Chapter 5B

(AST405) Lifetime data analysis

5 Inference Procedures for Log-location-scale Distributions

5.1 Log-normal and normal distributions

Log-normal distribution

• $T$ follows a log-normal distribution with location parameter $\mu$ and scale parameter $\sigma$ if $Y=\log T\sim \mathcal{N}(\mu ,{\sigma }^2)$

• The pdf and survivor function of log-normal distribution $$\begin{array}{rl}f(t;\mu ,\sigma )& =\frac{1}{\sigma t\sqrt{2\pi }}\exp [-\frac{1}{2}(\frac{\log t-\mu }{\sigma }{)}^2]\\ S(t;\mu ,\sigma )& =1-\mathrm{\Phi }(\frac{\log t-\mu }{\sigma })\end{array}$$

  • $\mu$ and $\sigma$ are the parameters of both normal and log-normal distributions

  • $\mathrm{\Phi }(\cdot )\to$ cumulative distribution function of standard normal distribution

---

• Log-normal distribution is a member of the log-location-scale family of distributions and the corresponding location-scale distribution is normal with $$\begin{array}{rl}{S}_0(z)& =1-\mathrm{\Phi }(z)\\ f_0(z)& =\frac{1}{\sqrt{2\pi }}{e}^{-z^2/2}=\varphi (z)\end{array}$$

  • $\varphi (\cdot )\to$ pdf of standard normal distribution

  • $z=(y-\mu )/\sigma$

---

• Density function of log-lifetime $$\begin{array}{rl}f(y;\mu ,\sigma )& =\frac{1}{\sigma }{f}_0(\frac{y-\mu }{\sigma })\\ & =\frac{1}{\sigma \sqrt{2\pi }}\exp [-\frac{1}{2}(\frac{y-\mu }{\sigma }{)}^2]\end{array}$$

• Survivor function of log-lifetime $$\begin{array}{r}S(y;\mu ,\sigma )=S_0(\frac{y-\mu }{\sigma })=1-\mathrm{\Phi }(\frac{y-\mu }{\sigma })\end{array}$$

Likelihood function normal distribution

• Data $$\{(t_i,{\delta }_i),i=1,\dots ,n\}$$

• Log-likelihood function $$\begin{array}{rl}\ell (\mu ,\sigma )& =\log \prod _{i=1}^n[(1/\sigma )f_0(z_i){]}^{{\delta }_i}[S_0(z_i){]}^{1-{\delta }_i}\\ & =-r\log \sigma +\sum _{i=1}^n{\delta }_i\log f_0(z_i)+\sum _{i=1}^n(1-{\delta }_i)\log S_0(z_i)\\ & =-r\log \sigma -\frac{1}{2}\sum _{i=1}^n{\delta }_i{z}_i^2+\sum _{i=1}^n(1-{\delta }_i)\log S_0(z_i)\end{array}$$

  • $z_i=(y_i-\mu )/\sigma$ and $y_i=\log t_i$

  • $r=\sum _{i=1}^n{\delta }_i$

---

• Elements of hessian matrix and score function depend on the followings $$\begin{array}{rl}\frac{\partial \log f_0(z)}{\partial z}& =-z\\ \frac{{\partial }^2\log f_0(z)}{\partial z^2}& =-1\\ \frac{\partial \log S_0(z)}{\partial z}& =-\frac{f_0(z)}{S_0(z)}\\ \frac{{\partial }^2\log S_0(z)}{\partial z^2}& =\frac{z{f}_0(z)}{S_0(z)}-[\frac{f_0(z)}{S_0(z)}{]}^2\end{array}$$

---

• MLEs $$\begin{array}{r}(\hat{\mu },\hat{\sigma }{)}^{\prime }={\arg max}_{\mathrm{\Theta }}\ell (\mu ,\sigma )\end{array}$$

  • Sampling distribution $$\begin{array}{r}(\hat{\mu },\hat{\sigma }{)}^{\prime }\sim \mathcal{N}((\mu ,\sigma {)}^{\prime },V)\end{array}$$ where $$\hat{V}=[-H(\hat{\mu },\hat{\sigma }){]}^{-1}$$

• Confidence intervals of parameters, quantiles, and survival probabilities can be obtained using the methods described for Weibull models

---

• Estimate of survivor function (Log-normal distribution) $$\begin{array}{rl}S(t;\hat{\mu },\hat{\sigma })& =1-\mathrm{\Phi }(\frac{\log t-\hat{\mu }}{\hat{\sigma }})\\ & =1-\mathrm{\Phi }(\hat{\psi })\end{array}$$ where $$\hat{\psi }={\mathrm{\Phi }}^{-1}(1-S(t;\hat{\mu },\hat{\sigma }))=\frac{\log t-\hat{\mu }}{\hat{\sigma }}$$

  • Standard error of $\hat{\psi }$ $$se(\hat{\psi })=\sqrt{{\mathbf{a}}^{\prime }\hat{V}\mathbf{a}}$$ where $$\mathbf{a}=(-1/\hat{\sigma },-\hat{\psi }/\hat{\sigma }{)}^{\prime }$$

Estimate of survivor function

• $(1-\alpha )100\mathrm{\%}$ CI of $S(t)$ $$\begin{array}{rl}L& <\psi <U\\ L& <{\mathrm{\Phi }}^{-1}(1-S(t;\mu ,\sigma ))<U\\ \mathrm{\Phi }(L)& <1-S(t;\mu ,\sigma )<\mathrm{\Phi }(U)\\ 1-\mathrm{\Phi }(U)& <S(t;\mu ,\sigma )<1-\mathrm{\Phi }(L)\end{array}$$ where $$\begin{array}{rl}L& =\hat{\psi }-z_{1-\alpha /2}se(\hat{\psi })\\ U& =\hat{\psi }+z_{1-\alpha /2}se(\hat{\psi })\end{array}$$

---

• LRT statistics based method of obtaining CI for survivor function is described with $$H_0:S(y_0)=S(\log t_0)=s_0$$

• The $100(1-\alpha )\mathrm{\%}$ CI for $S(t)$ can be obtained from the values of $s_0$ that satisfy $$\mathrm{\Lambda }(s_0)=2\ell (\hat{\mu },\hat{\sigma })-2\ell (\tilde{\mu },\tilde{\sigma })\le {\chi }_{(1),1-\alpha }^2$$

---

• Unrestricted and unrestricted MLEs are obtained as $$\begin{array}{rl}\text{unrestricted}(\hat{\mu },\hat{\sigma }{)}^{\prime }& ={\arg max}_{\mathrm{\Theta }}\ell (\mu ,\sigma )\\ \text{restricted}(\tilde{\mu },\tilde{\sigma }{)}^{\prime }& ={\arg max}_{\mathrm{\Theta }}\ell (y_0-\sigma {\mathrm{\Phi }}^{-1}(1-s_0),\sigma )\end{array}$$ where under $H_0$, we can show $$S(y_0)=1-\mathrm{\Phi }(\frac{y_0-\mu }{\sigma })=s_0\Rightarrow \mu =y_0-\sigma {\mathrm{\Phi }}^{-1}(1-s_0)$$

Quantiles

• The expression of estimate of $y_p$ $${\hat{y}}_p=\hat{\mu }+\hat{\sigma }{w}_p$$ where for normal distribution $$w_p=S_0^{-1}(1-p)={\mathrm{\Phi }}^{-1}(p)$$
• Standard error of ${\hat{y}}_p$
  $$se({\hat{y}}_p)=\sqrt{{\mathbf{a}}^{\prime }\hat{V}\mathbf{a}}$$ where $$\mathbf{a}=(1,w_p{)}^{\prime }$$

Homework

• Obtain the expressions of Wald-type and LRT based $100(1-\alpha )\mathrm{\%}$ confidence intervals of $y_p$

Example 5.3.1

• Data are available on lifetimes (in thousand miles) of 96 locomotive controls, of which were failed.

• The test was terminated after $135K$ miles, so 59 lifetimes were censored at $135K$.

---

dat_ex531

# A tibble: 96 × 2
    time status
   <dbl>  <int>
 1  22.5      1
 2  37.5      1
 3  46        1
 4  48.5      1
 5  51.5      1
 6  53        1
 7  54.5      1
 8  57.5      1
 9  66.5      1
10  68        1
# ℹ 86 more rows

---

dat_ex531 %>% 
  count(status)

# A tibble: 2 × 2
  status     n
   <int> <int>
1      0    59
2      1    37

---

Log-normal and normal model fit

mod_LN <- survreg(Surv(time, status) ~ 1, 
                  dist = "lognormal",
                  data = dat_ex531)

mod_N <- survreg(Surv(log(time), status) ~ 1, 
                dist = "gaussian",
                data = dat_ex531)

---

MLEs $(\hat{\mu },\log \hat{\sigma })$ and corresponding standard errors

tidy(mod_LN)

# A tibble: 2 × 5
  term        estimate std.error statistic p.value
  <chr>          <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)    5.19      0.129     40.3    0    
2 Log(scale)    -0.136     0.131     -1.04   0.297

---

Estimated variance of $(\hat{\mu },\log \hat{\sigma })$

mod_LN$var

            (Intercept) Log(scale)
(Intercept)  0.01657557 0.00983969
Log(scale)   0.00983969 0.01703353

Estimated variance $\mathrm{V}(\hat{\mu },\hat{\sigma })$ from $\mathrm{V}(\hat{\mu },\log \hat{\sigma })$

• $GV(\hat{\mu },\log \hat{\sigma })G^{\prime }$

           [,1]       [,2]
[1,] 0.01657557 0.00858735
[2,] 0.00858735 0.01297359

$$G=\left[\begin{array}{cc}1& 0\\ 0& \exp (\sigma )\end{array}\right]$$

---

95% Confidence intervals for $\mu$ and $\sigma$

par	lower	upper	lower	upper
$\mu$	4.942	5.447	5.000	5.400
$\sigma$	0.676	1.127	0.709	1.109

---

• Estimate of $S(80)$ $$\begin{array}{rl}S(80;\hat{\mu },\hat{\sigma })& =1-\mathrm{\Phi }(\frac{\log 80-\hat{\mu }}{\hat{\sigma }})\\ & =0.824\end{array}$$

  • $\hat{\mu }=5.195\text{and}\hat{\sigma }=0.873$

---

Comparison of the estimates of survivor function

`

---

Estimate and confidence interval of $S(80)$

parameter	est	lower	upper
$S(80)$	0.824	0.667	0.924

• Obtain LRT based 95% CI for $S(80)$

Quantiles

• General expression of $p$th quantile of log-lifetime ($\hat{\mu }=5.195$ and $\hat{\sigma }=0.873$) $${\hat{y}}_p=\hat{\mu }+\hat{\sigma }{w}_p$$
  • $w_p={\mathrm{\Phi }}^{-1}(p)$

---

Estimate and confidence intervals of different quantiles of locomotive controls lifetime (normal distribution)

$p$	$w_p$	${\hat{y}}_p$	se	lower	upper
0.25	-0.674	4.606	0.105	NA	NA
0.50	0.000	5.195	0.129	NA	NA
0.75	0.674	5.783	0.194	NA	NA

5.2 Log-logistic and logistic distributions

Log-logistic distribution

• $T$ follows a log-logistic distribution with parameters $\alpha$ (scale) and $\beta$ (shape) if $Y=\log T$ follows a logistic distribution with parameters $u$ (location) and $b$ (scale)

---

• The pdf, survivor, and hazard function of log-logistic distribution $$\begin{array}{rl}f(t;\alpha ,\beta )& =\frac{(\beta /\alpha )(t/\alpha {)}^{\beta -1}}{[1+(t/\alpha {)}^{\beta }{]}^2}\\ S(t;\alpha ,\beta )& =[1+(t/\alpha {)}^{\beta }{]}^{-1}\\ h(t;\alpha ,\beta )& =\frac{(\beta /\alpha )(t/\alpha {)}^{\beta -1}}{[1+(t/\alpha {)}^{\beta }]}\end{array}$$

Logistic distribution

• Log-logistic distribution is a member of the log-location-scale family of distributions and the corresponding location-scale distribution is logistic with $$\begin{array}{rl}{S}_0(z)& =\frac{1}{1+e^z}\\ f_0(z)& =\frac{e^z}{(1+e^z{)}^2}\end{array}$$

  • $z=(y-u)/b$

---

• Density function of log-lifetime $$\begin{array}{rl}f(y;u,b)& =\frac{1}{b}{f}_0(\frac{y-u}{b})\\ & =\frac{(1/b)\exp [(y-u)/b]}{\{1+\exp [(y-u)/b]{\}}^2}\end{array}$$

• Survivor function of log-lifetime $$\begin{array}{rl}S(y;u,b)& =S_0(\frac{y-u}{b})\\ & =\frac{1}{1+\exp [(y-u)/b]}\end{array}$$

---

• Data: $\{(t_i,{\delta }_i),i=1,\dots ,n\}$

• Log-likelihood function $$\begin{array}{rl}\ell (\mu ,\sigma )& =\log \prod _{i=1}^n[(1/b)f_0(z_i){]}^{{\delta }_i}[S_0(z_i){]}^{1-{\delta }_i}\\ & =-r\log b+\sum _{i=1}^n{\delta }_i\log f_0(z_i)+\sum _{i=1}^n(1-{\delta }_i)\log S_0(z_i)\\ & =-r\log b+\sum _{i=1}^n[{\delta }_i\{z_i-\log (1+e^{z_i})\}-\log (1+e^{z_i})]\end{array}$$

  • $z_i=(y_i-u)/b$ and $y_i=\log t_i$

  • $r=\sum _{i=1}^n{\delta }_i$

---

• Elements of hessian matrix and score function depend on the followings $$\begin{array}{rl}\frac{\partial \log f_0(z)}{\partial z}& =1-\frac{2e^z}{1+e^z}\\ \frac{{\partial }^2\log f_0(z)}{\partial z^2}& =-2f_0(z)\\ \frac{\partial \log S_0(z)}{\partial z}& =\frac{-e^z}{1+e^z}\\ \frac{{\partial }^2\log S_0(z)}{\partial z^2}& =\frac{-e^z}{(1+e^z{)}^2}\end{array}$$

---

• MLEs $$\begin{array}{r}(\hat{u},\hat{b}{)}^{\prime }={\arg max}_{\mathrm{\Theta }}\ell (u,b)\end{array}$$

• Sampling distribution $$\begin{array}{r}(\hat{u},\hat{b}{)}^{\prime }\sim \mathcal{N}((u,b{)}^{\prime },V)\end{array}$$ where $$\hat{V}=[-H(\hat{u},\hat{b}){]}^{-1}$$

• Confidence intervals of parameters, quantiles, and survival probabilities can be obtained using the methods described for Weibull models

---

• Estimate of survivor function (logistic distribution) $$\begin{array}{rl}{S}_0(\frac{y-\hat{u}}{\hat{b}})=S(y;\hat{u},\hat{b})& =\frac{1}{1+\exp [(y-\hat{u})/\hat{b}]}\\ \log [\frac{1-S(y)}{S(y)}]& =\frac{y-\hat{u}}{\hat{b}}=\hat{\psi }=S_0^{-1}(S(y))\end{array}$$

  • Standard error of $\hat{\psi }$ $$se(\hat{\psi })=\sqrt{{\mathbf{a}}^{\prime }\hat{V}\mathbf{a}},\text{where }\mathbf{a}=(-1/\hat{b},-\hat{\psi }/\hat{b}{)}^{\prime }$$

---

$(1-\alpha )100\mathrm{\%}$ CI of $S(t)$

$$\begin{array}{rl}L& <\psi <U\\ L& <\log \frac{1-S(y)}{S(y)}<U\\ \exp (L)& <\frac{1-S(y)}{S(y)}<\exp (U)\\ 1+\exp (L)& <1+\frac{1-S(y)}{S(y)}<1+\exp (L)\\ \frac{1}{1+\exp (U)}& <S(y)<\frac{1}{1+\exp (L)}\end{array}$$ where $$\begin{array}{rl}L& =\hat{\psi }-z_{1-\alpha /2}se(\hat{\psi })\\ U& =\hat{\psi }+z_{1-\alpha /2}se(\hat{\psi })\end{array}$$

Estimate of survivor function

• LRT statistics based method of obtaining CI for survivor function is described with $$H_0:S(y_0)=S(\log t_0)=s_0$$

• The $100(1-\alpha )\mathrm{\%}$ CI for $S(t)$ can be obtained from the values of $s_0$ that satisfy $$\mathrm{\Lambda }(s_0)\le {\chi }_{(1),1-\alpha }^2$$ where $$\begin{array}{r}\mathrm{\Lambda }(s_0)=2\ell (\hat{u},\hat{b})-2\ell (\tilde{u},\tilde{b})\end{array}$$

---

• Unrestricted and unrestricted MLEs are obtained as $$\begin{array}{rl}\text{unrestricted}(\hat{u},\hat{b}{)}^{\prime }& ={\arg max}_{\mathrm{\Theta }}\ell (u,b)\\ \text{restricted}(\tilde{u},\tilde{b}{)}^{\prime }& ={\arg max}_{\mathrm{\Theta }}\ell (y_0-b\log \{(1-s_0)/s_0\},b)\end{array}$$ where under $H_0$, we can show $$S(y_0)=s_0\Rightarrow u=y_0-b\log \frac{1-s_0}{s_0}$$

Quantiles

• The expression of estimate of $y_p$ $${\hat{y}}_p=\hat{u}+\hat{b}{w}_p$$ where for normal distribution $$w_p=S_0^{-1}(1-p)=\log \frac{p}{1-p}$$
• Standard error of ${\hat{y}}_p$
  $$se({\hat{y}}_p)=\sqrt{{\mathbf{a}}^{\prime }\hat{V}\mathbf{a}}$$ where $$\mathbf{a}=(1,w_p{)}^{\prime }$$

Homework

• Obtain the expressions of Wald-type and LRT based $100(1-\alpha )\mathrm{\%}$ confidence intervals of $y_p$

Example 5.3.1

• Data are available on lifetimes (in thousand miles) of 96 locomotive controls, of which were failed.

• The test was terminated after $135K$ miles, so 59 lifetimes were censored at $135K$.

---

dat_ex531

# A tibble: 96 × 2
    time status
   <dbl>  <int>
 1  22.5      1
 2  37.5      1
 3  46        1
 4  48.5      1
 5  51.5      1
 6  53        1
 7  54.5      1
 8  57.5      1
 9  66.5      1
10  68        1
# ℹ 86 more rows

---

dat_ex531 %>% 
  count(status)

# A tibble: 2 × 2
  status     n
   <int> <int>
1      0    59
2      1    37

---

Log-logistic and logistic model fit

mod_LL <- survreg(Surv(time, status) ~ 1, 
                  dist = "loglogistic",
                  data = dat_ex531)

mod_L <- survreg(Surv(log(time), status) ~ 1, 
                dist = "logistic",
                data = dat_ex531)

---

MLEs $(\hat{u},\log \hat{b})$

[1]  5.1206418 -0.8266704

Estimated variance of $(\hat{u},\log \hat{b})$

            (Intercept)  Log(scale)
(Intercept) 0.010490062 0.007837215
Log(scale)  0.007837215 0.022515937

---

MLEs of $(\hat{u},\hat{b})$

[1] 5.1206418 0.4375036

Estimated variance of $(\hat{u},\hat{b})$

            [,1]        [,2]
[1,] 0.010490062 0.003428809
[2,] 0.003428809 0.004309761

---

95% Confidence intervals for location and scale parameters

dist	par	est	lower	upper	lower	upper
Logistic	$u$	5.121	4.920	5.321	5.000	5.300
NA	$b$	0.438	0.326	0.587	0.360	0.559
Gaussian	$\mu$	5.195	4.942	5.447	5.000	5.400
NA	$\sigma$	0.873	0.676	1.127	0.709	1.109

---

• Estimate of $S(80)$ (log-logistic distribution) $$\begin{array}{rl}S(80;\hat{u},\hat{b})& =\frac{1}{1+\exp [(\log 80-\hat{u})/\hat{b}]}\\ & =0.844\end{array}$$

  • $\hat{u}=5.121\text{and}\hat{b}=0.438$

---

Comparison of the estimates of survivor function

`

---

Estimate and corresponding Wald-type confidence interval of the survival probability $S(80)$

dist	est	lower	upper
Log-logistic	0.844	0.566	0.957
Log-normal	0.824	0.667	0.924

• Obtain LRT based 95% CI for $S(80)$

Quantiles

• General expression of $p$th quantile of log-lifetime ($\hat{u}=5.121$ and $\hat{b}=0.438$) $${\hat{y}}_p=\hat{u}+\hat{b}{w}_p$$
  • $w_p=\log \frac{p}{1-p}$

---

Estimate and confidence intervals of different quantiles

dist	$p$	$w_p$	${\hat{y}}_p$	se	lower	upper
Logistic	0.25	-1.099	4.640	0.143	NA	NA
NA	0.50	0.000	5.121	0.102	NA	NA
NA	0.75	1.099	5.601	0.234	NA	NA
Gaussian	0.25	-0.674	4.826	0.101	NA	NA
NA	0.50	0.000	5.121	0.102	NA	NA
NA	0.75	0.674	5.416	0.177	NA	NA

Homework

• Analyze the locomotive control lifetimes using Weibull model and compare the results

5.3 Comparison of distributions

• Let $T_{ji}$ be the lifetime of $i$th subject of the $j$th group ($i=1,\dots ,n_j$, $j=1,\dots ,m$)

• Assume $T_{ji}$ follows a distribution of log-location-scale family with parameters ${\alpha }_j$ (scale) and ${\beta }_j$ (shape)

• The corresponding distribution of log-lifetime $Y_{ji}=\log T_{ji}$ is of a location-scale family distribution with parameters $u_j$ (location) and $b_j$ (scale) $$u_j=\log {\alpha }_j\text{and}{b}_j=(1/{\beta }_j)$$

Survivor functions

• The survivor function of $Y_{ji}=\log T_{ji}$ $$\begin{array}{r}{S}_j(y)=S_0(\frac{y-u_j}{b_j})\end{array}$$

• The survivor function of $T_{ji}$ $$\begin{array}{r}{S}_j(t)=S_0^{\star }[(t/{\alpha }_j{)}^{{\beta }_j}]\end{array}$$

  • $S_0^{\star }(x)=S_0(\log x)$

  • $u_j=\log {\alpha }_j$

  • $b_j=(1/{\beta }_j)$

---

• Comparison of several normal populations is a well-known problem in statistics, where equal population variances are assumed, and the comparisons are performed on the basis of equality of population means

Quantile

• General expression of the $p$th quantile of the $j$th population takes the form $$\begin{array}{r}{y}_{jp}=u_j+b_j{w}_p,j=1,\dots ,m\end{array}$$
  • $w_p=S_0^{-1}(1-p)$

Equality of two populations

• When the scales are not equal (i.e. $b_1\ne b_2$), the difference between the $p$th quantiles does depend on the probability $p$ $$\begin{array}{r}{y}_{1p}-y_{2p}=u_1-u_2+w_p(b_1-b_2)\end{array}$$

• Under the assumption of equality of the scales (i.e. $b_1=b_2$), difference between $p$th (log-lifetime) quantile of a pair of populations (say 1 and 2) is constant, i.e. it does not depend on the probability $p\in (0,1)$ $$\begin{array}{r}{y}_{1p}-y_{2p}=u_1-u_2\end{array}$$

---

• The difference between two log-lifetime quantiles can be expressed in terms of the ratio of lifetime quantiles $$\begin{array}{rl}{y}_{1p}-y_{2p}& =u_1-u_2\\ \log t_{1p}-\log t_{2p}& =\log {\alpha }_1-\log {\alpha }_2\\ t_{1p}/t_{2p}& ={\alpha }_1/{\alpha }_2\end{array}$$

• The ratio of the $p$th quantiles of two lifetime distributions does not depend on the probability $p$ when the corresponding shape parameters are equal $({\beta }_1={\beta }_2)$

---

• Equality of all quantiles of two distributions, i.e. $$y_{1p}=y_{2p}\mathrm{\forall }p\in (0,1),$$ corresponds to equality of two distributions, i.e. $$S_1(y)=S_2(y)$$

• Under the assumption of common scale (shape for lifetime) parameter, the null hypothesis of equality of two distributions can be expressed as $$\begin{array}{r}{H}_0:u_1-u_2=0\text{or}{H}_0:({\alpha }_1/{\alpha }_2)=1\end{array}$$

---

• Equality of two populations with survivor functions (say $S_1$ and $S_2$) can be expressed in terms of survivor functions

• Since $$y_{1p}=y_{2p}+u_1-u_2\text{or}{t}_{1p}=t_{2p}({\alpha }_1/{\alpha }_2),$$ the corresponding survivor functions can be expressed as $$\begin{array}{r}{S}_1(y+u_1-u_2)=S_2(y)\\ S_1(t({\alpha }_1/{\alpha }_2))=S_2(t)\end{array}$$

• That is, the survivor functions for $Y$ are translations of one another by an amount $(u_1-u_2)$ along the $y$-axis

Wald-type statistic

• Data $\{(t_{ji},{\delta }_{ji}),i=1,2\}$ and $y_{ji}=\log t_{ji}$

• Two populations can be compared in terms of $p$th quantile $$H_0:y_{1p}=y_{2p}$$

• Corresponding pivotal quantity $$Z_p=\frac{({\hat{y}}_{1p}-{\hat{y}}_{2p})-(y_{1p}-y_{2p})}{[\mathrm{var}({\hat{y}}_{1p})+\mathrm{var}(y_{2p}){]}^{1/2}}\sim \mathcal{N}(0,1)\text{under }{H}_0$$

  • The statistic $Z_p$ can be used to obtain confidence interval for $(y_{1p}-y_{2p})$

---

• To test $H_0:b_1=b_2$, the following pivotal quantity can be considered $$Z_b=\frac{(\log {\hat{b}}_1-\log {\hat{b}}_2)-(\log b_1-\log b_2)}{[\mathrm{var}(\log {\hat{b}}_1)+\mathrm{var}(\log {\hat{b}}_2){]}^{1/2}}\sim \mathcal{N}(0,1)\text{under }{H}_0$$

  • The statistic $Z_b$ can be used to obtain confidence interval for $(b_1/b_2)$

---

• When scales are equal, two populations can be compared with respect their location parameter $H_0:u_1=u_2$

• The corresponding pivotal quantity $$Z_u=\frac{({\hat{u}}_1-{\hat{u}}_2)-(u_1-u_2)}{[\mathrm{var}({\hat{u}}_1)+\mathrm{var}({\hat{u}}_2){]}^{1/2}}\sim \mathcal{N}(0,1)\text{under }{H}_0$$

  • The statistic $Z_u$ can be used to obtain confidence interval for $(u_1-u_2)$

---

• Wald statistic cannot be used to test $$H_0:u_1=u_2,b_1=b_2$$

LRT based inference

• Data $\{(t_{ji},{\delta }_{ji}),j=1,\dots ,m,i=1,\dots ,n_j\}$ and $y_{ji}=\log t_{ji}$

• Different tests and confidence intervals of interest

  1. $H_0:b_1=\cdots =b_m$

  2. Confidence interval for $(b_1/b_2)$

  3. Equality of several location parameters when scale parameters are equal $$\begin{array}{rl}{H}_0& :u_1=\cdots =u_m,b_1=\cdots =b_m\\ H_1& :\text{all }{u}_j\text{’s are not equal},b_1=\cdots =b_m\end{array}$$

  4. Confident interval for $(u_1-u_2)$ when $b_1=b_2$

  5. Confidence interval for $(y_{1p}-y_{2p})$ when $b_1\ne b_2$

Case 1

• Hypothesis of interest $$\begin{array}{}\text{(5.1)}& H_0:b_1=\cdots =b_m=b(\text{say})\end{array}$$

• Log-likelihood function $$\begin{array}{rl}\ell (u_1,\dots ,u_m,b_1,\dots ,b_m)& =\sum _{j=1}^m{\ell }_j(u_j,b_j)\end{array}$$

• Contribution to log-likelihood function for the $j$th population $$\begin{array}{r}{\ell }_j(u_j,b_j)=-r_j\log b_j+\sum _{i=1}^{n_j}[{\delta }_i\log f_0(z_{ji})+(1-{\delta }_{ji})\log S_0(z_{ji})]\end{array}$$

  • $r_j=\sum _i{\delta }_{ji}$

---

• LRT statistic $$\begin{array}{rl}\mathrm{\Lambda }& =2\ell ({\hat{u}}_1,\dots ,{\hat{u}}_m,{\hat{b}}_1,\dots ,{\hat{b}}_m)-2\ell ({\tilde{u}}_1,\dots ,{\tilde{u}}_m,\tilde{b},\dots ,\tilde{b})\end{array}$$

  • $\mathrm{\Lambda }\sim {\chi }_{(m-1)}^2$ under the null hypothesis defined in Equation 5.1

• MLEs

  • $({\hat{u}}_j,{\hat{b}}_j{)}^{\prime }={\arg max}_{\mathrm{\Theta }}{\ell }_j(u_j,b_j),j=1,\dots ,m$

  • $({\tilde{u}}_1,\dots ,{\tilde{u}}_m,\tilde{b},\dots ,\tilde{b}{)}^{\prime }={\arg max}_{\mathrm{\Theta }}\ell (u_1,\dots ,u_m,b,\dots ,b)$

Case 2

• To obtain confidence interval of $(b_1/b_2)$, consider $$H_0:(b_1/b_2)=a\Rightarrow H_0:b_1=a{b}_2$$

• $100(1-\alpha )\mathrm{\%}$ confidence interval of $(b_1/b_2)$ can be obtained from the range of $a$ values that satisfy $$\mathrm{\Lambda }(a)\le {\chi }_{(1),1-\alpha }^2,$$ where the LRT statistic $$\mathrm{\Lambda }(a)=2\ell ({\hat{u}}_1,{\hat{u}}_2,{\hat{b}}_1,{\hat{b}}_2)-2\ell ({\tilde{u}}_1,{\tilde{u}}_2,a{\tilde{b}}_2,{\tilde{b}}_2)$$

  • $({\hat{u}}_j,{\hat{b}}_j{)}^{\prime }={\arg max}_{\mathrm{\Theta }}{\ell }_j(u_j,b_j),j=1,2$

  • $({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{b}}_2{)}^{\prime }={\arg max}_{\mathrm{\Theta }}\ell (u_1,u_2,a{b}_2,b_2)$

Case 3

• Test equality of several location parameters when scale parameters are equal $$\begin{array}{rl}{H}_0& :u_1=\cdots =u_m,b_1=\cdots =b_m\\ H_1& :\text{all }{u}_j\text{’s are not equal},b_1=\cdots =b_m\end{array}$$

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• MLEs

  • under $H_0,$ $(u^{\star },b^{\star })={\arg max}_{\mathrm{\Theta }}\ell (u,\dots ,u,b,\dots ,b)$

  • under $H_1,$ $({\tilde{u}}_1,\dots ,{\tilde{u}}_m,\tilde{b})={\arg max}_{\mathrm{\Theta }}\ell (u_1,\dots ,u_m,b,\dots ,b)$

• LRT statistic $$\mathrm{\Lambda }=2\ell ({\tilde{u}}_1,\dots ,{\tilde{u}}_m,\tilde{b},\dots ,\tilde{b})-2\ell (u^{\star },\dots ,u^{\star },b^{\star },\dots ,b^{\star })$$

  • Under the null hypothesis, $\mathrm{\Lambda }$ follows ${\chi }_{(m-1)}^2$ distribution

Case 4

• To obtain a confidence interval of $(u_1-u_2)$ when $b_1=b_2$, consider the null and alternative hypothesis $$\begin{array}{r}{H}_0:u_1-u_2=\delta ,b_1=b_2\text{vs}{H}_1:u_1-u_2\ne \delta ,b_1=b_2\end{array}$$

• LRT statistic $$\mathrm{\Lambda }(\delta )=2\ell ({\tilde{u}}_1,{\tilde{u}}_2,\tilde{b},\tilde{b})-2\ell (u_2^{\star }+\delta ,u_2^{\star },b^{\star },b^{\star })$$

  • under $H_0,$ $(u^{\star },b^{\star })={\arg max}_{\mathrm{\Theta }}\ell (u,u,b,b)$

  • under $H_1,$ $({\tilde{u}}_1,{\tilde{u}}_2,\tilde{b})={\arg max}_{\mathrm{\Theta }}\ell (u_1,u_2,b,b)$

• $100(1-\alpha )$ confidence interval for $(u_1-u_2)$ can be obtained from the set of $\delta$ values that satisfy $\mathrm{\Lambda }(\delta )\le {\chi }_{(1),1-\alpha }^2$

Case 5

• When $b_1\ne b_2$, to obtain confidence interval for $(y_{1p}-y_{2p})$ consider the following hypothesis $$H_0:y_{1p}-y_{2p}=\mathrm{\Delta }\Rightarrow H_0:u_1-u_2=\mathrm{\Delta }+(b_2-b_1)w_p$$
  • $w_p=S_0^{-1}(1-p)$
• LRT statistic $$\mathrm{\Lambda }(\mathrm{\Delta })=2\ell ({\hat{u}}_1,{\hat{u}}_2,{\hat{b}}_1,{\hat{b}}_2)-2\ell ({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{b}}_1,{\tilde{b}}_2)$$

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• under $H_0$ $$({\tilde{u}}_1,{\tilde{u}}_2,{\tilde{b}}_1,{\tilde{b}}_2)={\arg max}_{\mathrm{\Theta }}\ell (u_2+\mathrm{\Delta }+(b_2-b_1)w_p,u_2,b_1,b_2)$$

• under $H_1$ $$({\hat{u}}_1,{\hat{u}}_2,{\hat{b}}_2,{\hat{b}}_1)={\arg max}_{\mathrm{\Theta }}\ell (u_1,u_2,b_1,b_2)$$

• $100(1-\alpha )$ confidence interval for $(y_{1p}-y_{2p})$ can be obtained from the set of $\mathrm{\Delta }$ values that satisfy $\mathrm{\Lambda }(\mathrm{\Delta })\le {\chi }_{(1),1-\alpha }^2$

Comparison of Weibull or extreme value distributions

• Assume $T_{ji}\sim \text{Weibull}({\alpha }_j,{\beta }_j)$ $(j=1,\dots ,m,i=1,\dots ,n_j)$

  • Data $\{(t_{ji},{\delta }_{ji}),j=1,\dots ,m,i=1,\dots ,n_j\}$

• Survivor function of Weibull distribution $$S_j(t)=\exp [-(t/{\alpha }_j{)}^{{\beta }_j}]$$

• Survivor function of extreme value distribution $$S_j(y)=\exp [-e^{(y-u_j)/b_j}]$$

  • $u_j=\log {\alpha }_j$

  • $b_j=1/{\beta }_j$

Example 5.4.1

• Data of the following table are on the time to breakdown of electrical insulating fluid subject to a constant voltage stress in a lifetest experiment

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Estimate of voltage-specific extreme value models

voltage	${\hat{u}}_j\pm se({\hat{u}}_j)$	${\hat{b}}_j\pm se({\hat{b}}_j)$
26	$6.862\pm 1.104$	$1.834\pm 0.885$
28	$5.865\pm 0.486$	$1.022\pm 0.474$
30	$4.351\pm 0.302$	$0.944\pm 0.303$
32	$3.256\pm 0.486$	$1.781\pm 0.254$
34	$2.503\pm 0.315$	$1.297\pm 0.211$
36	$1.457\pm 0.309$	$1.125\pm 0.221$
38	$0.001\pm 0.273$	$0.734\pm 0.367$

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Comparison of estimated survivor function

LRT (Case 1)

• Null hypothesis $$H_0:b_1=\cdots =b_7$$

• LRT statistic $$\begin{array}{rl}\mathrm{\Lambda }& =2\ell ({\hat{u}}_1,\dots ,{\hat{u}}_7,{\hat{b}}_1,\dots ,{\hat{b}}_7)-2\ell ({\tilde{u}}_1,\dots ,{\tilde{u}}_7,\tilde{b},\dots ,\tilde{b})\\ & =2(-132.181)-2(-136.578)\\ & =8.794\end{array}$$

  • p-value $$Pr({\chi }_{(6)}^2\ge \mathrm{\Lambda })=0.185$$ It does not provide enough evidence to reject the null hypothesis of equality of the scale parameters.

Confidence interval of $(b_1/b_2)$ (Case 2)

• Wald-type $$\begin{array}{rl}(\log {\hat{b}}_1-\log {\hat{b}}_2)& \pm z_{1-\alpha /2}se(\log {\hat{b}}_1-\log {\hat{b}}_2)\\ ({\hat{b}}_1/{\hat{b}}_2)& e^{\pm z_{1-\alpha /2}se(\log {\hat{b}}_1-\log {\hat{b}}_2)}\\ (1.834/1.022)& e^{\pm (1.96)(0.624)}\\ 0.529& <(b_1/b_2)<6.095\end{array}$$

  • Similarly confidence intervals for $(b_j/b_{j^{\prime }})$ $j>j^{\prime }$ can be obtained

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Estimate and 95% confidence interval of pair-wise comparisons of scale parameters $(b_j/b_{j^{\prime }})$

Case 3

• Equality of all location parameters when scales are equal $$H_0:u_1=\cdots =u_m,b_1=\cdots =b_m$$

• LRT statistic $$\begin{array}{rl}\mathrm{\Lambda }& =2\ell ({\tilde{u}}_1,\dots ,{\tilde{u}}_m,\tilde{b},\dots ,\tilde{b})-2\ell (u^{\star },\dots ,u^{\star },b^{\star },\dots ,b^{\star })\\ & =2(-136.578)-2(-176.584)\\ & =80.013\end{array}$$

  • p-value $P({\chi }_{(1)}^2>80.013)<.001\to$ There is a strong evidence against the assumption of equality of $m$ location parameters

Case 4

• Wald-type confidence interval of $(u_1-u_2)$ $$\begin{array}{rl}({\hat{u}}_1-{\hat{u}}_2)& \pm z_{1-\alpha /2}se({\hat{u}}_1-{\hat{u}}_2)\\ (6.862-4.351)& \pm (1.96)(1.206)\\ -1.367& <(u_1-u_2)<3.361\end{array}$$

  • There is no significant difference between $u_1$ and $u_2$

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Estimate and 95% confidence interval of pair-wise comparisons of location parameters $(u_j-u_{j^{\prime }})$

Case 5

• General expression of $p$th quantile of the group $j$ $$y_{jp}=u_j+b_j{w}_p,j=1,\dots ,m$$

• Difference of $p$th quantile between groups 1 and 2 $$y_{1p}-y_{2p}=u_1-u_2+(b_1-b_2)w_p$$

  • 95% confidence interval for the difference of median between groups 1 and 2 $$\begin{array}{rl}{\hat{y}}_{1m}-{\hat{y}}_{2m}& \pm z_{1-\alpha /2}se({\hat{y}}_{1m}-{\hat{y}}_{2m})\\ (6.19-5.491)& \pm (1.96)(1.291)\\ -1.831& <(y_{1m}-y_{2m})<3.231\end{array}$$

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Estimate and 95% confidence interval of pair-wise comparisons of medians $(y_{j,.5}-y_{j^{\prime },.5})$

Homework

• Analyse the breakdown time data using log-logistic and log-normal distributions and compare the results with that of Weibull distribution