Consider first the case of a single parameter, θ. Let l be the loglikelihood function for θ and let x be the data. The score is the derivative of the loglikelihood function with respect to θ:
The score test can be generalized to multiple parameters. Consider the vector of parameters θ. Then the test statistic for the score test of H0: is:
The test statistic is asymptotically Chisquare distribution with k degrees of freedom. Here k is the number of unbounded parameters.
Let be the value of where the algorithm terminates. Note that the relative gradient evaluated at is the score test statistic. A pvalue is calculated using a Chisquare distribution with k degrees of freedom. This pvalue gives an indication of whether the value of the unknown MLE is consistent with . The number of unbounded parameters listed in the Random Effects Covariance Parameter Estimates report equals k.
The standard random coefficient model specifies a random intercept and slope for each subject. Let yij denote the measurement of the jth observation on the ith subject. Then the random coefficient model can be written as follows:
with G and defined as above.
Assume that the sik are independent and identically distributed N(0, σs2) variables. Denote the number of treatment factors by t and the number of subjects by s. Then the distribution of eijk is N(0, Σ), where
Denote the block diagonal component of the covariance matrix Σ corresponding to the ikth subject within treatment by Σik. In other words, Σik = Var(yiksik). Because observations over time within a subject are not typically independent, it is necessary to estimate the variance of yijksik. Failure to account for the correlation leads to distorted inference. The following sections describe the structures available for Σik.
Here tj is the time of observation j. In this structure, observations taken at any given time have the same variance, . The parameter ρ, where 1 < ρ < 1, is the correlation between two observations that are one unit of time apart. As the time difference between observations increases, their covariance decreases because ρ is raised to a higher power. In many applications, AR(1) provides an adequate model of the within subject correlation, providing more power without sacrificing Type I error control.
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Here, the primary objective is to estimate spatial correlation. Consider the simple model . The spatial structure is modeled through the error term, ei. In general, the spatial correlation model can be defined as and.
Let si denote the location of yi, where si is specified by two coordinates. The coordinates could be latitude and longitude, row and column, indices of northsouth and eastwest, and so on. The spatial structure is typically restricted by assuming that the covariance is a function of the Euclidean distance, dij, between the locations si and sj.
If f(dij) does not depend on the direction, then the covariance structure is isotropic. If it does, then the structure is anisotropic. The spatial model functions available in JMP are shown below. The following formulas for isotropic structures are provided:
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For anisotropic models, the formulas are products of the isotropic formula with distance measured separately in each of the directions. Specifically, if c is the number of coordinates, then dij in the isotropic formulas is replaced by dijk. dijk is the absolute distance between the kth coordinate, k = 1,2,..., c, of the ith and jth observations in data table.
When the spatial process is secondorder stationary, these parameters define a semivariogram. Borrowed from geostatistics, the semivariogram is the standard tool for describing and estimating spatial variability. It measures spatial variability as a function of the distance, dij, between observations. Its key features are:
Defined as the distance at which the semivariogram reaches the sill. At distances less than the range, observations are spatially correlated. For distances greater than or equal to the range, spatial correlation is effectively zero. In spherical models, ρ is the range. In exponential models, 3ρ is the practical range. In Gaussian models, is the practical range. The practical range is defined as the distance where covariance is reduced to 95% of the sill.
In Completed Fit Model Launch Window Showing Repeated Structure Tab, the repeated effects covariance parameter estimates represent the various semivariogram features: