A confidence interval is a range of values that has a high probability of containing the parameter being estimated. The 95% confidence interval is constructed in such a way that 95% of such intervals will contain the parameter. Similarly, 99% of 99% confidence intervals contain the parameter. If the parameter being estimated were μ, the 95% confidence interval might look like the following:
12.5 μ 30.2
If other information about the value of the paramter is available, it should be taken into consideration when assessing the likelihood that the interval contains the parameter. As an extreme example, consider the case in which 1,000 studies estimating the value of μ in a certain population all resulted in estimates between 25 and 30. If one more study were conducted and if the 95% confidence interval on μ were computed (based on that one study) to be:
35 μ 45
then the probability that μ is between 35 and 45 is very low, the confidence interval not withstanding.
Friday, April 30, 2004
Estimation par intervalle de confiance
Cette nouvelle approche est souvent préférée dans la pratique car elle introduit la notion d'incertitude. On cherche à déterminer l'intervalle centré sur la valeur numérique estimée du paramèter inconnu contenant la valeur vraie avec un probabilité fixée a priori. Cette probabilité permet de s'adapter aux exigences de l'application.
L'intervalle est appelé intervalle de confiance et est le coefficient de confiance. Une estimation par intervalle de confiance sera d'autant meilleure que l'intervalle sera petit pour un coefficient de confiance grand.
La donnée de départ, outre l'échantillon, sera la connaissance de la loi de probabilité du paramètre à estimer. Comme il n'existe pas de résolution générale de ce problème, nous allons aborder successivement les cas les plus fréquents (estimation d'une proportion, d'une moyenne, d'une variance de loi normale).
Cette nouvelle approche est souvent préférée dans la pratique car elle introduit la notion d'incertitude. On cherche à déterminer l'intervalle centré sur la valeur numérique estimée du paramèter inconnu contenant la valeur vraie avec un probabilité fixée a priori. Cette probabilité permet de s'adapter aux exigences de l'application.
L'intervalle est appelé intervalle de confiance et est le coefficient de confiance. Une estimation par intervalle de confiance sera d'autant meilleure que l'intervalle sera petit pour un coefficient de confiance grand.
La donnée de départ, outre l'échantillon, sera la connaissance de la loi de probabilité du paramètre à estimer. Comme il n'existe pas de résolution générale de ce problème, nous allons aborder successivement les cas les plus fréquents (estimation d'une proportion, d'une moyenne, d'une variance de loi normale).
2. Theoretical background
Dynamic systems
The physiological behavior of some organs can be modeled using the dynamic systems theory. The heart, veins, arteries and capillaries may be treated as a biological system, mathematically representable, where variables and parameters interact regularly. This representation provides a powerful tool to estimate variables when some parameters are unknown. Another advantage of the dynamic systems theory is that we can handle various -apparently different- physical parameters (e.g., temperature, flow, concentration and voltage) without modifying the equation’s essence. In recent years, the study field for new concepts such as the biochemical circuit theory analysis is today a fundamental field of interest for many laboratories.
Figure 1. The blood circulation as a dynamic system
The physical elements treated in this medical context, are the perfusion parameters, i.e., regional cerebral blood flow rCBF, regional cerebral blood volume rCBV, mean transit time MTT. However, the same parameters and relationships can be deduced and applied in a different area such as the myocardial circulatory system. In the published literature, the notation is quite similar except for that the letter M (myocardial) replaces the letter C (cerebral), in the perfusion parameter abbreviations.
The human circulatory system is divided into two main parts with the heart acting as a double pump. This organ pumps of a special kind of fluid, which is the blood. The across variable in this case, the blood flow, leaves the left side of the pump (heart) and travels through arteries which gradually divide into capillaries. In the capillaries, the nutrients are released to the body cells. The blood then travels in veins back to the right side of the pump, and the whole process begins again.
One of the most common ways in which, this kind of process is analytically described is the black-box model. The biodynamic system has, therefore, the following components:
Parameters: the rCBF and rCBV.
Dependent variables: the arterial, tissular and venous concentration denoted Ca, Ct and Cv respectively
Independent variable: the time.
The Stewart-Hamilton physiological model
The Stewart-Hamilton model applied to MRI acquisition techniques can estimate perfusion parameters such as the regional blood flow rBF, mean transit time MTT or regional blood volume rBV, describing the tissue condition.
The concentration of tracer within a tissular volume, at a given time t, during the passage of a bolus injection of a contrast agent, is given by:
Where denotes the tissue density (i.e., the tissue mass per unit volume), Cartery is the arterial concentration and R(t) is the residue function and represents the tissular retention of the contrast agent. This biodynamic system has a single fluid storage element, hence, the residue function can be treated as the response of a circuit of first order. Solving the differential equation, we write:
Where MTT is the time taken by the blood to traverse the system, from the arterial entrance to the vascular exit.
Let
Assuming the tissular density being close to the water density, , then:
However, the concentration of the contrast agent is not directly measured in clinical practice; instead, MR images are used to determine signal intensity variations over time, which are related with the change in the spin-spin relaxation, also known as the transverse relaxation . The SI functions are obtained by selecting a pixel of a region of interest (ROI) in the MR gray-scale digital images.
Previous studies have shown the existence of a linear relation between the measured concentration C(t) and . On the other hand, the transverse relaxation and the signal intensity have an exponential dependence, therefore, we write:
Where TE is the sequence echo time, S(t) is the signal intensity over time and S0 is the baseline MR intensity.
Consequently, by measuring SItissue and SIaif we can estimate the regional blood flow using a deconvolution technique and evaluating the expression r(t) at zero.
Once the regional blood flow is estimated, the unknowns perfusion parameters, can be computed from the central volume theorem (Stewart, 1984; Meier and Zierler, 1954):
where
However, the concentration is measured from the MR digital images, therefore, the continuous time form of the equations is not considered. The Stewart-Hamilton model becomes a discrete convolution:
Moreover, the discrete convolution can be viewed as a matricial product:
Where N is the number of samples taken in the RM sequence.
This product can be compactly denoted as:
Where is unknown, then:
Hence, the deconvolution process becomes a linear algebra problem, specifically, a matrix inversion problem. However, this process, when working with noisy signals, could sometimes lead to determinants close to zero, the matrix A is said to be “ill-conditioned”, and consequently, unexpected results may be obtained.
Singular Value decomposition (SVD)
The singular value decomposition is a widely used technique to solve ill-conditioned problems with several applications, (e.g., in image compression, watermarking, image filtering). The SVD can be written as:
Every real matrix A can be decomposed into a product of three matrices of the form:
Where, U and V are orthogonal matrices and S is a diagonal matrix, whose elements are the singular values of the original matrix. with .
Consequently, the inverse can be expressed as:
Where the elements of W, denoted w i,j, are of the form:
Depending on the signal-to-noise ratio of the signal intensity function, a tolerance threshold PSVD is set. The values of W corresponding to values where S is less than PSVD are set to zero.
Typically the PSVD is given as a percentage generally varying between 20 and 30%. The general principle is the higher the noise, the higher the PSVD.
Adaptive threshold
Liu et al. (2) showed that the PSVD selection had a significant influence in the shape of the residue function and an apparently inaccuracy in the rCBF estimation. An oscillation index O, was proposed by Østergaard et al. (1) to measure the distortion in the residue function as follows:
Where f is the scaled estimated residue function, fmax is the maximum amplitude of f, and L is the number of sample points.
ARMA
this expression can be written as follows:
Dynamic systems
The physiological behavior of some organs can be modeled using the dynamic systems theory. The heart, veins, arteries and capillaries may be treated as a biological system, mathematically representable, where variables and parameters interact regularly. This representation provides a powerful tool to estimate variables when some parameters are unknown. Another advantage of the dynamic systems theory is that we can handle various -apparently different- physical parameters (e.g., temperature, flow, concentration and voltage) without modifying the equation’s essence. In recent years, the study field for new concepts such as the biochemical circuit theory analysis is today a fundamental field of interest for many laboratories.
Figure 1. The blood circulation as a dynamic system
The physical elements treated in this medical context, are the perfusion parameters, i.e., regional cerebral blood flow rCBF, regional cerebral blood volume rCBV, mean transit time MTT. However, the same parameters and relationships can be deduced and applied in a different area such as the myocardial circulatory system. In the published literature, the notation is quite similar except for that the letter M (myocardial) replaces the letter C (cerebral), in the perfusion parameter abbreviations.
The human circulatory system is divided into two main parts with the heart acting as a double pump. This organ pumps of a special kind of fluid, which is the blood. The across variable in this case, the blood flow, leaves the left side of the pump (heart) and travels through arteries which gradually divide into capillaries. In the capillaries, the nutrients are released to the body cells. The blood then travels in veins back to the right side of the pump, and the whole process begins again.
One of the most common ways in which, this kind of process is analytically described is the black-box model. The biodynamic system has, therefore, the following components:
Parameters: the rCBF and rCBV.
Dependent variables: the arterial, tissular and venous concentration denoted Ca, Ct and Cv respectively
Independent variable: the time.
The Stewart-Hamilton physiological model
The Stewart-Hamilton model applied to MRI acquisition techniques can estimate perfusion parameters such as the regional blood flow rBF, mean transit time MTT or regional blood volume rBV, describing the tissue condition.
The concentration of tracer within a tissular volume, at a given time t, during the passage of a bolus injection of a contrast agent, is given by:
Where denotes the tissue density (i.e., the tissue mass per unit volume), Cartery is the arterial concentration and R(t) is the residue function and represents the tissular retention of the contrast agent. This biodynamic system has a single fluid storage element, hence, the residue function can be treated as the response of a circuit of first order. Solving the differential equation, we write:
Where MTT is the time taken by the blood to traverse the system, from the arterial entrance to the vascular exit.
Let
Assuming the tissular density being close to the water density, , then:
However, the concentration of the contrast agent is not directly measured in clinical practice; instead, MR images are used to determine signal intensity variations over time, which are related with the change in the spin-spin relaxation, also known as the transverse relaxation . The SI functions are obtained by selecting a pixel of a region of interest (ROI) in the MR gray-scale digital images.
Previous studies have shown the existence of a linear relation between the measured concentration C(t) and . On the other hand, the transverse relaxation and the signal intensity have an exponential dependence, therefore, we write:
Where TE is the sequence echo time, S(t) is the signal intensity over time and S0 is the baseline MR intensity.
Consequently, by measuring SItissue and SIaif we can estimate the regional blood flow using a deconvolution technique and evaluating the expression r(t) at zero.
Once the regional blood flow is estimated, the unknowns perfusion parameters, can be computed from the central volume theorem (Stewart, 1984; Meier and Zierler, 1954):
where
However, the concentration is measured from the MR digital images, therefore, the continuous time form of the equations is not considered. The Stewart-Hamilton model becomes a discrete convolution:
Moreover, the discrete convolution can be viewed as a matricial product:
Where N is the number of samples taken in the RM sequence.
This product can be compactly denoted as:
Where is unknown, then:
Hence, the deconvolution process becomes a linear algebra problem, specifically, a matrix inversion problem. However, this process, when working with noisy signals, could sometimes lead to determinants close to zero, the matrix A is said to be “ill-conditioned”, and consequently, unexpected results may be obtained.
Singular Value decomposition (SVD)
The singular value decomposition is a widely used technique to solve ill-conditioned problems with several applications, (e.g., in image compression, watermarking, image filtering). The SVD can be written as:
Every real matrix A can be decomposed into a product of three matrices of the form:
Where, U and V are orthogonal matrices and S is a diagonal matrix, whose elements are the singular values of the original matrix. with .
Consequently, the inverse can be expressed as:
Where the elements of W, denoted w i,j, are of the form:
Depending on the signal-to-noise ratio of the signal intensity function, a tolerance threshold PSVD is set. The values of W corresponding to values where S is less than PSVD are set to zero.
Typically the PSVD is given as a percentage generally varying between 20 and 30%. The general principle is the higher the noise, the higher the PSVD.
Adaptive threshold
Liu et al. (2) showed that the PSVD selection had a significant influence in the shape of the residue function and an apparently inaccuracy in the rCBF estimation. An oscillation index O, was proposed by Østergaard et al. (1) to measure the distortion in the residue function as follows:
Where f is the scaled estimated residue function, fmax is the maximum amplitude of f, and L is the number of sample points.
ARMA
this expression can be written as follows: