are given. An RCS measurement system is designed by improving conventional radars. These systems measure two Luneburg lens reflectors of different sizes on the lake. The average RCS of the big Luneburg lens reflector is obtained, and the error is calculated to be 2.73% compared with its actual value, which verifies the rationality of the proposed measurement method and system. The system measures the RCS of a ship in the open sea and obtains its circumferential RCS distribution. The effect of the error in the ship's position calculation on the RCS measurement results is analysed, and the uncertainty range of the error is obtained.

Misclassification into exposure categories formed from a continuous variable arises from measurement error in the continuous variable. Examples and mathematical results are presented to show that if the measurement error is nondifferential (independent of disease status), the resulting misclassification will often be differential, even in cohort studies. The degree and direction of differential misclassification vary with the exposure distribution, the category definitions, the measurement error distribution, and the exposure-disease relation. Failure to recognize the likelihood of differential misclassification may lead to incorrect conclusions about the effects of measurement error on estimates of relative risk when categories are formed from continuous variables, such as dietary intake. Simulations were used to examine some effects of nondifferential measurement error. Under the conditions used, nondifferential measurement error reduced relative risk estimates, but not to the degree predicted by the assumption of nondifferential misclassification. When relative risk estimates were corrected using methods appropriate for nondifferential misclassification, the "corrected" relative risks were almost always higher than the true relative risks, sometimes considerably higher. The greater the measurement error, the more inaccurate was the correction. The effects of exposure measurement errors need more critical evaluation.

SDSC SHAR, ISRO equipped with ground-based Precision Radars for tracking the launch vehicles for range safety operations and trajectory computations. The Design of Radar parameters, waveform generation and Receiver signal processing [4] always appears to be challenging owing to the wide effects it can cause on the performance of radars [3]. The radars under consideration are Tracking Radars, intended for track Launch Vehicle. It is designed for skin mode and Transponder mode operation which calls for accurate estimation of Signal strength to achieve sufficient link margin for a target complex Radar Cross Section (RCS) [6] at slant range. In radar technology, scattering phenomenology is quantified by the target parameter RCS, can be modelled as random process. Probability density functions techniques are used to describe the target scattering and RCS estimation. Receiver that employs an envelope detector followed by a threshold decision technique [5]. The input to the receiver is composed of radar echo s(t) and additive zero mean white Gaussian noise n(t), with variance 2 . The input noise is assumed to be spatially incoherent and uncorrelated with signal. The IF filter output is a complex random variable that is composed of either noise alone or noise plus target return signal. The noise quadrature components nI(t) and nQ(t) are uncorrelated zero mean low pass Gaussian noise with equal variance 2 . The Joint Probability Density Functions of random variables is Rician PDF if target is available or Rayleigh PDF if noise alone present. Variations in radar-target geometry [8], target vibration, radar frequency, and polarization changes can lead to variations in target RCS, resulting in fluctuating targets having Swerling models [9] of target RCS fluctuation, The effect of these models on radar detection, signal strength estimation for a given radar parameters are considered.

The USU LadarSIM software package is a ladar system engineering tool that has recently been enhanced to include the modeling of the radiometry of Ladar beam footprints. This paper will discuss our validation of the radiometric model and present a practical approach to future validation work. In order to validate complicated and interrelated factors affecting radiometry, a systematic approach had to be developed. Data for known parameters were first gathered then unknown parameters of the system were determined from simulation test scenarios. This was done in a way to isolate as many unknown variables as possible, then build on the previously obtained results. First, the appropriate voltage threshold levels of the discrimination electronics were set by analyzing the number of false alarms seen in actual data sets. With this threshold set, the system noise was then adjusted to achieve the appropriate number of dropouts. Once a suitable noise level was found, the range errors of the simulated and actual data sets were compared and studied. Predicted errors in range measurements were analyzed using two methods: first by examining the range error of a surface with known reflectivity and second by examining the range errors for specific detectors with known responsivities. This provided insight into the discrimination method and receiver electronics used in the actual system.

This paper considers optimization and implementation of the pulse shape for Orthogonal Frequency Division Multiplexing (OFDM) systems over doubly dispersive channel. The proposed pulse shape is expressed as a linear combination of the most localized Hermite waveforms. In our previous works, we have proposed pulses optimization based on an exact expression of the Signal to Interference Ratio (SIR). In order to accelerate the optimization procedure, we propose an approximate expression of the SIR using Taylor series of interference and desired signal mean powers. For a performance evaluation of the optimized OFDM system, we propose an efficient implementation of both the modulator and the demodulator of the system and an efficient method to find out the resulting simulated SIR. Simulations results, obtained for different values of the dispersion factor, show that the SIR computed through system level simulation matches the maximum achievable SIR obtained through numerical optimization.

A measurement error (ME) is a component of any study involving the use of actual measurements, but is often not recognized or is ignored. The consequences of ME on models can be severe, affecting estimates of tree and stand attributes and model parameters. Although correction methods do exist for countering the effects of ME, the use of these methods requires knowledge of the distribution of the errors. A new method for modeling error distributions, called the two-stage error distribution (TSED) method, is presented here. This method is compared with traditional methods for error modeling through an example using diameter and height ME. Comparisons between the fitted error distribution surfaces and the empirical error surface are based on a dissimilarity measure. The results indicate that the TSED method produces a much more accurate and precise characterization of the ME distribution than do traditional methods when a high percentage of errors is identical. In other cases, the TSED method works as well as the most accurate form of the traditional method. The TSED method is also expected to perform better at characterizing asymmetric distributions. It is therefore more adaptable than traditional methods and is being proposed for error modeling in the future. FOR. SCI. 50(6):743–756.

The aims of this study were to determine the size of the medial retropharyngeal lymph nodes in apparently healthy dogs using ultrasonography and to investigate relationships between body weight (1.8-59 kg), age (1.0-15 years), and medial retropharyngeal lymph node sizes (width, height, and length). The sample population consisted of 100 apparently healthy, volunteered, adult dogs. The data were normally distributed, thus mean, SD, and Pearson's correlation were used. Repeatability of ultrasound measures was assessed as the percentage of differences between duplicate measures that were within 2 SDs of the differences: all measures were at least 93% repeatable (differences typically were < or = 0.25 cm and always < 1 cm). No difference between sexes was observed. The medial retropharyngeal lymph node increased in size with increased body weight (r = 0.46 to 0.59) and decreased in size with increased age (r = -0.30 to -0.50). Although statistically significant, the actual variation is not likely clinically important due to the small range of sizes, measurement error, and various combinations of age and body weight. Therefore, regardless of body weight or age, the average width is 1.0 cm, height is 0.5 cm, and length is 2.5 cm and maximum width is 2 cm, height is 1 cm, and length is 5 cm. Based on the maximal difference between duplicate measures (with some exception), any change > or = 0.4 cm with width or height, or > or = 1.0 cm in length, in a follow up measurement probably represents a true biological change rather than measurement error.

The development of a novel RFID tag operating at 869 MHz and its performance evaluation using simulated Radar Cross Section (RCS) are presented. The RCS variations of the tag antenna are evaluated over azimuth and elevation angular ranges for three cases of load: short, matched load and open. The tag exhibits appreciably good differential RCS between its switching states over wide azimuth angular ranges and over the entire elevation angular ranges. The measured read range variations over the azimuth and elevation angular ranges are also presented to validate the simulated results.

Errors-in-variables (EIV) regression is widely used in econometric models. The statistical analysis becomes challenging when the regression function is discontinuous and the distribution of measurement error is unknown. In the literature, most existing jump regression methods either assume that there is no measurement error involved or require that jumps are explicitly detected before the regression function can be estimated. In some applications, however, the ultimate goal is to estimate the regression function and to preserve the jumps in the process of estimation. In this paper, we are concerned with reconstructing jump regression curve from data that involve measurement error. We propose a direct jump-preserving method that does not explicitly detect jumps. The challenge of restoring jump structure masked by measurement error is handled by local clustering. Theoretical analysis shows that the proposed curve estimator is statistically consistent. A numerical comparison with an existing jump regression method highlights its jump-preserving property. Finally, we demonstrate our method by an application to a health tax policy study in Australia.

1. The distribution of the random measurement error depends in a general case on the distribution of the measured quantity. This relationship becomes essential and should be taken into account when the root-mean-square error of the measuring device and the root-mean-square deviation of the measured quantity are of the same order. In order to account for the above relationship for normal distribution laws of the measured quantity and the measuring device error, it is necessary to apply to the measurement results correction mδ/y according to (10), and to determine the root-mean-square measurement error from (8). 2. If the root-mean-square deviation of the measured quantity is of a higher order, the effect of its distribution becomes unessential. In such a case, the distribution of the measurement error virtually coincides with that of the measuring device error, and it is no longer necessary to account for the distribution of the measured quantity.

Clinical practice involves measuring quantities for a variety of purposes, such as aiding diagnosis, predicting future patient outcomes, and serving as endpoints in studies or randomized trials. Measurements are almost always prone to various sorts of errors, which cause the measured value to differ from the true value; accordingly, studies investigating measurement error frequently appear in this and other journals. The importance of measurement error depends upon the context in which the measurements in question are to be used. For example, a certain degree of measurement error may be acceptable if measurements are to be used as an outcome in a comparative study such as a clinical trial, but the same measurement errors may be unacceptably large to make measurements usable in individual patient management, such as screening or risk prediction. In the past 20 years many papers have been published advocating how studies of measurement error should be analyzed, with a paper by Bland and Altman1 being one of the most cited and well known examples. There has been much controversy concerning the choice of parameter to be estimated and reported, and consequently confusion surrounding the meaning and interpretation of results from studies investigating measurement error. In this paper we first distinguish between the general concepts of agreement and reliability to aid researchers in considering which are relevant for their particular application. We then review the statistical methods that can be used to investigate and quantify agreement and reliability, dealing separately with the different types of measurement error study, while emphasizing the largely common techniques that should be used for data analysis. We reiterate that the judgment of whether agreement or reliability are acceptable must be related to the clinical application, and cannot be proven by a statistical test. We highlight the fact that reliability depends on the population in which measurements are made, and not just on the measurement errors of the measurement method. We discuss the advantages of method comparison studies making at least two measurements with each measurement method on each subject. A key advantage is that the cause of a correlation between paired differences and means in the so-called Bland–Altman plot can be determined, in contrast to when only a single measurement is made with each method. Throughout the paper, we try to emphasize that calculated values of agreement and reliability from measurement error studies are estimates of parameters, and as such we should report such estimates with CIs to indicate the uncertainty with which they have been estimated. We restrict our attention to measurements of a continuous quantity; alternative methods are required for categorical data2. One difficulty in the measurement error field is the number of different terms used to describe studies of measurement error. The terms 'agreement', 'reliability', 'reproducibility' and 'repeatability' are used with varying degrees of consistency in the medical literature. We first make clear the distinction between the statistical concepts of agreement and reliability3. Agreement quantifies how close two measurements made on the same subject are, and is measured on the same scale as the measurements themselves. Two measurements of the same subject may differ for a number of reasons, depending on the conditions under which the measurements were made. In a method comparison study there will be differences because of inherent variability in each of the measurement methods, as well as potentially a bias between the measurements from the methods. If the measurements are made by different observers or raters, differences may be due to biases between the observers. Agreement between measurements is a characteristic of the measurement method(s) involved, which does not depend on the population in which measurements are made, unless bias or measurement precision varies with the true value being measured. One popular way of quantifying agreement is to estimate the 95% limits of agreement, as proposed by Bland and Altman1. These limits are defined such that we expect that, in the long run, 95% of future differences between measurements made on the same subject will lie within the limits. If reliability is high, measurement errors are small in comparison to the true differences between subjects, so that subjects can be relatively well distinguished (in terms of the quantity being measured) on the basis of the error-prone measurements. Conversely, if measurement errors tend to be large compared with the true differences between subjects, reliability will be low because differences between measurements of two subjects could be due purely to error rather than to a genuine difference in their true values. The reliability parameter is also known as an intraclass correlation (ICC), as it equals the correlation between any two measurements made on the same subject. Reliability takes values between zero and one, with a value of one corresponding to zero measurement error and a value of zero meaning that all the variability in measurements is due to measurement error. As a dimensionless quantity, it is arguably quite difficult to interpret, and deciding what value constitutes sufficiently high reliability is often made in a subjective fashion. Repeatability of measurements refers to the variation in repeat measurements made on the same subject under identical conditions4. This means that measurements are made by the same instrument or method, the same observer (or rater) if human input is required, and that the measurements are made over a short period of time, over which the underlying value can be considered to be constant. Variability in measurements made on the same subject in a repeatability study can then be ascribed only to errors due to the measurement process itself. Reproducibility refers to the variation in measurements made on a subject under changing conditions4. The changing conditions may be due to different measurement methods or instruments being used, measurements being made by different observers or raters, or measurements being made over a period of time, within which the 'error-free' level of the variable could undergo non-negligible change. The first type of study we consider is a repeatability study, in which we investigate and quantify the repeatability of measurements made by a single instrument or method, and in which the conditions of measurement remain constant. The second type of study we consider is a method comparison study, in which measurements are made using two measurement methods on a sample of subjects. The use of two different methods means that this is a reproducibility study, but the term method comparison is used as it clearly communicates the changing conditions under which measurements have been made. In contrast to a repeatability study, systematic bias may exist between measurements made by two different methods, and their measurement errors may have different SDs. The last type of study we consider is one in which measurements are made by different observers or raters. Again this is a type of reproducibility study. As with method comparison studies, biases may exist between observers, and their measurement SDs may differ. As we discuss below (see Measurements with observers or raters), if interest lies in quantifying the measurement error characteristics of the particular observers in one's study, exactly the same analysis methods should be used as for a method comparison study. In order to investigate the repeatability of measurements, a repeatability study must, for an appropriately selected sample, make at least two measurements per subject under identical conditions. This means that the measurements must be made by the same measurement method, or the same observer or rater. The objective is to then quantify the agreement and reliability of measurements made by that particular method or observer. If the differences between two measurements made on a subject are approximately Normally distributed, in the long run we expect the absolute difference between two measurements on a subject to differ by no more than the repeatability coefficient on 95% of occasions. To estimate the within-subject SD, we can fit a one-way analysis of variance (ANOVA) model to the data containing the repeat measurements made on subjects. ANOVAs can be fitted in all modern statistical packages. ANOVA models partition variability in data into that which can be ascribed to differences between groups, and that remaining to within groups. For a repeatability study, the groups are defined by the subjects under measurement, and this must be specified in the statistical package used. The ANOVA model estimates how much variation in measurements can be attributed to differences in the true, or 'error-free' values of subjects, with the remainder constituting measurement error. Fitting the ANOVA model results in estimates of the between-subject and within-subject SDs (or alternatively the corresponding variances, which are the SDs squared). The estimate of the within-subject SD can be used in the above formula to give an estimate of the repeatability coefficient. We illustrate with a study by Järvelä et al., who investigated the repeatability of measurements of flow volume made by one observer using three-dimensional (3D) power Doppler ultrasonography from 22 ovaries5. The estimated repeatability coefficient was 3.97, meaning that the absolute difference between any two future measurements made by that particular observer on a particular subject/unit are estimated to be no greater than 3.97 on 95% of occasions. It is important to note that the repeatability of another observer may be different, because of differences in the training and ability of observers. Because the repeatability coefficient calculated is an estimate, it is important to calculate a CI for it to indicate how precisely it has been estimated. A CI may be given automatically by statistical software, but in the Appendix we review how a 95% CI for the within-subject SD can be calculated. Such a CI can be used to find a CI for the repeatability coefficient by multiplying the CI limits by . If the CI for the within-subject variance is given by software instead, the limits must first be square rooted to give a CI for the within-subject SD. The ANOVA model assumes that the measurement errors are statistically independent of the true 'error-free' value, and that the SD of the errors is constant throughout the range of 'error-free' values. Sometimes the SD of errors increases with the true value being measured. This should be checked by plotting paired differences between measurements against their mean, the so called Bland–Altman plot. We illustrate this later in the context of a method comparison study (see Plotting the data) and describe how such heterogeneous errors can often be dealt with by making a log transformation (see Non-uniform variability in measurement of the repeatability coefficient on the differences between measurements being approximately Normally This can be checked by a or plot of the paired differences in measurements on each subject. made in the of the CI for the within-subject SD is that the measurement errors are If this is in the may be to The reliability of a measurement method is often of interest when measurements are to be used to between subjects or groups of subjects. For example, if we have a choice of two measurement methods that could be used to an outcome in a clinical or study, using the method with reliability will give greater statistical power to differences between groups, for a given sample In this we describe how measurements from a single method can be used to estimate the reliability in a given we discuss the use of reliability to measurement methods (see Reliability in method comparison and different observers (see in The same ANOVA model as above (see can be used to estimate statistical will automatically give the estimated which is the of we can the estimates of the and within-subject SDs into the formula given above (see Agreement and to estimate the For the flow data of Järvelä et an estimated of was This means that of the variability in measurements of the flow was estimated to be due to genuine differences in flow between with the remaining being due to errors in the measurement process and the observer Because the measurements were all made by one the reliability may be to as As with the repeatability the calculated is an estimate, and a 95% CI should be statistical give a but in the Appendix we review the of a 95% CI for the For the flow data of Järvelä et the estimate of a 95% of to Because we use the same ANOVA model to estimate reliability as we describe for agreement, the same of 95% CIs for the estimate on of the true 'error-free' values. The reliability of a measurement method depends upon the of the population in which the measurements are made. the of reliability given above (see Agreement and we that the of true 'error-free' values in the measured by the between-subject SD, the value of agreement between repeat measurements is a characteristic of the method or instrument the of measurement errors is the range of true reliability depends on the of measurement errors and the true in the population in which measurements are made. The is with a a method for measuring has a within-subject SD of which does not with the underlying value being measured. we a study to estimate the reliability, and we sample from a heterogeneous population in which the between-subject SD in true is 20 This give a reliability or of . we sample our subjects from a more in which the SD of true values is the same as the variability of the within-subject error. In this the reliability is to . If studies report only an estimate of the reliability (ICC), can only make use of the estimate if the population in which the to use such measurements has We that report estimates of and within-subject SD, in to the In this can whether the measurement method will be sufficiently for their application, in which the between subjects may be we use a measurement method or in clinical we must that the measurements it are to by the measurement method used, that measurements made with the method are using the method. If the measurements from the two methods are sufficiently close and the of a patient on their basis be the the method could the method in clinical because the method is or to The as to what constitutes depends on how the measurements are to be used. If the two measurements are made on the same or we can quantify their To investigate and quantify the agreement between measurements made by two methods, we must at a a sample of subjects using the measurement method and the proposed method. The data from such a study of of measurements from each with the containing the measurements from the two measurement methods. we making two measurements on each subject using each method because the one measurement per method is the most common we by analysis. The first to such a is to plot the The plot is of measurements from the method against from the method (or If measurements were from we expect the to lie on the of the of can be used to if there is if more data are above the than this that the method on the measurements on the of fit for this plot and report a statistical for whether the of the plot from the of As by Bland and in a paper in this we expect the to have a than one if the method on the any measurement and so a of the that the true is to one is not a it the same as a plot with the of of the between the measurements from two methods is often more by plotting the difference in a measurements from the two methods against the of their measurements, as first by and this is to as the Bland–Altman and is frequently in of measurement error We illustrate the plot using an from the literature. et volume measurement in with and volume was measured using and also by We have approximately their plot of differences against means and it is as using only data from the in volume measured by three-dimensional and by against their from study by et the SD and the SD, each with 95% CI The plot the difference between calculated using and against the of two values. et not their plot or of results to indicate which measurements were from It is important to be and always the same measurement from the The the of the paired from zero an estimate of the bias between the two methods. which measurements have been from which to which method, on measurements. The indicate the estimated limits of agreement their CI which we describe The plot can be used to the differences between measurements made by the two methods. The variability of the differences between the two methods how well the methods If the variability of the paired differences is the range of measurements, and there is no between the difference and mean, we can quantify the agreement between the methods by the limits of the of the limits of agreement, we illustrate how the for limits of agreement may be and how they can be dealt on we expect the difference in volume as measured by the two methods to lie between and for 95% of future measurements. It is important to that the limits of agreement calculated in this are just and as with any type of estimate it is to quantify and report how precisely the limits are estimated of 95% The estimated limits of agreement indicate how large the between the measurements will be on 95% of occasions. The whether the methods sufficiently well must then be made depending on the context in which the measurements will be used. In contrast to the repeatability which assumes no bias between measurements, the limits of agreement method this The of the paired differences whether on one method to or measurements to the measurements of the second method, which we to as a bias between the methods. In the data in from et al., there was of a bias between the two methods, because the difference between the methods was close to at We can a statistical to whether there is of bias by a of the differences paired of the measurements from each we the that the true of the differences is corresponding to no bias between the methods. For the data in this that there was no of a bias between and made using The limits of agreement method assumes that the SD of the differences is throughout the range of measurements. Sometimes this will not be the In frequently the SD of the differences will with the In this for small values the limits of agreement will be than for values the limits will be of the plot of differences against from the study by et that the variability of the differences may be when the value being measured is The SD of the paired differences below the of the means was it was for above the This can often be by of the measurements of the two If the difference in the of the measurements results in constant we can calculate the limits of agreement and CIs for the limits in the Because the difference of two is to the of the we can all the estimates made on the log scale by Because of the limits of agreement to the of one measurements to the For the data in the of the difference in measurements was with SD The limits of agreement on the log scale are to the of with 95% limits of agreement for the of to The 95% CIs can be calculated on the log scale to the which measurements were made by which method, we cannot whether the is for measurements by to by or a plot of the of the measurements by the two methods to their mean, using a with the estimated 95% limits of agreement and their 95% This that the of a SD for the paired differences is more of measured by three-dimensional (3D) and by against of measurements, on data published by et the SD and the SD, each with 95% CI to log transformation can also be used to try to For example, et estimate limits of agreement using paired differences in measurements as a of the of the two measurements. The key is that a transformation should be used which that are approximately Normally and have approximately SD the range of It is for there to be an between the paired differences and of this is in a study by et al., who the agreement between as measured using and the calculated using in We have the data from their Bland–Altman plot of the difference between the volume and volume against the mean, and it in There to be a between the paired differences and differences between volume measured by three-dimensional and by from study by et correlation the SD and the SD, each with 95% CI We can a statistical to the for a whether the correlation coefficient between the paired differences and means from zero or by of the differences against the The is known as in the context of the SD of measurements from two methods, which for the data in is statistically for an between paired differences and how should we than models for the difference as a of the we should be given to the cause of the A between the paired differences and means when the SD of measurements from one method from the SD of measurements from the other method. There are at least two of an between paired differences and The first is that there is between the difference in measurements from the two methods and true value being the bias between methods over the range of true values. The second is that the within-subject SDs of the two methods differ. This will in the of changing bias if a method has or measurement errors than the method. This is and when a or measurement method is compared with a If the cause of the correlation is within-subject SDs and we were to plot the paired differences against the true value being there be no The between the Bland–Altman plot and a plot against true values because the paired means measurement errors, in contrast to the true values. as by and with only one measurement per method per we cannot which of two is the To we must make an that the within-subject SDs of the two methods are so that the correlation is attributed to changing or that the two within-subject SDs are and that there in no between bias and true value we may have a of For the of we with the from et by that the correlation is due to a difference in the within-subject SDs of the two methods. A correlation between differences and means will when the method with measurement error is from measurements of the method with error. For the data in the correlation is the errors than the measured by which and to the volume measured by as the true that such measurements were to no an that there is no changing we can calculate 95% limits of agreement, as by et al., because the limits of agreement method does not that the two within-subject SDs are In order to from the data whether changing bias is we must make at least two measurements with each method on each subject. This study has often been but is we describe how such studies can be to whether a correlation between paired differences and means is due to changing bias between the methods (see with two measurements from each As reliability may be a parameter with which to two different measurement methods. To estimate each reliability, we must make at least two measurements of each subject with each of the two methods. The repeat measurements from each method can then be as two repeatability studies (see estimates of each reliability, which can be advantage of using reliability to measurement methods is that it can be used to methods when their measurements are given on different or as the reliability is a dimensionless Because reliability depends on the of the true values in the population (see Reliability depends on population it is that reliability are compared only if they have been estimated from the same report a single reliability estimate from method comparison studies in which it that only a single measurement per subject per method is It appear that the estimates are using a one-way ANOVA as above (see The one-way ANOVA model assumes that there are no systematic biases between measurements within subjects, and that the within-subject SDs are for all measurements measurements are from two different methods, of may be bias may exist between the methods and their within-subject SDs may differ. If is the model is and the estimates of within and between subject SD are estimates of different The reliability estimate then has a different interpretation from the We against the estimate from the one-way ANOVA to data from two different measurement methods. As making two measurements with each method an of whether bias between the methods is constant or whether the measurement error SDs differ. the measurements from each method can be separately as two repeatability studies, using the methods estimates of the repeatability coefficient and reliability for each method. The Bland–Altman plot of paired differences against means can be by using the of a two measurements from each method in of the single As with the Bland–Altman plot on a single measurement from each method, an correlation between differences and means could be due to changing bias or a difference in measurement error between the methods. We describe a to whether any bias between measurements made by a method with the as measured by the method. We by the of the two measurements made using the method and we by the measurement made on subject by the measurement method. A plot of against can then be used to investigate whether the bias between the and methods with the as measured by the method. It is that is against rather than against because the a to the common measurement error in and It is for the same that plotting paired differences between measurements from two methods against the measurements from one method is in method comparison between values of and that the bias between the methods with the true value as measured by the method. A statistical of this can be by a for with as the The for the that the in this from zero the of against the of constant If of bias changing with true value is we may be in the at which the bias between the methods We can estimate this by the estimate from the above model by our estimate of the This quantity is an estimate of the in bias method for a in the true value as measured by the method. The choice to against is We could have against and have a different and estimate of the at which the bias This choice is because we will two different and because it means the is statistically The model underlying is a type of and as such can be fitted using software for The is to and for measurements are made by observers or raters. measurements made by two different observers are than are two measurements made by the same observer. as with two methods, measurements from two observers may differ due to bias between the observers observer and their measurement errors may also have different SDs. For example, measurements from an observer who can make more measurements will have a SD than made by a observer. If measurements in the future are to be made by different observers, we to describe and quantify the differences between such measurements in order to whether differences are genuine or may be due to measurement error. As with a method comparison study, the way to study this is for each observer to make at least two measurements of a sample of subjects. The of such a study and the type of statistical analysis that are should be by whether interest lies in a particular of observers, or whether we are in a population of observers or raters. If future measurements are to be made by a of observers, the measurement error study should each of observers making measurements of a sample of subjects, with at least two measurements per observer per subject. We can then use exactly the same methods as for method different observers as different measurement methods. If each observer two or more measurements on each we can whether there is bias or changing bias between observers. We can the bias of future measurements (in by measurements using the corresponding estimated We can also estimate repeatability and reliability for each using the one-way ANOVA model (see It may be of interest to which observers are more and if differences in reliability can be related to observer such as of or If we are to that biases between observers are we can fit a so-called model to such a for a subject and observer The estimates indicate the and for bias between observers. such models can be to the measurement error SD to differ between observers. The observers in a measurement error study can often be considered as a sample of observers from a population of observers who may be used in future studies or clinical In this we are not in the particular observers in the measurement error study, but only in the that they the population of observers. In this it is important that a number of observers is used in the measurement error study. For example, if the measurement error study only two observers, can be the population of observers, because we have a sample of just Because the observers in the measurement error study are considered a sample from the population of observers, we such a study using a model that the observer as a We use a with subject and observer Fitting a model estimates of the between-subject SD, SD and the measurement error SD to be the same for different The SD variability in measurements due to observers, or biases between the observers. such estimates one can calculate the SD of the difference in two measurements, made by the same observer or by two different observers. the SD is the will be than the This is because biases between observers to make measurements from different observers it is more difficult to distinguish between subjects on the basis of measurements made by two different observers than if the subjects been measured by the same observer. In this paper we have distinguished between the concepts of agreement and parameter is to the as they describe different characteristics of the measurement The choice of what to report in a particular study should be by how measurements are to be used in the and also by the fact that may to use a measurement method in a different We have the fact that the reliability of a method depends not only on the of the measurement errors but on the of true values in the population in which measurements are made. As measurement techniques potentially may be used in a variety of clinical and different it is to report estimates of and within-subject SDs. We have which methods we are for the analysis of repeatability studies, method comparison studies, and studies with measurements made by different observers or raters. In we not a single reliability coefficient should be used for method comparison If the reliability of two methods are to be each reliability should be estimated by making at least two measurements on each subject with each measurement method. We have how an between paired differences and means may not be by changing bias between two methods. Such an may also be by a difference in the measurement error but with only one measurement per subject per method it is not to which of is the comparison studies should make at least two measurements per subject per method. This an of the of any between paired differences and means for measurements made by the two methods, and also the repeatability and reliability of each method to be estimated. measurements an observer or measurement error studies must use an number of observers if interest lies in making a population of observers. we results for the of CIs for two in the one-way ANOVA in which two measurements are from each

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