Simulation cases

The boxplots of thep-values and of the variances ofX_iβandZ_ib_iare given in this Section for the Normal, Poisson, Bernoulli and Binomial distributions and for the three different cases described in Section 2.4.1.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Normal distribution in case 1.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Normal distribution in case 2.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Normal distribution in case 3.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 _● the right for the Poisson distribution in case 1.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Poisson distribution in case 2.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Poisson distribution in case 3.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Bernoulli distribution in case 1.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Bernoulli distribution in case 2.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 ^● the right for the Bernoulli distribution in case 3.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Binomial distribution in case 1.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Binomial distribution in case 2.

●

xc1.1 xc1.2 xc2.1 xf1.1 xf1.1 X xc1.2 xf3.12 xf3.13 xf3.14

0.00.20.40.60.81.0 the right for the Binomial distribution in case 3.

In this Section, we give for the Normal, Poisson, Bernoulli and Binomial distributions the expressions of the joint log-likelihood function ^Pm_i=1l^joint_i (θ,b_i | y_i), of the joint log-likelihood function under the saturated model ^Pm_i=1l^joint_s,i (φ,b_i | y_i) and of the joint pop-ulation deviance ^Pm_i=1d^joint_i (y_i,µ_i), from which we derive the formula of D_rand given in equation (2.9) in Section 2.4.2.2.

B.4.1 Normal distribution

When the conditional distribution of y_i given b_i is the Normal distribution, the GLMM defined in Section 2.2 is such that g(µ_i) =µ_i =η_i. Furthermore,y_i |b_i ∼ N(µ_i, σ²I_ni), with the n_i×n_i identity matrixI_ni, b_i ∼ N(0,D) and the variance of the responsesy_i is Σ_i =Z_iDZ⁰_i+σ²I_ni.

The joint log-likelihood function is

m

X

i=1

l_i^joint(θ,b_i |y_i) =

m

X

i=1

log

s(σ²I_ni)⁻¹

2π exp− 1

2(y_i−µ_i)⁰(σ²I_ni)⁻¹(y_i−µ_i)

+ log

sD⁻¹

2π exp− 1

2b⁰_iD⁻¹b_i

!

.

The joint log-likelihood function under the saturated model is

m

X

i=1

l^joint_s,i (φ,b_i |y_i) =

m

X

i=1



log

s(σ²I_ni)⁻¹ 2π

+ log

sD⁻¹

2π exp−1

2b⁰_iD⁻¹b_i



.

The population joint deviance is

m

X

i=1

d^joint_i (y_i,µ_i) =

m

X

i=1

−1

2(y_i −µ_i)⁰(σ²I_ni)⁻¹(y_i−µ_i).

Thus, the formula of D_rand is Drand = 1−

Pm

i=1(y_i−yˆ_i)⁰(y_i −yˆ_i)

Pm

i=1(y_i−yˆ_i0)⁰(y_i −yˆ_i0).

B.4.2 Poisson distribution

When the conditional distribution of y_i given b_i is the Poisson distribution, the GLMM defined in Section 2.2 is such that g(µ_i) = log(µ_i), µ_i = exp(η_i), y_i | b_i ∼ Poisson(µ_i) and b_i ∼ N(0,D).

The joint log-likelihood function is

m

X

i=1

l^joint_i (θ,b_i |y_i) =

m

X

i=1

y_ilog(µ_i)−log(y_i!)−µ_i + log

sD⁻¹

2π exp−1

2b⁰_iD⁻¹b_i

!

.

The joint log-likelihood function under the saturated model is

m

X

i=1

l_s,i^joint(φ,b_i |y_i) =

m

X

i=1

y_ilog(y_i)−log(y_i!)−y_i + log

sD⁻¹

2π exp−1

2b⁰_iD⁻¹b_i

!

.

The population joint deviance is

m

X

i=1

d^joint_i (y_i,µ_i) =

m

X

i=1

y_ilogy_i µ_i

+

m

X

i=1

(y_i−µ_i).

Thus, the formula of D_rand is D_rand = 1−

Pm

i=1y_ilog^y_yˆⁱ

i

+^Pm_i=1(y_i−yˆ_i)

Pm

i=1y_ilog_yˆ^yi

i0

+^Pm_i=1(y_i−yˆ_i0)

B.4.3 Bernoulli distribution

When the conditional distribution of y_i givenb_i is the Bernoulli distribution, the GLMM defined in Section 2.2 is such thatg(µ_i) = logit(µ_i) = log(µ_i/(1−µ_i)),µ_i = exp(η_i)/(1 + exp(η_i)), y_i |b_i ∼Bernoulli(µ_i) and b_i ∼ N(0,D).

The joint log-likelihood function is

m

X

i=1

l^joint_i (θ,b_i |y_i) =

m

X

i=1

y_ilog(µ_i) + (1−y_i) log(1−µ_i) + log

sD⁻¹

2π exp−1

2b⁰_iD⁻¹b_i

!

.

The joint log-likelihood function under the saturated model is

m

X

i=1

l_s^joint(φ,b_i |y_i) =

m

X

i=1

y_ilog(y_i) + (1−y_i) log(1−y_i) + log

sD⁻¹

2π exp−1

2b⁰_iD⁻¹b_i

!

.

The population joint deviance is

m

X

i=1

d^joint_i (y_i,µ_i) =

m

X

i=1

y_ilogy_i µ_i

+

m

X

i=1

(1−y_i) log1−y_i 1−µ_i

.

Thus, the formula of D_rand is D_rand = 1−

Pm

i=1y_ilog^y_yˆⁱ

i

+^Pm_i=1(1−y_i) log^1−y_1−ˆyⁱ

i

Pm

i=1y_ilog_ˆy^yi

i0

+^Pm_i=1(1−y_i) log_1−ˆ^1−y_yⁱ

i0

When the conditional distribution ofy_i givenb_i is the Binomial distribution, the GLMM defined in Section 2.2 is such thaty_i |b_i ∼Binomial(n_i,p_i) andb_i ∼ N(0,D). Therefore, E(y_i | b_i) =n_ip_i and g(p_i) = logit(p_i) = log(p_i/(1−p_i)), p_i = exp(η_i)/(1 + exp(η_i)).

We prefer to equivalently work with ˜y_i =y_i/n_i, for which E(˜y_i |b_i) = p_i =µ_i. The joint log-likelihood function is

m

X

i=1

l_i^joint(θ,b_i |y˜_i) =

m

X

i=1

log n_i n_iy˜_i

!

+n_iy˜_ilog(µ_i) +n_i(1−y˜_i) log(1−µ_i) + log

sD⁻¹

2π exp− 1

2b⁰_iD⁻¹b_i

!

.

The joint log-likelihood function under the saturated model is

m

X

i=1

l_s,i^joint(φ,b_i |y˜_i) =

m

X

i=1

log n_i n_iy˜_i

!

+n_iy˜_ilog(˜y_i) +n_i(1−y˜_i) log(1−y˜_i) + log

sD⁻¹

2π exp−1

2b⁰_iD⁻¹b_i

!

.

The population joint deviance is

m

X

i=1

d^joint_i (˜y_i,µ_i) =

m

X

i=1

n_iy˜_ilogy˜_i µ_i

+

m

X

i=1

n_i(1−y˜_i) log1−y˜_i 1−µ_i

.

Thus, the formula of D_rand is D_rand = 1−

P_m

i=1n_iy˜_ilog_yˆ^y˜i

i/ni

+^Pm_i=1n_i(1−y˜_i) log_1−ˆ^1−˜_y^yi

i/ni

Pm

i=1n_iy˜_ilog_ˆy^y˜i

i0/ni

+^Pm_i=1n_i(1−y˜_i) log_1−ˆ^1−˜_y^yi

i0/ni

B.5 Simulation study results

We give in this Section the Tables containing the results of the simulation study presented in Section 2.4 for all the considered measures, except PRD, the marginal r_c and the conditional r_c that are not adapted for model selection.

Table B.1: Normal distribution - case 1. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 70 1(0) 0.492(0.489) 5 0.997(0.033) 0.085(0.248) 0 0.999(0.008) 0.467(0.464)

mAIC^R 0 1(0) 0(0) 98 1(0) 0.49(0.501) 0 1(0) 0.03(0.031)

mAIC^{F R} 68 1(0) 0.508(0.49) 66 0.947(0.123) 0.607(0.448) 3 0.987(0.031) 0.514(0.465) mAICC^F_N 81 1(0) 0.597(0.475) 5 0.997(0.033) 0.095(0.258) 0 0.999(0.008) 0.566(0.45)

mAICC^R_N 0 1(0) 0(0) 114 1(0) 0.57(0.496) 0 1(0) 0.035(0.03)

mAICC^{F R}_N 82 1(0) 0.643(0.464) 65 0.937(0.131) 0.652(0.427) 6 0.984(0.033) 0.644(0.439) mAICC^F_m 149 1(0) 0.981(0.036) 10 0.995(0.041) 0.185(0.334) 0 0.999(0.01) 0.933(0.037)

mAICC^R_m 0 1(0) 0(0) 111 1(0) 0.555(0.498) 0 1(0) 0.034(0.031)

mAICC^{F R}_m 157 1(0) 0.989(0.023) 33 0.85(0.166) 0.805(0.307) 10 0.963(0.042) 0.978(0.03) mCAIC^F_N 186 1(0) 0.997(0.013) 4 0.995(0.041) 0.145(0.296) 0 0.999(0.01) 0.944(0.017)

mCAIC^R_N 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

mCAIC^{F R}_N 186 1(0) 0.997(0.013) 9 0.742(0.14) 0.937(0.143) 2 0.935(0.035) 0.993(0.016) mCAIC^F_m 175 1(0) 0.993(0.022) 5 0.995(0.041) 0.153(0.304) 0 0.999(0.01) 0.941(0.023)

mCAIC^R_m 0 1(0) 0(0) 160 1(0) 0.8(0.401) 0 1(0) 0.049(0.025)

mCAIC^{F R}_m 175 1(0) 0.994(0.017) 20 0.8(0.164) 0.87(0.238) 5 0.95(0.041) 0.986(0.022) BIC^F_N 186 1(0) 0.997(0.013) 4 0.995(0.041) 0.145(0.296) 0 0.999(0.01) 0.944(0.017)

BIC^R_N 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

BIC^{F R}_N 186 1(0) 0.997(0.013) 9 0.742(0.14) 0.937(0.143) 2 0.935(0.035) 0.993(0.016) BIC^F_m 175 1(0) 0.992(0.023) 5 0.995(0.041) 0.15(0.302) 0 0.999(0.01) 0.941(0.024)

BIC^R_m 0 1(0) 0(0) 160 1(0) 0.8(0.401) 0 1(0) 0.049(0.025)

BIC^{F R}_m 175 1(0) 0.994(0.017) 20 0.8(0.164) 0.87(0.238) 5 0.95(0.041) 0.986(0.022) BICh 184 1(0) 0.996(0.014) 14 0.807(0.165) 0.84(0.267) 6 0.952(0.041) 0.987(0.021)

PRDpen,nullmodel2

mAIC^F 101 1(0) 0.728(0.423) 3 1(0) 0.085(0.234) 0 1(0) 0.689(0.401)

mAIC^R 0 1(0) 0(0) 138 1(0) 0.69(0.464) 0 1(0) 0.042(0.028)

mAIC^{F R} 99 1(0) 0.735(0.421) 96 0.943(0.126) 0.763(0.379) 28 0.986(0.031) 0.737(0.397)

mAICC^F_N 109 1(0) 0.773(0.396) 4 1(0) 0.093(0.246) 0 1(0) 0.732(0.375)

mAICC^R_N 0 1(0) 0(0) 162 1(0) 0.81(0.393) 0 1(0) 0.05(0.024)

mAICC^{F R}_N 108 1(0) 0.78(0.393) 96 0.938(0.13) 0.788(0.356) 31 0.985(0.032) 0.781(0.369)

mAICC^F_m 160 1(0) 0.984(0.036) 5 1(0) 0.102(0.257) 0 1(0) 0.93(0.035)

mAICC^R_m 0 1(0) 0(0) 158 1(0) 0.79(0.408) 0 1(0) 0.048(0.025)

mAICC^{F R}_m 172 1(0) 0.993(0.02) 56 0.857(0.165) 0.87(0.245) 37 0.964(0.041) 0.985(0.023)

mCAIC^F_N 196 1(0) 0.999(0.006) 1 1(0) 0.078(0.219) 0 1(0) 0.943(0.012)

mCAIC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mCAIC^{F R}_N 196 1(0) 0.999(0.006) 15 0.718(0.121) 0.973(0.091) 11 0.93(0.03) 0.998(0.008)

mCAIC^F_m 187 1(0) 0.997(0.015) 4 1(0) 0.1(0.252) 0 1(0) 0.942(0.016)

mCAIC^R_m 0 1(0) 0(0) 183 1(0) 0.915(0.28) 0 1(0) 0.056(0.017)

mCAIC^{F R}_m 186 1(0) 0.997(0.013) 38 0.785(0.16) 0.935(0.166) 28 0.946(0.04) 0.993(0.016)

BIC^F_N 196 1(0) 0.999(0.006) 1 1(0) 0.078(0.219) 0 1(0) 0.943(0.012)

BIC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

BIC^{F R}_N 196 1(0) 0.999(0.006) 15 0.718(0.121) 0.973(0.091) 11 0.93(0.03) 0.998(0.008)

BIC^F_m 187 1(0) 0.997(0.015) 4 1(0) 0.1(0.252) 0 1(0) 0.942(0.016)

BIC^R_m 0 1(0) 0(0) 183 1(0) 0.915(0.28) 0 1(0) 0.056(0.017)

BIC^{F R}_m 185 1(0) 0.996(0.013) 38 0.785(0.16) 0.935(0.166) 28 0.946(0.04) 0.993(0.017) BIC_h 195 1(0) 0.999(0.007) 30 0.807(0.165) 0.89(0.212) 27 0.952(0.041) 0.992(0.015)

m.Drand 0 1(0) 0(0) 137 1(0) 0.685(0.466) 0 1(0) 0.042(0.029)

D_rand 0 1(0) 0.004(0.063) 6 1(0) 0.03(0.171) 0 1(0) 0.006(0.06)

Prand 0 1(0) 0(0) 4 1(0) 0.02(0.14) 0 1(0) 0.001(0.009)

R²_m 1 1(0) 0.005(0.071) 109 0.998(0.024) 0.55(0.499) 0 1(0.006) 0.038(0.075)

R²_c 4 1(0) 0.07(0.248) 28 1(0) 0.147(0.351) 0 1(0) 0.075(0.236)

mrc,a 3 1(0) 0.015(0.122) 118 0.987(0.11) 0.605(0.49) 0 0.997(0.028) 0.051(0.121) crc,a 27 0.861(0.108) 0.914(0.262) 0 0.792(0.162) 0.755(0.377) 0 0.844(0.121) 0.904(0.261) mAIC 173 1(0) 0.993(0.021) 100 0.853(0.166) 0.963(0.166) 75 0.963(0.041) 0.991(0.022) mAICCN 176 1(0) 0.993(0.02) 97 0.848(0.166) 0.963(0.166) 75 0.962(0.042) 0.992(0.021) mAICCm 190 1(0) 0.998(0.012) 70 0.792(0.162) 0.988(0.084) 60 0.948(0.04) 0.997(0.012)

mCAICN 200 1(0) 1(0) 19 0.698(0.098) 1(0) 19 0.925(0.024) 1(0)

mCAICm 195 1(0) 0.999(0.007) 43 0.742(0.14) 0.997(0.033) 38 0.935(0.035) 0.999(0.007) cAIC 157 0.996(0.031) 0.979(0.079) 19 0.912(0.147) 0.453(0.467) 7 0.975(0.042) 0.946(0.083)

BICN 200 1(0) 1(0) 19 0.698(0.098) 1(0) 19 0.925(0.024) 1(0)

BICm 195 1(0) 0.999(0.007) 44 0.743(0.141) 0.997(0.033) 39 0.936(0.035) 0.999(0.007)

BICh 200 1(0) 1(0) 39 0.742(0.14) 0.987(0.087) 39 0.935(0.035) 0.999(0.005)

Table B.2: Poisson distribution, case 1. The exponents , and indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 171 1(0) 0.989(0.03) 8 1(0) 0.06(0.224) 0 1(0) 0.932(0.027)

mAIC^R 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mAIC^{F R} 169 1(0) 0.99(0.025) 91 0.84(0.167) 0.965(0.155) 64 0.96(0.042) 0.989(0.027)

mAICC^F_N 171 1(0) 0.989(0.03) 8 1(0) 0.06(0.224) 0 1(0) 0.932(0.027)

mAICC^R_N 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mAICC^{F R}_N 169 1(0) 0.99(0.025) 90 0.837(0.167) 0.967(0.153) 63 0.959(0.042) 0.989(0.027)

mAICC^F_m 191 1(0) 0.998(0.009) 7 1(0) 0.042(0.195) 0 1(0) 0.939(0.004)

mAICC^R_m 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mAICC^{F R}_m 189 1(0) 0.998(0.01) 62 0.78(0.158) 0.99(0.057) 52 0.945(0.04) 0.997(0.01)

mCAIC^F_N 199 1(0) 1(0.003) 1 1(0) 0.005(0.071) 0 1(0) 0.939(0.001)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 200 1(0) 1(0) 16 0.693(0.091) 1(0) 16 0.923(0.023) 1(0)

mCAIC^F_m 197 1(0) 0.999(0.005) 3 1(0) 0.015(0.122) 0 1(0) 0.939(0.002)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 198 1(0) 1(0.004) 33 0.723(0.126) 0.998(0.024) 31 0.931(0.031) 0.999(0.004)

BIC^F_N 199 1(0) 1(0.003) 1 1(0) 0.005(0.071) 0 1(0) 0.939(0.001)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 200 1(0) 1(0) 16 0.693(0.091) 1(0) 16 0.923(0.023) 1(0)

BIC^F_m 197 1(0) 0.999(0.005) 3 1(0) 0.015(0.122) 0 1(0) 0.939(0.002)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 198 1(0) 1(0.004) 33 0.723(0.126) 0.998(0.024) 31 0.931(0.031) 0.999(0.004)

BICh 200 1(0) 1(0) 30 0.728(0.13) 0.988(0.061) 30 0.932(0.032) 0.999(0.004)

PRDpen,nullmodel2

mAIC^F 99 1(0) 0.705(0.437) 79 0.993(0.047) 0.475(0.48) 29 0.998(0.012) 0.691(0.409)

mAIC^R 10 1(0) 0.068(0.249) 147 1(0) 0.742(0.436) 9 1(0) 0.109(0.239)

mAIC^{F R} 100 1(0) 0.703(0.441) 116 0.942(0.127) 0.812(0.367) 49 0.985(0.032) 0.71(0.421) mAICC^F_N 102 1(0) 0.743(0.415) 77 0.993(0.047) 0.472(0.478) 29 0.998(0.012) 0.727(0.388)

mAICC^R_N 10 1(0) 0.068(0.249) 147 1(0) 0.742(0.436) 9 1(0) 0.109(0.239)

mAICC^{F R}_N 102 1(0) 0.749(0.412) 117 0.942(0.127) 0.82(0.358) 49 0.985(0.032) 0.754(0.394) mAICC^F_m 130 0.997(0.033) 0.958(0.126) 76 0.988(0.061) 0.522(0.464) 39 0.995(0.029) 0.931(0.122)

mAICC^R_m 10 1(0) 0.068(0.249) 150 1(0) 0.757(0.428) 9 1(0) 0.11(0.239)

mAICC^{F R}_m 136 0.997(0.033) 0.972(0.08) 114 0.92(0.143) 0.893(0.26) 67 0.978(0.042) 0.967(0.078) mCAIC^F_N 163 0.99(0.057) 0.96(0.172) 69 0.983(0.073) 0.502(0.464) 44 0.988(0.046) 0.932(0.164) mCAIC^R_N 13 1(0) 0.083(0.274) 170 0.998(0.024) 0.862(0.343) 11 1(0.006) 0.131(0.26) mCAIC^{F R}_N 162 0.988(0.061) 0.964(0.158) 79 0.842(0.167) 0.927(0.222) 49 0.952(0.056) 0.962(0.152) mCAIC^F_m 144 0.998(0.024) 0.899(0.277) 73 0.987(0.065) 0.498(0.469) 42 0.995(0.024) 0.874(0.263)

mCAIC^R_m 10 1(0) 0.068(0.249) 160 1(0) 0.807(0.393) 9 1(0) 0.113(0.238)

mCAIC^{F R}_m 150 0.998(0.024) 0.903(0.278) 99 0.888(0.158) 0.89(0.28) 64 0.971(0.042) 0.902(0.268) BIC^F_N 163 0.99(0.057) 0.96(0.172) 69 0.983(0.073) 0.502(0.464) 44 0.988(0.046) 0.932(0.164) BIC^R_N 13 1(0) 0.083(0.274) 170 0.998(0.024) 0.862(0.343) 11 1(0.006) 0.131(0.26) BIC^{F R}_N 162 0.988(0.061) 0.964(0.158) 79 0.842(0.167) 0.927(0.222) 49 0.952(0.056) 0.962(0.152) BIC^F_m 144 0.998(0.024) 0.899(0.277) 73 0.987(0.065) 0.498(0.469) 42 0.995(0.024) 0.874(0.263)

BIC^R_m 10 1(0) 0.068(0.249) 160 1(0) 0.807(0.393) 9 1(0) 0.113(0.238)

BIC^{F R}_m 150 0.998(0.024) 0.903(0.278) 100 0.89(0.157) 0.89(0.28) 65 0.971(0.042) 0.902(0.268) BIC_h 161 0.99(0.057) 0.95(0.197) 92 0.887(0.158) 0.893(0.254) 63 0.964(0.054) 0.946(0.19) mDrand 101 1(0) 0.505(0.501) 60 0.495(0.501) 0.805(0.397) 0 0.874(0.125) 0.523(0.483) D_rand 0 0.986(0.055) 0.065(0.247) 11 0.978(0.082) 0.12(0.326) 0 0.984(0.062) 0.068(0.247)

Prand 0 1(0) 0(0) 10 1(0) 0.05(0.218) 0 1(0) 0.003(0.013)

R²_m 0 1(0) 0(0) 122 1(0) 0.61(0.489) 0 1(0) 0.037(0.03)

R²_c 20 0.98(0.069) 0.361(0.463) 73 0.993(0.066) 0.375(0.485) 4 0.983(0.054) 0.361(0.443) mrc,a 155 1(0) 0.779(0.415) 18 0.225(0.419) 0.865(0.343) 0 0.806(0.105) 0.785(0.405) crc,a 9 0.994(0.035) 0.134(0.334) 27 0.99(0.057) 0.195(0.384) 0 0.993(0.04) 0.138(0.322) mAIC 169 1(0) 0.99(0.025) 91 0.84(0.167) 0.965(0.155) 64 0.96(0.042) 0.989(0.027) mAICCN 169 1(0) 0.99(0.025) 90 0.837(0.167) 0.967(0.153) 63 0.959(0.042) 0.989(0.027) mAICCm 189 1(0) 0.998(0.01) 62 0.78(0.158) 0.99(0.057) 52 0.945(0.04) 0.997(0.01)

mCAICN 200 1(0) 1(0) 16 0.693(0.091) 1(0) 16 0.923(0.023) 1(0)

mCAICm 198 1(0) 1(0.004) 33 0.723(0.126) 0.998(0.024) 31 0.931(0.031) 0.999(0.004)

cAIC NA NA(NA) NA(NA) NA NA(NA) NA(NA) NA NA(NA) NA(NA)

BICN 200 1(0) 1(0) 16 0.693(0.091) 1(0) 16 0.923(0.023) 1(0)

BICm 198 1(0) 1(0.004) 33 0.723(0.126) 0.998(0.024) 31 0.931(0.031) 0.999(0.004)

BICh 200 1(0) 1(0) 30 0.728(0.13) 0.988(0.061) 30 0.932(0.032) 0.999(0.004)

Table B.3: Bernoulli distribution, case 1. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 167 0.998(0.024) 0.988(0.03) 3 1(0) 0.065(0.211) 0 0.999(0.018) 0.932(0.028)

mAIC^R 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mAIC^{F R} 174 0.99(0.057) 0.993(0.023) 26 0.49(0.245) 0.992(0.078) 7 0.865(0.079) 0.993(0.022) mAICC^F_N 170 0.998(0.024) 0.989(0.029) 3 1(0) 0.058(0.202) 0 0.999(0.018) 0.932(0.027)

mAICC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mAICC^{F R}_N 176 0.99(0.057) 0.994(0.021) 25 0.485(0.243) 0.992(0.078) 8 0.864(0.078) 0.994(0.02) mAICC^F_m 186 0.995(0.041) 0.997(0.016) 5 1(0) 0.045(0.191) 0 0.996(0.03) 0.938(0.015)

mAICC^R_m 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

mAICC^{F R}_m 185 0.982(0.076) 0.999(0.006) 11 0.413(0.181) 1(0) 7 0.84(0.077) 0.999(0.006) mCAIC^F_N 107 0.847(0.167) 1(0.003) 92 1(0) 0.463(0.499) 0 0.885(0.125) 0.967(0.031)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 91 0.82(0.167) 1(0.003) 1 0.338(0.053) 1(0) 0 0.7(0.127) 1(0.003)

mCAIC^F_m 167 0.948(0.121) 1(0.004) 32 1(0) 0.163(0.369) 0 0.961(0.091) 0.948(0.022)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 163 0.942(0.127) 1(0.004) 2 0.36(0.102) 1(0) 0 0.796(0.102) 1(0.004) BIC^F_N 107 0.847(0.167) 1(0.003) 92 1(0) 0.463(0.499) 0 0.885(0.125) 0.967(0.031)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 91 0.82(0.167) 1(0.003) 1 0.338(0.053) 1(0) 0 0.7(0.127) 1(0.003)

BIC^F_m 167 0.948(0.121) 1(0.004) 32 1(0) 0.163(0.369) 0 0.961(0.091) 0.948(0.022)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 163 0.942(0.127) 1(0.004) 2 0.36(0.102) 1(0) 0 0.796(0.102) 1(0.004)

BICh 101 0.837(0.167) 1(0.003) 12 0.533(0.203) 0.998(0.024) 6 0.761(0.169) 1(0.003)

PRDpen,nullmodel2

mAIC^F 159 0.997(0.033) 0.975(0.104) 27 0.952(0.151) 0.333(0.438) 17 0.985(0.045) 0.936(0.1)

mAIC^R 0 1(0) 0(0) 197 1(0) 0.985(0.122) 0 1(0) 0.06(0.007)

mAIC^{F R} 166 0.99(0.057) 0.98(0.103) 67 0.602(0.308) 0.993(0.074) 39 0.893(0.091) 0.981(0.097) mAICC^F_N 162 0.997(0.033) 0.981(0.077) 26 0.952(0.151) 0.332(0.436) 18 0.985(0.045) 0.941(0.079)

mAICC^R_N 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

mAICC^{F R}_N 169 0.99(0.057) 0.986(0.075) 63 0.59(0.304) 0.993(0.074) 38 0.89(0.09) 0.986(0.071) mAICC^F_m 185 0.997(0.033) 0.996(0.017) 24 0.943(0.164) 0.323(0.438) 20 0.983(0.047) 0.955(0.031)

mAICC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mAICC^{F R}_m 183 0.983(0.073) 0.998(0.011) 37 0.487(0.263) 1(0) 28 0.859(0.086) 0.998(0.01) mCAIC^F_N 104 0.842(0.167) 1(0.003) 74 0.868(0.263) 0.615(0.474) 12 0.848(0.161) 0.976(0.029)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 84 0.808(0.165) 1(0.003) 3 0.345(0.084) 1(0) 1 0.693(0.127) 1(0.003) mCAIC^F_m 166 0.95(0.119) 0.999(0.009) 38 0.92(0.209) 0.392(0.466) 18 0.943(0.118) 0.962(0.03)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 154 0.927(0.138) 1(0.004) 18 0.4(0.195) 1(0) 15 0.795(0.121) 1(0.004) BIC^F_N 104 0.842(0.167) 1(0.003) 74 0.868(0.263) 0.615(0.474) 12 0.848(0.161) 0.976(0.029)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 84 0.808(0.165) 1(0.003) 3 0.345(0.084) 1(0) 1 0.693(0.127) 1(0.003)

BIC^F_m 166 0.95(0.119) 0.999(0.009) 38 0.92(0.209) 0.392(0.466) 18 0.943(0.118) 0.962(0.03)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 154 0.927(0.138) 1(0.004) 18 0.4(0.195) 1(0) 15 0.795(0.121) 1(0.004)

BICh 96 0.828(0.167) 1(0.003) 24 0.542(0.233) 0.998(0.024) 18 0.757(0.175) 1(0.003)

mD_rand 0 1(0) 0(0) 159 1(0) 0.795(0.405) 0 1(0) 0.049(0.025)

Drand 0 1(0) 0(0) 8 1(0) 0.04(0.196) 0 1(0) 0.002(0.012)

Prand 0 1(0) 0(0) 6 1(0) 0.03(0.171) 0 1(0) 0.002(0.01)

R²_m 0 1(0) 0(0) 81 1(0) 0.405(0.492) 0 1(0) 0.025(0.03)

R²_c 0 1(0) 0.009(0.089) 24 1(0) 0.12(0.326) 0 1(0) 0.016(0.085)

mrc,a 0 1(0) 0(0) 68 1(0) 0.34(0.475) 0 1(0) 0.021(0.029)

crc,a 21 0.971(0.075) 0.298(0.452) 12 0.968(0.098) 0.162(0.366) 0 0.97(0.077) 0.29(0.438) mAIC 174 0.99(0.057) 0.993(0.023) 26 0.49(0.245) 0.992(0.078) 7 0.865(0.079) 0.993(0.022) mAICCN 176 0.99(0.057) 0.994(0.021) 25 0.485(0.243) 0.992(0.078) 8 0.864(0.078) 0.994(0.02) mAICCm 185 0.982(0.076) 0.999(0.006) 11 0.413(0.181) 1(0) 7 0.84(0.077) 0.999(0.006)

mCAICN 91 0.82(0.167) 1(0.003) 1 0.338(0.053) 1(0) 0 0.7(0.127) 1(0.003)

mCAICm 163 0.942(0.127) 1(0.004) 2 0.36(0.102) 1(0) 0 0.796(0.102) 1(0.004)

BICN 91 0.82(0.167) 1(0.003) 1 0.338(0.053) 1(0) 0 0.7(0.127) 1(0.003)

BICm 163 0.942(0.127) 1(0.004) 2 0.36(0.102) 1(0) 0 0.796(0.102) 1(0.004)

BICh 101 0.837(0.167) 1(0.003) 12 0.533(0.203) 0.998(0.024) 6 0.761(0.169) 1(0.003)

Table B.4: Binomial distribution, case 1. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 175 1(0) 0.992(0.025) 5 1(0) 0.062(0.214) 0 1(0) 0.935(0.022)

mAIC^R 0 1(0) 0(0) 188 1(0) 0.94(0.238) 0 1(0) 0.058(0.015)

mAIC^{F R} 174 1(0) 0.992(0.023) 85 0.833(0.167) 0.948(0.204) 63 0.958(0.042) 0.989(0.027)

mAICC^F_N 180 1(0) 0.994(0.02) 4 1(0) 0.057(0.204) 0 1(0) 0.937(0.016)

mAICC^R_N 0 1(0) 0(0) 189 1(0) 0.945(0.229) 0 1(0) 0.058(0.014)

mAICC^{F R}_N 178 1(0) 0.994(0.019) 84 0.83(0.167) 0.953(0.192) 64 0.958(0.042) 0.991(0.021)

mAICC^F_m 190 1(0) 0.998(0.012) 2 1(0) 0.033(0.157) 0 1(0) 0.938(0.008)

mAICC^R_m 0 1(0) 0(0) 192 1(0) 0.96(0.196) 0 1(0) 0.059(0.012)

mAICC^{F R}_m 188 1(0) 0.997(0.01) 71 0.793(0.162) 0.985(0.107) 60 0.948(0.041) 0.997(0.012)

mCAIC^F_N 200 1(0) 1(0) 0 1(0) 0.003(0.047) 0 1(0) 0.939(0.003)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 200 1(0) 1(0) 14 0.692(0.088) 0.998(0.024) 14 0.923(0.022) 1(0.001)

mCAIC^F_m 199 1(0) 1(0.003) 0 1(0) 0.007(0.066) 0 1(0) 0.939(0.003)

mCAIC^R_m 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

mCAIC^{F R}_m 200 1(0) 1(0) 38 0.732(0.132) 0.998(0.024) 38 0.933(0.033) 1(0.001)

BIC^F_N 200 1(0) 1(0) 0 1(0) 0.003(0.047) 0 1(0) 0.939(0.003)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 200 1(0) 1(0) 14 0.692(0.088) 0.998(0.024) 14 0.923(0.022) 1(0.001)

BIC^F_m 199 1(0) 1(0.003) 0 1(0) 0.007(0.066) 0 1(0) 0.939(0.003)

BIC^R_m 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

BIC^{F R}_m 200 1(0) 1(0) 38 0.732(0.132) 0.998(0.024) 38 0.933(0.033) 1(0.001)

BICh 200 1(0) 1(0) 34 0.737(0.136) 0.983(0.093) 34 0.934(0.034) 0.999(0.006)

PRDpen,nullmodel2

mAIC^F 150 1(0) 0.948(0.185) 30 0.992(0.052) 0.305(0.412) 19 0.998(0.013) 0.908(0.175)

mAIC^R 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

mAIC^{F R} 148 1(0) 0.958(0.158) 105 0.89(0.157) 0.925(0.215) 68 0.973(0.039) 0.956(0.148) mAICC^F_N 157 1(0) 0.969(0.125) 31 0.992(0.052) 0.307(0.414) 21 0.998(0.013) 0.929(0.122)

mAICC^R_N 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

mAICC^{F R}_N 155 1(0) 0.97(0.124) 105 0.885(0.159) 0.94(0.182) 72 0.971(0.04) 0.968(0.117) mAICC^F_m 179 1(0) 0.993(0.023) 29 0.988(0.061) 0.327(0.415) 24 0.997(0.015) 0.952(0.031)

mAICC^R_m 0 1(0) 0(0) 186 1(0) 0.93(0.256) 0 1(0) 0.057(0.016)

mAICC^{F R}_m 182 1(0) 0.995(0.019) 90 0.845(0.167) 0.965(0.131) 78 0.961(0.042) 0.993(0.02) mCAIC^F_N 199 1(0) 1(0.003) 29 0.988(0.061) 0.303(0.414) 29 0.997(0.015) 0.957(0.025)

mCAIC^R_N 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mCAIC^{F R}_N 200 1(0) 1(0) 43 0.74(0.138) 0.998(0.024) 43 0.935(0.035) 1(0.001)

mCAIC^F_m 195 1(0) 0.999(0.01) 28 0.988(0.061) 0.305(0.412) 27 0.997(0.015) 0.956(0.025)

mCAIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mCAIC^{F R}_m 197 1(0) 0.999(0.009) 73 0.798(0.163) 0.987(0.087) 71 0.95(0.041) 0.998(0.01) BIC^F_N 199 1(0) 1(0.003) 29 0.988(0.061) 0.303(0.414) 29 0.997(0.015) 0.957(0.025)

BIC^R_N 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

BIC^{F R}_N 200 1(0) 1(0) 43 0.74(0.138) 0.998(0.024) 43 0.935(0.035) 1(0.001)

BIC^F_m 195 1(0) 0.999(0.01) 28 0.988(0.061) 0.305(0.412) 27 0.997(0.015) 0.956(0.025)

BIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

BIC^{F R}_m 197 1(0) 0.999(0.009) 73 0.798(0.163) 0.987(0.087) 71 0.95(0.041) 0.998(0.01)

BICh 200 1(0) 1(0) 61 0.803(0.164) 0.958(0.138) 61 0.951(0.041) 0.997(0.008)

mD_rand 1 1(0) 0.005(0.071) 129 0.995(0.071) 0.65(0.478) 0 0.999(0.018) 0.044(0.074)

Drand 0 1(0) 0.004(0.063) 9 1(0) 0.045(0.208) 0 1(0) 0.007(0.06)

Prand 0 1(0) 0(0) 5 1(0) 0.025(0.157) 0 1(0) 0.002(0.01)

R²_m 0 1(0) 0(0) 80 1(0) 0.4(0.491) 0 1(0) 0.024(0.03)

R²_c 35 1(0) 0.388(0.468) 66 1(0) 0.33(0.471) 9 1(0) 0.384(0.441)

mrc,a 1 1(0) 0.005(0.071) 111 1(0) 0.555(0.498) 1 1(0) 0.039(0.075)

crc,a 31 0.85(0.104) 0.914(0.271) 1 0.775(0.157) 0.77(0.384) 0 0.831(0.117) 0.906(0.269) mAIC 174 1(0) 0.992(0.023) 85 0.833(0.167) 0.948(0.204) 63 0.958(0.042) 0.989(0.027) mAICCN 178 1(0) 0.994(0.019) 84 0.83(0.167) 0.953(0.192) 64 0.958(0.042) 0.991(0.021) mAICCm 188 1(0) 0.997(0.01) 71 0.793(0.162) 0.985(0.107) 60 0.948(0.041) 0.997(0.012)

mCAICN 200 1(0) 1(0) 14 0.692(0.088) 0.998(0.024) 14 0.923(0.022) 1(0.001)

mCAICm 200 1(0) 1(0) 38 0.732(0.132) 0.998(0.024) 38 0.933(0.033) 1(0.001)

BICN 200 1(0) 1(0) 14 0.692(0.088) 0.998(0.024) 14 0.923(0.022) 1(0.001)

BICm 200 1(0) 1(0) 38 0.732(0.132) 0.998(0.024) 38 0.933(0.033) 1(0.001)

BICh 200 1(0) 1(0) 34 0.737(0.136) 0.983(0.093) 34 0.934(0.034) 0.999(0.006)

Table B.5: Normal distribution - case 2. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 93 1(0) 0.665(0.454) 4 0.978(0.082) 0.135(0.317) 0 0.995(0.021) 0.633(0.432)

mAIC^R 0 1(0) 0(0) 106 1(0) 0.53(0.5) 0 1(0) 0.032(0.031)

mAIC^{F R} 103 1(0) 0.69(0.449) 46 0.893(0.156) 0.64(0.449) 1 0.973(0.039) 0.687(0.427) mAICC^F_N 108 1(0) 0.763(0.402) 4 0.972(0.093) 0.165(0.344) 0 0.993(0.023) 0.726(0.384)

mAICC^R_N 0 1(0) 0(0) 120 1(0) 0.6(0.491) 0 1(0) 0.037(0.03)

mAICC^{F R}_N 118 1(0) 0.793(0.387) 42 0.873(0.162) 0.697(0.42) 1 0.968(0.041) 0.787(0.369) mAICC^F_m 152 1(0) 0.981(0.038) 5 0.958(0.111) 0.24(0.391) 0 0.99(0.028) 0.935(0.044)

mAICC^R_m 0 1(0) 0(0) 117 1(0) 0.585(0.494) 0 1(0) 0.036(0.03)

mAICC^{F R}_m 165 1(0) 0.991(0.021) 26 0.795(0.163) 0.872(0.265) 4 0.949(0.041) 0.984(0.026) mCAIC^F_N 193 1(0) 0.998(0.008) 2 0.953(0.116) 0.213(0.384) 0 0.988(0.029) 0.95(0.023)

mCAIC^R_N 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

mCAIC^{F R}_N 195 1(0) 0.999(0.007) 1 0.712(0.114) 0.953(0.13) 0 0.928(0.029) 0.996(0.011) mCAIC^F_m 176 1(0) 0.993(0.023) 3 0.955(0.114) 0.237(0.391) 0 0.989(0.029) 0.946(0.03)

mCAIC^R_m 0 1(0) 0(0) 162 1(0) 0.81(0.393) 0 1(0) 0.05(0.024)

mCAIC^{F R}_m 175 1(0) 0.994(0.016) 17 0.758(0.149) 0.927(0.171) 2 0.94(0.037) 0.99(0.019) BIC^F_N 193 1(0) 0.998(0.008) 2 0.953(0.116) 0.213(0.384) 0 0.988(0.029) 0.95(0.023)

BIC^R_N 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

BIC^{F R}_N 195 1(0) 0.999(0.007) 1 0.712(0.114) 0.953(0.13) 0 0.928(0.029) 0.996(0.011) BIC^F_m 175 1(0) 0.992(0.024) 3 0.955(0.114) 0.233(0.39) 0 0.989(0.029) 0.946(0.031)

BIC^R_m 0 1(0) 0(0) 162 1(0) 0.81(0.393) 0 1(0) 0.05(0.024)

BIC^{F R}_m 175 1(0) 0.994(0.016) 17 0.758(0.149) 0.927(0.171) 2 0.94(0.037) 0.99(0.019) BICh 188 1(0) 0.997(0.012) 12 0.762(0.151) 0.908(0.198) 5 0.94(0.038) 0.992(0.017)

PRDpen,nullmodel2

mAIC^F 34 1(0) 0.287(0.44) 45 1(0) 0.245(0.423) 2 1(0) 0.284(0.41)

mAIC^R 0 1(0) 0(0) 152 1(0) 0.76(0.428) 0 1(0) 0.047(0.026)

mAIC^{F R} 35 1(0) 0.293(0.444) 141 0.995(0.041) 0.77(0.402) 17 0.999(0.01) 0.322(0.418)

mAICC^F_N 42 1(0) 0.39(0.47) 44 1(0) 0.243(0.421) 2 1(0) 0.381(0.438)

mAICC^R_N 0 1(0) 0(0) 160 1(0) 0.8(0.401) 0 1(0) 0.049(0.025)

mAICC^{F R}_N 43 1(0) 0.396(0.473) 141 0.99(0.057) 0.792(0.383) 22 0.998(0.014) 0.42(0.445) mAICC^F_m 126 1(0) 0.964(0.05) 18 0.995(0.041) 0.198(0.359) 7 0.999(0.01) 0.917(0.053)

mAICC^R_m 0 1(0) 0(0) 157 1(0) 0.785(0.412) 0 1(0) 0.048(0.025)

mAICC^{F R}_m 125 1(0) 0.967(0.046) 126 0.952(0.118) 0.875(0.276) 66 0.988(0.029) 0.962(0.047) mCAIC^F_N 193 1(0) 0.997(0.017) 12 0.99(0.057) 0.19(0.348) 10 0.998(0.014) 0.947(0.026)

mCAIC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mCAIC^{F R}_N 188 1(0) 0.996(0.019) 105 0.872(0.163) 0.963(0.133) 94 0.968(0.041) 0.994(0.019) mCAIC^F_m 155 1(0) 0.942(0.197) 17 0.993(0.047) 0.202(0.36) 8 0.998(0.012) 0.897(0.184)

mCAIC^R_m 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

mCAIC^{F R}_m 158 1(0) 0.941(0.208) 117 0.918(0.144) 0.913(0.227) 83 0.98(0.036) 0.939(0.199) BIC^F_N 193 1(0) 0.997(0.017) 12 0.99(0.057) 0.19(0.348) 10 0.998(0.014) 0.947(0.026)

BIC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

BIC^{F R}_N 188 1(0) 0.996(0.019) 105 0.872(0.163) 0.963(0.133) 94 0.968(0.041) 0.994(0.019) BIC^F_m 154 1(0) 0.942(0.197) 17 0.993(0.047) 0.205(0.361) 8 0.998(0.012) 0.897(0.184)

BIC^R_m 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

BIC^{F R}_m 158 1(0) 0.941(0.208) 116 0.918(0.144) 0.908(0.236) 82 0.98(0.036) 0.939(0.199) BIC_h 184 1(0) 0.993(0.025) 101 0.92(0.143) 0.885(0.235) 88 0.98(0.036) 0.987(0.027)

mDrand 0 1(0) 0.004(0.063) 139 1(0) 0.695(0.462) 0 1(0) 0.047(0.063)

D_rand 0 0.933(0.102) 0.3(0.459) 0 0.9(0.153) 0.3(0.459) 0 0.925(0.115) 0.3(0.459)

Prand 0 1(0) 0(0) 0 1(0) 0(0) 0 1(0) 0(0)

R²_m 0 1(0) 0(0) 112 1(0) 0.56(0.498) 0 1(0) 0.034(0.03)

R²_c 0 1(0) 0.014(0.111) 15 1(0) 0.075(0.264) 0 1(0) 0.017(0.11)

mrc,a 8 1(0) 0.058(0.23) 116 0.962(0.19) 0.62(0.487) 0 0.99(0.047) 0.092(0.22)

crc,a 8 0.803(0.071) 0.977(0.141) 0 0.705(0.107) 0.922(0.246) 0 0.779(0.08) 0.973(0.142) mAIC 166 1(0) 0.991(0.023) 84 0.828(0.167) 0.958(0.18) 55 0.957(0.042) 0.989(0.025) mAICCN 166 1(0) 0.991(0.023) 84 0.827(0.167) 0.96(0.179) 54 0.957(0.042) 0.989(0.025) mAICCm 188 1(0) 0.997(0.01) 64 0.783(0.159) 0.983(0.11) 52 0.946(0.04) 0.997(0.012)

mCAICN 200 1(0) 1(0) 18 0.693(0.102) 1(0) 18 0.923(0.026) 1(0)

mCAICm 198 1(0) 1(0.004) 32 0.722(0.124) 0.998(0.024) 30 0.93(0.031) 0.999(0.004) cAIC 166 1(0) 0.986(0.033) 11 0.897(0.155) 0.452(0.469) 5 0.974(0.039) 0.954(0.046)

BICN 200 1(0) 1(0) 18 0.693(0.102) 1(0) 18 0.923(0.026) 1(0)

BICm 198 1(0) 1(0.004) 32 0.722(0.124) 0.998(0.024) 30 0.93(0.031) 0.999(0.004)

BICh 200 1(0) 1(0) 28 0.722(0.124) 0.988(0.084) 28 0.93(0.031) 0.999(0.005)

Table B.6: Poisson distribution, case 2. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 177 1(0) 0.991(0.026) 16 1(0) 0.13(0.311) 11 1(0) 0.939(0.029)

mAIC^R 0 1(0) 0(0) 178 1(0) 0.89(0.314) 0 1(0) 0.054(0.019)

mAIC^{F R} 179 1(0) 0.993(0.022) 60 0.77(0.178) 0.968(0.162) 41 0.943(0.045) 0.992(0.023)

mAICC^F_N 179 1(0) 0.993(0.024) 16 1(0) 0.13(0.311) 11 1(0) 0.94(0.027)

mAICC^R_N 0 1(0) 0(0) 178 1(0) 0.89(0.314) 0 1(0) 0.054(0.019)

mAICC^{F R}_N 181 1(0) 0.994(0.022) 59 0.763(0.176) 0.983(0.11) 41 0.941(0.044) 0.993(0.022)

mAICC^F_m 193 1(0) 0.998(0.013) 13 1(0) 0.105(0.285) 11 1(0) 0.943(0.019)

mAICC^R_m 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

mAICC^{F R}_m 197 1(0) 0.999(0.009) 35 0.72(0.155) 0.988(0.084) 32 0.93(0.039) 0.998(0.01) mCAIC^F_N 199 0.998(0.024) 1(0) 11 1(0) 0.088(0.264) 10 0.999(0.018) 0.944(0.016)

mCAIC^R_N 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mCAIC^{F R}_N 200 1(0) 1(0) 4 0.618(0.135) 1(0) 4 0.905(0.034) 1(0)

mCAIC^F_m 200 1(0) 1(0) 11 1(0) 0.088(0.264) 11 1(0) 0.944(0.016)

mCAIC^R_m 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mCAIC^{F R}_m 200 1(0) 1(0) 19 0.668(0.151) 0.997(0.033) 19 0.917(0.038) 1(0.002)

BIC^F_N 199 0.998(0.024) 1(0) 11 1(0) 0.088(0.264) 10 0.999(0.018) 0.944(0.016)

BIC^R_N 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

BIC^{F R}_N 200 1(0) 1(0) 4 0.618(0.135) 1(0) 4 0.905(0.034) 1(0)

BIC^F_m 200 1(0) 1(0) 11 1(0) 0.088(0.264) 11 1(0) 0.944(0.016)

BIC^R_m 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

BIC^{F R}_m 200 1(0) 1(0) 19 0.67(0.149) 0.997(0.033) 19 0.918(0.037) 1(0.002)

BICh 200 1(0) 1(0) 19 0.705(0.107) 0.993(0.047) 19 0.926(0.027) 1(0.003)

PRDpen,nullmodel2

mAIC^F 40 1(0) 0.338(0.462) 69 0.988(0.077) 0.433(0.475) 5 0.997(0.019) 0.343(0.438)

mAIC^R 2 1(0) 0.024(0.151) 112 1(0) 0.567(0.495) 2 1(0) 0.057(0.147)

mAIC^{F R} 41 1(0) 0.343(0.465) 86 0.983(0.093) 0.552(0.469) 7 0.996(0.023) 0.356(0.44) mAICC^F_N 44 1(0) 0.376(0.472) 68 0.988(0.077) 0.435(0.473) 6 0.997(0.019) 0.38(0.448)

mAICC^R_N 2 1(0) 0.024(0.151) 112 1(0) 0.567(0.495) 2 1(0) 0.057(0.147)

mAICC^{F R}_N 44 1(0) 0.368(0.471) 87 0.982(0.096) 0.568(0.464) 7 0.995(0.024) 0.38(0.446) mAICC^F_m 90 1(0) 0.689(0.443) 51 0.98(0.092) 0.425(0.453) 15 0.995(0.023) 0.672(0.425)

mAICC^R_m 2 1(0) 0.024(0.151) 116 1(0) 0.587(0.491) 2 1(0) 0.058(0.147)

mAICC^{F R}_m 97 1(0) 0.768(0.398) 85 0.967(0.12) 0.668(0.41) 24 0.992(0.03) 0.762(0.379) mCAIC^F_N 109 0.997(0.033) 0.698(0.448) 51 0.975(0.1) 0.45(0.453) 21 0.991(0.035) 0.683(0.431)

mCAIC^R_N 2 1(0) 0.024(0.151) 128 1(0) 0.647(0.477) 2 1(0) 0.062(0.146)

mCAIC^{F R}_N 104 0.997(0.033) 0.697(0.448) 83 0.96(0.128) 0.655(0.425) 34 0.988(0.04) 0.695(0.431) mCAIC^F_m 92 0.998(0.024) 0.631(0.47) 54 0.98(0.092) 0.43(0.458) 15 0.994(0.029) 0.619(0.448)

mCAIC^R_m 2 1(0) 0.024(0.151) 121 1(0) 0.612(0.486) 2 1(0) 0.06(0.147)

mCAIC^{F R}_m 88 0.998(0.024) 0.636(0.469) 79 0.965(0.122) 0.63(0.429) 22 0.99(0.035) 0.636(0.448) BIC^F_N 109 0.997(0.033) 0.698(0.448) 51 0.975(0.1) 0.45(0.453) 21 0.991(0.035) 0.683(0.431)

BIC^R_N 2 1(0) 0.024(0.151) 128 1(0) 0.647(0.477) 2 1(0) 0.062(0.146)

BIC^{F R}_N 104 0.997(0.033) 0.697(0.448) 83 0.96(0.128) 0.655(0.425) 34 0.988(0.04) 0.695(0.431) BIC^F_m 92 0.998(0.024) 0.631(0.47) 54 0.98(0.092) 0.43(0.458) 15 0.994(0.029) 0.619(0.448)

BIC^R_m 2 1(0) 0.024(0.151) 121 1(0) 0.612(0.486) 2 1(0) 0.06(0.147)

BIC^{F R}_m 88 0.998(0.024) 0.636(0.469) 79 0.965(0.122) 0.63(0.429) 22 0.99(0.035) 0.636(0.448) BICh 108 0.998(0.024) 0.693(0.45) 67 0.965(0.118) 0.585(0.436) 24 0.99(0.034) 0.686(0.43) mD_rand 11 1(0) 0.055(0.229) 93 0.945(0.229) 0.52(0.501) 0 0.986(0.057) 0.083(0.224) Drand 0 0.836(0.098) 0.754(0.431) 9 0.753(0.147) 0.795(0.401) 0 0.815(0.11) 0.757(0.426) Prand 2 0.988(0.051) 0.111(0.31) 46 0.982(0.076) 0.302(0.454) 0 0.986(0.057) 0.123(0.299)

R²_m 1 1(0) 0.027(0.156) 84 1(0) 0.42(0.495) 0 1(0) 0.051(0.145)

R²_c 29 0.981(0.078) 0.313(0.453) 90 0.927(0.209) 0.56(0.498) 1 0.967(0.09) 0.328(0.43) mrc,a 83 1(0) 0.424(0.494) 59 0.587(0.493) 0.71(0.455) 0 0.897(0.123) 0.441(0.479) crc,a 6 0.928(0.104) 0.429(0.49) 35 0.892(0.157) 0.523(0.493) 0 0.919(0.117) 0.434(0.477) mAIC 179 1(0) 0.993(0.022) 60 0.77(0.178) 0.968(0.162) 41 0.943(0.045) 0.992(0.023) mAICCN 181 1(0) 0.994(0.022) 59 0.763(0.176) 0.983(0.11) 41 0.941(0.044) 0.993(0.022) mAICCm 197 1(0) 0.999(0.009) 35 0.72(0.155) 0.988(0.084) 32 0.93(0.039) 0.998(0.01)

mCAICN 200 1(0) 1(0) 4 0.618(0.135) 1(0) 4 0.905(0.034) 1(0)

mCAICm 200 1(0) 1(0) 19 0.668(0.151) 0.997(0.033) 19 0.917(0.038) 1(0.002)

BICN 200 1(0) 1(0) 4 0.618(0.135) 1(0) 4 0.905(0.034) 1(0)

BICm 200 1(0) 1(0) 19 0.67(0.149) 0.997(0.033) 19 0.918(0.037) 1(0.002)

BICh 200 1(0) 1(0) 19 0.705(0.107) 0.993(0.047) 19 0.926(0.027) 1(0.003)

Table B.7: Bernoulli distribution, case 2. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 167 0.998(0.024) 0.984(0.076) 5 1(0) 0.068(0.222) 0 0.999(0.018) 0.928(0.071)

mAIC^R 0 1(0) 0(0) 194 1(0) 0.97(0.171) 0 1(0) 0.059(0.01)

mAIC^{F R} 179 0.997(0.033) 0.99(0.072) 30 0.475(0.251) 0.993(0.074) 12 0.866(0.069) 0.991(0.068) mAICC^F_N 174 0.998(0.024) 0.992(0.025) 5 1(0) 0.065(0.218) 0 0.999(0.018) 0.935(0.022)

mAICC^R_N 0 1(0) 0(0) 194 1(0) 0.97(0.171) 0 1(0) 0.059(0.01)

mAICC^{F R}_N 180 0.995(0.041) 0.996(0.016) 28 0.468(0.246) 0.993(0.074) 12 0.863(0.07) 0.996(0.015) mAICC^F_m 187 0.997(0.033) 0.997(0.014) 5 1(0) 0.045(0.191) 0 0.998(0.025) 0.939(0.012)

mAICC^R_m 0 1(0) 0(0) 197 1(0) 0.985(0.122) 0 1(0) 0.06(0.007)

mAICC^{F R}_m 187 0.987(0.065) 0.999(0.007) 11 0.405(0.18) 0.998(0.024) 6 0.841(0.069) 0.999(0.007)

mCAIC^F_N 119 0.865(0.164) 1(0) 81 1(0) 0.405(0.492) 0 0.899(0.123) 0.964(0.03)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 105 0.842(0.167) 1(0) 1 0.337(0.047) 1(0) 1 0.715(0.126) 1(0)

mCAIC^F_m 182 0.973(0.091) 1(0.004) 17 1(0) 0.088(0.283) 0 0.98(0.068) 0.944(0.017)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 167 0.948(0.121) 1(0.004) 6 0.37(0.133) 1(0) 4 0.804(0.1) 1(0.004)

BIC^F_N 119 0.865(0.164) 1(0) 81 1(0) 0.405(0.492) 0 0.899(0.123) 0.964(0.03)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 105 0.842(0.167) 1(0) 1 0.337(0.047) 1(0) 1 0.715(0.126) 1(0)

BIC^F_m 182 0.973(0.091) 1(0.004) 17 1(0) 0.088(0.283) 0 0.98(0.068) 0.944(0.017)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 167 0.948(0.121) 1(0.004) 6 0.37(0.133) 1(0) 4 0.804(0.1) 1(0.004)

BICh 119 0.865(0.164) 1(0) 8 0.547(0.189) 0.998(0.024) 8 0.785(0.168) 1(0.001)

PRDpen,nullmodel2

mAIC^F 148 1(0) 0.934(0.218) 36 0.908(0.203) 0.477(0.464) 17 0.977(0.051) 0.906(0.205)

mAIC^R 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

mAIC^{F R} 156 0.995(0.041) 0.934(0.228) 87 0.653(0.322) 0.99(0.1) 46 0.91(0.09) 0.937(0.214) mAICC^F_N 152 1(0) 0.954(0.172) 37 0.908(0.203) 0.485(0.463) 18 0.977(0.051) 0.925(0.16)

mAICC^R_N 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

mAICC^{F R}_N 159 0.993(0.047) 0.959(0.172) 82 0.642(0.319) 0.993(0.074) 46 0.905(0.091) 0.961(0.161) mAICC^F_m 177 0.993(0.047) 0.993(0.025) 27 0.888(0.222) 0.467(0.463) 21 0.967(0.073) 0.96(0.037)

mAICC^R_m 0 1(0) 0(0) 198 1(0) 0.99(0.1) 0 1(0) 0.061(0.006)

mAICC^{F R}_m 183 0.988(0.061) 0.997(0.015) 56 0.553(0.298) 0.998(0.024) 47 0.88(0.093) 0.997(0.015) mCAIC^F_N 120 0.867(0.164) 1(0) 47 0.785(0.302) 0.628(0.469) 16 0.846(0.175) 0.977(0.029)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 93 0.822(0.167) 1(0) 12 0.377(0.161) 1(0) 10 0.71(0.139) 1(0)

mCAIC^F_m 179 0.973(0.091) 0.998(0.014) 32 0.873(0.24) 0.497(0.468) 24 0.948(0.104) 0.967(0.032)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 160 0.938(0.13) 0.999(0.009) 29 0.442(0.238) 1(0) 26 0.814(0.125) 0.999(0.008) BIC^F_N 120 0.867(0.164) 1(0) 47 0.785(0.302) 0.628(0.469) 16 0.846(0.175) 0.977(0.029)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 93 0.822(0.167) 1(0) 12 0.377(0.161) 1(0) 10 0.71(0.139) 1(0)

BIC^F_m 179 0.973(0.091) 0.998(0.014) 32 0.873(0.24) 0.497(0.468) 24 0.948(0.104) 0.967(0.032)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 160 0.938(0.13) 0.999(0.009) 29 0.443(0.239) 1(0) 26 0.815(0.125) 0.999(0.008) BICh 111 0.852(0.166) 1(0) 48 0.602(0.269) 0.998(0.024) 41 0.789(0.179) 1(0.001)

mD_rand 0 1(0) 0(0) 174 1(0) 0.87(0.337) 0 1(0) 0.053(0.021)

Drand 0 0.991(0.044) 0.044(0.205) 9 0.988(0.061) 0.08(0.272) 0 0.99(0.047) 0.047(0.204)

Prand 0 1(0) 0(0) 6 1(0) 0.03(0.171) 0 1(0) 0.002(0.01)

R²_m 0 1(0) 0(0) 73 1(0) 0.365(0.483) 0 1(0) 0.022(0.03)

R²_c 1 1(0) 0.005(0.071) 19 1(0) 0.095(0.294) 0 1(0) 0.011(0.068)

mrc,a 0 1(0) 0(0) 54 1(0) 0.27(0.445) 0 1(0) 0.017(0.027)

crc,a 15 0.943(0.097) 0.403(0.485) 19 0.938(0.13) 0.303(0.452) 0 0.942(0.1) 0.397(0.476) mAIC 179 0.997(0.033) 0.99(0.072) 30 0.475(0.251) 0.993(0.074) 12 0.866(0.069) 0.991(0.068) mAICCN 180 0.995(0.041) 0.996(0.016) 28 0.468(0.246) 0.993(0.074) 12 0.863(0.07) 0.996(0.015) mAICCm 187 0.987(0.065) 0.999(0.007) 11 0.405(0.18) 0.998(0.024) 6 0.841(0.069) 0.999(0.007)

mCAICN 105 0.842(0.167) 1(0) 1 0.337(0.047) 1(0) 1 0.715(0.126) 1(0)

mCAICm 167 0.948(0.121) 1(0.004) 6 0.37(0.133) 1(0) 4 0.804(0.1) 1(0.004)

BICN 105 0.842(0.167) 1(0) 1 0.337(0.047) 1(0) 1 0.715(0.126) 1(0)

BICm 167 0.948(0.121) 1(0.004) 6 0.37(0.133) 1(0) 4 0.804(0.1) 1(0.004)

BICh 119 0.865(0.164) 1(0) 8 0.547(0.189) 0.998(0.024) 8 0.785(0.168) 1(0.001)

Table B.8: Binomial distribution, case 2. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 178 1(0) 0.992(0.024) 4 1(0) 0.05(0.194) 0 1(0) 0.935(0.021)

mAIC^R 0 1(0) 0(0) 188 1(0) 0.94(0.238) 0 1(0) 0.058(0.015)

mAIC^{F R} 177 1(0) 0.993(0.021) 81 0.828(0.167) 0.947(0.204) 60 0.957(0.042) 0.991(0.022)

mAICC^F_N 180 1(0) 0.993(0.022) 4 1(0) 0.05(0.194) 0 1(0) 0.936(0.019)

mAICC^R_N 0 1(0) 0(0) 188 1(0) 0.94(0.238) 0 1(0) 0.058(0.015)

mAICC^{F R}_N 177 1(0) 0.993(0.021) 78 0.822(0.167) 0.952(0.193) 57 0.955(0.042) 0.991(0.022)

mAICC^F_m 194 1(0) 0.998(0.01) 2 1(0) 0.02(0.128) 0 1(0) 0.938(0.008)

mAICC^R_m 0 1(0) 0(0) 190 1(0) 0.95(0.218) 0 1(0) 0.058(0.013)

mAICC^{F R}_m 194 1(0) 0.999(0.007) 61 0.773(0.156) 0.995(0.041) 55 0.943(0.039) 0.998(0.007)

mCAIC^F_N 200 1(0) 1(0) 0 1(0) 0(0) 0 1(0) 0.939(0)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 200 1(0) 1(0) 15 0.692(0.094) 0.998(0.024) 15 0.923(0.024) 1(0.001)

mCAIC^F_m 200 1(0) 1(0) 0 1(0) 0(0) 0 1(0) 0.939(0)

mCAIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mCAIC^{F R}_m 200 1(0) 1(0) 34 0.727(0.128) 0.997(0.033) 34 0.932(0.032) 1(0.002)

BIC^F_N 200 1(0) 1(0) 0 1(0) 0(0) 0 1(0) 0.939(0)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 200 1(0) 1(0) 15 0.692(0.094) 0.998(0.024) 15 0.923(0.024) 1(0.001)

BIC^F_m 199 1(0) 1(0.003) 0 1(0) 0.003(0.047) 0 1(0) 0.939(0)

BIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

BIC^{F R}_m 200 1(0) 1(0) 34 0.727(0.128) 0.997(0.033) 34 0.932(0.032) 1(0.002)

BICh 200 1(0) 1(0) 34 0.728(0.13) 0.995(0.041) 34 0.932(0.032) 1(0.002)

PRDpen,nullmodel2

mAIC^F 95 1(0) 0.77(0.394) 67 0.988(0.061) 0.503(0.455) 18 0.997(0.015) 0.753(0.37)

mAIC^R 0 1(0) 0(0) 143 1(0) 0.715(0.453) 0 1(0) 0.044(0.028)

mAIC^{F R} 94 1(0) 0.785(0.383) 117 0.962(0.107) 0.807(0.348) 41 0.99(0.027) 0.787(0.363) mAICC^F_N 99 1(0) 0.808(0.362) 66 0.988(0.061) 0.505(0.452) 20 0.997(0.015) 0.789(0.34)

mAICC^R_N 0 1(0) 0(0) 144 1(0) 0.72(0.45) 0 1(0) 0.044(0.028)

mAICC^{F R}_N 104 1(0) 0.827(0.35) 116 0.953(0.116) 0.83(0.324) 45 0.988(0.029) 0.827(0.331) mAICC^F_m 147 1(0) 0.98(0.037) 56 0.98(0.079) 0.517(0.438) 30 0.995(0.02) 0.952(0.041)

mAICC^R_m 0 1(0) 0(0) 150 1(0) 0.75(0.434) 0 1(0) 0.046(0.027)

mAICC^{F R}_m 157 1(0) 0.985(0.032) 98 0.913(0.147) 0.887(0.235) 67 0.978(0.037) 0.979(0.033) mCAIC^F_N 190 1(0) 0.997(0.014) 50 0.972(0.093) 0.522(0.434) 44 0.993(0.023) 0.968(0.027)

mCAIC^R_N 0 1(0) 0(0) 194 1(0) 0.97(0.171) 0 1(0) 0.059(0.01)

mCAIC^{F R}_N 188 1(0) 0.997(0.012) 77 0.825(0.167) 0.97(0.096) 67 0.956(0.042) 0.995(0.013) mCAIC^F_m 176 1(0) 0.992(0.024) 49 0.972(0.093) 0.517(0.434) 37 0.993(0.023) 0.963(0.033)

mCAIC^R_m 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

mCAIC^{F R}_m 179 1(0) 0.994(0.019) 85 0.867(0.164) 0.935(0.156) 69 0.967(0.041) 0.991(0.02) BIC^F_N 190 1(0) 0.997(0.014) 50 0.972(0.093) 0.522(0.434) 44 0.993(0.023) 0.968(0.027)

BIC^R_N 0 1(0) 0(0) 194 1(0) 0.97(0.171) 0 1(0) 0.059(0.01)

BIC^{F R}_N 188 1(0) 0.997(0.012) 77 0.825(0.167) 0.97(0.096) 67 0.956(0.042) 0.995(0.013) BIC^F_m 176 1(0) 0.992(0.024) 49 0.972(0.093) 0.517(0.434) 37 0.993(0.023) 0.963(0.033)

BIC^R_m 0 1(0) 0(0) 182 1(0) 0.91(0.287) 0 1(0) 0.056(0.018)

BIC^{F R}_m 179 1(0) 0.994(0.019) 85 0.868(0.163) 0.933(0.157) 69 0.967(0.041) 0.99(0.02) BICh 187 1(0) 0.996(0.017) 72 0.87(0.163) 0.903(0.191) 62 0.968(0.041) 0.99(0.02) mD_rand 1 1(0) 0.005(0.071) 132 0.995(0.071) 0.665(0.473) 0 0.999(0.018) 0.045(0.074) Drand 0 0.993(0.038) 0.03(0.171) 5 0.99(0.057) 0.055(0.229) 0 0.993(0.043) 0.032(0.171)

Prand 0 1(0) 0(0) 6 1(0) 0.03(0.171) 0 1(0) 0.002(0.01)

R²_m 0 1(0) 0(0) 82 1(0) 0.41(0.493) 0 1(0) 0.025(0.03)

R²_c 31 1(0) 0.35(0.458) 56 1(0) 0.28(0.45) 2 1(0) 0.345(0.432)

mrc,a 2 1(0) 0.014(0.118) 109 1(0) 0.545(0.499) 1 1(0) 0.047(0.113)

crc,a 20 0.819(0.087) 0.967(0.171) 0 0.728(0.13) 0.862(0.323) 0 0.796(0.097) 0.96(0.171) mAIC 177 1(0) 0.993(0.021) 81 0.828(0.167) 0.947(0.204) 60 0.957(0.042) 0.991(0.022) mAICCN 177 1(0) 0.993(0.021) 78 0.822(0.167) 0.952(0.193) 57 0.955(0.042) 0.991(0.022) mAICCm 194 1(0) 0.999(0.007) 61 0.773(0.156) 0.995(0.041) 55 0.943(0.039) 0.998(0.007)

mCAICN 200 1(0) 1(0) 15 0.692(0.094) 0.998(0.024) 15 0.923(0.024) 1(0.001)

mCAICm 200 1(0) 1(0) 34 0.727(0.128) 0.997(0.033) 34 0.932(0.032) 1(0.002)

BICN 200 1(0) 1(0) 15 0.692(0.094) 0.998(0.024) 15 0.923(0.024) 1(0.001)

BICm 200 1(0) 1(0) 34 0.727(0.128) 0.997(0.033) 34 0.932(0.032) 1(0.002)

BICh 200 1(0) 1(0) 34 0.728(0.13) 0.995(0.041) 34 0.932(0.032) 1(0.002)

Table B.9: Normal distribution - case 3. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 74 1(0) 0.512(0.49) 5 0.997(0.033) 0.085(0.248) 0 0.999(0.008) 0.486(0.464)

mAIC^R 0 1(0) 0(0) 99 1(0) 0.495(0.501) 0 1(0) 0.03(0.031)

mAIC^{F R} 71 1(0) 0.538(0.488) 68 0.947(0.123) 0.623(0.441) 4 0.987(0.031) 0.543(0.462) mAICC^F_N 86 1(0) 0.618(0.471) 5 0.997(0.033) 0.095(0.258) 1 0.999(0.008) 0.586(0.447)

mAICC^R_N 0 1(0) 0(0) 116 1(0) 0.58(0.495) 0 1(0) 0.036(0.03)

mAICC^{F R}_N 86 1(0) 0.663(0.457) 65 0.933(0.134) 0.665(0.421) 7 0.983(0.033) 0.663(0.433) mAICC^F_m 149 1(0) 0.981(0.036) 8 0.995(0.041) 0.178(0.325) 1 0.999(0.01) 0.932(0.037)

mAICC^R_m 0 1(0) 0(0) 112 1(0) 0.56(0.498) 0 1(0) 0.034(0.03)

mAICC^{F R}_m 158 1(0) 0.989(0.023) 35 0.85(0.166) 0.812(0.302) 11 0.963(0.042) 0.978(0.029) mCAIC^F_N 184 0.994(0.035) 0.997(0.012) 7 0.993(0.047) 0.162(0.317) 1 0.994(0.031) 0.946(0.019)

mCAIC^R_N 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

mCAIC^{F R}_N 172 0.983(0.059) 0.997(0.013) 10 0.742(0.14) 0.938(0.142) 2 0.923(0.059) 0.993(0.016) mCAIC^F_m 174 0.999(0.016) 0.993(0.022) 6 0.995(0.041) 0.155(0.308) 1 0.998(0.015) 0.941(0.024)

mCAIC^R_m 0 1(0) 0(0) 162 1(0) 0.81(0.393) 0 1(0) 0.05(0.024)

mCAIC^{F R}_m 172 0.997(0.027) 0.994(0.017) 21 0.798(0.163) 0.88(0.222) 5 0.947(0.046) 0.987(0.021) BIC^F_N 184 0.994(0.035) 0.997(0.012) 7 0.993(0.047) 0.162(0.317) 1 0.994(0.031) 0.946(0.019)

BIC^R_N 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

BIC^{F R}_N 172 0.983(0.059) 0.997(0.013) 10 0.742(0.14) 0.938(0.142) 2 0.923(0.059) 0.993(0.016) BIC^F_m 174 0.999(0.016) 0.993(0.022) 6 0.995(0.041) 0.155(0.308) 1 0.998(0.015) 0.941(0.024)

BIC^R_m 0 1(0) 0(0) 162 1(0) 0.81(0.393) 0 1(0) 0.05(0.024)

BIC^{F R}_m 172 0.997(0.027) 0.994(0.017) 21 0.798(0.163) 0.88(0.222) 5 0.947(0.046) 0.987(0.021) BICh 182 0.997(0.027) 0.996(0.013) 14 0.805(0.165) 0.845(0.261) 5 0.949(0.046) 0.987(0.02)

PRDpen,nullmodel2

mAIC^F 117 1(0) 0.787(0.39) 5 1(0) 0.092(0.248) 0 1(0) 0.744(0.369)

mAIC^R 0 1(0) 0(0) 139 1(0) 0.695(0.462) 0 1(0) 0.043(0.028)

mAIC^{F R} 116 1(0) 0.799(0.383) 85 0.925(0.14) 0.777(0.363) 28 0.981(0.035) 0.797(0.361)

mAICC^F_N 127 1(0) 0.855(0.326) 5 1(0) 0.095(0.251) 0 1(0) 0.809(0.309)

mAICC^R_N 0 1(0) 0(0) 161 1(0) 0.805(0.397) 0 1(0) 0.049(0.024)

mAICC^{F R}_N 128 1(0) 0.868(0.315) 77 0.915(0.146) 0.79(0.341) 29 0.979(0.036) 0.863(0.297)

mAICC^F_m 169 1(0) 0.99(0.027) 4 1(0) 0.103(0.255) 0 1(0) 0.935(0.025)

mAICC^R_m 0 1(0) 0(0) 156 1(0) 0.78(0.415) 0 1(0) 0.048(0.025)

mAICC^{F R}_m 174 1(0) 0.993(0.019) 49 0.835(0.167) 0.887(0.228) 32 0.959(0.042) 0.987(0.022) mCAIC^F_N 189 0.991(0.044) 0.999(0.005) 8 1(0) 0.1(0.265) 0 0.993(0.033) 0.944(0.016)

mCAIC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mCAIC^{F R}_N 184 0.987(0.053) 0.999(0.006) 14 0.713(0.116) 0.977(0.085) 7 0.918(0.047) 0.998(0.008) mCAIC^F_m 186 0.999(0.016) 0.997(0.013) 3 1(0) 0.095(0.244) 0 0.999(0.012) 0.942(0.014)

mCAIC^R_m 0 1(0) 0(0) 184 1(0) 0.92(0.272) 0 1(0) 0.056(0.017)

mCAIC^{F R}_m 183 0.998(0.022) 0.996(0.013) 33 0.775(0.157) 0.94(0.152) 22 0.942(0.042) 0.993(0.016) BIC^F_N 189 0.991(0.044) 0.999(0.005) 8 1(0) 0.1(0.265) 0 0.993(0.033) 0.944(0.016)

BIC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

BIC^{F R}_N 184 0.987(0.053) 0.999(0.006) 14 0.713(0.116) 0.977(0.085) 7 0.918(0.047) 0.998(0.008) BIC^F_m 186 0.999(0.016) 0.997(0.013) 3 1(0) 0.095(0.244) 0 0.999(0.012) 0.942(0.014)

BIC^R_m 0 1(0) 0(0) 183 1(0) 0.915(0.28) 0 1(0) 0.056(0.017)

BIC^{F R}_m 183 0.998(0.022) 0.996(0.013) 33 0.775(0.157) 0.94(0.152) 22 0.942(0.042) 0.993(0.016) BIC_h 193 0.998(0.022) 0.999(0.007) 26 0.795(0.163) 0.895(0.21) 22 0.947(0.044) 0.993(0.014)

mDrand 0 1(0) 0(0) 137 1(0) 0.685(0.466) 0 1(0) 0.042(0.029)

D_rand 0 1(0) 0.004(0.063) 6 1(0) 0.03(0.171) 0 1(0) 0.006(0.06)

Prand 0 1(0) 0(0) 4 1(0) 0.02(0.14) 0 1(0) 0.001(0.009)

R²_m 0 1(0) 0(0) 102 1(0) 0.51(0.501) 0 1(0) 0.031(0.031)

R²_c 5 1(0) 0.074(0.253) 34 1(0) 0.173(0.378) 0 1(0) 0.08(0.24)

mrc,a 1 1(0) 0.005(0.071) 120 0.997(0.047) 0.605(0.49) 0 0.999(0.012) 0.042(0.074) crc,a 35 0.883(0.111) 0.881(0.304) 0 0.825(0.167) 0.675(0.413) 0 0.869(0.125) 0.868(0.302) mAIC 173 1(0) 0.993(0.021) 100 0.853(0.166) 0.963(0.166) 75 0.963(0.041) 0.991(0.022) mAICCN 176 1(0) 0.993(0.02) 97 0.848(0.166) 0.963(0.166) 75 0.962(0.042) 0.992(0.021) mAICCm 190 1(0) 0.998(0.012) 70 0.792(0.162) 0.988(0.084) 60 0.948(0.04) 0.997(0.012)

mCAICN 182 0.98(0.064) 1(0) 21 0.702(0.102) 1(0) 15 0.91(0.049) 1(0)

mCAICm 190 0.994(0.035) 0.999(0.007) 44 0.743(0.141) 0.997(0.033) 36 0.932(0.041) 0.999(0.007) cAIC 144 0.983(0.059) 0.977(0.079) 25 0.923(0.141) 0.435(0.469) 6 0.968(0.058) 0.944(0.083)

BICN 182 0.98(0.064) 1(0) 21 0.702(0.102) 1(0) 15 0.91(0.049) 1(0)

BICm 190 0.994(0.035) 0.999(0.007) 45 0.745(0.142) 0.997(0.033) 37 0.932(0.041) 0.999(0.007) BICh 195 0.994(0.035) 1(0) 40 0.743(0.141) 0.987(0.087) 37 0.932(0.041) 0.999(0.005)

Table B.10: Poisson distribution, case 3. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 177 1(0) 0.991(0.026) 7 1(0) 0.055(0.213) 0 1(0) 0.934(0.024)

mAIC^R 0 1(0) 0(0) 192 1(0) 0.96(0.196) 0 1(0) 0.059(0.012)

mAIC^{F R} 175 1(0) 0.993(0.021) 83 0.817(0.166) 0.978(0.13) 59 0.954(0.042) 0.992(0.021)

mAICC^F_N 179 1(0) 0.993(0.024) 7 1(0) 0.055(0.213) 0 1(0) 0.935(0.022)

mAICC^R_N 0 1(0) 0(0) 192 1(0) 0.96(0.196) 0 1(0) 0.059(0.012)

mAICC^{F R}_N 176 1(0) 0.993(0.02) 82 0.813(0.166) 0.983(0.11) 59 0.953(0.041) 0.993(0.019)

mAICC^F_m 192 1(0) 0.997(0.016) 4 1(0) 0.023(0.148) 0 1(0) 0.937(0.015)

mAICC^R_m 0 1(0) 0(0) 196 1(0) 0.98(0.14) 0 1(0) 0.06(0.009)

mAICC^{F R}_m 194 1(0) 0.998(0.01) 66 0.778(0.158) 0.998(0.024) 60 0.945(0.039) 0.998(0.01)

mCAIC^F_N 182 0.98(0.064) 1(0) 18 1(0) 0.093(0.29) 0 0.985(0.048) 0.944(0.018)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 172 0.969(0.077) 1(0) 12 0.682(0.09) 1(0) 9 0.897(0.06) 1(0)

mCAIC^F_m 195 0.997(0.027) 1(0.004) 4 1(0) 0.027(0.154) 0 0.998(0.02) 0.94(0.008)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 189 0.991(0.044) 0.999(0.005) 35 0.725(0.127) 1(0) 28 0.925(0.041) 0.999(0.005)

BIC^F_N 182 0.98(0.064) 1(0) 18 1(0) 0.093(0.29) 0 0.985(0.048) 0.944(0.018)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 172 0.969(0.077) 1(0) 12 0.682(0.09) 1(0) 9 0.897(0.06) 1(0)

BIC^F_m 196 0.998(0.022) 1(0.004) 3 1(0) 0.022(0.138) 0 0.998(0.017) 0.94(0.007)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 189 0.991(0.044) 0.999(0.005) 35 0.725(0.127) 1(0) 28 0.925(0.041) 0.999(0.005) BICh 193 0.992(0.041) 1(0) 33 0.725(0.127) 0.997(0.033) 29 0.925(0.04) 1(0.002)

PRDpen,nullmodel2

mAIC^F 140 1(0) 0.877(0.304) 37 0.995(0.041) 0.257(0.416) 13 0.999(0.01) 0.839(0.279)

mAIC^R 2 1(0) 0.014(0.118) 169 1(0) 0.845(0.363) 1 1(0) 0.065(0.109)

mAIC^{F R} 138 1(0) 0.877(0.304) 111 0.897(0.155) 0.895(0.291) 63 0.974(0.039) 0.878(0.291) mAICC^F_N 146 1(0) 0.898(0.277) 34 0.995(0.041) 0.248(0.408) 13 0.999(0.01) 0.859(0.256)

mAICC^R_N 2 1(0) 0.014(0.118) 169 1(0) 0.845(0.363) 1 1(0) 0.065(0.109)

mAICC^{F R}_N 144 1(0) 0.894(0.284) 107 0.888(0.158) 0.9(0.284) 64 0.972(0.039) 0.895(0.274) mAICC^F_m 174 1(0) 0.985(0.076) 35 0.995(0.041) 0.253(0.411) 20 0.999(0.01) 0.94(0.071)

mAICC^R_m 2 1(0) 0.014(0.118) 170 1(0) 0.85(0.358) 1 1(0) 0.066(0.109)

mAICC^{F R}_m 176 1(0) 0.986(0.075) 96 0.855(0.166) 0.945(0.205) 75 0.964(0.041) 0.984(0.072) mCAIC^F_N 179 0.989(0.049) 0.992(0.072) 37 0.992(0.052) 0.287(0.424) 22 0.99(0.04) 0.948(0.069)

mCAIC^R_N 2 1(0) 0.014(0.118) 189 1(0) 0.945(0.229) 1 1(0) 0.071(0.107)

mCAIC^{F R}_N 173 0.98(0.064) 0.992(0.073) 45 0.753(0.161) 0.962(0.177) 35 0.923(0.064) 0.99(0.07) mCAIC^F_m 181 0.999(0.016) 0.975(0.141) 33 0.993(0.047) 0.258(0.41) 20 0.998(0.017) 0.931(0.13)

mCAIC^R_m 2 1(0) 0.014(0.118) 183 1(0) 0.915(0.28) 1 1(0) 0.07(0.107)

mCAIC^{F R}_m 180 0.996(0.031) 0.98(0.123) 81 0.822(0.167) 0.957(0.19) 67 0.952(0.05) 0.979(0.116) BIC^F_N 179 0.989(0.049) 0.992(0.072) 37 0.992(0.052) 0.287(0.424) 22 0.99(0.04) 0.948(0.069)

BIC^R_N 2 1(0) 0.014(0.118) 189 1(0) 0.945(0.229) 1 1(0) 0.071(0.107)

BIC^{F R}_N 173 0.98(0.064) 0.992(0.073) 45 0.755(0.158) 0.962(0.177) 35 0.924(0.064) 0.99(0.07) BIC^F_m 181 0.999(0.016) 0.975(0.141) 33 0.993(0.047) 0.258(0.41) 20 0.998(0.017) 0.931(0.13)

BIC^R_m 2 1(0) 0.014(0.118) 183 1(0) 0.915(0.28) 1 1(0) 0.07(0.107)

BIC^{F R}_m 180 0.996(0.031) 0.98(0.123) 81 0.822(0.167) 0.957(0.19) 67 0.952(0.05) 0.979(0.116) BICh 185 0.997(0.027) 0.986(0.101) 69 0.823(0.167) 0.932(0.212) 60 0.953(0.048) 0.983(0.096)

mD_rand 94 1(0) 0.47(0.5) 64 0.53(0.5) 0.79(0.408) 0 0.883(0.125) 0.49(0.482)

Drand 1 0.987(0.053) 0.07(0.255) 8 0.978(0.082) 0.108(0.31) 0 0.985(0.06) 0.072(0.254)

Prand 0 1(0) 0(0) 5 1(0) 0.025(0.157) 0 1(0) 0.002(0.01)

R²_m 0 1(0) 0(0) 100 1(0) 0.5(0.501) 0 1(0) 0.031(0.031)

R²_c 12 0.986(0.056) 0.295(0.438) 66 0.993(0.066) 0.34(0.475) 3 0.988(0.045) 0.298(0.417) mrc,a 153 1(0) 0.769(0.422) 17 0.235(0.425) 0.85(0.358) 0 0.809(0.106) 0.774(0.412) crc,a 16 0.993(0.038) 0.184(0.381) 25 0.99(0.057) 0.198(0.381) 0 0.993(0.041) 0.185(0.367) mAIC 175 1(0) 0.993(0.021) 83 0.817(0.166) 0.978(0.13) 59 0.954(0.042) 0.992(0.021) mAICCN 176 1(0) 0.993(0.02) 82 0.813(0.166) 0.983(0.11) 59 0.953(0.041) 0.993(0.019) mAICCm 194 1(0) 0.998(0.01) 66 0.778(0.158) 0.998(0.024) 60 0.945(0.039) 0.998(0.01)

mCAICN 172 0.969(0.077) 1(0) 12 0.682(0.09) 1(0) 9 0.897(0.06) 1(0)

mCAICm 189 0.991(0.044) 0.999(0.005) 35 0.725(0.127) 1(0) 28 0.925(0.041) 0.999(0.005)

BICN 172 0.969(0.077) 1(0) 12 0.682(0.09) 1(0) 9 0.897(0.06) 1(0)

BICm 189 0.991(0.044) 0.999(0.005) 35 0.725(0.127) 1(0) 28 0.925(0.041) 0.999(0.005) BICh 193 0.992(0.041) 1(0) 33 0.725(0.127) 0.997(0.033) 29 0.925(0.04) 1(0.002)

Table B.11: Bernoulli distribution, case 3. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 172 0.995(0.041) 0.992(0.024) 9 1(0) 0.082(0.251) 0 0.996(0.03) 0.936(0.022)

mAIC^R 0 1(0) 0(0) 191 1(0) 0.955(0.208) 0 1(0) 0.058(0.013)

mAIC^{F R} 175 0.99(0.057) 0.995(0.016) 34 0.5(0.261) 0.993(0.074) 13 0.868(0.077) 0.995(0.016) mAICC^F_N 174 0.995(0.041) 0.992(0.024) 9 1(0) 0.075(0.244) 0 0.996(0.03) 0.936(0.022)

mAICC^R_N 0 1(0) 0(0) 191 1(0) 0.955(0.208) 0 1(0) 0.058(0.013)

mAICC^{F R}_N 177 0.99(0.057) 0.996(0.016) 33 0.497(0.259) 0.993(0.074) 14 0.867(0.076) 0.996(0.015) mAICC^F_m 181 0.992(0.052) 0.997(0.011) 12 1(0) 0.087(0.266) 1 0.994(0.039) 0.941(0.011)

mAICC^R_m 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mAICC^{F R}_m 178 0.977(0.084) 0.998(0.009) 19 0.432(0.211) 1(0) 9 0.841(0.083) 0.998(0.008) mCAIC^F_N 91 0.831(0.16) 1(0.003) 110 1(0) 0.55(0.499) 1 0.873(0.12) 0.972(0.03)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 89 0.818(0.164) 1(0) 1 0.348(0.077) 1(0) 0 0.701(0.125) 1(0)

mCAIC^F_m 158 0.936(0.129) 1(0.003) 43 1(0) 0.215(0.412) 1 0.952(0.096) 0.952(0.025)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 156 0.929(0.136) 1(0.003) 3 0.365(0.114) 1(0) 1 0.788(0.108) 1(0.003)

BIC^F_N 91 0.831(0.16) 1(0.003) 110 1(0) 0.55(0.499) 1 0.873(0.12) 0.972(0.03)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 89 0.818(0.164) 1(0) 1 0.348(0.077) 1(0) 0 0.701(0.125) 1(0)

BIC^F_m 158 0.936(0.129) 1(0.003) 43 1(0) 0.215(0.412) 1 0.952(0.096) 0.952(0.025)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 156 0.929(0.136) 1(0.003) 3 0.365(0.114) 1(0) 1 0.788(0.108) 1(0.003)

BICh 105 0.848(0.163) 1(0.003) 14 0.555(0.201) 1(0) 7 0.775(0.164) 1(0.003)

PRDpen,nullmodel2

mAIC^F 164 0.995(0.041) 0.987(0.033) 21 0.995(0.053) 0.205(0.366) 11 0.995(0.033) 0.939(0.038)

mAIC^R 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mAIC^{F R} 171 0.99(0.057) 0.992(0.024) 50 0.548(0.289) 0.993(0.074) 25 0.88(0.084) 0.992(0.023) mAICC^F_N 167 0.995(0.041) 0.989(0.03) 22 0.995(0.053) 0.21(0.37) 11 0.995(0.033) 0.941(0.035)

mAICC^R_N 0 1(0) 0(0) 195 1(0) 0.975(0.157) 0 1(0) 0.06(0.01)

mAICC^{F R}_N 175 0.99(0.057) 0.994(0.021) 45 0.537(0.281) 0.993(0.074) 24 0.877(0.082) 0.994(0.02) mAICC^F_m 181 0.992(0.052) 0.997(0.011) 25 0.992(0.07) 0.237(0.386) 13 0.992(0.043) 0.95(0.021)

mAICC^R_m 0 1(0) 0(0) 196 1(0) 0.98(0.14) 0 1(0) 0.06(0.009)

mAICC^{F R}_m 177 0.976(0.085) 0.998(0.009) 31 0.475(0.251) 0.998(0.024) 20 0.851(0.089) 0.998(0.008) mCAIC^F_N 92 0.832(0.16) 1(0.003) 94 0.95(0.17) 0.602(0.475) 4 0.861(0.138) 0.975(0.029)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 86 0.813(0.164) 1(0) 2 0.352(0.09) 1(0) 0 0.698(0.124) 1(0)

mCAIC^F_m 159 0.937(0.127) 1(0.003) 46 0.982(0.101) 0.342(0.448) 10 0.948(0.102) 0.959(0.027)

mCAIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_m 155 0.928(0.136) 1(0.003) 7 0.378(0.144) 1(0) 5 0.791(0.111) 1(0.003) BIC^F_N 92 0.832(0.16) 1(0.003) 94 0.95(0.17) 0.602(0.475) 4 0.861(0.138) 0.975(0.029)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 86 0.813(0.164) 1(0) 2 0.352(0.09) 1(0) 0 0.698(0.124) 1(0)

BIC^F_m 160 0.939(0.126) 1(0.003) 46 0.985(0.09) 0.337(0.446) 10 0.95(0.099) 0.959(0.027)

BIC^R_m 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_m 155 0.928(0.136) 1(0.003) 7 0.378(0.144) 1(0) 5 0.791(0.111) 1(0.003)

BICh 99 0.838(0.164) 1(0.003) 19 0.557(0.214) 1(0) 11 0.768(0.166) 1(0.003)

mD_rand 3 1(0) 0.015(0.122) 164 0.985(0.122) 0.835(0.372) 0 0.996(0.03) 0.065(0.118)

Drand 0 1(0) 0.005(0.068) 12 1(0) 0.063(0.242) 0 1(0) 0.008(0.068)

Prand 0 1(0) 0(0) 9 1(0) 0.045(0.208) 0 1(0) 0.003(0.013)

R²_m 0 1(0) 0(0) 72 1(0) 0.36(0.481) 0 1(0) 0.022(0.029)

R²_c 0 1(0) 0.004(0.063) 21 1(0) 0.105(0.307) 0 1(0) 0.011(0.062)

mrc,a 0 1(0) 0(0) 71 1(0) 0.355(0.48) 0 1(0) 0.022(0.029)

crc,a 20 0.98(0.064) 0.281(0.442) 13 0.975(0.088) 0.16(0.358) 0 0.979(0.068) 0.274(0.424) mAIC 175 0.99(0.057) 0.995(0.016) 34 0.5(0.261) 0.993(0.074) 13 0.868(0.077) 0.995(0.016) mAICCN 177 0.99(0.057) 0.996(0.016) 33 0.497(0.259) 0.993(0.074) 14 0.867(0.076) 0.996(0.015) mAICCm 178 0.977(0.084) 0.998(0.009) 19 0.432(0.211) 1(0) 9 0.841(0.083) 0.998(0.008)

mCAICN 89 0.818(0.164) 1(0) 1 0.348(0.077) 1(0) 0 0.701(0.125) 1(0)

mCAICm 156 0.929(0.136) 1(0.003) 3 0.365(0.114) 1(0) 1 0.788(0.108) 1(0.003)

BICN 89 0.818(0.164) 1(0) 1 0.348(0.077) 1(0) 0 0.701(0.125) 1(0)

BICm 156 0.929(0.136) 1(0.003) 3 0.365(0.114) 1(0) 1 0.788(0.108) 1(0.003)

BICh 105 0.848(0.163) 1(0.003) 14 0.555(0.201) 1(0) 7 0.775(0.164) 1(0.003)

Table B.12: Binomial distribution, case 3. The exponents ^F, ^R and ^{F R} indicate which value for k is considered: dim(β), dim((τ⁰, φ)⁰) and dim(θ), respectively. The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities.

Fixed part Random part Model

Measure freqF sensF specF freqR sensR specR freq sens spec

PRDpen,nullmodel1

mAIC^F 176 1(0) 0.992(0.024) 1 1(0) 0.055(0.188) 0 1(0) 0.935(0.02)

mAIC^R 0 1(0) 0(0) 192 1(0) 0.96(0.196) 0 1(0) 0.059(0.012)

mAIC^{F R} 175 1(0) 0.993(0.02) 87 0.837(0.167) 0.955(0.182) 65 0.959(0.042) 0.991(0.021)

mAICC^F_N 179 1(0) 0.993(0.021) 1 1(0) 0.052(0.183) 0 1(0) 0.936(0.018)

mAICC^R_N 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mAICC^{F R}_N 176 1(0) 0.994(0.018) 86 0.833(0.167) 0.957(0.181) 64 0.958(0.042) 0.992(0.02)

mAICC^F_m 190 1(0) 0.998(0.012) 1 1(0) 0.032(0.148) 0 1(0) 0.938(0.007)

mAICC^R_m 0 1(0) 0(0) 193 1(0) 0.965(0.184) 0 1(0) 0.059(0.011)

mAICC^{F R}_m 193 1(0) 0.998(0.008) 66 0.788(0.161) 0.982(0.112) 59 0.947(0.04) 0.997(0.01) mCAIC^F_N 185 0.983(0.059) 1(0) 15 1(0) 0.075(0.264) 0 0.988(0.044) 0.943(0.016)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 181 0.979(0.065) 1(0) 16 0.693(0.091) 1(0) 11 0.908(0.049) 1(0)

mCAIC^F_m 195 0.997(0.027) 1(0.004) 3 1(0) 0.022(0.138) 0 0.998(0.02) 0.94(0.007)

mCAIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mCAIC^{F R}_m 192 0.994(0.035) 0.999(0.005) 37 0.733(0.134) 0.995(0.041) 30 0.929(0.037) 0.999(0.006)

BIC^F_N 185 0.983(0.059) 1(0) 15 1(0) 0.075(0.264) 0 0.988(0.044) 0.943(0.016)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 181 0.979(0.065) 1(0) 16 0.693(0.091) 1(0) 11 0.908(0.049) 1(0)

BIC^F_m 195 0.997(0.027) 1(0.004) 3 1(0) 0.022(0.138) 0 0.998(0.02) 0.94(0.007)

BIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

BIC^{F R}_m 192 0.994(0.035) 0.999(0.005) 37 0.733(0.134) 0.995(0.041) 30 0.929(0.037) 0.999(0.006) BICh 196 0.996(0.031) 1(0) 32 0.732(0.132) 0.985(0.09) 29 0.93(0.037) 0.999(0.006)

PRDpen,nullmodel2

mAIC^F 167 1(0) 0.988(0.031) 28 0.995(0.041) 0.248(0.395) 17 0.999(0.01) 0.942(0.035)

mAIC^R 0 1(0) 0(0) 187 1(0) 0.935(0.247) 0 1(0) 0.057(0.015)

mAIC^{F R} 164 1(0) 0.989(0.028) 99 0.862(0.165) 0.947(0.196) 70 0.965(0.041) 0.986(0.029) mAICC^F_N 170 1(0) 0.989(0.03) 27 0.995(0.041) 0.24(0.391) 17 0.999(0.01) 0.943(0.034)

mAICC^R_N 0 1(0) 0(0) 187 1(0) 0.935(0.247) 0 1(0) 0.057(0.015)

mAICC^{F R}_N 167 1(0) 0.99(0.026) 99 0.86(0.165) 0.952(0.184) 71 0.965(0.041) 0.988(0.026) mAICC^F_m 187 1(0) 0.997(0.015) 25 0.995(0.041) 0.22(0.382) 19 0.999(0.01) 0.949(0.023)

mAICC^R_m 0 1(0) 0(0) 192 1(0) 0.96(0.196) 0 1(0) 0.059(0.012)

mAICC^{F R}_m 190 1(0) 0.998(0.009) 82 0.818(0.166) 0.978(0.116) 75 0.955(0.042) 0.997(0.012) mCAIC^F_N 186 0.986(0.055) 1(0.003) 35 0.995(0.041) 0.25(0.41) 22 0.988(0.042) 0.954(0.025)

mCAIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

mCAIC^{F R}_N 182 0.98(0.064) 1(0) 21 0.702(0.102) 1(0) 17 0.91(0.051) 1(0)

mCAIC^F_m 196 0.998(0.022) 1(0.004) 25 0.995(0.041) 0.203(0.376) 22 0.997(0.019) 0.951(0.023)

mCAIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

mCAIC^{F R}_m 193 0.996(0.031) 0.999(0.005) 54 0.758(0.149) 0.998(0.024) 48 0.936(0.041) 0.999(0.005) BIC^F_N 186 0.986(0.055) 1(0.003) 35 0.995(0.041) 0.25(0.41) 22 0.988(0.042) 0.954(0.025)

BIC^R_N 0 1(0) 0(0) 200 1(0) 1(0) 0 1(0) 0.061(0)

BIC^{F R}_N 182 0.98(0.064) 1(0) 21 0.702(0.102) 1(0) 17 0.91(0.051) 1(0)

BIC^F_m 196 0.998(0.022) 1(0.004) 25 0.995(0.041) 0.203(0.376) 22 0.997(0.019) 0.951(0.023)

BIC^R_m 0 1(0) 0(0) 199 1(0) 0.995(0.071) 0 1(0) 0.061(0.004)

BIC^{F R}_m 193 0.996(0.031) 0.999(0.005) 54 0.758(0.149) 0.998(0.024) 48 0.936(0.041) 0.999(0.005) BICh 196 0.997(0.027) 1(0.003) 48 0.76(0.15) 0.98(0.114) 46 0.938(0.041) 0.999(0.008)

mD_rand 2 1(0) 0.01(0.1) 137 0.99(0.1) 0.695(0.462) 0 0.998(0.025) 0.052(0.1)

Drand 0 1(0) 0(0) 8 1(0) 0.04(0.196) 0 1(0) 0.002(0.012)

Prand 0 1(0) 0(0) 5 1(0) 0.025(0.157) 0 1(0) 0.002(0.01)

R²_m 0 1(0) 0(0) 79 1(0) 0.395(0.49) 0 1(0) 0.024(0.03)

R²_c 33 1(0) 0.396(0.467) 55 1(0) 0.275(0.448) 5 1(0) 0.388(0.442)

mrc,a 0 1(0) 0.005(0.068) 110 1(0) 0.553(0.497) 0 1(0) 0.038(0.071)

crc,a 34 0.87(0.11) 0.916(0.262) 2 0.808(0.165) 0.715(0.402) 0 0.855(0.123) 0.903(0.26) mAIC 175 1(0) 0.993(0.02) 87 0.837(0.167) 0.955(0.182) 65 0.959(0.042) 0.991(0.021) mAICCN 176 1(0) 0.994(0.018) 86 0.833(0.167) 0.957(0.181) 64 0.958(0.042) 0.992(0.02) mAICCm 193 1(0) 0.998(0.008) 66 0.788(0.161) 0.982(0.112) 59 0.947(0.04) 0.997(0.01)

mCAICN 181 0.979(0.065) 1(0) 16 0.693(0.091) 1(0) 11 0.908(0.049) 1(0)

mCAICm 192 0.994(0.035) 0.999(0.005) 37 0.733(0.134) 0.995(0.041) 30 0.929(0.037) 0.999(0.006)

BICN 181 0.979(0.065) 1(0) 16 0.693(0.091) 1(0) 11 0.908(0.049) 1(0)

BICm 192 0.994(0.035) 0.999(0.005) 37 0.733(0.134) 0.995(0.041) 30 0.929(0.037) 0.999(0.006) BICh 196 0.996(0.031) 1(0) 32 0.732(0.132) 0.985(0.09) 29 0.93(0.037) 0.999(0.006)

List of acronyms

A overall model Adequacy

A&S overall model Adequacy and model Selection AIC Akaike Information Criterion

ALASSO Adaptive Least Absolute Shrinkage ANOVA ANalysis Of VAriance

BIC Baysesian Information Criterion

BIC_h Baysesian Information Criterion withλ_n^?k= log(m)k_R+ log(N)k_F BICm Baysesian Information Criterion withλn^? = log(m)

BIC_N Baysesian Information Criterion withλ_n^? = log(N) BLUP Best Linear Unbiased Predictor

cAIC conditional Akaike Information Criterion df degrees of freedom

DIC Deviance Information Criterion F model adequacy due to Fixed effects

F&S model adequacy due to Fixed effects and model Selection freq frequency of choosing the correct model

freq_F frequency of choosing the correct Fixed part freq_R frequency of choosing the correct Random part GEE Generalized Estimating Equations

GH Gaussian-Hermite

GIC Generalized Information Criterion GLM Generalized Linear Model

GLMM Generalized Linear Mixed-effects Model GOF Goodness-Of-Fit

IC Information Criterion LASSO Least Absolute Shrinkage LL marginal Log-Likelihood LMM Linear Mixed-effects Model LOESS LOcal regrESSion

LRT Likelihood Ratio Test

mAIC marginal Akaike Information Criterion

or highly parameterized models

mAICC_m marginal Akaike Information Criterion Corrected for small samples or highly parameterized models withλ_n^? = 2(m/(m−k−1)) mAICC_N marginal Akaike Information Criterion Corrected for small samples

or highly parameterized models withλ_n^? = 2(N/(N −k−1))

MC Monte Carlo

mCAIC marginal Consistent Akaike Information Criterion

mCAIC_m marginal Consistent Akaike Information Criterion with λ_n^? = log(m+ 1)

mCAIC_N marginal Consistent Akaike Information Criterion with λ_n^? = log(N + 1)

MCMC Markov Chain Monte Carlo

ML Maximum Likelihood

NELS National Education Longitudinal Study NLMM NonLinear Mixed-effects Model

OLS Ordinary Least Squares PQL Penalized Quasi-Likelihood

PRD Proportional Reduction in Deviance

PRD(n₀) Proportional Reduction in Deviance obtained by considering the null modeln₀ (eithern₀ = 1 or n₀ = 2)

PRD_pen penalized Proportional Reduction in Deviance

PRD^IC_pen(n₀) penalized Proportional Reduction in Deviance corresponding to the Information Criterion and obtained by considering the null model n₀ (either n₀ = 1 or n₀ = 2)

REML REstricted Maximum Likelihood

S model Selection

SCAD Smoothly Clipped Absolute Deviation SEM Structural Equation Model

sens sensitivity of the entire model sens_F sensitivity of the Fixed part sens_R sensitivity of the Random part spec specificity of the entire model spec_F specificity of the Fixed part spec_R specificity of the Random part

SS Sums of Squares

SSVS Stochastic Search Variable Selection UCLA University of California, Los Angeles

WAIC Widely Applicable or Watanabe-Akaike Information Criterion

1.1 Characteristics of the considered measures. A=overall model adequacy;

F=model adequacy due to fixed effects; S=model selection; Abs=absolute;

Rel=relative; EV=explained variation; C=concordance coefficient between observed and predicted values; IC=information criterion. m.=marginal;

c.=conditional. . . 9 1.2 Measures for models of home radon levels data. For the measures computed

with several null models, (0) indicates that the considered null model is model 0 and (2) indicates that the considered null model is model 2. The dashed line separates null models. m.=marginal; c.=conditional. . . 21 1.3 Simulation cases. The original values of the parameters are those of case

1, in bold. . . 24 1.4 Pearson correlations matrix for measures in categories A and A&S

com-puted after estimation of model 6 in simulation case 1 (original values).

For the measures computed with several null models, (0) indicates that the considered null model is model 0 and (2) indicates that the considered null model is model 2. c.=conditional. . . 25 1.5 Pearson correlations matrix for the measures in category F&S computed

af-ter estimation of model 6 in simulation case 1 (original values). m.=marginal. 26 1.6 Pearson correlations matrix for the measures in category S computed after

estimation of model 6 in simulation case 1 (original values). . . 26 1.7 Results of the analyses of variance: the factors are in the columns and

the dependent variables are in the lines. The cells contain the partial η² effect size measure, partial η² smaller than 0.001 are indicated by the × symbol and non significant results (5% threshold) are in parentheses. For the measures computed with several null models, (0) indicates that the considered null model is model 0 and (2) indicates that the considered null model is model 2. The dashed line separates null models. m.=marginal;

c.=conditional. . . 27 1.8 Number of times that the measure chooses model o (o = 1,2,3,4,5,6) in

simulation cases 1 and 2. m.=marginal; c.=conditional. . . 30 2.1 List of the considered penalties for PRD_pen. . . 44 2.2 Description of the population (pop.) and the 15 alternative models. . . 54 2.3 Parameter values of the population model in the 4×3 different setups.

Three parameters are required for xf3.1 because this factor has 4 levels.

The parameter σ² concerns only the Normal distribution. . . 55 2.4 Number of parameters (p= dim(β), q = dim(τ)), theoretical sensitivities

and specificities of the population (pop.) model and of the 15 alternatives. 60

(2) corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 64 2.6 Poisson distribution, case 1. (1) corresponds to the null model 1 and (2)

corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 65 2.7 Bernoulli distribution, case 1. (1) corresponds to the null model 1 and

(2) corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 66 2.8 Binomial distribution, case 1. (1) corresponds to the null model 1 and

(2) corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 66 2.9 Normal distribution - case 2. (1) corresponds to the null model 1 and

(2) corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 68 2.10 Poisson distribution, case 2. (1) corresponds to the null model 1 and (2)

corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 69 2.11 Bernoulli distribution, case 2. (1) corresponds to the null model 1 and

(2) corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 69 2.12 Binomial distribution, case 2. (1) corresponds to the null model 1 and

(2) corresponds to the null model 2. The different versions of PRD_pen are computed with k = dim(θ). The columns sens_F, spec_F, sens_R, spec_R, sens and spec, contain the mean, and the standard deviation in parentheses, of the sensitivities and specificities. . . 69 2.12 Binomial distribution, case 2. (1) corresponds to the null model 1 and