Test for lack-of-fit

An F-test for lack of fit is based on the ratio of the lack of fit sum to the pure error sum of squares divided by their corresponding degrees of freedom ... 

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,... 

An appropriate test for lack of fit is the / -test described in Section 6.5. 

It is important in this case to keep in mind the distinction between statistical significance and practical significance. If, in a practical sense, the residuals are small enough to be considered acceptable for the particular application, it is not necessary to test for lack of fit. 

Is there a better equation Any given model equation can be tested for lack of fit . 

Such a situation will often arise if the model does indeed fit the data well, and if the measurement process is highly precise. Recall that the F-test for lack of fit compares the variance due to lack of fit with the variance due to purely experimental uncertainty. The reference point of this comparison is the precision with which measurements can be made. Thus, although the lack of fit might be so small as to be of no practical importance, the F-test for lack of fit will show that it is statistically significant if the estimated variance due to purely experimental uncertainty is relatively very small. 

To test for lack-of-fit, a classification variable is added to the model representing the expected mass group ... 

F test for statistical significance of the regression—MS i/MS, F test for lack of fit=MSLf/MSp5... 

To demonstrate this we use the simple example that was introduced in chapter 4, that of solubility in a mixed surfactant system. The treatment is in two stages, the first being a intuitive rather than mathematical demonstration of testing for lack of fit and curvature of a response surface. Then, in section Il.B, we will carry out a more detailed, statistical analysis of the same process, showing how prediction confidence limits are calculated and the use of ANOVA in validating a model. 

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

For data that resemble Pattern C, the researcher needs to up the power scale of X (x, r , etc.) or down the power scale of y y/y, log y, etc.) to linearize the data. For reasons previously discussed, it is recommended to transform the y values only, leaving the x values in the original form. In addition, once the data have been reexpressed, plot them to help determine visually if the reexpression adequately linearized them. If not, the next lower power transformation should be used, on the y value in this case. Once the data are reasonably linear, as determined visually, the F test for lack of fit can be used. Again, the smaller the Fc value, the better. If, say, the data are not quite linearized by y/y but are slightly curved in the opposite direction with the log y transformation, pick the reexpression with the smaller F value in the lack-of-fit test. 

EXPLORATORY DATA ANALYSIS TO DETERMINE THE LINEARITY OF A REGRESSION LINE WITHOUT USING THE Fc TEST FOR LACK OF FIT... 

A relatively simple and effective way to determine if a selected reexpression procedure linearizes the data can be completed with EDA pencil-paper techniques (Figure 2.28). It is known as the method of half-slopes in EDA parlance. In practice, it is suggested, when reexpressing a data set to approximate a straight line, that this EDA procedure be used rather than the Fc- test for lack of fit. 

Tests for goodness of fit. While a linear model can be fit to any set of data, a straight line may not be the best model. It is possible in the regression analysis to check for lack of fit, that is, the inability of the model to adequately describe the relationship between X and Y. This statistical test requires that two or more independent observations (measurements) must be made at each level of X. Independent here means that unique responses must be obtained. Determining the white blood cell count from the same sample three times does not provide three independent observations. Although the test for lack of fit cannot indicate what the appropriate model would be, it can enable the experimenter to assess the validity of the assumed model. This is why it is frequently requested by statisticians that multiple observations be obtained for the various levels of the X variable. 

We return now to the data in duphcate in Table 5.7. We already know that the linear model is inadequate for this temperature range but we are going to adjust it anyway, to show how the F-test for lack of fit works. Once again, we start by using matrix Eq. (5.12) to find the regression equation, taking care to match the corresponding elements of the X and y matrices, which now have 18 rows instead of 9. We then write, from Table 5.7,... 

The cubic model cannot be tested for lack of fit, since it has as many parameters as there are distinct runs. It is indeed better at describing the interior of the triangle than the quadratic model, but we do not have a way to decide whether the cubic model is the most adequate to represent the data over the entire experimental region. Consequently, we can spht the total variance about the average into only two contributions, one for the regression and the other for the residuals. In other words, the ANOVA in this case will be similar to the one of Table 5.2. The results are in Table 7.2. 

For the tabulated critical F value, we obtain f (0.95 1, 1) = 161 (cf. Table A.4). The F test for lack-of-fit is not significant, since the calculated F value is smaller than the critical value. The significance level is only 0.779, as the p value shows in Table 6.3. 

In summary, polynomials of order 2 (quadratic) provide the analyst with a readily available means of matching points from a curved calibration to a function suitable for interpolation. Tests for lack of fit based on residuals are available in much the same way as for straight-line calibration. Extra caution is necessary to avoid lack of fit, and even small extrapolations beyond the calibrated range are unwise. It is unlikely that polynomials of order greater than quadratic would be appropriate for analytical calibration. 

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