The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

190 Solutions to Selected Exercises

ρ=

{}−+

{}−+

=

=

∑

∑

RX RY n n

RX n n R

i ii

n

i i

n

()() ([ ])

() ([ ]) (

1

2

2

1

2

12

12

/

/ i YYnni

n

)([])

.

2

1

122

{}∑= −+/

In the case of no ties, this formula simplifi es to

(^) ρ=− 16 Tnn/[ (^2 − 1 )},
where TRXRY=∑in= 1 [( ) ()]ii−^2.
Since we have ties, we cannot use the simplifi ed formula.
So in this example, the estimate
ρ=∑{()()}[{() }{()^12 iii=== 1 RX RY− 507 /5∑^12 ii 1 RX^2 − 507 ∑^12 ii 1 RY^2 − 507
535 0 507 0 649 0 507 0 649 0 507 0
28 142^2

}]

(. .)(. .)(. .)

()

=− − −

==

/

/228 20 164 0 001389/,..=

The correlation is negligible, meaning there is practically no relationship between

order of birth and aggressiveness of the twins. A statistical test would show that

the correlation is not statistically signifi cantly different from 0. So the twins tend

to be similar in the amount of aggressiveness shown in their scores.

Chapter 10

2. If the survival function S ( t ) = 1 − t / b for 0 ≤ t ≤ b , where b is a fi xed positive
   constant, and S ( t ) = 0 for t > b , calculate the hazard function. When is the
   hazard function lowest? Is there a highest rate?
   F ( t ) = 1 − S ( t ) = 1 − ( 1 − t / b ) = t / b for 0 ≤ t ≤ b in this case, and F ( t ) = 1 for
   t > b. Now f ( t ) = dF ( t )/ dt = 1 / b for all 0 ≤ t ≤ b and f ( t ) = 0 otherwise. Now the
   hazard function is defi ned as h ( t ) = f ( t )/ S ( t ) = 1/[ bS ( t )] = 1 / b (1 − t / b ) = 1/( b − t )
   for 0 ≤ t ≤ b. At t = 0, the hazard function is 1/ b , and that is its lowest value. For
   b > t > 0 h ( t ) = 1/( b − t ) > 1 / b , since b > b − t. So the hazard function is increas-
   ing. But as t → b , b − t → 0, and h ( t ) = 1/( b − t ) → ∞. So there is no maximum
   value for the hazard function.


3. Suppose the survival time since a bone marrow transplant for eight patients
   who received the transplant is 3, 4.5, 6, 11, 18.5 20, 26, and 35. No observa-
   tions were censored.
   (a) What is the median survival time for these patients?
   (b) What is the mean survival time?
   (c) Construct a life table where each interval is 5 months.

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