HOMEWORK FOR EXST 7036
(Chapter 4)
Student Name: Tanza Erlambang
4.1)A study used logistic regression to determine characteristics associated with Y =
whether a cancer patient achieved remission (1= yes). The most important explanatory
variable was a labeling index (LI) that measures proliferative activity of cells after a
patient receives an injection of triated thymidine. It represents the percentage of cells
that are “labeled”. Table 4.8 shows the grouped data. Sofware reports Table 4.9 for a
logistic regression model using LI to predict π = P(Y=1)
a.
Show how software obtained
π
= 0.068 when LI = 8
Answer
:
π
=e
α+β(LI)
/[1+e
α+β(LI)
]=e
3.777+0.145(8)
/ [1+e
3.777+0.145(8)
]= 0.073/1.073= 0.068
b
. Show that
π
= 0.50 when LI= 26.0
Answer
: at 
α
/
β
= 3.777/ 0.145 = 26.0
c.
Show that the rate of change in
π
is 0.009 when LI = 8 and is 0.036 when LI = 26
Answer:
At LI =8,
π
=0.068 →rate of change=
βπ
(1
π
)= 0.145(0.068)(10.068)= 0.0092
At LI =26,
π
=0.5 →rate of change=
βπ
(1
π
)= 0.145(0.5)(10.5)= 0.036
d
. The lower quartile and upper quartile for LI are 14 and 28. Show that
π
increase by 0.42
from 0.15 to 0.57 between those values.
Answer:
LI= 14 →
π
=e
α+β(LI)
/[1+e
α+β(LI)
]=e
3.777+0.145(14)
/[1+e
3.777+0.145(14)
]=0.1743/1.1743= 0.148
LI= 28 →
π
=e
α+β(LI)
/[1+e
α+β(LI)
]=e
3.777+0.145(28)
/[1+e
3.777+0.145(28)
]=1.327/2.327= 0.57
The increase between upper and lower quartile= 0.57 0.148 = 0.422
1
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e
. When LI increase by 1, show the estimate odds of remission multiply by 1.16.
Answer
: e
β
= e
0.145
= 1.16
4.2) Refer to the previous exercise. Using information from Table 4.9:
a.
Conduct a Wald test for the LI effect. Interpret
Answer
: Wald Statistics : (β/SE)
2
= (0.145/ 0.059)
2
= 6.04, → chisquare = z
2
= 6.04 with
df=1 → see Chisquare Table: p= 0.01. Thus, HO: β= 0 is rejected→there is effect of
treatment on patient.
b
. Construct a Wald CI for the odds ratio corresponding to a 1unit increase in LI. Interpret
Answer
: Odds ratio = e
β
= e
0.145
= 1.16
CI → e
β
± Z
α/2
(SE) = 1.16 ± 1.96 (0.059), thus the CI is ( 1.044, 1.276)
So, odds of remission at LI = x + 1 are estimated between 1.044 and 1.276 times the odds of
remission at LI= x.
c.
Conduct a likelihoodratio test for the LI effect. Interpret
Answer
: Likelihoodratio statistics → See Table 4.9 → is (8.30), with df=1 → p<0.01
→there is effect of treatment on patient.
d
. Construct the likelihoodratio CI for the odds ratio. Interpret
Answer
: From Table 4.9 (under column “Likelihood ratio 95% Conf.Limits”), we find the
lower and upper limits limits (0.0425 and 0.2846). Then, exponentiating : e
0.0425
and e
02846
are
(1.04, 1.33)→ Thus, the odds of remission at LI=x+1 are estimated to fall between 1.04 and
1.33 times the odds of remission at LI=x
4.9) For the horseshoe crab data, fit a logistic regression model for the probability of a
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 Spring '10
 BrianMarx
 Regression Analysis, Logit, Likelihoodratio test, Parameter Estimates Parameter, Confidence Error Limits

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