Chapter 6, Problem 7RQ

Summary Introduction

To calculate: The probability that a person enrolled in the study will survive till the end of third year.

Introduction: Life table shows the probability of a person either diseased or health dying before their next birthday. It is also known as mortality table or actuarial table.

Explanation of Solution

In order to find the probability that a person enrolled in the study will survive till the end of third year column 7 for all years must be found that is denoted with question mark.

(1)	(2)	(3)	(4)	(5)	(6)	(7)
Interval since beginning of treatment (Months)	Alive at Beginning of interval	Died during interval	Withdrew during interval	Effective number exposed to Risk of Dying during interval	Proportion who died during interval	Proportion who did not die during interval
x	lx	dx	wx	l’x	qx	px
1-12	248	96	27			?
13-24	125	55	13			?
25-36	57	55	2			?

In order to find column 7, column 5 and column 6 for all years must be calculated.

The formula to calculate effective number of people exposed to risk of dying during interval (column5):

$\begin{array}{l}\text{Effective number of people exposed to risk of dying during interval}\\ \text{=No}\text{.}\text{of}\text{people}\text{alive}\text{at}\text{beginning}\text{of}\text{interval-}\left(\frac{\text{1}}{\text{2}}\text{×No}\text{.}\text{of}\text{people}\text{who}\text{withdrew}\text{}\text{during}\text{experiment}\right)\\ l{\text{'}}_x=l_x-\left(\frac{1}{2}\times w_x\right)\end{array}$

The proportion of people who died during interval is given by the below formula (column 6):

$\begin{array}{l}\text{Proportion of people who died during interval}\\ \text{=}\frac{\text{Number}\text{of}\text{people}\text{who}\text{died}\text{during}\text{the}\text{interval}}{\text{Effective number of people exposed to risk of dying during interval}}\\ q_x=\frac{d_x}{l{\text{'}}_x}\end{array}$

The proportion of people who did not die during interval is by the below formula (column 7):

$\begin{array}{l}\text{Proportion of people who did not die during interval=1-Proportion of people who died during interval}\\ {\text{p}}_{\text{x}}{\text{=1-q}}_{\text{x}}\end{array}$

Given:

For first year

l_x=248

d_x=96

w_x=27

Substitute the values in the given formula for column 5:

$\begin{array}{c}l{\text{'}}_x=l_x-\left(\frac{1}{2}\times w_x\right)\\ =248-\left(\frac{1}{2}\times 27\right)\\ =248-13.5\\ =234.5\end{array}$

Substituting the values in the given formula for column 6:

$\begin{array}{c}{q}_x=\frac{d_x}{l{\text{'}}_x}\\ =\frac{96}{234.5}\\ =0.409\end{array}$

Substituting values in the given formula for column 7:

$\begin{array}{c}{\text{p}}_{\text{x1}}{\text{=1-q}}_{\text{x}}\\ =1-0.409\\ =0.591\end{array}$

For second year

l_x=125

d_x=55

w_x=13

Substitute the values in the given formula for column 5:

$\begin{array}{c}l{\text{'}}_x=l_x-\left(\frac{1}{2}\times w_x\right)\\ =125-\left(\frac{1}{2}\times 13\right)\\ =125-6.5\\ =118.5\end{array}$

Substituting the values in the given formula for column 6:

$\begin{array}{c}{q}_x=\frac{d_x}{l{\text{'}}_x}\\ =\frac{55}{118.5}\\ =0.464\end{array}$

Substituting values in the given formula for column 7:

$\begin{array}{c}{\text{p}}_{\text{x2}}{\text{=1-q}}_{\text{x}}\\ =1-0.464\\ =0.536\end{array}$

For third year

l_x=57

d_x=55

w_x=2

Substitute the values in the given formula for column 5:

$\begin{array}{c}l{\text{'}}_x=l_x-\left(\frac{1}{2}\times w_x\right)\\ =57-\left(\frac{1}{2}\times 2\right)\\ =57-1\\ =56\end{array}$

Substituting the values in the given formula for column 6:

$\begin{array}{c}{q}_x=\frac{d_x}{l{\text{'}}_x}\\ =\frac{55}{56}\\ =0.982\end{array}$

Substituting values in the given formula for column 7:

$\begin{array}{c}{\text{p}}_{\text{x3}}{\text{=1-q}}_{\text{x}}\\ =1-0.982\\ =0.018\end{array}$

Add all the calculated values to the given table:

(1)	(2)	(3)	(4)	(5)	(6)	(7)
Interval since beginning of treatment (Months)	Alive at Beginning of interval	Died during interval	Withdrew during interval	Effective number exposed to Risk of Dying during interval	Proportion who died during interval	Proportion who did not die during interval
x	lx	dx	wx	l’x	qx	px
1-12	248	96	27	234.5	0.409	0.591
13-24	125	55	13	118.5	0.464	0.536
25-36	57	55	2	56	0.982	0.018

The probability that a person enrolled in the study will survive till the end of third year is:

$\begin{array}{c}{p}_x=p_{x1}\times p_{x2}\times p_{x3}\\ =0.591\times 0.536\times 0.018\\ =0.0057\\ =0.006\end{array}$

Conclusion

The probability that a person enrolled in the study will survive till the end of third year is 0.006.