Abstract
The general form of the model equation is E{μ} = f(X 1, X 2, … , β 0, β 1, …) where f() stands for some algebraic expression. For modeling one has to choose some specific f() and the question was how to do so. Even if the right f() is not known one can still make decent estimates of E{μ}. However, to say what change in E{μ} is caused by a some change in a predictor variable, not knowing the right f() can be a problem. The task of fashioning the f() out of predictor variables and parameters is difficult. Road safety data can be represented by many different and footloose functions, and what the modeler chooses matters in terms of what estimates are produced. Limited guidance is offered.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Hundred data points were generated such that the X 1 are uniformly distributed in between 2 and 10 m, the X 2 between 3 and 8 m, the hypotenuse Y was computed by the Pythagorean theorem, and a \( \mathcal{N} \) (0, 0.01 m) random error was added to each measurement.
- 2.
Seeing the objects and having measured their dimensions the researcher could have reasoned that if both X 1 and X 2 are very short Y must also be so and therefore, the intercept β 0 should be set to 0. Doing so would remove a logical blemish but make for a slightly worse fit.
- 3.
Were the Pythagorean model used the estimates of β 1, β 2, and β 3 for the same data would have been 1.98, 1.98 and 0.50. That is, if one had the right f(.) then regression would yield the right parameters.
- 4.
The concept of simplicity is an esthetic one and has no agreed upon measure.
- 5.
William of Ockham’s (c. 1287–1347) razor says something like: “All things being equal, the simplest solution tends to be the best one.” The “tends to be” means that simplest is not always the best. Besides, what makes the linear model “simpler”? Some object to Ockham’s razor, because it seems to imply that of two alternative representations of nature the simpler one is more likely to be true. There is no empirical evidence for such a belief.
- 6.
This leads to another important point which will be elaborated on in Sect. 10.9, namely, that by choosing the functional form the modeler predetermines how the dependent variable will change due to a change in a predictor variable.
- 7.
The same distinction was made in Sect. 6.7 where the purposes of regressions were discussed. The three purposes were: (1) to summarize or to describe a body of data; (2) to predict the value of a dependent variable Y (here E{μ}) from a set of independent variables X 1, … , X n ; (3) To predict the change in the value of the dependent variable from an intervention that changes the value of X 1, etc. The legitimacy of purpose (3) is widely questioned.
- 8.
I always found Henri Theil’s comforting aphorism that “Models are to be used, not believed” puzzling; can one use a model without believing that what it tells us is approximately true? Perhaps what Theil (1971) meant was that one need not believe that the function and its parameters are right if the predictions produced by the model are close to the mark. In the road safety context Theil’s cryptic statement applies only to practical applications perspective. If the model is to be used to tell by how much E{μ} is likely to change as a results of some change in a predictor variable, one must believe to have the substantively correct function and parameters.
- 9.
The differential is \( dY=\frac{Y}{X_i}{\beta}_id{X}_i \) .
- 10.
- 11.
The model Y = β 0 + β 1 X 1 + β 2 X 2 +⋯ is linear in both parameters and variables. The model \( Y={\beta}_0+{\beta}_1\sqrt{X_1+{X}_2} \) is linear in parameters but not in variables. The model \( Y={\beta}_0\sqrt{X_1^{\beta_1}+{X}_2^{\beta_2}} \) is not linear in either parameters or variables.
- 12.
Why exactly is it desirable to make fi a function of a single predictor variable? According to Lau doing so allows the parameter β i to be interpreted as a measure of the effect on the dependent variable of a change in the predictor variable X i . This convenience-motivated simplification is an assumption about reality. In road safety it means, e.g., the safety effect of lane widening does not depend on traffic flow, shoulder design, or any other predictor variable.
- 13.
The subject of interaction will be discussed in Sect. 10.9.
- 14.
One of the main limitation of the multiplicative model is that road segments are never homogeneous in their traits. The presence of a narrow bridge, a horizontal curve or of a driveway affects the safety of only a part of the segment and as such should be accounted for by an addition, not a multiplication. For discussion see Hauer (2004).
- 15.
Specifically, the fit minimizing ∑∣residuals∣ from Fig. 9.13 was used.
- 16.
For detail see Appendix K.
- 17.
To download these spreadsheets go to http://extras.springer.com/ and enter the ISBN of this book. Look in the “Spreadsheets” folder for “Chapter 9, Alternative objectives, 13-year panel data, L, AADT, Terrain, Common scale parameter and Chapter 10 Absolute residuals, Power x Line.xls or xlsx.”
- 18.
To download this spreadsheet go to http://extras.springer.com/ and enter the ISBN of this book. Look in the “Spreadsheets” folder for “Chapter 10. The visualization tool.xls or xlsx.”
- 19.
The addition of a positive intercept can raise the entire basic function by a fixed amount.
- 20.
The Hoerl function is used mainly for curve-fitting. It arises in physics where it represents the dependence of the strong nuclear force on the distance between particles and, in inverse form, in the vapor pressure curve. However, here it is not representing any theory. The most that can be said is that if the embryonic theory in Appendix J has some substance then the Hoerl function could approximate the general shape which that theory suggests.
- 21.
- 22.
See Sect. 4.2.
- 23.
See Sect. 4.3.
- 24.
In SPF modeling when for some subset of units the sum of squared residuals is smaller than the sum of fitted values, the crash counts are less widely dispersed around the fitted value than what is consistent with the Poisson distribution; such as model is most likely “overfitted. ”
- 25.
The AIC is based on information theory, the amount of information lost when a given model is used to represent the process that generated the data. The BIC is based on comparing the posterior probability of the data of models the prior probability of which is assumed to be the same. The BIC tends to be the more stringent of the two criteria.
- 26.
The current estimates of β Rolling and β Mountainous are in Fig. 10.6.
- 27.
See Sect. 3.6.
- 28.
To download these spreadsheets go to http://extras.springer.com/ and enter the ISBN of this book. Look in the “Spreadsheets” folder for “Chap. 10. Flat, Power x Line.xls or xlsx,” “Chap. 10. Rolling, Power x Line.xls or xlsx” and “Chap. 10. Mountainous, Power x Line.xls or xlsx.”
- 29.
While both AIC and BIC use the maximized log-likelihood the values in the rightmost column of Table 10.4 and in Fig. 10.6 are based parameter estimates obtained by minimizing the sum of absolute residuals. Even so, increase of the log-likelihood of 129 is close to what would obtain if all parameters were estimated by maximizing likelihood.
- 30.
One might be inclined to conduct a statistical test of significance with the “all-parameters-are-the-same” assumption as the null hypothesis. This inclination is best resisted. The problem is that such a statistical test does not answer the question of interest; it does not say what the probability of the aforementioned assumption to be (approximately) true is. The statistical test only speaks about the probability of obtaining parameter estimate differences equal to or larger than those in Table 10.4 if the null hypothesis was true. If this probability is small (say, less than 0.05) then the “no difference” hypothesis “rejected”; otherwise it is “not rejected.” If the “no difference” hypothesis is not rejected then, in spite of one’s better judgment, one is stuck with the “no-difference” hypothesis even though the “yes-difference” hypotheses are more plausible It makes no common sense to proceed on the basis an un-rejected “no difference” assumption when a “yes-difference” assumption is better supported by the data. For discussion see, e.g., Edwards (1976, pp. 179, 180), Harlow et al. (1997), Hauer (1983).
- 31.
See Sect. 1.5.
- 32.
See Sect. 2.3.
- 33.
- 34.
- 35.
To download this spreadsheet go to http://extras.springer.com/ and enter the ISBN of this book. Look in the “Spreadsheets” folder for “Chap. 10. Terrain as a function of X1 and X2.xls or xlsx.”
- 36.
Thus, e.g., having a 10′ lane on a rural two-lane road instead of a 12′ lane is expected to increase the number of accidents by a factor of 1.3 when AADT = 2,000 and by a factor of 1.11 when AADT = 1,000 (AASHTO 2010, Figs. 13.1 and 13.5).
- 37.
For the general model equation E{μ} = f(X 1, X 2, … , β 0, β 1, …), the elasticity of function f with respect to variable X i is defined as \( {\varepsilon}_{f,{X}_i}\equiv \frac{\partial f}{\partial {X}_i}\ \frac{X_i}{f} \).
- 38.
Thus, e.g., Chen and Persaud (2014, p. 132) assert that in their model equation \( E\left\{\mu \right\}={\beta}_0{\mathrm{AADT}}^{\beta_1}{\mathrm{e}}^{\beta_2{X}_2}{\mathrm{e}}^{\beta_3{X}_3}\dots \) the factors \( {\mathrm{e}}^{\beta_2{X}_2},\ {\mathrm{e}}^{\beta_3{X}_3}, \dots \) “are technically CM-Functions” for the X 2, X 3, … Since theirs is a multiplicative model equation in which each factor is a function of only one predictor variable, as shown in Appendix L, the assertion is open to question.
- 39.
See Appendix L.
- 40.
Whether CMFs obtained from such model equations can be trusted is a different question as discussed in Sect. 6.7.
- 41.
For detail see Appendix M.
- 42.
See, e.g., Chen and Persaud (2014, Sect. 4).
- 43.
In this specific case it produces suspicious results for low-volume short road segments. Thus, e.g., for a 0.2 miles long segment with AADT = 100 it predicts fewer accident in mountainous than in rolling terrain.
References
AASHTO (The American Association of State Highway and Transportation Officials) (2010) Highway safety manual, 1st edn
Chen Y, Persaud B (2014) Methodology to develop crash modification functions for road safety treatments with fully specified and hierarchical models. Accid Anal Prev 70:131–139
Chiou Y-C, Fu C (2013) Modeling crash frequency and severity using multinomial-generalized Poisson model with error components. Accid Anal Prev 50:73–82
Edwards AWF (1976) Likelihood. Cambridge University Press, Cambridge
Harlow LL, Mulaik SA, Steiger JH (1997) What if there were no significance tests? Lawrence Erlbaum, London
Hauer E (1983) Reflections on methods of statistical inference in research on the effect of safety countermeasures. Accid Anal Prev 15(4):275–285
Hauer E (2004) Statistical safety modelling, Transportation research record 1897. National Academies Press, Washington, DC, pp 81–87
Hoerl AE (1950) Fitting curves to data. In: Perry JH (ed) Chemical engineer’s handbook. McGraw-Hill, New York (Chapter 20)
Jaccard J, Turrisi R (2003) Interaction effects in multiple regression, 2nd edn. Sage University papers on quantitative applications in the social sciences, 07-072. Sage, Thousand Oaks
Lau LJ (1986) Functional forms in econometric model building. In: Griliches Z, Intriligator MD (eds) Handbook of econometrics, vol III. North-Holland, Amsterdam
Lord D, Bonneson JA (2007) Development of accident modification factors for rural frontage road segments in Texas. Transport Res Rec 2023:20–27
Lord D (2008) Methodology for estimating the variance and confidence intervals for the estimate of the product of baseline models and AMFs. Accid Anal Prev 40:1013–1017
Lord D, Kuo P-F, Geedipally SR (2010) Comparison of application of product of baseline models and accident-modification factors and models with covariates. Transport Res Rec 2147:113–122
Poch M, Mannering F (1996) Negative binomial analysis of intersection-accident frequencies. J Transport Eng 122:105–113
Shankar V, Mannering F, Barfield W (1995) Effect of roadway geometrics and environmental factors on rural freeway accident frequencies. Accid Anal Prev 27(3):371–389
Theil H (1971) Principles of econometrics. Wiley, New York
Wood GR (2005) Confidence and prediction intervals for generalized linear accident models. Accid Anal Prev 37:267–273
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hauer, E. (2015). Choosing the Function Behind the Data. In: The Art of Regression Modeling in Road Safety. Springer, Cham. https://doi.org/10.1007/978-3-319-12529-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-12529-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12528-2
Online ISBN: 978-3-319-12529-9
eBook Packages: EngineeringEngineering (R0)