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# PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

## 82 normality assumption

82 normality assumption of any linear regression model. There- fore they do not use the term "regression" for the direct linear relationship. They conclude that the explanatory model is better than the direct linear relationship for the analysis of queue acceptance. 4.4.2 The Comparison of Models byMaher and Dowse Maher and Dowse (1982) compared six models of predicting gap-acceptance parameters. They included four simple linear models, the Armitage and McDonald method, and a method using Maximum Likelihood Estimates (4LE) which they developed. The four simple linear models were regressions of N on T, and T on N, firstly using all the data, and secondly, excluding the rejected gaps. As to the applicability of the term regression they comment that the model assumptions, of either (i) independent errors with zero mean and constant variance, or (ii) normally distributed errors, do not hold in these cases. They con- clude that any special status which least squares regression might hold as a method is inappropriate, but the validity of any method of estimating ci. and depends on the assumptions made about the underlying mechanism of gap-acceptance. They tested the six methods for unbiassedness and efficiency. A method is unbiassed if the estimator has a mean (or expected) value of 9, i.e. E() = 0. A method is asymptotically unbiassed if E()-- 0 as, the sample size, n^. The most efficient one is that which has minimum mean squared error, i.e. E(-0) 2 is minimum. The relative efficiency of two unbiassed estimators is the ratio of their

83 mean squared errors or variances, Var( 1 )/Var( 2 ). The efficiency of an estimator depends on the statistical assumptions made. If one can be confident of the model assumed then the best estimator can be used, if not a robust or insensitive to model assumption estimator should be used. The disadvantage of any MLE method is that specific probabilistic model assumptions need to be made, the form of the estimates being specific to that model. Furthermore the estimates need to be calculated by means of some numerical iterative scheme for maximising the likelihood function. In their comparison for bias they conclude that three of the six methods are not seriously biassed; the MLE method, Armitage and McDonald's, and the linear model assuming T as the dependent, N as the independent variable while excluding all rejected gaps. Comparing the relative efficiencies, they conclude that the MILE method is the most efficient, while Armitage and McDonald's method was more efficient than the simple linear model. 4.5 The Development of a Simple Linear Model As has been suggested by previous research of Pearson and Ferreri (1961), Uber (1978) and Cooper and Wennell (1978) there can be a direct linear relationship between the size of the gap and the number of vehicles entering during the gap. Their findings are described in more detail in section 4.4. Here, the development of such a linear model is described. 4.5.1 Theoretical Aspects of Linear Regression Mood and Graybill (1963) define a simple linear model

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