Simple linear regression model

The simple linear regression model which has a single response variable, a single independent variable and two unknown parameters. 

This problem corresponds to the simple linear regression model (w= 1, n= 1, p=2). Taking as Q,=l (all data points are weighed equally) Equations 3.19a and 3.19b become... 

The transformed response values were regressed on the transformed amount values using the simple linear regression model and ordinary least squares estimation. The standard deviation of the response values (about the regression line) was calculated, and plots were formed of the transformed response values and of the residuals versus transformed amounts. 

The simple linear regression model will be assumed throughout this section That is. 

Amount Transformation. Step 2. The amount transformation was performed in a way similar to that of response by use of a power series but for a different reason. In this case linearity was desired in order to use a simple linear regression model. This transformation therefore required a test for satisfactory conformity. One can use a variety of criteria including the correlation coefficient or visual examination of the plot of rgsiduals verses amount. We chose the F test for lack of fit,... 

Assessing linearity is an important aspect in calibration work since lack-of-fit will usually lead to biased results. When a simple linear regression model is chosen, the more general test of goodness-of-fit becomes a test of linearity. 

An example of a calibration curve for Cd with a Zeeman electrothermal atomization atomic absorption spectrometry (ET-AAS) is presented in Figure 6.1. A simple linear regression model is fitted through the data points. The response of the ET-AAS is placed on the ordinate and the concentration of the injected standard solutions on the abscissa. The concentration of the unknown samples can be calculated back as X = (Yt — a)/b. 

Just like simple linear regression model, the method of least squares was used to estimate the coefficients. 

Continuous toxicity data can be generally described as using a regression-type model as depicted in Figure 5.14. This is a simple linear regression model using only one parameter to describe the toxicity. The resulting expression used to describe the relationship between toxicity and the parameter is a typical linear equation ... 

Comparison of the relative carbonyl oxygen separations obtained from FITMOL with the Na+,K+-ATPase inhibition activities given in Table II revealed a striking correspondence. A simple linear regression model was used to test the relationship between oxygen separation and the I50 data for Na+,K+-ATPase... 

In this example one can see that the log-normal model, Fig. 4, is a better fit to the data than the normal distribution, Fig. 3. The parameter estimates for the median and standard deviation of each sample can be found from a straightforward application of the simple linear regression model expressed by equation (3) ... 

Buonaccorsi (1995) present equations for using Option 2 or 3 for the simple linear regression model. In summary, measurement error is not a problem if the goal of the model is prediction, but keep in mind the assumption that the predictor data set must have the same measurement error distribution as the modeling data set. The problem with using option 2 is that there are three variance terms to deal with the residual variance of the model, a2, the uncertainty in 9, and the measurement error in the sample to be predicted. For complex models, the estimation of a corrected a2 may be difficult to obtain. 

The F test for lack of fit of the simple linear regression model is easUy expressed in the six-step procedure. 

Note The null hypothesis for the lack of fit is that the simple linear regression model cannot be rejected at the specific a level. 

This author would elect to use the simple linear regression model to approximate the antimicrobial activity, but would collect more data sets not only to see if the Hq hypothesis would continue to be rejected, but also if the extra variable (182) model would be adequate for the new data. In statistics, data-pattem chasing can be an endless pursuit with no conclusion ever reached. 

If the simple linear regression model, in the researcher s opinion, does not model the data properly, then there are several options ... 

Multiple linear regression is a direct extension of simple linear regression. In simple linear regression models, only one jr predictor variable is present, but in multiple linear regression, there are k predictor values, x X2,..., For example, a two-variable predictor model is presented in the following equation ... 

Because Fc = 0.08 < Ft, 5.47, we cannot reject the Ho hypothesis at a = 0.025. The researcher probably already knew that relative humidity did not influence the stability data substantially, but the calculation was included because it was a variable. The researcher now uses a simple linear regression model. 

The reader is directed to Appendix II for a review of matrices and application of matrix algebra. Once that is completed, we will look at examples of Studentized and jackknifed residuals applied to data from simple linear regression models and then discuss rescaling of residuals as it applies to model leveraging due to outliers. 

The time series shows a. stationary pattern until a homogeneously oscillating flow rate around a constant level. However, an additive outlier occurs at time index 128. To analyse the time dependency pattern of the Naphtha time series, the effect of the outlier has to be removed. Therefore, the outlier s effect (i.e. the raise beyond the mean level) is estimated by modelling a simple linear regression model and the corresponding value of the time series is corrected by this estimate. For the corrected Naphtha time series, ACF and PACF are calculated (Figure 2.13a and Figure 2.13b). 

In analytical chemistry we always try to arrange that Equation [8.19a] provides an adeqnate model for the relationship between the instrumental response (Y) and the concentration (or amount) of analyte (x) injected directly into the instrument (instrumental cahbration) or used to spike a blank matrix (method calibration, see Section 8.5). When analytical chemists speak of a linear calibration equation they refer to Equation [8.19a], a simple linear regression model that is linear in both the fitting parameters and also in the independent variable Equation [8.19b] might be referred to as a non-linear calibration equation by a chemist, although to a statistician it is an example of a simple linear regression model, i.e., it is hnear in aU of the fitting parameters. 

We have already seen in Chapter 1 an example of a simple linear regression model (eqn 1.2, Fig. 1.2) in which anaesthetic activity was related to the hydrophobicity parameter, ti. How was the equation derived If we consider the data shown plotted in Fig. 6.1, it is fairly obvious that a straight line can be fitted through the points. A line is shown on the figure and is described by the well-known equation for a straight line. 

The relationship between the V(N) and V(01) population values and N-Ol bond length is shown in Fig. 19.12. The simple linear regression model applied to the population values shows a negative slope along with the elongation of the r(N-01) distance The longer N-Ol bond has smaller amount of electron density... 

Suppose we have the following nine observations from the simple linear regression model where we know the standard deviation (T = 1. 

In (eco)toxicological QSAR studies, the molecular descriptor of choice is the n-octanol/water partition coefficient (log P), generally used in a simple regression equation. However, sometimes a simple linear regression model is inadequate to model properly the dependence of biological activity (BA) on logF. For example, fish exposed to very hydrophobic chemicals for a limited test duration have insufficient time to achieve a pseudo-steady state partitioning equilibrium between the toxicant concentration in aqueous circumambient phase and the hydrophobic site of action within the fish. Hansch initiated the use of a parabolic model in log P (equation 1) to overcome... 

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