Linear model with intercept

Linear Model With Intercept. There are two distinct linear first-order regression models that are generally encountered in analytical calibration. The non-zero intercept model is the most familiar, and it is given by Equation 1. 

Table Il-a. Formulation of Regression Analysis Table Using The Calibration Data of Table I and A Linear Model With Intercept...

FIGURE 5.2 Diagram of three different types of linear models with n standards. Left the simplest model has a slope and no intercept. The center model adds a nonzero intercept. The right model is typically noted in the literature as the multiple linear regression (MLR) model because it uses more than one response variable, and n>(m+ 1) with an intercept term and n> m without an intercept term. This model is shown with a nonzero intercept. 

Inspection of these tables shows that the LOF test results are very similar to those for the models with intercepts. Comparison of Tables Il-b and Ill-b reveals that the SS residuals are somewhat larger for the zero intercept models than for the models with an intercept. This difference can be used to test the hypothesis that the intercept is zero. First, it must be demonstrated that the LOF is not significant since it would not make good sense to test the zero intercept hypothesis for linear models shown not to fit the data. Furthermore, the SS error and SS(LOF) should not be combined as SS residuals when LOF is significant. These requirements are met by the Case I results. To test the hypothesis that the intercept does not differ significantly from zero, calculate ... 

As measure for the quality of fit, the F value for the goodness-of-fit is calculated by the linear model with the intercept, b, by... 

According to the last column in Table 5.9, the stepwise approach has resulted in a model with 22 variables plus an intercept that are statistically significant at greater than tlie 95% confidence level. Considering that there are a small number of sources of variation in tire mixtures (e.g., only two components) and that the data are known to behave linearly, a 22-variabIe model is probably excessive. 

It should be noted that the system of linear equations expressed by (54) represents the response variable from the three batches. The required condition for the three batches to have the same intercept is that 5i = 52 = 0. The three batches will have the same slope if and only if Ai = A2 = 0. Thus, the problem for testing poolabil-ity reduces to fit the regression model (54) and test the following hypotheses hy. 5i = 52 = 0 and h2 = A2 = 0. Therefore, if the null hypothesis h2 is not rejected at the 0.25 significance level, it implies that the slopes of the three batches are the same, that is, Pi = p2 = = p. Similarly, if the null hypothesis hi is not rejected, the intercepts of the three batches are the same, that is, ai = a2 = a3 = a. If that is the case, the shelf life is determined by a model with a single intercept and a single slope. 

Prerequisites for the Calibration Types. It depends on the design of the analytical procedure as to which regression parameters are meaningful and which results are acceptable. In other words, the model to be used for quantitation must be justified. For a singlepoint calibration (external standardization), a linear function, zero intercept, and the homogeneity of variances are required. The prerequisites for a linear multiple-point calibration are a linear function and in case of an unweighted calibration also the homogeneity of variances. A non-linear calibration requires only a continuous function. With respect to the 100%... 

Before a solution to the problem is presented, it is necessary to examine what happens when the measurement error in x is ignored and the SLR model applies. When a classical error model applies, the effect of measurement error in x is attenuation of the slope and corresponding inflation of the intercept. To illustrate this, consider the linear model Y = x + 10, where x is a set of triplicate measurements at 50, 100, 250, 500, 750, 1000. Y is not measured with error, only x has error. Figure 2.8 plots the resulting least squares fit with increasing measurement error in x. As the measurement error variance increases, the slope of the line decreases with increasing intercept. Sometimes the attenuation is so severe that bias correction techniques must be used in place of OLS estimates. 

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.

There are many situations where a linear model is desired (i.e., y = bx + c). The optimum values of the first-order coefficient b (i.e., slope) and the zeroth-order coefficient c (i.e., intercept) can be calculated from a subset of the information provided above for a second-order polynomial model. It is not necessary to minimize the error with respect to the second-order coefficient a. Furthermore, a = 0 in the other two linear equations. Hence, equations (7-29) reduce to ... 

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