4.1.2. PROYECTOS NEOCLÁSICOS PARA EL PALACIO REAL NUEVO DE MADRID
4.1.2.2. Primeros proyectos de Sacchetti
Figure 22.1. Testing whether there is a signifi cant departure from a linear regression. Both a linear and second - order (quadratic) polynomial are fi tted to the data and analysis of variance is used to determine whether a signifi cant proportion of the remaining variance after fi tting the linear relationship is accounted for by the quadratic polynomial.
260000
mean square of the deviation from a curvilinear regression is signifi cant, then the curved relationship is a better fi t to the data than the straight line. Second - order regressions will often work well for estimation and interpolation within the range of the data even if the actual relationship between Y and X is not strictly quadratic. Extrapolation beyond the data for estimation, however, is extremely risky. If several values of Y are available at each X , then the goodness of fi t of the line can be tested more rigorously (Snedecor &
Cochran, 1980 ).
22.4.2 Interpretation
The analysis of variance of the data is shown in Table 22.1 , and both regression lines are shown in Figure 22.1 . The difference in the SS as a result of fi tting a second - order polynomial compared with a linear regression is calculated and its mean square tested against the mean square deviation from a second - order polynomial. A value of F = 5.93 was obtained, which is signifi cant at the 5% level of probability ( P < 0.05). Hence, the second - order polynomial is a signifi cantly better fi t to the data than the linear regression.
22.5 SCENARIO B : FITTING A GENERAL POLYNOMIAL - TYPE CURVE The human disease Creutzfeldt – Jakob disease (CJD) is caused by unusual proteinaceous infectious agents called prions (Will et al., 1996 ). Characteristic of the brain pathology of CJD is the development of vacuoles ( spongiform change ) within the cerebral cortex, resulting from the death of neurons. An investigator wished to determine if the extent of the vacuolation varied across an area of cerebral cortex from pia mater to white matter.
116 NONLINEAR REGRESSION: FITTING A GENERAL POLYNOMIAL-TYPE CURVE
The specifi c objectives were to determine which cortical laminae were signifi cantly affected and, therefore, which aspects of brain processing were likely to be impaired (Armstrong et al., 2002b, 2002c ). To obtain the data, fi ve traverses from the pia mater to the edge of the white matter were located at random within an area of cortex. The vacuoles were counted in 50 × 250 μ m sample fi elds, the larger dimension of the fi eld being located parallel with the surface of the pia mater. An eyepiece micrometer was used as the sample fi eld and was moved down each traverse one step at a time from the pia mater to the white matter. Histological features of the section were used to correctly position the fi eld. Counts from the fi ve traverses were added together to study the vertical distribution of lesions across the cerebral cortex.
22.6 DATA
The data comprise estimates of the density of vacuoles ( Y ) at different distances below the pia mater of the brain ( X ) and are presented in Table 22.2 .
ANALYSIS 117
Figure 22.2. Fitting a fourth - order polynomial curve to the distribution of vacuoles in an area of cerebral cortex in a case of Creutzfeldt – Jakob disease (CJD). The distribution is bimodal with a peak of vacuole density in the upper cortex close to the surface of the pia mater and a second peak in the lower cortex. polynomial, the regression coeffi cients, standard errors (SE), values of t , and the residual mean square were obtained. From these statistics, a judgment can be made as to whether a polynomial of suffi ciently high degree has been fi tted to the data. Hence, at each stage, the reduction in the SS is tested for signifi cance as each term is added. The analysis is continued by fi tting successively higher order polynomials until a nonsignifi cant value of F is obtained. As a precaution, it is usually good practice to check the next order polyno-mial after a nonsignifi cant term has been fi tted. This analysis may be included within the regression option within statistical software but in some packages can be found within the general linear modeling option.
22.7.2 Interpretation
The analysis (Table 22.2 ) suggests that the linear, quadratic, and cubic polynomials were not signifi cant. However, the quartic polynomial was signifi cant ( F = 11.67, P < 0.01), suggesting a complex curved relationship between the distribution of the vacuoles and distance below the pia mater (Fig. 22.2 ) consistent with the vacuoles affecting specifi c cortical laminae. Incidentally, the fi t to the fi fth - order polynomial (not shown) was not signifi cant.
There are various strategies that can be employed to decide which polynomial curve actually fi ts the data best, and these depend on the objective of the study. First, as each polynomial is fi tted, the reduction in the SS is tested for signifi cance. The analysis is then continued by fi tting successively higher order polynomials until a nonsignifi cant value of F is obtained. The fi nal polynomial giving a signifi cant F is then chosen as the “ best ” fi t
118 NONLINEAR REGRESSION: FITTING A GENERAL POLYNOMIAL-TYPE CURVE
to the data. Second, it may be obvious that a simple relationship such as a linear or qua-dratic polynomial would not fi t the data and that a more complex curve is required. In this case, examination of the value of multiple correlation coeffi cent ( R 2 ) (see Statnote 25 ) may give an indication of the correct polynomial to fi t. Subsequently, F tests can be used to choose the most parsimonious model. This is the approach we adopted in the present scenario. Third, the objective may be to obtain the best possible predictions of Y from X. Hence, polynomial curves varying up to the 10th order can be fi tted to the data and the curve of best fi t selected on the basis of visual inspection and the highest possible regression coeffi cient obtained.
22.8 CONCLUSION
In some circumstances, there may be no scientifi c model of the relationship between X and Y that can be specifi ed in advance, and the objective may be to provide a curve of best fi t for descriptive or predictive purposes. In such an example, the fi tting of successive polynomials may be the best approach. There are various strategies to decide on the polynomial of best fi t depending on the objectives of the investigation.