## The Wald Test and Cramér–Rao Bound for Misspecified Models in Electromagnetic Source Analysis

Citations: | 1 - 1 self |

### BibTeX

@MISC{Waldorp_thewald,

author = {Lourens J. Waldorp and Hilde M. Huizenga and Raoul P. P. P. Grasman},

title = {The Wald Test and Cramér–Rao Bound for Misspecified Models in Electromagnetic Source Analysis},

year = {}

}

### OpenURL

### Abstract

Abstract—By using signal processing techniques, an estimate of activity in the brain from the electro- or magneto-encephalogram (EEG or MEG) can be obtained. For a proper analysis, a test is required to indicate whether the model for brain activity fits. A problem in using such tests is that often, not all assumptions are satisfied, like the assumption of the number of shells in an EEG. In such a case, a test on the number of sources (model order) might still be of interest. A detailed analysis is presented of the Wald test for these cases. One of the advantages of the Wald test is that it can be used when not all assumptions are satisfied. Two different, previously suggested, Wald tests in electromagnetic source analysis (EMSA) are examined: a test on source amplitudes and a test on the closeness of source pairs. The Wald test is analytically studied in terms of alternative hypotheses that are close to the null hypothesis (local alternatives). It is shown that the Wald test is asymptotically unbiased, that it has the correct level and power, which makes it appropriate to use in EMSA. An accurate estimate of the Cramér–Rao bound (CRB) is required for the use of the Wald test when not all assumptions are satisfied. The sandwich CRB is used for this purpose. It is defined for nonseparable least squares with constraints required for the Wald test on amplitudes. Simulations with EEG show that when the sensor positions are incorrect, or the number of shells is incorrect, or the conductivity parameter is incorrect, then the CRB and Wald test are still good, with a moderate number of trials. Additionally, the CRB and Wald test appear robust against an incorrect assumption on the noise covariance. A combination of incorrect sensor positions and noise covariance affects the possibility of detecting a source with small amplitude. Index Terms—Approximate model, constrained optimization, Fisher information with constraints, model checking, parameter covariance, separable least squares, source localization. I.

### Citations

709 |
Advanced Econometrics
- Amemiya
- 1985
(Show Context)
Citation Context ...ions are incorrect. In practice, neither nor are known and have to be estimated in order to obtain an estimate of the CRB. For ,an unbiased and asymptotically consistent estimate is (e.g., [12], [39]–=-=[41]-=-) when . This is a common and practical choice if the interest is not in noise characteristics. Other estimates of the noise covariance matrix based on parametric models for the noise covariance (e.g.... |

601 |
Aspects of multivariate statistical theory
- Muirhead
- 1982
(Show Context)
Citation Context ...positions are incorrect. In practice, neither nor are known and have to be estimated in order to obtain an estimate of the CRB. For ,an unbiased and asymptotically consistent estimate is (e.g., [12], =-=[39]-=-–[41]) when . This is a common and practical choice if the interest is not in noise characteristics. Other estimates of the noise covariance matrix based on parametric models for the noise covariance ... |

534 |
Testing Statistical Hypotheses
- Lehmann
- 1959
(Show Context)
Citation Context ...ility that is rejected given that is true is at most . If, in addition, the test rejects when is true with probability 1, which is known as the power of the test, then it is asymptotically consistent =-=[47]-=-. The Wald test can be seen to be asymptotically consistent because for under , , and for alternatives under , .As stated before, most statistical tests are asymptotically consistent; therefore, local... |

526 |
Maximum likelihood estimation of mis-speci¯ed models
- White
- 1982
(Show Context)
Citation Context ...erivativesofthemodel.Inastudy on categorical time series, it was found that the sandwich estimator gives very precise estimates of the parameter variance in spite of possible modeling errors [29]. In =-=[30]-=-, it was shown that the Wald test using the sandwich estimator is a level test (i.e., it rejects the null hypothesis when it is true with probability not grater than : typically 0.05), even when the m... |

396 |
Nonlinear Regression
- Seber, Wild
- 2003
(Show Context)
Citation Context ...seen that if is continuous and if the first-order partial derivative matrix has full row-rank , then the covariance matrix of is , where is the covariance matrix of . The Wald test is then defined as =-=[38]-=- If are normal, then under , the Wald test is asymptotically distributed [38]. Only linear hypotheses of the form are considered. Informally, a test is asymptotically consistent if it can distinguish ... |

165 |
der Vaart, Asymptotic Statistics
- van
- 1998
(Show Context)
Citation Context ...his problem is defined for -estimators, which refers to maximizing a general function of parameters. An example is Huber’s function, which is designed to minimize the influence of extreme data points =-=[33]-=-. The paper is organized as follows. First, the NSLS method is briefly described, followed by the reparameterization and incorporating the ensuing constraints in the CRB. Then, the Wald test is descri... |

113 |
Magnetoencephalography—Theory, instrumentation, and applications to noninvasive studies of the working human
- Hämäläinen, Hari, et al.
- 1993
(Show Context)
Citation Context ...of the activity in the brain from the electro- or magneto-encephaolgram (EEG or MEG) measured at or near the scalp [1]. The inverse problem in which an estimate is obtained relies on many assumptions =-=[2]-=-, [3]. Some examples in EEG are the number of shells [4], their conductivities [5], [6], Manuscript received October 8, 2004; revised March 2, 2005. This work was supported by The Netherlands Organiza... |

96 |
Multiple dipole modeling and localization from spatio-temporal meg data
- Mosher, Lewis, et al.
- 1992
(Show Context)
Citation Context ...rely statistical-based techniques like the residual variance [9], sequential tests based on singular value decompositions [13], or multivariate testing [14] to signal processing techniques like MUSIC =-=[15]-=-, [16], to biophysical model-based approaches [17], [18]. However, these methods do not explicitly incorporate the fact that some of the assumptions could be incorrect. In the present paper, the notio... |

90 |
Dynamic statistical parametric mapping: combining fMRI and MEG for high-resolution imaging of cortical activity
- Dale, Liu, et al.
- 2000
(Show Context)
Citation Context ...ance [9], sequential tests based on singular value decompositions [13], or multivariate testing [14] to signal processing techniques like MUSIC [15], [16], to biophysical model-based approaches [17], =-=[18]-=-. However, these methods do not explicitly incorporate the fact that some of the assumptions could be incorrect. In the present paper, the notion of an approximate (incorrect) model used in the statis... |

79 |
Fundamentals of dipole source potential analysis. In Auditory evoked magnetic fields and electric potential. advances in audiology (Vol
- Scherg
- 1990
(Show Context)
Citation Context ...matrix, then the noise is white (i.e., uncorrelated and homoscedastic). The model can be extended to capture more than a single time sample, where either just the dipole locations are fixed over time =-=[35]-=- or both the locations and orientations are fixed over time [36], [37]. Modeling error arises if the true mean cannot be attained with the function used in the regression for any . A model for the mea... |

66 | Linear and nonlinear models for the analysis of repeated measurements - VONESH, CHINCHILLI - 1997 |

54 |
Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics
- ENGLE
- 1984
(Show Context)
Citation Context ...evaluate the performance of a statistical test, it is more informative to look at alternative hypotheses that are close to the null hypothesis, since this makes it harder to distinguish the two [33], =-=[46]-=-. An example is misspecifying an amplitude parameter of a source by a small amount. The alternative hypothesis is then referred to as a local alternative. A local alternative for the parameter , or eq... |

51 |
Using Least Squares to Approximate Unknown Regression Functions
- White
(Show Context)
Citation Context ...e for using the Wald test in the way described above is that a good estimate of the Cramér–Rao bound (CRB) of the parameters is available, even for approximate models. It has been shown before (e.g., =-=[23]-=-, [24]) that using the Hessian matrix in least squares (LS), or the Fisher information matrix in maximum likelihood (ML), is inappropriate if the model is approximate. This means that the CRB, as desc... |

39 |
A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres”, Phys
- Zhang
- 1995
(Show Context)
Citation Context ...ding conductivities are 1, 0.80, and 1 m . So-called Berg-parameters were estimated to approximate the three shell EEG truncated series (to 70 terms) with the above-mentioned radii and conductivities =-=[49]-=-, [50]. The dipoles have locations with eccentricity 0.81, both with orientation and amplitude 1. Noise distributed as is added to the sensors. The matrix is defined as , where is the Euclidean distan... |

24 | Electrical neuroimaging based on biophysical constraints
- Peralta, Murray, et al.
- 2004
(Show Context)
Citation Context ...l variance [9], sequential tests based on singular value decompositions [13], or multivariate testing [14] to signal processing techniques like MUSIC [15], [16], to biophysical model-based approaches =-=[17]-=-, [18]. However, these methods do not explicitly incorporate the fact that some of the assumptions could be incorrect. In the present paper, the notion of an approximate (incorrect) model used in the ... |

22 | Source localization using recursively applied and projected (rap) music
- Mosher, Leahy
- 1999
(Show Context)
Citation Context ...tatistical-based techniques like the residual variance [9], sequential tests based on singular value decompositions [13], or multivariate testing [14] to signal processing techniques like MUSIC [15], =-=[16]-=-, to biophysical model-based approaches [17], [18]. However, these methods do not explicitly incorporate the fact that some of the assumptions could be incorrect. In the present paper, the notion of a... |

22 |
Error bounds for eeg and meg dipole source localization
- Mosher, Spencer, et al.
- 1993
(Show Context)
Citation Context ...hat using the Hessian matrix in least squares (LS), or the Fisher information matrix in maximum likelihood (ML), is inappropriate if the model is approximate. This means that the CRB, as described in =-=[25]-=-, is not appropriate if, for example, the assumed number of shells in EEG isincorrect.Therefore,adifferentestimatorfortheCRBshouldbe used: the so-called sandwich estimator. The sandwich estimator is n... |

17 |
A fast method to compute the potential in the multi sphere model
- Munck, Peters
- 1993
(Show Context)
Citation Context ...o-encephaolgram (EEG or MEG) measured at or near the scalp [1]. The inverse problem in which an estimate is obtained relies on many assumptions [2], [3]. Some examples in EEG are the number of shells =-=[4]-=-, their conductivities [5], [6], Manuscript received October 8, 2004; revised March 2, 2005. This work was supported by The Netherlands Organization for Scientific Research (NWO). The associate editor... |

14 |
Instrumental variable approach to array processing in spatially correlated noise fields
- Stoica, Viberg, et al.
- 1994
(Show Context)
Citation Context ...priate if, for example, the assumed number of shells in EEG isincorrect.Therefore,adifferentestimatorfortheCRBshouldbe used: the so-called sandwich estimator. The sandwich estimator is not new (e.g., =-=[26]-=-–[28]). The essential difference between the traditional CRB and the sandwich is that the latter involves no assumption on the model being correct. The expression is consequently slightly more involve... |

14 |
Estimated generalized least squares electromagnetic source analysis based on a parametric noise covariance model
- Waldorp, Huizenga, et al.
- 2002
(Show Context)
Citation Context ... This is a common and practical choice if the interest is not in noise characteristics. Other estimates of the noise covariance matrix based on parametric models for the noise covariance (e.g., [34], =-=[42]-=-) could also be used, provided they are consistent for , which requires that the model of the noise covariance is correct. Using a consistent estimate like may pose a problem in finite samples, becaus... |

13 |
Mathematical dipoles are adequate to describe realistic generators of human brain activity
- Munck, Dijk, et al.
- 1988
(Show Context)
Citation Context ...m.huizenga@uva.nl; grasman@psy.uva.nl). Digital Object Identifier 10.1109/TSP.2005.853213 1053-587X/$20.00 © 2005 IEEE their shapes, the type of source (i.e., a dipole or quadrupole, etc.; see, e.g., =-=[7]-=- and [8]), and the number of sources (see [9] when dipoles are used). A model comprised of several such assumptions is only an approximation of what is really going on. In [7], it is shown that dipole... |

12 |
Sensitivity distributions of EEG and MEG measurements
- Malmivuo, Suihko, et al.
- 1997
(Show Context)
Citation Context ...that dipoles are a good approximation of extended sources, but in [9] and [10, Ch. 3], it is shown that using the incorrect number of dipoles can have serious effects on localization. In addition, in =-=[11]-=-, it is shown that the distance between the brain and skull has a large effect on sensitivity of both EEG and MEG. In addition, statistical assumptions, like the distribution of the noise, can affect ... |

11 |
Simulation studies of multiple dipole neuromagnetic source localization: Model order and limits of source resolution
- Supek, Aine
- 1993
(Show Context)
Citation Context ...al Object Identifier 10.1109/TSP.2005.853213 1053-587X/$20.00 © 2005 IEEE their shapes, the type of source (i.e., a dipole or quadrupole, etc.; see, e.g., [7] and [8]), and the number of sources (see =-=[9]-=- when dipoles are used). A model comprised of several such assumptions is only an approximation of what is really going on. In [7], it is shown that dipoles are a good approximation of extended source... |

11 |
The Lagrange Multiplier Test
- Silvey
- 1959
(Show Context)
Citation Context ...se the model requires these or because of identifiability, then the CRB in (4) has to be modified. If is not identifiable, then is singular [44], and therefore, the CRB in (4) cannot be estimated. In =-=[45]-=-, a method to incorporate constraints into the CRB is described for maximum likelihood. For more general estimation methods ( -estimators), a slightly different version is required. This version is st... |

9 | Array processing in correlated noise fields based on instrumental variables and subspace fitting - Viberg, Stoica, et al. - 1995 |

9 | Lopes da Silva, “A random dipole model for spontaneous brain activity - Munck, Vijn, et al. - 1992 |

9 | Spatiotemporal EEG/MEG source analysis based on a parametric noise covariance model
- Huizenga, Munck, et al.
- 2002
(Show Context)
Citation Context .... The model can be extended to capture more than a single time sample, where either just the dipole locations are fixed over time [35] or both the locations and orientations are fixed over time [36], =-=[37]-=-. Modeling error arises if the true mean cannot be attained with the function used in the regression for any . A model for the mean is then said to be approximate or incorrect. This is, for example, t... |

7 |
Estimating and testing the sources of evoked potentials
- Huizenga, Molenaar
- 1994
(Show Context)
Citation Context .... I. INTRODUCTION IN electromagnetic source analysis (EMSA), an estimate is obtained of the activity in the brain from the electro- or magneto-encephaolgram (EEG or MEG) measured at or near the scalp =-=[1]-=-. The inverse problem in which an estimate is obtained relies on many assumptions [2], [3]. Some examples in EEG are the number of shells [4], their conductivities [5], [6], Manuscript received Octobe... |

7 |
Equivalent source estimation of scalp potential fields contaminated by heteroscedastic and correlated noise
- Huizenga, Molenaar
- 1995
(Show Context)
Citation Context ...ain and skull has a large effect on sensitivity of both EEG and MEG. In addition, statistical assumptions, like the distribution of the noise, can affect the accuracy of the dipole parameters greatly =-=[12]-=-. Even if some of the assumptions like the number of shells in EEG may be incorrect, it can still be of interest to know how many sources are present (model order). Many methods to detect sources and ... |

7 | Model selection in electromagnetic source analysis with an application to VEF’s
- Waldorp, Huizenga, et al.
- 2002
(Show Context)
Citation Context ...ypotheses that are very “similar” (local alternatives). The Wald test has been used before in similar settings [19], [20]. Two Wald tests are used in the context of EMSA, which have been suggested in =-=[21]-=- for a single time sample and in [22] for a spatio-temporal model. The first is a test on source amplitudes. The idea is that if a source has (nearly) zero amplitude, it should be considered inactive.... |

7 | Making correct statistical inferences using a wrong probability model
- Golden
- 1995
(Show Context)
Citation Context ...rpartialderivativesofthemodel.Inastudy on categorical time series, it was found that the sandwich estimator gives very precise estimates of the parameter variance in spite of possible modeling errors =-=[29]-=-. In [30], it was shown that the Wald test using the sandwich estimator is a level test (i.e., it rejects the null hypothesis when it is true with probability not grater than : typically 0.05), even w... |

7 |
nonlinear least squares: the variable projection method and its applications, Inverse Problems
- Separable
- 2003
(Show Context)
Citation Context ...ld test still is a level- test. Since the biophysical function in EEG and MEG has both a nonlinear and a linear part, nonlinear separable least squares (NSLS) is used to estimate the parameters [31], =-=[32]-=-. In [28], the asymptotic CRB of all estimated parameters in NSLS is derived, although constraints are not incorporated. Using the Wald test on amplitudes requires a reparameterization that in turn re... |

6 |
Lopes da Silva, “In vivo measurement of the Brain and Skull resisitivities using and EIT based method and the combined analysis of SEF/SEP data
- Gonçalves, Verbunt, et al.
(Show Context)
Citation Context ...generated by two dipoles in a spherical head model with three isotropic and concentric shells on 61 sensors (see Fig. 1). The radii for the brain, skull, and skin are, respectively, 0.87, 0.92, and 1 =-=[48]-=-, and the corresponding conductivities are 1, 0.80, and 1 m . So-called Berg-parameters were estimated to approximate the three shell EEG truncated series (to 70 terms) with the above-mentioned radii ... |

4 |
Spatial resolution of neuronal generators based on
- RD, Biscay-Lirio
- 1993
(Show Context)
Citation Context ...e activity in the brain from the electro- or magneto-encephaolgram (EEG or MEG) measured at or near the scalp [1]. The inverse problem in which an estimate is obtained relies on many assumptions [2], =-=[3]-=-. Some examples in EEG are the number of shells [4], their conductivities [5], [6], Manuscript received October 8, 2004; revised March 2, 2005. This work was supported by The Netherlands Organization ... |

4 |
Estimating brain conductivities and dipole source signals with EEG arrays
- Gutierrez, Nehorai, et al.
- 2004
(Show Context)
Citation Context ...asured at or near the scalp [1]. The inverse problem in which an estimate is obtained relies on many assumptions [2], [3]. Some examples in EEG are the number of shells [4], their conductivities [5], =-=[6]-=-, Manuscript received October 8, 2004; revised March 2, 2005. This work was supported by The Netherlands Organization for Scientific Research (NWO). The associate editor coordinating the review of thi... |

4 |
Determining the number of independent sources of the EEG. a simulation study on information criteria. Brain Topography
- Knosche, Berends, et al.
(Show Context)
Citation Context ...ces and methods to determine the model order exist. These methods vary from purely statistical-based techniques like the residual variance [9], sequential tests based on singular value decompositions =-=[13]-=-, or multivariate testing [14] to signal processing techniques like MUSIC [15], [16], to biophysical model-based approaches [17], [18]. However, these methods do not explicitly incorporate the fact th... |

4 |
Statistical maps for EEG dipolar source localization
- Bénar, Gunn, et al.
(Show Context)
Citation Context ...he model order exist. These methods vary from purely statistical-based techniques like the residual variance [9], sequential tests based on singular value decompositions [13], or multivariate testing =-=[14]-=- to signal processing techniques like MUSIC [15], [16], to biophysical model-based approaches [17], [18]. However, these methods do not explicitly incorporate the fact that some of the assumptions cou... |

3 |
Model selection in spatio-temporal electromagnetic sources analysis
- Waldorp, Huizenga, et al.
- 2005
(Show Context)
Citation Context ...cal alternatives). The Wald test has been used before in similar settings [19], [20]. Two Wald tests are used in the context of EMSA, which have been suggested in [21] for a single time sample and in =-=[22]-=- for a spatio-temporal model. The first is a test on source amplitudes. The idea is that if a source has (nearly) zero amplitude, it should be considered inactive. The other is a test on source “close... |

3 |
and detection of misspecified nonlinear regression models
- “Consequences
- 1981
(Show Context)
Citation Context ...using the Wald test in the way described above is that a good estimate of the Cramér–Rao bound (CRB) of the parameters is available, even for approximate models. It has been shown before (e.g., [23], =-=[24]-=-) that using the Hessian matrix in least squares (LS), or the Fisher information matrix in maximum likelihood (ML), is inappropriate if the model is approximate. This means that the CRB, as described ... |

3 |
de Munck, “The estimation of time varying dipoles on the basis of evoked potentials
- C
- 1990
(Show Context)
Citation Context ...astic). The model can be extended to capture more than a single time sample, where either just the dipole locations are fixed over time [35] or both the locations and orientations are fixed over time =-=[36]-=-, [37]. Modeling error arises if the true mean cannot be attained with the function used in the regression for any . A model for the mean is then said to be approximate or incorrect. This is, for exam... |

2 |
On the forward and the inverse problem for EEG and MEG
- Peters, Munck
- 1990
(Show Context)
Citation Context ...G) measured at or near the scalp [1]. The inverse problem in which an estimate is obtained relies on many assumptions [2], [3]. Some examples in EEG are the number of shells [4], their conductivities =-=[5]-=-, [6], Manuscript received October 8, 2004; revised March 2, 2005. This work was supported by The Netherlands Organization for Scientific Research (NWO). The associate editor coordinating the review o... |

2 | A combined order selection and parameter estimation algorithm for undamped exponentials
- Ying, Sabharwal, et al.
- 2000
(Show Context)
Citation Context ...se it tests parameters directly and has asymptotically relatively large power for hypotheses that are very “similar” (local alternatives). The Wald test has been used before in similar settings [19], =-=[20]-=-. Two Wald tests are used in the context of EMSA, which have been suggested in [21] for a single time sample and in [22] for a spatio-temporal model. The first is a test on source amplitudes. The idea... |

2 |
solutions for inverse methods
- Mosher, Leahy, et al.
- 1999
(Show Context)
Citation Context ...onductivities are 1, 0.80, and 1 m . So-called Berg-parameters were estimated to approximate the three shell EEG truncated series (to 70 terms) with the above-mentioned radii and conductivities [49], =-=[50]-=-. The dipoles have locations with eccentricity 0.81, both with orientation and amplitude 1. Noise distributed as is added to the sensors. The matrix is defined as , where is the Euclidean distance bet... |

1 |
et al., “MEG source imaging using multipolar expansions
- Mosher, Leahy, et al.
- 1999
(Show Context)
Citation Context ...ga@uva.nl; grasman@psy.uva.nl). Digital Object Identifier 10.1109/TSP.2005.853213 1053-587X/$20.00 © 2005 IEEE their shapes, the type of source (i.e., a dipole or quadrupole, etc.; see, e.g., [7] and =-=[8]-=-), and the number of sources (see [9] when dipoles are used). A model comprised of several such assumptions is only an approximation of what is really going on. In [7], it is shown that dipoles are a ... |

1 | The Statistical Approach to Electromagnetic Source Localization - Huizenga - 1995 |

1 |
statistic for model order selection in superposition models
- Sabharwal, Potter, et al.
- 2002
(Show Context)
Citation Context ... because it tests parameters directly and has asymptotically relatively large power for hypotheses that are very “similar” (local alternatives). The Wald test has been used before in similar settings =-=[19]-=-, [20]. Two Wald tests are used in the context of EMSA, which have been suggested in [21] for a single time sample and in [22] for a spatio-temporal model. The first is a test on source amplitudes. Th... |

1 |
Large sample properties of separable nonlinear least squares estimators
- Mahata, Söderström
- 2004
(Show Context)
Citation Context ...e if, for example, the assumed number of shells in EEG isincorrect.Therefore,adifferentestimatorfortheCRBshouldbe used: the so-called sandwich estimator. The sandwich estimator is not new (e.g., [26]–=-=[28]-=-). The essential difference between the traditional CRB and the sandwich is that the latter involves no assumption on the model being correct. The expression is consequently slightly more involved tha... |

1 |
The differentiation of pseudo-inverses and nolinear least squares problems whose variables separate
- Golub, Pereyra
- 1973
(Show Context)
Citation Context ...the Wald test still is a level- test. Since the biophysical function in EEG and MEG has both a nonlinear and a linear part, nonlinear separable least squares (NSLS) is used to estimate the parameters =-=[31]-=-, [32]. In [28], the asymptotic CRB of all estimated parameters in NSLS is derived, although constraints are not incorporated. Using the Wald test on amplitudes requires a reparameterization that in t... |

1 |
A note on the efficiency of sandwich covariance extimation
- Kauermann, Carroll
- 2001
(Show Context)
Citation Context ...est can be avoided by using the sandwich estimate. An estimate of is given by when . This estimate is an extension of the commonly used version for the case of white noise, i.e., (see, e.g., [23] and =-=[43]-=-). From the estimates of the noise covariance above, three estimators of the CRB can be constructed: two from the familiar one and one from the sandwich . First of all, to compute the CRB, the estimat... |