Decomposing the deviance in GLMMs, with applications in marine - PowerPoint PPT Presentation

Decomposing the deviance in GLMM’s, with applications in marine ecology Mariangela SCIANDRA, Gianfranco LOVISON sciandra@dssm.unipa.it, lovison@unipa.it Dipartimento di Scienze Statistiche e Matematiche ‘S. Vianelli’ Universit` a di Palermo GRASPA Conference 2008 SIENA - March, 27-28 2008

Outline

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements 2. Partitioning variation: Whittaker’s extension

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements 2. Partitioning variation: Whittaker’s extension 3. Partitioning variation: which extension to mixed models?

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements 2. Partitioning variation: Whittaker’s extension 3. Partitioning variation: which extension to mixed models? • Partitioning the Penalized Quasi-Likelihood

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements 2. Partitioning variation: Whittaker’s extension 3. Partitioning variation: which extension to mixed models? • Partitioning the Penalized Quasi-Likelihood • Partitioning the h -Likelihood

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements 2. Partitioning variation: Whittaker’s extension 3. Partitioning variation: which extension to mixed models? • Partitioning the Penalized Quasi-Likelihood • Partitioning the h -Likelihood 4. Partitioning the Penalized Quasi-Likelihood for given random effects: an application in Marine Ecology

Outline 1. Partitioning variation: Newton and Spurrel’s regression elements 2. Partitioning variation: Whittaker’s extension 3. Partitioning variation: which extension to mixed models? • Partitioning the Penalized Quasi-Likelihood • Partitioning the h -Likelihood 4. Partitioning the Penalized Quasi-Likelihood for given random effects: an application in Marine Ecology 5. Open problems: how to quantify the contribution of random effects?

1. Partitioning variation: Newton and Spurrel’s regression elements

1. Partitioning variation: Newton and Spurrel’s regression elements How, and at what extent, can we uniquely attribute variation in the response variable to each explanatory variable in classical linear regression?

1. Partitioning variation: Newton and Spurrel’s regression elements How, and at what extent, can we uniquely attribute variation in the response variable to each explanatory variable in classical linear regression? Newton, Spurrel (1967) = ⇒ regression elements

1. Partitioning variation: Newton and Spurrel’s regression elements How, and at what extent, can we uniquely attribute variation in the response variable to each explanatory variable in classical linear regression? Newton, Spurrel (1967) = ⇒ regression elements Let: y = β 0 + ǫ G (:) = rss ( ∅ ) y = β 0 + β 1 x 1 + ǫ G (: 1) = rss ( x 1 ) y = β 0 + β 2 x 2 + ǫ G (: 2) = rss ( x 2 ) y = β 0 + β 1 x 1 + β 2 x 2 + ǫ G (: 21) = rss ( x 1 , x 2 )

1. Partitioning variation: Newton and Spurrel’s regression elements How, and at what extent, can we uniquely attribute variation in the response variable to each explanatory variable in classical linear regression? Newton, Spurrel (1967) = ⇒ regression elements Let: y = β 0 + ǫ G (:) = rss ( ∅ ) y = β 0 + β 1 x 1 + ǫ G (: 1) = rss ( x 1 ) y = β 0 + β 2 x 2 + ǫ G (: 2) = rss ( x 2 ) y = β 0 + β 1 x 1 + β 2 x 2 + ǫ G (: 21) = rss ( x 1 , x 2 ) Then: G (1 :) = ss ( x 1 ) = G (:) − G (: 1) variation that can be attributed to x 1 ignoring x 2 G (1 : 2) = ss ( x 1 | x 2 ) = G (: 2) − G (: 12) variation that can be attributed to x 1 adjusting for x 2 G (2 :) = ss ( x 2 ) = G (:) − G (: 2) variation that can be attributed to x 2 ignoring x 1 G (2 : 1) = ss ( x 2 | x 1 ) = G (: 1) − G (: 12) variation that can be attributed to x 2 adjusting for x 1 G (12 :) = G (1 :) − G (1 : 2) = G (2 :) − G (2 : 1) variation that can be equally well be attributed to either x 1 or to x 2

G (: 12) , G (1 : 2) , G (2 : 1) , G (12 :) are called regression elements

G (: 12) , G (1 : 2) , G (2 : 1) , G (12 :) are called regression elements Notice: G (:) = G (: 12) + G (1 : 2) + G (2 : 1) + G (12 :) i.e. the regression elements provide an additive decomposition (i.e. a parti- tion) of the total variation in y .

2. Partitioning variation: Whittaker’s extension

2. Partitioning variation: Whittaker’s extension Whittaker(1984) gave a general theoretical framework to Newton and Spurrel’s regression elements, showing how to extend them to more complex (fixed ef- fects) models.

2. Partitioning variation: Whittaker’s extension Whittaker(1984) gave a general theoretical framework to Newton and Spurrel’s regression elements, showing how to extend them to more complex (fixed ef- fects) models. Let: K = { 1 , 2 , . . . , k } the index set of K variables L = {∅ , 1 , 2 , . . . , k, 12 , . . . , 123 , . . . , 12 ..k } the power set of K (a binary lattice) S ( a ) , for a ∈ L a set function L �→ ℜ monotone on L , i.e. S ( ia ) ≤ S ( a ), where ia contains the integers in a plus i

2. Partitioning variation: Whittaker’s extension Whittaker(1984) gave a general theoretical framework to Newton and Spurrel’s regression elements, showing how to extend them to more complex (fixed ef- fects) models. Let: K = { 1 , 2 , . . . , k } the index set of K variables L = {∅ , 1 , 2 , . . . , k, 12 , . . . , 123 , . . . , 12 ..k } the power set of K (a binary lattice) S ( a ) , for a ∈ L a set function L �→ ℜ monotone on L , i.e. S ( ia ) ≤ S ( a ), where ia contains the integers in a plus i Define: G (: a ) = S ( a ) ∀ a ∈ L

2. Partitioning variation: Whittaker’s extension Whittaker(1984) gave a general theoretical framework to Newton and Spurrel’s regression elements, showing how to extend them to more complex (fixed ef- fects) models. Let: K = { 1 , 2 , . . . , k } the index set of K variables L = {∅ , 1 , 2 , . . . , k, 12 , . . . , 123 , . . . , 12 ..k } the power set of K (a binary lattice) S ( a ) , for a ∈ L a set function L �→ ℜ monotone on L , i.e. S ( ia ) ≤ S ( a ), where ia contains the integers in a plus i Define: G (: a ) = S ( a ) ∀ a ∈ L Then, the additive elements of S ( · ) on L are denoted by where: { a, b } is a partition of the full index set K G ( a : b ) and are obtained through the recursion: G ( ia : b ) = G ( a : b ) − G ( a : ib ) ∀ a, b ∈ L and i ∈ K

Within this general approach, Newton and Spurrel’s regression elements are just a special case of additive elements, with S ( a ) = rss ( a )

Within this general approach, Newton and Spurrel’s regression elements are just a special case of additive elements, with S ( a ) = rss ( a ) Whittaker proposed to use S ( a ) = deviance ( a ) to attribute portions of variation to explanatory variables in the linear predictor of a GLM.

Within this general approach, Newton and Spurrel’s regression elements are just a special case of additive elements, with S ( a ) = rss ( a ) Whittaker proposed to use S ( a ) = deviance ( a ) to attribute portions of variation to explanatory variables in the linear predictor of a GLM. Notice that, since: deviance ( a ) = 2 ℓ ( saturated ) − 2 ℓ ( a ) the additive elements result to be differences in − 2 ℓ ( · ). E.g.: G (1 : 23) = G (: 23) − G (: 123) = dev (23) − dev (123) = 2 ℓ ( saturated ) − 2 ℓ (23) − [2 ℓ ( saturated ) − 2 ℓ (123)] = − 2 ℓ (23) − [ − 2 ℓ (123)] Consequently, Whittaker called them additive likelihood elements.

3. Partitioning variation: which extension to mixed models?

3. Partitioning variation: which extension to mixed models? In ecological applications, often unexplained heterogeneity and between-units dependence are taken into account through random effects within mixed mod- els (LMM, GLMM or HGLM).

3. Partitioning variation: which extension to mixed models? In ecological applications, often unexplained heterogeneity and between-units dependence are taken into account through random effects within mixed mod- els (LMM, GLMM or HGLM). Our objective is to extend Whittaker additive likelihood elements to mixed models.

3. Partitioning variation: which extension to mixed models? In ecological applications, often unexplained heterogeneity and between-units dependence are taken into account through random effects within mixed mod- els (LMM, GLMM or HGLM). Our objective is to extend Whittaker additive likelihood elements to mixed models. However ....extension of the idea of additive elements to mixed models is problematic:

3. Partitioning variation: which extension to mixed models? In ecological applications, often unexplained heterogeneity and between-units dependence are taken into account through random effects within mixed mod- els (LMM, GLMM or HGLM). Our objective is to extend Whittaker additive likelihood elements to mixed models. However ....extension of the idea of additive elements to mixed models is problematic: • choice of likelihood to use: