Template:Lognormal distribution rank regression on Y

Rank Regression on Y

Performing a rank regression on Y requires that a straight line be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized.

The least squares parameter estimation method, or regression analysis, was discussed in Chapter 3 and the following equations for regression on Y were derived, and are again applicable:

[math]\displaystyle{ \hat{a}=\bar{y}-\hat{b}\bar{x}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}}{N} }[/math]

and:

[math]\displaystyle{ \hat{b}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}-\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,x_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}} \right)}^{2}}}{N}} }[/math]

In our case the equations for [math]\displaystyle{ {{y}_{i}} }[/math] and [math]\displaystyle{ x_{i} }[/math] are:

[math]\displaystyle{ {{y}_{i}}={{\Phi }^{-1}}\left[ F(T_{i}^{\prime }) \right] }[/math]

and:

[math]\displaystyle{ {{x}_{i}}=T_{i}^{\prime } }[/math]

where the [math]\displaystyle{ F(T_{i}^{\prime }) }[/math] is estimated from the median ranks. Once [math]\displaystyle{ \widehat{a} }[/math] and [math]\displaystyle{ \widehat{b} }[/math] are obtained, then [math]\displaystyle{ \widehat{\sigma } }[/math] and [math]\displaystyle{ \widehat{\mu } }[/math] can easily be obtained from Eqns. (aln) and (bln).

The Correlation Coefficient

The estimator of [math]\displaystyle{ \rho }[/math] is the sample correlation coefficient, [math]\displaystyle{ \hat{\rho } }[/math] , given by:

[math]\displaystyle{ \hat{\rho }=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,({{x}_{i}}-\overline{x})({{y}_{i}}-\overline{y})}{\sqrt{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{({{x}_{i}}-\overline{x})}^{2}}\cdot \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{({{y}_{i}}-\overline{y})}^{2}}}} }[/math]

Example 2

Fourteen units were reliability tested and the following life test data were obtained:

Table 9.1 - Life Test Data for Example 2
Data point index	Time-to-failure
1	5
2	10
3	15
4	20
5	25
6	30
7	35
8	40
9	50
10	60
11	70
12	80
13	90
14	100

Assuming the data follow a lognormal distribution, estimate the parameters and the correlation coefficient, [math]\displaystyle{ \rho }[/math] , using rank regression on Y.

Solution to Example 2

Construct Table 9.2, as shown next.

[math]\displaystyle{ \overset{{}}{\mathop{\text{Table 9}\text{.2 - Least Squares Analysis}}}\, }[/math] [math]\displaystyle{ \begin{matrix} N & T_{i} & F(T_{i}) & {T_{i}}'& y_{i} & {{T_{i}}'}^{2} & y_{i}^{2} & T_{i} y_{i} \\ \text{1} & \text{5} & \text{0}\text{.0483} & \text{1}\text{.6094}& \text{-1}\text{.6619} & \text{2}\text{.5903} & \text{2}\text{.7619} & \text{-2}\text{.6747} \\ \text{2} & \text{10} & \text{0}\text{.1170} & \text{2.3026}& \text{-1.1901} & \text{5.3019} & \text{1.4163} & \text{-2.7403} \\ \text{3} & \text{15} & \text{0}\text{.1865} & \text{2.7080}&\text{-0.8908} & \text{7.3335} & \text{0.7935} & \text{-2.4123} \\ \text{4} & \text{20} & \text{0}\text{.2561} & \text{2.9957} &\text{-0.6552} & \text{8.9744} & \text{0.4292} & \text{-1.9627} \\ \text{5} & \text{25} & \text{0}\text{.3258} & \text{3.2189}& \text{-0.4512} & \text{10.3612} & \text{0.2036} & \text{-1.4524} \\ \text{6} & \text{30} & \text{0}\text{.3954} & \text{3.4012}& \text{-0.2647} & \text{11.5681} & \text{0.0701} & \text{-0.9004} \\ \text{7} & \text{35} & \text{0}\text{.4651} & \text{3.5553} & \text{-0.0873} & \text{12.6405} & \text{-0.0076}& \text{-0.3102} \\ \text{8} & \text{40} & \text{0}\text{.5349} & \text{3.6889}& \text{0.0873} & \text{13.6078} & \text{0.0076} & \text{0.3219} \\ \text{9} & \text{50} & \text{0}\text{.6046} & \text{3.912} & \text{0.2647} & \text{15.3039} & \text{0.0701} &\text{1.0357} \\ \text{10} & \text{60} & \text{0}\text{.6742} & \text{4.0943} & \text{0.4512} & \text{16.7637} & \text{0.2036}&\text{1.8474} \\ \text{11} & \text{70} & \text{0}\text{.7439} & \text{4.2485} & \text{0.6552} & \text{18.0497}& \text{0.4292} & \text{2.7834} \\ \text{12} & \text{80} & \text{0}\text{.8135} & \text{4.382} & \text{0.8908} & \text{19.2022} & \text{0.7935} & \text{3.9035} \\ \text{13} & \text{90} & \text{0}\text{.8830} & \text{4.4998} & \text{1.1901} & \text{20.2483}&\text{1.4163} & \text{5.3552} \\ \text{14} & \text{100}& \text{1.9517} & \text{4.6052} & \text{1.6619} & \text{21.2076} &\text{2.7619} & \text{7.6533} \\ \sum_{}^{} & \text{ } & \text{ } & \text{49.222} & \text{0} & \text{183.1531} & \text{11.3646} & \text{10.4473} \\ \end{matrix} }[/math]

The median rank values ( [math]\displaystyle{ F({{T}_{i}}) }[/math] ) can be found in rank tables or by using the Quick Statistical Reference in Weibull++ .

The [math]\displaystyle{ {{y}_{i}} }[/math] values were obtained from the standardized normal distribution's area tables by entering for [math]\displaystyle{ F(z) }[/math] and getting the corresponding [math]\displaystyle{ z }[/math] value ( [math]\displaystyle{ {{y}_{i}} }[/math] ).

Given the values in the table above, calculate [math]\displaystyle{ \widehat{a} }[/math] and [math]\displaystyle{ \widehat{b} }[/math] using Eqns. (aaln) and (bbln):

[math]\displaystyle{ \begin{align} & \widehat{b}= & \frac{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime }{{y}_{i}}-(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime })(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,{{y}_{i}})/14}{\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime 2}-{{(\underset{i=1}{\overset{14}{\mathop{\sum }}}\,T_{i}^{\prime })}^{2}}/14} \\ & & \\ & \widehat{b}= & \frac{10.4473-(49.2220)(0)/14}{183.1530-{{(49.2220)}^{2}}/14} \end{align} }[/math]

or:

[math]\displaystyle{ \widehat{b}=1.0349 }[/math]

and:

[math]\displaystyle{ \widehat{a}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\widehat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,T_{i}^{\prime }}{N} }[/math]

or:

[math]\displaystyle{ \widehat{a}=\frac{0}{14}-(1.0349)\frac{49.2220}{14}=-3.6386 }[/math]

Therefore, from Eqn. (bln):

[math]\displaystyle{ {{\sigma }_{{{T}'}}}=\frac{1}{\widehat{b}}=\frac{1}{1.0349}=0.9663 }[/math]

and from Eqn. (aln):

[math]\displaystyle{ {\mu }'=-\widehat{a}\cdot {{\sigma }_{{{T}'}}}=-(-3.6386)\cdot 0.9663 }[/math]

or:

[math]\displaystyle{ {\mu }'=3.516 }[/math]

The mean and the standard deviation of the lognormal distribution are obtained using Eqns. (mean) and (sdv):

[math]\displaystyle{ \overline{T}=\mu ={{e}^{3.516+\tfrac{1}{2}{{0.9663}^{2}}}}=53.6707\text{ hours} }[/math]

and:

[math]\displaystyle{ {{\sigma }_{T}}=\sqrt{({{e}^{2\cdot 3.516+{{0.9663}^{2}}}})({{e}^{{{0.9663}^{2}}}}-1)}=66.69\text{ hours} }[/math]

The correlation coefficient can be estimated using Eqn. (RHOln):

[math]\displaystyle{ \widehat{\rho }=0.9754 }[/math]

The above example can be repeated using Weibull++ , using RRY.

The mean can be obtained from the QCP and both the mean and the standard deviation can be obtained from the Function Wizard.