Table 1 Proposed models

M1

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \gamma_{ij}\)

Conventional

M2

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \gamma_{ij}\)

Conventional

M3

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \mathop \sum \limits_{k = 1}^{p} x_{ijk} \beta_{k} + \gamma_{ij}\)

Conventional

M4

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \mathop \sum \limits_{k = 1}^{p} x_{ijk} \beta_{k} + \gamma_{ij}\)

Conventional

M5

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian B-splines

M6

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian Fourier

M7

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian B-splines basis

M8

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian Fourier basis

M9

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \mathop \sum \limits_{k = 1}^{p} x_{ijk} \beta_{k} + \mathop \sum \limits_{k = 1}^{p} x_{ijk} \beta_{ki} + \gamma_{ij}\)

Conventional

M10

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \mathop \sum \limits_{k = 1}^{p} x_{ijk} \beta_{k} + \mathop \sum \limits_{k = 1}^{p} x_{ijk} \beta_{ki} + \gamma_{ij}\)

Conventional

M11

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{2i} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian B-splines basis

M12

\(\hat{\mu }_{ij} = E_{i} + g_{j} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{2i} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian Fourier basis

M13

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{2i} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian B-splines basis

M14

\(\hat{\mu }_{ij} = E_{i} + g_{j} + gE_{ij} + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{1} \left( k \right)dk + \mathop \int \limits_{392}^{851} x_{ij} \left( k \right)\beta_{2i} \left( k \right)dk + \gamma_{ij}\)

Functional Bayesian Fourier basis