Slope covariance with intercept

By way of illustration, the regression parameters of a straight line with slope = 1 and intercept = 0 are recursively estimated. The results are presented in Table 41.1. For each step of the estimation cycle, we included the values of the innovation, variance-covariance matrix, gain vector and estimated parameters. The variance of the experimental error of all observations y is 25 10 absorbance units, which corresponds to r = 25 10 au for all j. The recursive estimation is started with a high value (10 ) on the diagonal elements of P and a low value (1) on its off-diagonal elements. 

The truth probably lies between these two extremes, and this is where the random-effects model F is useful. Like the summary-measures approach, with which it has conceptually much in common, this also has two levels. However, these two levels are not treated as two stages but are handled together in one model and fitting process. The individual intercept and slopes, and ft, form the first level. These are considered to be random drawings from a population with a Normal population of such values with parameters and the covariance between slope and... 

Finally, the effect of the position of the third experiment on the covariance associated with hg nd h, is seen in Figure 8.6 to equal zero at x,3 = 0. If the third experiment is located at x, < 0, then the estimates of the slope and intercept vary together in the same way (the covariance is positive see Section 7.4). If the third experiment is located at x, > 0, the estimates of the slope and intercept vary together in opposite ways (the covariance is negative). 

Figure 6.5 Simulated time-effect data where the intercept was normally distributed with a mean of 100 and a standard deviation of 60. The effect was linear over time within an individual with a slope of 1.0. No random error was added to the model—the only predictor in the model is time and it is an exact predictor. The top plot shows the data pooled across 50 subjects. Solid line is the predicted values under the simple linear model pooled across observations. Dashed line is the 95% confidence interval. The coefficient of determination for this model was 0.02. The 95% confidence interval for the slope was —1.0, 3.0 with a point estimate of 1.00. The bottom plot shows how the data in the top plot extended to an individual. The bottom plot shows perfect correspondence between effect and time within an individual The mixed effects coefficient of determination for this data set was 1.0, as it should be. This example was meant to illustrate how the coefficient of determination using the usual linear regression formula is invalid in the mixed effects model case because it fails to account for between-subject variability and use of such a measure results in a significant underestimation of the predictive power of a covariate.

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