## 4 Quantitative Methods for the Detection of AD
### General Linear Models and Statistical Parametric Mapping  
The general linear model (GLM) is a simple method for generating predictions from data. The GLM will model the response variable (\(Y\)) in terms of a weighted linear combination of the explanatory variables (\(X\)) and uniquely defined response parameters (\(\beta\)), plus a normally distributed error term (\(\epsilon\)) in the form  
\[Y_{M}=\beta_{1}X_{m1}+\ \ldots\ +\ \beta_{\ell}X_{m\ell}\ +\ \ldots\ +\ \beta_{L}X_{ML}+\epsilon_{M} \tag{1}\]  
where there are \(M\) modeled responses and \(L\) total explanatory variables [64]. The explanatory variables may be continuous or discrete and may be any combination of features (e.g., FDG uptake in the precuneus) or covariates (e.g., age and sex). For the case of neuroimaging studies where each voxel is a feature, the number of observations is always less than the number of features/covariates, and so the GLM is an underdetermined system and cannot be explicitly solved. The model parameters are then estimated using the method of ordinary least squares,  
\[\epsilon=Y-\beta X=Y-\hat{Y}, \tag{2}\]  
where the hat notation refers to the estimated model values and the bold notation indicates row or column vectors of values. Under the condition that \(\mathbf{X}^{T}\mathbf{X}\) is invertible, we may uniquely solve for the least squares estimate of the parameters as  
\[\hat{\mathbf{\beta}}=\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{Y}. \tag{3}\]  
If the parameter estimates are normally distributed, any arbitrary linear combination of these will also be normally distributed. Consider then that for some vector of weights \(c\) known as the contrast vector, a hypothesis based on Student's \(t\)-distribution for \(M-L\) degrees of freedom and using the model parameters may be written as:  
\[T=\frac{\epsilon^{T}\hat{\mathbf{\beta}}-\epsilon^{T}\beta}{\sqrt{\hat{\sigma^{2} }\epsilon^{T}\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\epsilon}}, \tag{4}\]  
where \(\hat{\sigma^{2}}\) is the estimated variance of the errors, defined as:  
\[\hat{\sigma^{2}}=\frac{\epsilon^{T}\epsilon}{M-L} \tag{5}\]  
and the null hypothesis requires that \(\epsilon^{T}\beta=0\). A similar expression may be obtained for generating \(F\)-statistics [64]. The extremely successful application of the GLM to the domain of neuroimaging has been enshringed in the method of statistical parametric mapping (SPM). Developedby [PERSON] and colleagues, it is the most widely used neuroimaging analytic tool that has been used for research purposes since the 1990s [76; 77]. It allows users to run voxel-wise GLM analyses and provides options for multiple comparison corrections that are tailored for neuroimaging data. The output of this mass univariate analysis is a statistical parametric map such as that seen in Figure 2--an image where voxel values are assumed to be distributed under a known probability density function (e.g., \(t\)-distribution). These maps are the primary output of SPM and are used as evidence for or against the null hypothesis at the level of the voxel [76; 77]. This method of generating parametric maps has resulted in the eponymous software package known as \"SPM\", of which the latest version as of publication is SPM12 (Wellcome Trust Centre for Neuroimaging, London, UK). In SPM and many other neuroimaging analytic software programs, the images are rotated in a common direction and are spatially normalized with respect to a common stereotactic space (e.g., Montreal Neurological Institute stereotactic coordinates) to ensure spatial registration at the level of the voxel. These may also be co-registered in cases where more than one mode of neuroimaging data are available. Images are then usually smoothed with an isotropic Gaussian filter to ensure the robustness of the registration process and increase the signal-to-noise ratio of the data. Smoothing also ensures that the neighboring voxels are interdependent. Smoothing is also a necessary condition for the multiple comparison correction with Gaussian field randoms [78].