Statistical Tests. Matthieu de Lapparent

Transcription

1 Statistical Tests Matthieu de Lapparent Transport and Mobility Laboratory, School of Architecture, Civil and Environmental Engineering, Ecole Polytechnique Fédérale de Lausanne Transport and Mobility Laboratory Decision-Aid Methodologies 1 / 65

2 Introduction Introduction Modeling Difficult to determine the most appropriate model specification A good fit does not imply a good model Formal testing is necessary, but not sufficient No clear-cut rules can be given Good modeling = good (subjective) judgment + good analysis Wilkinson (1999) The grammar of graphics. Springer... some researchers who use statistical methods pay more attention to goodness of fit than to the meaning of the model... Statisticians must think about what the models mean, regardless of fit, or they will promulgate nonsense. Transport and Mobility Laboratory Decision-Aid Methodologies 2 / 65

3 Introduction Introduction Hypothesis testing Four steps step 1: State the hypotheses H 0 null hypothesis H 1 alternative hypothesis step 2: Set the criteria for a decision step 3: Compute a test statistic step 4: Make a decision Step 1: Analogy with a court trial H 0 : defendant is presumed innocent until proved guilty H 0 is accepted, unless the data argue strongly to the contrary Transport and Mobility Laboratory Decision-Aid Methodologies 3 / 65

4 Introduction Introduction Step 2: Criterion for a decision Court-room: criterion is to show guilt beyond reasonable doubt Implies defining the level of significance α Step 3: Test statistic Determine the likelihood of obtaining a sample outcome if the H 0 hypothesis were true How far we accept to be from the H 0 Step 4: Decide Decide if null is retained or rejected Gives the probability value (p-value of obtaining an outcome, given that the H 0 is true) Transport and Mobility Laboratory Decision-Aid Methodologies 4 / 65

5 Introduction Introduction Possible decision outcomes Accept H 0 Reject H 0 H 0 is true Correct (1-α) Type I error (prob. α) H 0 is false Type II error (prob. β) Correct (1-β) Relations For a given sample size N, there is a trade-off between α and β. Only way to reduce both types of error probabilities is to increase N. π = 1 β is the power of the test, that is, the probability of correctly rejecting H 0. Researcher directly controls Type I errors by fixing α Transport and Mobility Laboratory Decision-Aid Methodologies 5 / 65

6 Case study Summary of case-studies Netherlands mode choice Intercity travelers Choice between train & car 228 respondents Revealed preference data with self-reported trip characteristics Swissmetro Travelers St. Gallen - Geneva Choice between train, car & swissmetro 441 respondents Stated preference (swissmetro is a non-existing mag-lev train) Transport and Mobility Laboratory Decision-Aid Methodologies 6 / 65

7 Case study Informal tests Informal tests Sign of the coefficient Do the estimated parameters have the right sign? Example: Netherlands Mode Choice Case Robust Coeff. Asympt. Parameter estimate std. error t-stat p-value ASC car β cost β time Transport and Mobility Laboratory Decision-Aid Methodologies 7 / 65

8 Case study Informal tests Informal tests Value of trade-offs Are the trade-offs reasonable? How much are we ready to pay for a marginal improvement of the level-of-service? Example: reduction of travel time The increase in cost must be exactly compensated by the reduction of travel time β cost (C + C) + β time (T T ) +... = β cost C + β time T +... Therefore, C T = β time β cost Transport and Mobility Laboratory Decision-Aid Methodologies 8 / 65

9 Case study Informal tests Informal tests Value of trade-offs: example with Netherlands data In general: Trade-off: Units: V / x V / x C 1/Hour 1/Guilder = Guilder Hour Parameter Coeff. Guilders Euros CHF ASC car β cost β time (/Hour) Transport and Mobility Laboratory Decision-Aid Methodologies 9 / 65

10 Case study t-tests t-test Question Is the parameter θ significantly different from a given value θ? Statistic H 0 : θ = θ H 1 : θ θ Under H 0, if ˆθ is normally distributed with known variance σ 2 Therefore ˆθ θ σ N(0, 1). P( 1.96 ˆθ θ σ 1.96) = 0.95 = Transport and Mobility Laboratory Decision-Aid Methodologies 10 / 65

11 Case study t-tests t-test H 0 can be rejected at the 5% level (α = 0.05) if ˆθ θ σ Comments If ˆθ asymptotically normal If variance unknown A t test should be used with n degrees of freedom. When n 30, the Student t distribution is well approximated by a N(0, 1) Transport and Mobility Laboratory Decision-Aid Methodologies 11 / 65

12 Case study t-tests t-test Swissmetro: model specification Car Train Swissmetro ASC car ASC train β cost cost cost cost β time time time time β headway 0 headway headway Transport and Mobility Laboratory Decision-Aid Methodologies 12 / 65

13 Case study t-tests t-test Swissmetro: coefficient estimates Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost β headway β time H 0 : β cost = 0: rejected at the 5% level H 0 : β headway = 0: rejected at the 5% level H 0 : β time = 0: rejected at the 5% level Transport and Mobility Laboratory Decision-Aid Methodologies 13 / 65

14 Case study t-tests t-test Comparing two coefficients H 0 : β 1 = β 2. The t statistic is given by β 1 β 2 var( β 1 β 2 ) var( β 1 β 2 ) = var( β 1 ) + var( β 2 ) 2 cov( β 1, β 2 ) Transport and Mobility Laboratory Decision-Aid Methodologies 14 / 65

15 Case study t-tests t-test Comparing two coefficients Example: alternative specific or generic coefficients? Below alternative specific time Car Train Swissmetro ASC car ASC train β cost cost cost cost β time car time 0 0 β time train 0 time 0 β time Swissmetro 0 0 time β headway 0 headway headway Transport and Mobility Laboratory Decision-Aid Methodologies 15 / 65

16 Case study t-tests t-test Swissmetro: coefficient estimates (alternative specific time) Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost β headway β time car β time Swissmetro β time train Transport and Mobility Laboratory Decision-Aid Methodologies 16 / 65

17 Case study t-tests t-test Variance-covariance matrix Parameter 1 Parameter 2 Covariance Correlation t-stat β time car β time train 7.57e β time car β time Swissmetro 1.38e β time Swissmetro β time train 1.47e H 0 : β time car = β time train var( β t.car β t.train ) = var( β t.car ) + var( β t.train ) 2 cov( β t.car, β t.train ) = = Transport and Mobility Laboratory Decision-Aid Methodologies 17 / 65

18 Case study t-tests t-test H 0 : β time car = β time train β t.car β t.train var( β t.car β = t.train ) We can reject the H 0 of parameter equality ( ) = What about β time car = β time metro and β time metro = β time train? Homework to calculate the t-ratios for these parameter differences! Transport and Mobility Laboratory Decision-Aid Methodologies 18 / 65

19 Likelihood ratio test Likelihood ratio test Comparing two models Used for nested hypotheses One model is a special case of another obtained from a set of linear restrictions on the parameters H 0 : the restricted model is the true model Statistic under H 0 2(L( ˆβR) L( ˆβU)) χ 2 (K U K R ) L( ˆβ R ) is the log likelihood of the restricted model L( ˆβ U ) is the log likelihood of the unrestricted model K R is the number of parameters in the restricted model K U is the number of parameters in the unrestricted model Transport and Mobility Laboratory Decision-Aid Methodologies 19 / 65

20 Likelihood ratio test Likelihood ratio test Test of parameters being equal to zero: Netherlands Statistic Unrestricted model: 3 parameters: β time, β cost, ASC car. Final log likelihood: Restricted model Restrictions: β time = β cost = 0 1 parameter: ASC car. Final log likelihood: Test: 2( ) = χ 2, 2 degrees of freedom, 95% quantile: 5.99 H 0 is rejected The unrestricted model is preferred. Transport and Mobility Laboratory Decision-Aid Methodologies 20 / 65

21 Likelihood ratio test Test of generic attributes Likelihood ratio test Test of generic attributes: Swissmetro Restricted model: Car Train Swissmetro ASC car ASC train β cost cost cost cost β time time time time β headway 0 headway headway Restrictions: β time car = β time train = β time Swissmetro Unrestricted model: Car Train Swissmetro ASC car ASC train β cost cost cost cost β time car time 0 0 β time train 0 time 0 β time Swissmetro 0 0 time β headway 0 headway headway Transport and Mobility Laboratory Decision-Aid Methodologies 21 / 65

22 Likelihood ratio test Test of generic attributes Likelihood ratio test Test of generic attributes: Swissmetro Statistic Restricted model: Final log likelihood: parameters Unrestricted model: Final log likelihood: parameters -2( ) = χ 2, 2 degrees of freedom, 95% quantile: 5.99 Reject the restrictions (H 0 ) The alternative specific specification is preferred Transport and Mobility Laboratory Decision-Aid Methodologies 22 / 65

23 Likelihood ratio test Test of taste variation Test of taste variations Segmentation Classify the data into G groups. Size of group g: N g. The same specification is considered for each group. A different set of parameters is estimated for each group. Restrictions: β 1 = β 2 =... = β G where β g is the vector of coefficients of market segment g. Statistic: 2 L N ( β) G L Ng ( β g ) g=1 χ 2 with G g=1 K g K degrees of freedom. In general, G g=1 K g K = (G 1)K. Transport and Mobility Laboratory Decision-Aid Methodologies 23 / 65

24 Likelihood ratio test Test of taste variation Test of taste variations Segmentation according to income: Swissmetro Unrestricted model: a different set of parameters for each income group 1: [0 50], 2: [50 100], 3:[100 ], 4: unknown (KCHF) Restricted model: same parameters across income groups Hypothesis H 0 the true parameters are the same across income classes Transport and Mobility Laboratory Decision-Aid Methodologies 24 / 65

25 Likelihood ratio test Test of taste variation Estimation results by income groups Estimation procedure Divide the sample into 4 subsets, corresponding to the income groups Estimate the restricted model on each of the samples separately Add up the log likelihoods Group Log likelihood Sample size Total Transport and Mobility Laboratory Decision-Aid Methodologies 25 / 65

26 Likelihood ratio test Test of taste variation Different taste across income groups? Test of taste variations Restricted model: 7 parameters Final log likelihood: Unrestricted model: 7 4 = 28 parameters Final log likelihood: Statistic Likelihood ratio test gives: χ 2, 21 degrees of freedom, 95% quantile: > hence H 0 is rejected There is evidence of taste variation per income group Transport and Mobility Laboratory Decision-Aid Methodologies 26 / 65

27 Test of nonlinear specifications Piecewise linear specification Nonlinear specifications Consider a variable x of the model (travel time, say) Unrestricted model: V is a nonlinear function of x Restricted model: V is a linear function of x We consider the following nonlinear specifications: Piecewise linear Power series Box-Cox transforms For each case, the linear specification is obtained using simple restrictions on the nonlinear specification Transport and Mobility Laboratory Decision-Aid Methodologies 27 / 65

28 Test of nonlinear specifications Piecewise linear specification Piecewise linear specification Model procedure Partition the range of values of x into M intervals [a m, a m+1 ], m = 1,..., M For example, the partition [0 500], [ ], [1000 ] corresponds to M = 3, a 1 = 0, a 2 = 500, a 3 = 1000, a 4 = + The slope of the utility function may vary across intervals Therefore, there will be M parameters instead of 1 The function must be continuous Transport and Mobility Laboratory Decision-Aid Methodologies 28 / 65

29 Test of nonlinear specifications Piecewise linear specification Piecewise linear specification Linear specification: V i = βx i + Piecewise linear specification M V i = β m x im + m=1 where that is x im = max(0, min(x a m, a m+1 a m )) 0 if x < a m x im = x a m if a m x < a m+1 a m+1 a m if a m+1 x Transport and Mobility Laboratory Decision-Aid Methodologies 29 / 65

30 Test of nonlinear specifications Piecewise linear specification Piecewise linear specification Example: M = 3, a 1 = 0, a 2 = 500, a 3 = 1000, a 4 = + x x 1 x 2 x Transport and Mobility Laboratory Decision-Aid Methodologies 30 / 65

31 Test of nonlinear specifications Piecewise linear specification Piecewise linear specification: restricted model Test of piecewise specification Restricted model Unrestricted model Car Train Swissmetro ASC car ASC train β cost cost cost cost β time time time time β headway 0 headway headway Car Train Swissmetro ASC car ASC train β cost cost cost cost β time,1 time 1 time 1 time 1 β time,2 time 2 time 2 time 2 β time,3 time 3 time 3 time 3 β headway 0 headway headway Transport and Mobility Laboratory Decision-Aid Methodologies 31 / 65

32 Test of nonlinear specifications Piecewise linear specification Piecewise linear specification Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost β headway β time, β time, β time, Transport and Mobility Laboratory Decision-Aid Methodologies 32 / 65

33 Test of nonlinear specifications Piecewise linear specification Piecewise linear specification 0 Piecewise linear Utility Time Transport and Mobility Laboratory Decision-Aid Methodologies 33 / 65

34 Test of nonlinear specifications Piecewise linear specification Likelihood ratio test Test of piecewise linear specification for time Restricted model: 5 parameters Final log likelihood: Unrestricted model: 7 parameters Final log likelihood: Statistic LR Test: χ 2, 2 degrees of freedom, 95% quantile: 5.99 H 0 is rejected The linear specification is rejected Transport and Mobility Laboratory Decision-Aid Methodologies 34 / 65

35 Test of nonlinear specifications Power series Power series Idea If the utility function is nonlinear in x, it can be approximated by a polynomial of degree M Linear specification: V i = βx i + Power series M V i = β m xi m + m=1 Transport and Mobility Laboratory Decision-Aid Methodologies 35 / 65

36 Test of nonlinear specifications Power series Power series: restricted model Test of power series specification for time Restricted model Unrestricted model Car Train Swissmetro ASC car ASC train β cost cost cost cost β time time time time β headway 0 headway headway Car Train Swissmetro ASC car ASC train β cost cost cost cost β time,1 time time time β time,2 time 2 /10 5 time 2 /10 5 time 2 /10 5 β time,3 time 3 /10 5 time 3 /10 5 time 3 /10 5 β headway 0 headway headway Transport and Mobility Laboratory Decision-Aid Methodologies 36 / 65

37 Test of nonlinear specifications Power series Power series: unrestricted model Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost β headway β time, β time, β time, Transport and Mobility Laboratory Decision-Aid Methodologies 37 / 65

38 Test of nonlinear specifications Power series Power series: M=3 0 Power series -1-2 Utility Time Transport and Mobility Laboratory Decision-Aid Methodologies 38 / 65

39 Test of nonlinear specifications Power series Likelihood ratio test Test of power series specification for time Restricted model: 5 parameters Final log likelihood: Unrestricted model: 7 parameters Final log likelihood: Statistic LR Test: χ 2, 2 degrees of freedom, 95% quantile: 5.99 H 0 is rejected The linear specification is rejected Transport and Mobility Laboratory Decision-Aid Methodologies 39 / 65

40 Test of nonlinear specifications Box-Cox Box-Cox transform Definition Let x > 0 be a positive variable Its Box-Cox transform is defined as x λ 1 if λ 0 B(x, λ) = λ ln x if λ = 0. where λ R is a parameter. Continuity x λ 1 lim = ln x. λ 0 λ Transport and Mobility Laboratory Decision-Aid Methodologies 40 / 65

41 Test of nonlinear specifications Box-Cox Box-Cox transform Linear specification V i = βx i + Box-Cox specification V i = βb(x, λ) + Properties Convex if λ > 1 Linear if λ = 1 Concave if λ < 1 Transport and Mobility Laboratory Decision-Aid Methodologies 41 / 65

42 Test of nonlinear specifications Box-Cox Box-Cox specification Test of Box-Cox transformation on time Restricted model Unrestricted model Car Train Swissmetro ASC car ASC train β cost cost cost cost β time time time time β headway 0 headway headway Car Train Swissmetro ASC car ASC train β cost cost cost cost β time B(time,λ) B(time,λ) B(time,λ) β headway 0 headway headway λ Note: specification tables are not designed for nonlinear specifications. Transport and Mobility Laboratory Decision-Aid Methodologies 42 / 65

43 Test of nonlinear specifications Box-Cox Box-Cox: unrestricted model Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost β headway β time λ Transport and Mobility Laboratory Decision-Aid Methodologies 43 / 65

44 Test of nonlinear specifications Box-Cox Box-Cox transform 2 Box-Cox Utility Time Transport and Mobility Laboratory Decision-Aid Methodologies 44 / 65

45 Test of nonlinear specifications Box-Cox Likelihood ratio test Test of Box-Cox specification for time Restricted model: 5 parameters Final log likelihood: Unrestricted model: 6 parameters Final log likelihood: Statistic LR Test: χ 2, 1 degree of freedom, 95% quantile: 3.84 H 0 is rejected The linear specification is rejected Also possible to employ t-test to compare Box-Cox to linear Transport and Mobility Laboratory Decision-Aid Methodologies 45 / 65

46 Test of nonlinear specifications Box-Cox Comparison of nonlinear time specifications Linear Piecewise linear Power series Box-Cox -4-6 Utility Time Transport and Mobility Laboratory Decision-Aid Methodologies 46 / 65

47 Non nested hypotheses Non nested hypotheses Nested hypotheses Restricted and unrestricted models Linear restrictions H 0 : restricted model is correct Test: likelihood ratio test Non nested hypotheses Need to compare two models None of them is a restriction of the other Likelihood ratio test cannot be used Two possible tests: Cox composite model Horowitz test ρ 2 Transport and Mobility Laboratory Decision-Aid Methodologies 47 / 65

48 Non nested hypotheses Cox test Cox test We want to test model 1 against model 2 We generate a composite model C such that both models 1 and 2 are restricted cases of model C. We test model 1 against C using the likelihood ratio test We test model 2 against C using the likelihood ratio test Possible outcomes: Only one of the two models is rejected. Keep the other. Both models are rejected. Better models should be developed. Both models are accepted. Use another test. Model C Model 1 Model 2 Transport and Mobility Laboratory Decision-Aid Methodologies 48 / 65

49 Non nested hypotheses Cox test Cox test Models M 1 : U in = + βx in + + ε (1) in M 2 : U in = + θlog(x) in + + ε (2) in M C : U in = + βx in + θlog(x) in + + ε in. Testing M 1 against M C Restrictions: θ = 0 Testing M 2 against M C Restrictions: β = 0 Transport and Mobility Laboratory Decision-Aid Methodologies 49 / 65

50 Non nested hypotheses Cox test Non nested models: estimates for model 1 Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost car β cost Swissmetro β cost train β gen. abo β headway β time car β time Swissmetro β time train Transport and Mobility Laboratory Decision-Aid Methodologies 50 / 65

51 Non nested hypotheses Cox test Non nested models: estimates for model 2 Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β log cost car β cost Swissmetro β cost train β gen. abo β headway β time car β time Swissmetro β time train Transport and Mobility Laboratory Decision-Aid Methodologies 51 / 65

52 Non nested hypotheses Cox test Non nested models Log likelihood # parameters Model 1 (linear car cost) Model 2 (log car cost) The fit of model 1 is better But we cannot apply a likelihood ratio test We estimate a composite model Transport and Mobility Laboratory Decision-Aid Methodologies 52 / 65

53 Non nested hypotheses Cox test Non nested models: estimates of the composite model Robust Parameter Coeff. Asympt. number Description estimate std. error t-stat p-value 1 ASC car ASC train β cost car β log cost car β cost Swissmetro β cost train β gen. abo β headway β time car β time Swissmetro β time train Transport and Mobility Laboratory Decision-Aid Methodologies 53 / 65

54 Non nested hypotheses Cox test Non nested models Test 1: model 1 vs. composite Restricted model (linear cost): 10 parameters Final log likelihood: Unrestricted model (Composite): 11 parameters Final log likelihood: Test: 1.58 χ 2, 1 degree of freedom, 95% quantile: 3.84 H 0 cannot be rejected Model 1 cannot be rejected Transport and Mobility Laboratory Decision-Aid Methodologies 54 / 65

55 Non nested hypotheses Cox test Non nested models Test 2: model 2 vs. composite Restricted model (log cost): 10 parameters Final log likelihood: Unrestricted model (Composite): 11 parameters Final log likelihood: Test: χ 2, 1 degree of freedom, 95% quantile: 3.84 H 0 can be rejected Model 2 can be rejected Overall conclusion: model 1 (linear car cost) is preferred over model 2 (log car cost). Transport and Mobility Laboratory Decision-Aid Methodologies 55 / 65

56 Non nested hypotheses Adjusted likelihood ratio index Adjusted likelihood ratio index Likelihood ratio index ρ 2 = 1 L( ˆβ) L(0) ρ 2 = 0: trivial model, equal probabilities ρ 2 = 1: perfect fit. Adjusted likelihood ratio index ρ 2 is increasing with the number of parameters. A higher fit (that is a higher ρ 2 ) does not mean a better model. An adjustment is needed. ρ 2 = 1 L( ˆβ) K L(0) Transport and Mobility Laboratory Decision-Aid Methodologies 56 / 65

57 Non nested hypotheses Adjusted likelihood ratio index ρ 2 test (Horowitz) Compare model 0 and model 1. where We expect that the best model corresponds to the best fit. We will be wrong if M 0 is the true model and M 1 produces a better fit. What is the probability that this happens? If this probability is low, M 0 can be rejected. ( P( ρ 2 1 ρ 2 0 > z M 0 ) Φ ) 2zL(0) + (K 1 K 0 ) ρ l 2 is the adjusted likelihood ratio index of model l = 0, 1 K l is the number of parameters of model l Φ is the standard normal CDF. Transport and Mobility Laboratory Decision-Aid Methodologies 57 / 65

58 Non nested hypotheses Adjusted likelihood ratio index ρ 2 test (Horowitz) Back to the example: ρ 2 # parameters Model 0 (log car cost) Model 1 (linear car cost) ( P( ρ 2 1 ρ 2 0 > z M 0 ) Φ ) 2zL(0) + (K 1 K 0 ) ( P( ρ 2 1 ρ 2 0 > M 0 ) Φ ) 2z( ) + (10 10) P( ρ 1 2 ρ 0 2 > M 0 ) Φ ( 3.73) 0 Therefore, M 0 can be rejected, and the linear specification is preferred. Transport and Mobility Laboratory Decision-Aid Methodologies 58 / 65

59 Non nested hypotheses Adjusted likelihood ratio index ρ 2 test (Horowitz) In practice, if the sample is large enough (i.e. more than 250 observations) if the values of the ρ 2 differ by 0.01 or more the model with the lower ρ 2 is almost certainly incorrect Transport and Mobility Laboratory Decision-Aid Methodologies 59 / 65

60 Further tests Outlier analysis Outlier analysis Procedure Apply the model on the sample Examine observations where the predicted probability is the smallest for the observed choice Test model sensitivity to outliers, as a small probability has a significant impact on the log likelihood Potential causes of low probability: Coding or measurement error in the data Model misspecification Unexplainable variation in choice behavior Transport and Mobility Laboratory Decision-Aid Methodologies 60 / 65

61 Further tests Outlier analysis Outlier analysis Coding or measurement error in the data Look for signs of data errors Correct or remove the observation Model misspecification Seek clues of missing variables from the observation Keep the observation and improve the model Unexplainable variation in choice behavior Keep the observation Avoid over fitting of the model to the data Transport and Mobility Laboratory Decision-Aid Methodologies 61 / 65

62 Further tests Market segments Market segments Procedure Compare predicted vs. observed shares per segment Let N g be the set of sampled individuals in segment g Observed share for alt. i and segment g S g (i) = n N g y in /N g Predicted share for alt. i and segment g Ŝ g (i) = n N g P n (i)/n g Transport and Mobility Laboratory Decision-Aid Methodologies 62 / 65

63 Further tests Market segments Market segments Note: With a full set of constants for segment g: y in = P n (i) n N g n N g Do not saturate the model with constants Transport and Mobility Laboratory Decision-Aid Methodologies 63 / 65

64 Further tests Market segments Conclusions Tests are designed to check meaningful hypotheses Do not test hypotheses that do not make sense Do not apply the tests blindly Always use your judgment. Transport and Mobility Laboratory Decision-Aid Methodologies 64 / 65

65 Appendix 90%, 95% and 99% of the χ 2 distribution with K degrees of freedom K 90% 95% 99% K 90% 95% 99% Transport and Mobility Laboratory Decision-Aid Methodologies 65 / 65