Linear models, confidence intervals

Confidence limits for the parameter estimates define the region where values of bj are not significantly different from the optimal value at a certain probability level 1-a with all other parameters kept at their optimal values estimated. The confidence limits are a measure of uncertainty in the optimal estimates the broader the confidence limits the more uncertain are the estimates. These intervals for linear models are given by... 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

The corresponding (l-a)100% confidence interval for the multiresponse linear model is... 

Next let us turn our attention to models described by a set of ordinary differential equations. We are interested in establishing confidence intervals for each of the response variables y, j=l,...,/w at any time t=to. The linear approximation of the output vector at time to,... 

This suggests that a plot of r0 vs pA (Fig. 4) or r0/pA vs pA should be linear. It is very difficult to reject this model on the basis of data curvature, even though it is evident that some curvature could exist in Fig. 4. However, Eq. (16) demands that Fig. 4 also exhibit a zero intercept. In fact, the 99.99% confidence interval on the intercept of a least-square line through the data does not contain zero. Hence the model could be rejected with 99.99% certainty. 

Figure 1. Plots showing the Calibration Process. A. Response transformation to constant variance Examples showing a. too little, b. appropriate, and c. too much transformation power. B. Amount Transformation in conforming to a (linear) model. C. Construction of p. confidence bands about the regressed line, q. response error bounds and intersection of these to determine r. the estimated amount interval.

Analysis of variance appropriate for a crossover design on the pharmacokinetic parameters using the general linear models procedures of SAS or an equivalent program should be performed, with examination of period, sequence and treatment effects. The 90% confidence intervals for the estimates of the difference between the test and reference least squares means for the pharmacokinetic parameters (AUCo-t, AUCo-inf, Cmax should be calculated, using the two one-sided t-test procedure). 

The linear regression model is inadequate with 95% confidence. Since the linear model is neither symmetrical nor adequate and since the application of the method of steepest ascent would lead to a one-factor optimization (b2 is by far the greatest), a new FRFE 24 1 has been designed with doubled variation intervals for X3 X3 and X4. 

Despite the fact that it is difficult to define a relationship between PK parameters and measures of hepatic function, the most appropriate statistical approach is to calculate geometric means and 95% confidence intervals to compare the healthy and impaired groups (see example). Investigation of the relationships between hepatic functional abnormalities and selected PK parameters using linear and non linear models in order to derive dose recommendations are an appropriate alternative, yet, in spite of many constraints. 

The test results are evaluated using the Shooman plot. The discovery rate is plotted versus the total number of defects discovered (Fig. 5). A regression linear fit curve is calculated and plotted together with maximum and minimum fits which by definition have a confidence interval of 5%. From the Shooman reliability model (Fig. 5), the number of remaining defects can be estimated. This information is useful to forecast the number of test cycles that are still necessary and a possible release date. 

The double bootstrap was a method originally suggested by Efron (15) as a way to improve on the bootstrap bias correction of the apparent error rate of a linear discrimination rule. It is simply a bootstrap iteration (i.e., taking resamples from each bootstrap resample). The double bootstrap has been useful in improving the accuracy of confidence intervals but it substantially increases computation time and most likely increases the incidence of unsuccessfully terminated runs. It has been applied to linear models but not to PM modeling. 

The model used for accuracy (1) is shown in the previous section (Example Protocol section IIB) recall that accuracy for each data point is defined as Accuracy = (Observed mass/Ex-pected mass)X100%. The assay-specific averages for accuracy, the across-assay average, standard error of the across-assay average, Satterthwaite degrees of freedom, and 95% confidence intervals are shown in Table 9. Note that because Satterthwaite degrees of freedom are a linear combination of sums of squares, they can assume noninteger values. 

FIGURE 3.1 Linear regression for Example 3.25. Filled circles represent the experimental data and the linear model is shown with the solid line. The dashed lines mark the bounds of the 95% confidence interval on predictions for the linear model. 

Neither. These tell you about the linear relation between y and x, true, but in analytical chemistry you are rarely testing the linear model. The standard error of the regression (Sy/X) is a useful number to quote, or calculate 95% confidence intervals on parameters and estimated concentrations of test solutions. Plot residuals against concentration if you are concerned about curvature or heteroscedacity. (Sections 5.3.2, 5.5)... 

Donaldson and Schnabel (1987) used Monte Carlo simulation to determine which of the variance estimators was best in constructing approximate confidence intervals. They conclude that Eq. (3.47) is best because it is easy to compute, and it gives results that are never worse and sometimes better than the other two, and is more stable numerically than the other methods. However, their simulations also show that confidence intervals obtained using even the best methods have poor coverage probabilities, as low as 75% for a 95% confidence interval. They go so far as to state confidence intervals constructed using the linearization method can be essentially meaningless (Donaldson and Schnabel, 1987). Based on their results, it is wise not to put much emphasis on confidence intervals constructed from nonlinear models. 

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters... 

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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