5.1 Loss functions, posterior estimates and posterior risks
For the Bayesian analysis of the model under study, we used different loss functions. Loss function shows the difference between the parameter and the estimate. It is used for the estimation of the parameters. For the accuracy of the method used to analyze the data set, loss function plays an important role. Under the loss function the Bayesian estimator minimizes the expected loss based on the posterior distribution. The expected value calculated is considered as Bayes estimates (BEs) while the expected loss is represented by the posterior risk (PR). Smaller value of the posterior risk is the indication of higher reliability. In this section we present the derivation of BEs and PRs under various loss functions using the Uniform and Jeffreys priors. We use four loss functions, namely the Squared Error Loss Function (SELF), Quadratic Loss Function (QLF), DeGroot Loss Function (DLF) and Precautionary Loss Function (PLF), (Sindhu et al 2019). The Bayes estimators along with their posterior risks are used (Ullah et al 2020).
The BEs and PRs using various loss functions under Uniform and Jeffreys priors are evaluated and are given in Tables 2 and 3. Results show that PRs for the parameters of \(\phi\) under SELF are smaller as compared to those for the remaining loss functions, i.e., DLF, PLF and QLF. The results clearly show that SELF is the most suitable loss function for the estimation of model’s parameters \(\phi\).
Table 2
BEs and PRs (in parentheses) under different loss functions using uniform prior
Parameters
|
Loss Functions
|
SELF
|
PLF
|
QLF
|
DLF
|
\({\gamma _1}\)
|
0.2028
(0.00082)
|
0.2031
(0.00902)
|
0.2027
(0.00181)
|
0.2032
(0.00200)
|
\({\gamma _2}\)
|
0.2176
(0.00015)
|
0.2177
(0.00687)
|
0.2169
(0.00335)
|
0.2183
(0.00314)
|
\({\gamma _3}\)
|
0.1872
(0.00016)
|
0.1875
(0.00843)
|
0.1872
(0.00158)
|
0.1880
(0.00450)
|
\({\gamma _4}\)
|
0.1936
(0.00013)
|
0.1933
(0.00656)
|
0.1935
(0.00172)
|
0.1942
(0.00339)
|
\({\gamma _5}\)
|
0.1988
(0.00029)
|
0.1984
(0.00343)
|
0.1997
(0.00195)
|
0.1963
(0.00175)
|
\(\varepsilon\)
|
0.1810
(0.0015)
|
0.1810
(0.00804)
|
0.1809
(0.04522)
|
0.1813
(0.04205)
|
Table 3
BEs & PRs (in parentheses) under different loss functions using the Jeffreys prior
Parameters
|
Loss Functions
|
SELF
|
PLF
|
QLF
|
DLF
|
\({\gamma _1}\)
|
0.2026
(0.00076)
|
0.2028
(0.00388)
|
0.2024
(0.00172)
|
0.2030
(0.00191)
|
\({\gamma _2}\)
|
0.2171
(0.00047)
|
0.2174
(0.00697)
|
0.2170
(0.00340)
|
0.2177
(0.00320)
|
\({\gamma _3}\)
|
0.1875
(0.00035)
|
0.1879
(0.00845)
|
0.1878
(0.00448)
|
0.1883
(0.00449)
|
\({\gamma _4}\)
|
0.1942
(0.00037)
|
0.1942
(0.00640)
|
0.1940
(0.00371)
|
0.1945
(0.00329)
|
\({\gamma _5}\)
|
0.1986
(0.00066)
|
0.1972
(0.00334)
|
0.1988
(0.00187)
|
0.1965
(0.00167)
|
\(\varepsilon\)
|
0.1839
(0.00338)
|
0.1877
(0.00769)
|
0.1837
(0.04314)
|
0.1917
(0.0405)
|
The posterior estimates for the parameters \(\phi\) using uniform prior under SELF are 0.2026, 0.22171, 0.1875, 0.11942, 0.1986 and 0.1839, respectively. Form the estimates, it is clear that the ranking of the ecological factors is as: \({\text{ED}} \to TO \to BI \to LI \to PY{\text{,}}\) indicating that Edaphic factor (ED) is preferred the most and Limiting factor (LI) is the least preferred factor.
5.2 The preference probabilities
The probability that defines the chance of preferring the ecological factor Tj over Tk in a single comparison. We denote preference probability by pi.ij. The posterior estimates are used to calculate preference probabilities. Since the posterior estimates obtained under SELF have minimum posterior risks, so these estimates are used to find the preference probabilities for the worth parameters \(\phi\) based on the noninformative priors and are given in Table 4. The value p1.12 = 0.4104 for the factors pair (TO, ED) that indicates that TO has 41.04% probability of being preferred against ED and it is p2.12= 0.5070 indicating 50.70% preference in the favor of ED against TO and there are 8.26% probability that none of the ecological factor will be preferred. In the similar ways, we can interpret the remaining preference probabilities.
Table 4
The preference probabilities using the uniform prior
|
(1,2)
|
(1,3)
|
(1,4)
|
(1,5)
|
(2,3)
|
(2,4)
|
(2,5)
|
(3,4)
|
(3,5)
|
(4,5)
|
pj.jk
|
0.4104
|
0.5135
|
0.4904
|
0.4722
|
0.5614
|
0.5387
|
0.5206
|
0.4354
|
0.4174
|
0.4403
|
pk.jk
|
0.5070
|
0.4039
|
0.4267
|
0.4448
|
0.3574
|
0.3794
|
0.397
|
0.4816
|
0.4999
|
0.4767
|
po.jk
|
0.0826
|
0.0826
|
0.0829
|
0.0830
|
0.0812
|
0.0819
|
0.0824
|
0.0830
|
0.0827
|
0.0830
|
The preference probabilities using the Jeffreys prior
|
pj.jk
|
0.4107
|
0.5111
|
0.487
|
0.4716
|
0.558
|
0.5343
|
0.519
|
0.4338
|
0.4182
|
0.4425
|
pk.jk
|
0.5054
|
0.4051
|
0.4289
|
0.4442
|
0.3595
|
0.3824
|
0.3973
|
0.4820
|
0.4974
|
0.4733
|
po.jk
|
0.0839
|
0.0838
|
0.0841
|
0.0842
|
0.0825
|
0.0833
|
0.0837
|
0.0842
|
0.0844
|
0.0842
|
When we observe the ranking order, we found that there exists complete coordination between the posterior estimates and the preference probabilities.
5.3 The Predictive Probabilities
The predictive probabilities describes the preference of ecological factor Tj over Tk in a single future comparison of the two factors (Tj, Tk). The predictive probability of these two factors (T1, T2) is denoted by p12 and can be calculated as;
\({P_{{\text{12}}}}=\int_{0}^{1} {\int_{0}^{{1 - {\gamma _1}}} {\int_{0}^{{1 - {\gamma _1} - {\gamma _2}}} {\int_{0}^{{1 - {\gamma _1} - {\gamma _2} - {\gamma _3}}} {\int_{0}^{\infty } {{\theta _{1.12}}p(} } } } } \phi \left| {a)d\varepsilon d{\gamma _4}d{\gamma _3}d{\gamma _2}d{\gamma _1}} \right.\) , \({\gamma _j} \geqslant 0,i=1,2,...5.{\text{ }}\sum\limits_{{j=1}}^{5} {{\gamma _j} \leqslant 1,} {\text{ }}\gamma >0,{\text{ }}\varepsilon >0\).
Where \({\theta _{1.12}}\) is the model preference probability given in (5), similarly the predictive probabilities of P21 and P012 can be calculated as following;
\({P_{{\text{21}}}}=\int_{0}^{1} {\int_{0}^{{1 - {\gamma _1}}} {\int_{0}^{{1 - {\gamma _1} - {\gamma _2}}} {\int_{0}^{{1 - {\gamma _1} - {\gamma _2} - {\gamma _3}}} {\int_{0}^{\infty } {{\theta _{2.12}}p(} } } } } \phi \left| {a)d\varepsilon d{\gamma _4}d{\gamma _3}d{\gamma _2}d{\gamma _1}} \right.\) , \({\gamma _j} \geqslant 0,j=1,2,...5.{\text{ }}\sum\limits_{{j=1}}^{5} {{\gamma _j} \leqslant 1,} {\text{ }}\gamma >0,{\text{ }}\varepsilon >0\),
\({P_{012}}=\int_{0}^{1} {\int_{0}^{{1 - {\gamma _1}}} {\int_{0}^{{1 - {\gamma _1} - {\gamma _2}}} {\int_{0}^{{1 - {\gamma _1} - {\gamma _2} - {\gamma _3}}} {\int_{0}^{\infty } {{\theta _{0.12}}p(} } } } } \phi \left| {a)d\varepsilon d{\gamma _4}d{\gamma _3}d{\gamma _2}d{\gamma _1}} \right.\) , \({\gamma _j} \geqslant 0,j=1,2,...5.{\text{ }}\sum\limits_{{j=1}}^{5} {{\gamma _j} \leqslant 1,} {\text{ }}\gamma >0,{\text{ }}\varepsilon >0\).
Where \({\theta _{2.12}}\)and \({\theta _{0.12}}\)are defined in (6) and (7) respectively. The predictive probabilities under both noninformative priors are obtained and given in Table 5.
Table 5
The predictive probabilities using the uniform prior.
|
(1,2)
|
(1,3)
|
(1,4)
|
(1,5)
|
(2,3)
|
(2,4)
|
(2,5)
|
(3,4)
|
(3,5)
|
(4,5)
|
pjk
|
0.4109
|
0.5133
|
0.4900
|
0.4713
|
0.5596
|
0.5370
|
0.5188
|
0.4350
|
0.4164
|
0.4394
|
pjk
|
0.5055
|
0.4033
|
0.4262
|
0.4445
|
0.3586
|
0.3803
|
0.3978
|
0.4814
|
0.5000
|
0.4766
|
pojk
|
0.0836
|
0.0834
|
0.0838
|
0.0842
|
0.0818
|
0.0827
|
0.0834
|
0.0836
|
0.0836
|
0.084
|
The predictive probabilities using the Jeffreys prior.
|
pjk
|
0.4122
|
0.5117
|
0.4887
|
0.4709
|
0.5570
|
0.5347
|
0.5173
|
0.4356
|
0.4178
|
0.4401
|
pjk
|
0.5047
|
0.4054
|
0.4280
|
0.4455
|
0.3617
|
0.3831
|
0.3998
|
0.4814
|
0.4991
|
0.4761
|
pojk
|
0.0831
|
0.0829
|
0.0833
|
0.0836
|
0.0813
|
0.0822
|
0.0829
|
0.083
|
0.0831
|
0.0838
|
The predictive probability p12 predicts the future preference in the contest of ecological factors in their single future comparisons. The predictive probability for preferring the ecological factor TO on ED is 0.4109, which indicates that there are 41.09% chances that the ecological factor TO will be preferred to ED in the single future comparison. The remaining predictive probabilities can be also interpreted on the same lines. From the results obtained, it is evident that the predictive probabilities do agree with the ranking order given by the posterior means under both of the priors. Similar results are observed on the basis of the preference probabilities.
5.4 Bayesian Hypothesis Testing
Bayesian hypothesis testing is a simple and straightforward procedure. The posterior probabilities are calculated, and decision between the hypotheses is directly made. For the comparison of the worth of any two ecological factors Tj and Tk, we consider the following hypotheses:
\({H_{jk}}:{\text{ }}{\gamma _j} \geqslant {\gamma _k}{\text{ Vs }}{H_{kj}}:{\text{ }}{\gamma _k}>{\gamma _j}\) , \(j \ne k{\text{ }}1,2,...,5\).
We represent posterior probability for \({H_{jk}}\)by \({p_{jk}}=P({\gamma _j}>{\gamma _k})\) and for \({H_{kj}}\), we use \({q_{kj}}=1 - {p_{jk}}\), so the posterior probability for p12 is H12 which is defined as:
\({p_{12}}=p({H_{12}})={\text{p}}({\gamma _1} \geqslant {\gamma _2})={\text{p}}({\gamma _1} - {\gamma _2} \geqslant 0)\)
\({\text{p}}(\delta >0\left| Y \right.)=\int\limits_{{\delta =0}}^{1} {\int\limits_{{\varphi =\delta }}^{{\frac{{1+\delta }}{2}}} {\int\limits_{{{\gamma _3}=0}}^{{1+\delta - 2\varphi }} {\int\limits_{{{\gamma _4}=0}}^{{1 - \delta - (\delta - \varphi ) - {\gamma _3}}} {\int\limits_{{\varepsilon =0}}^{\infty } {p\left( {\delta ,\varphi ,{\gamma _3},{\gamma _{4,}}\varepsilon \left| {\text{y}} \right.} \right)} } } } } {\text{ }}d\varepsilon {\text{ d}}{\gamma _4}{\text{d}}{\gamma _{\text{3}}}d\varphi d\delta\) \({\text{ where }}\delta ={\gamma _1} - {\gamma _2}{\text{, }}\varphi {\text{ = }}{\gamma _1}\) and\({q_{12}}=p({H_{21}})=1 - p({H_{12}})\)
The hypothesis with higher probability will be accepted. The posterior probabilities of the hypotheses \({H_{jk}}{\text{ and }}{H_{kj}}{\text{ (j<k=1,2}}...{\text{,5)}}\) are computed for the noninformative priors and given in Table 6. The posterior probability p12 = 0.0735 for the ecological factors pair (TO, ED) indicates that the probability for the ecological factor TO is very small, so we shall reject the null hypothesis and accept the alternative hypothesis indicating a higher preference for the factor ED.
Table 6
The posterior probabilities using the uniform prior.
Pairs
|
p12
|
p13
|
p14
|
p15
|
p23
|
p24
|
p25
|
p34
|
p35
|
p45
|
Hjk
|
0.0735
|
0.5450
|
0.3257
|
0.1844
|
0.7532
|
0.7165
|
0.6842
|
0.2026
|
0.0655
|
0.0783
|
Hkj
|
0.9265
|
0.4550
|
0.6743
|
0.8156
|
0.2468
|
0.2835
|
0.3158
|
0.7974
|
0.9345
|
0.9217
|
The posterior probabilities using the Jeffreys prior.
|
Hjk
|
0.0719
|
0.5292
|
0.3125
|
0.1754
|
0.7342
|
0.6980
|
0.6674
|
0.1991
|
0.0638
|
0.0755
|
Hkj
|
0.9281
|
0.4708
|
0.6875
|
0.8246
|
0.2658
|
0.3020
|
0.3326
|
0.8009
|
0.9362
|
0.9245
|
From the results, we see that the hypotheses H21, H13, H41, H51 H23, H24, H25, H34, H35 and H54 are accepted while all the remaining are rejected. On the same lines we can interpret the remaining probabilities. Our results also indicate that ecological factor ED is preferred the most and ecological factor LI is preferred least, same ranking is observed through the posterior estimates which shows complete coordination among the results.
5.5 Plausibility of the model
In order to check the plausibility of the new developed model, we use the Chi-square test of goodness of fit. Let \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} _{jk}}\), \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} _{kj}}\)and \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} _{0.jk}}\)denote the expected frequencies that can be obtained as \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} _{jk}}={n_{jk}}{\theta _{j.jk}}\), \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} _{kj}}={n_{kj}}{\theta _{k.kj}}\) and \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} _{o.jk}}={n_{o.jk}}{\theta _{o.jk}}\), where \({\theta _{j.jk}}\), \({\theta _{k.jk}}\)and \({\theta _{o.jk}}\) are defined in (5), (6) and (7), respectively. We define the following hypotheses.
\({\chi ^2}=\sum\limits_{{j<k}}^{5} {\left[ {\frac{{{{\left\{ {{n_{jk}} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} }_{jk}}} \right\}}^2}}}{{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} }_{jk}}}}+\frac{{{{\left\{ {{n_{kj}} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} }_{kj}}} \right\}}^2}}}{{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} }_{kj}}}}+\frac{{{{\left\{ {{n_{o.jk}} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} }_{ojk}}} \right\}}^2}}}{{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{n} }_{o.jk}}}}} \right]}\) with degrees of freedom, \(m(m - 2)\).
The obtained value of the Chi Square statistic for the Extended Weibull PC model is 7.227, with p-value is 0.94946. So according to decision rule, we conclude that the model under study is plausible and fit for the PC data.