Accounting Chapter 14 Homework We include only age and gender in the model along

434 CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DAT

A

14.40 Since we are controlling for covariates, we need to use a Cox proportional hazards model. We note that,

due to the low number of events, this model cannot be fit while using “side” as a covariate. Because of

——————————————————————————

_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]

————-+—————————————————————-

_Igroup_1 | 2.452794 1.844971 1.19 0.233 .5615579 10.71341

Cox regression — Breslow method for ties

No. of subjects = 57 Number of obs = 57

——————————————————————————

14.41 For this question, we use the new definition of success, and use for our “Time” variable, the time of the

second consecutive follow-up visit at which an improvement of at least 50% was reported. Selected

values from this new data is shown below.

Obs ID group side gender age Success Time

1 1 3 L 1 48 1 2

2 2 2 0 3

3 3 3 1 62 0 2

4 4 3 1 42 1 3

CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DATA 435

——————————————————————————

70 total obs.

1 obs. end on or before enter()

Log-rank test for equality of survivor functions

| Events Events

group | observed expected

——+————————-

1 | 13 12.31

Log-rank test for equality of survivor functions

| Events Events

group | observed expected

——+————————-

2 | 1 6.46

436 CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DAT

A

Log-rank test for equality of survivor functions

| Events Events

group | observed expected

——+————————-

1 | 13 15.71

14.42 Again fitting a Cox proportional hazards regression model to our new data (using age and gender as

covariates), we find no significant treatment differences, though the comparison between the Botox

group and the placebo group (Group 3 vs. Group 2) approaches significance (p=0.09), and estimates that

Botox injections are approximately 6 (CI: 0.8- 47) times more likely to result in a “success” at any given

time, relative to a placebo injection.

. xi: stcox i.group age gender

i.group _Igroup_1-3 (naturally coded; _Igroup_1 omitted)

——————————————————————————

_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]

CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DATA 437

——————————————————————————

_t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]

————-+—————————————————————-

_Igroup_1 | 3.334792 3.479857 1.15 0.248 .4313598 25.78089

14.43 The estimated incidence rate in the ASA group 34 50,000 68

events per 105 person-years compared

14.44 To compare the incidence rates we have

 

285 50,000 250,000 285 50,000 250,000 39.58 5.

14.46 To obtain a 95% CI for RR we first compute 11

ln 0.183

34 251

se RR



ªº

§·



¨¸

«»

©¹

¬¼

. Then, a 95% CI for

438 CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DAT

A

14.47 In this case, we have

We have

12

2, 20,

aa

14.48 The estimated mortality rate for current smokers is 3602/420,761 = 8.56 per 1000 person-years.

14.50 To compare the mortality rates we have

















(889 3602) 124,095 420,761 (4491) 124,095 420,761 789.9

14.51 The estimated rate ratio for former smokers who quit 20+ years ago vs. current smokers is

14.52 Yes, the age-adjusted rate ratio of 0.34 is lower than the crude rate ratio of 0.59 because age is a positive

CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DATA 439

14.54 For each group, the 10-year survival probability is estimated by

1

ˆ(10) (1 )

j

d

SS



.

14.56 For our log-rank test, we have the test statistic

We note that at the first time point (1990), there are no observed failures in either group, and so the data

from 1990 do not contribute to the log-rank test. The remaining data is shown below.

440 CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DAT

A

14.57 The hazard rate in this study is the instantaneous probability of death from COPD for any person who is

currently alive at a given time.

14.58 The proportional hazards model is written

h(t) h

0

(t)exp(

E

j

x

j

)

¦

, where h(t) refers to the hazard rate

for a particular person with covariates (x) at a given time t. Here, h0(t) refers to a baseline hazard

function, and coefficients Ⱦj describe the effect of each individual covariate xj on the overall hazard.

“Proportional hazards” refers to the assumption that the effect of any given covariate on the hazard

14.59 The hazard ratio is estimated to be exp(0.166) = 1.18, with a 95% confidence interval of

14.60 For this comparison, we assume the observed effect of smoking for men and women are both normally

CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DATA 441

So then,

14.62 In order to have 90% for this type of study we would require

14.63 For this analysis, we will use the nonparametric log-rank test. The resulting Kaplan-Meier curves are

Distribution Analysis: foluptm by pmh

Comparison of Survival Curves

442 CHAPTER 14/HYPOTHESIS TESTING: PERSON-TIME DAT

A

14.64 For this analysis, we calculate the median duration of use among the current PMH users to be 48 months

for estrogen and 24 months for estrogen + progesterone. We then create indicator variables “longEst” and

Distribution Analysis: foluptm_3 by longEst_3

Variable: foluptm_3

Distribution Analysis: foluptm_3 by longEst_3

Variable: foluptm_3

longEst_3 = 1

Distribution Analysis: foluptm_3 by longEst_3

Distribution Analysis: foluptm_3 by longPro_3

Variable: foluptm_3

longPro_3 = 0

Distribution Analysis: foluptm_3 by longPro_3

Variable: foluptm_3

Distribution Analysis: foluptm_3 by longPro_3

Comparison of Survival Curves

14.65 Below, we use STATA to compare the two groups with respect to all available covariates, using two-

sample binomial tests and non-parametric Wilcoxon rank-sum test for all other variables, using the .prtest

and .ranksum commands. First, we make sure to recode the values in column ‘afb’ so that values of 98

are changed to missing, using the following code.