Testing Likelihood ratio test Michel Bierlaire Introduction to - PowerPoint PPT Presentation

Testing Likelihood ratio test Michel Bierlaire Introduction to choice models

Applications of the likelihood ratio test

Benchmarking Unrestricted model Restricted model Equal probability model V in = β 1 x ink + · · · V in = 0 V jn = β 2 x jnk + · · · V jn = 0 . . . . . . Restrictions β k = 0 , ∀ k

Benchmarking Log likelihood of the unrestricted Log likelihood of the restricted model model L ( � β ) P in = 1 / J n , ∀ i ∈ C n , ∀ n N � L (0) = − log( J n ) n =1 Statistic − 2( L (0) − L ( � β )) ∼ χ 2 K

Benchmarking revisited Unrestricted model Restricted model Only alternative specific constants V in = β 1 x ink + · · · V in = β i V jn = β 2 x jnk + · · · V jn = β j . . . . . . Restrictions All coefficients but the constants are constrained to zero.

Benchmarking revisited Log likelihood of the unrestricted Log likelihood of the restricted model model L ( � β ) P in = N i / N ∀ i ∈ C , ∀ n J � L ( c ) = N i log( N i / N ) i =1 Statistic − 2( L ( c ) − L ( � β )) ∼ χ 2 d with d = K − J + 1

Benchmarking Classical output of estimation software Summary statistics Number of observations = 2544 L (0) = − 2794 . 870 L ( c ) = − 2203 . 160 L (ˆ β ) = − 1640 . 525 − 2[ L (0) − L (ˆ β )] = 2308 . 689

Test of generic attributes Unrestricted model Restricted model Alternative specific Generic V in = β 1 i x ink + · · · V in = β 1 x ink + · · · V jn = β 1 j x jnk + · · · V jn = β 1 x jnk + · · · . . . . . . Restriction β 1 i = β 1 j = · · ·

Test of generic attributes Log likelihood of the unrestricted Log likelihood of the restricted model model L ( � L ( � β AS ) β G ) Statistic − 2( L ( � β G ) − L ( � β AS )) ∼ χ 2 d with d = K AS − K G

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.

Test of taste variations N 1 N 2 N 3 N 4 N � G g =1 L N g ( � L N 1 ( � β 1 ) L N 2 ( � L N 3 ( � L N 4 ( � β g ) β 2 ) β 3 ) β 4 )

Test of taste variations Unrestricted model Restricted model Group specific coefficients Generic coefficients � G V in = β 1 x ink + · · · V in = ( δ ng β 1 g ) x ink + · · · V jn = β 2 x jnk + · · · g =1 . . G . � V jn = ( δ ng β 2 g ) x jnk + · · · g =1 . . . Restrictions β k 1 = β k 2 = · · · = β kG , ∀ k .

Test of taste variations Log likelihood of the unrestricted Log likelihood of the restricted model model � G L N ( � β ) L N g ( � β g ) g =1 Statistic � � G G � � L N ( � L N g ( � β g ) ∼ χ 2 − 2 β ) − d with d = K − K = ( G − 1) K . g =1 g =1

Tests of nonlinear specifications Unrestricted model Restricted model Power series Linear specification � L V in = β 1 x ink + · · · x ink ℓ V in = β 1 ℓ + · · · x ref V jn = β 2 x jnk + · · · ℓ =1 . . V jn = β 2 x jnk + · · · . . . . Restrictions β 12 = β 13 = · · · = β 1 L = 0

Power series V in x ink

Test of nonlinear specifications Log likelihood of the unrestricted Log likelihood of the restricted model model L ( � L ( � β U ) β R ) Statistic � � L ( � β R ) − L ( � ∼ χ 2 − 2 β U ) d with d = L − 1

Notes ◮ Usually not behaviorally meaningful ◮ Danger of overfitting ◮ Polynomials are most of the time inappropriate for extrapolation due to oscillation ◮ Other nonlinear specifications can be used for testing ◮ Piecewise linear ◮ Box-Cox