Advance Epidemiology II Survival Analysis: Cox Proportional Hazard Model

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Advance Epidemiology II Survival AnalysisCox
Proportional Hazard Model
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• John Graunt (1662)- Natural and Political
  Observations upon the Bill of Mortality
• Report on registrations of births and deaths in
  London
• Edmund Halley (1687-1691)
• Developed first life table using data from a
  population in Poland
• s.e. formula for the survival probability
  developed in 1926
• Biological, epidemiological, medical and industry
  research have stimulated this field in the last
  60 years
• In the last 30 year the application of survival
  analysis has increased in clinical research

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• Survival analysis in biomedical research deals
  with
• Estimation of failure time distributions
• Comparison of survival of different groups and
  estimation of treatment effect
• Prognostic evaluation of different
  factors/variables
• Univariate
• Multivariate

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• The term "survival analysis" is slightly
  misleading
• It is used when the objective is to study the
  time elapsed from a particular starting point to
  the occurrence of an event
• may not be related to survival and death
• Time to death
• Time to relapse
• Time to remission
• Time to disease to develop
• Time to symptom to develop

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• A common problem in this type of data is
  censoring
• time-to-event is not observed
• Survival Analysis accounts for the fact that we
  almost never observe the event of interest in all
  subjects
• We do not know if they will develop the event of
  interest only that they have not presented the
  event of interest at the end of the study
• Censoring / survival time censored / censored data

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• Types of censoring
• Type I censoring
• Experimenter terminates observation at certain
  time point of follow-up
• (2) Type II censoring
• Experimenter terminates observation when a number
  r of failures/events is reached
• In this case failures/events are not a random
  variable

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• In clinical and epidemiological research
  censoring is caused by time restriction (Type I
  censoring)
• Two major sources of censoring
• lost to follow-up/drop-outs
• patient is alive at the last contact but his
  subsequent survival status is unknown
• time to last contact censoring time
• (2) patients/subjects are alive at the time of
  study closure
• time to study closure censoring time

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• If the study period were long enough to observe
  the survival time of all subjects (e.g., some
  animal experiments) a more common method of
  analysis for continuous variables would be
  appropriate
• t-test
• ordinary least square regression

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• In studies of human subjects there is often
  censoring and the outcome cannot be analyzed by
  the usual methods for continuous data
• Subjects are often censored at different times
• leading to unequal follow-up time
• Analysis of the probability of survival during
  the study period as a dichotomous variable (alive
  vs dead), e.g. by a Chi-square test, would fail
  to account for this non-comparability between
  subjects

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Time (months)
0
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12
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Patient Accrual
Observation Period
Diagram showing patients entering the study at
different times and the observation of known (
?) and censored ( ?) survival times
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• Different ways to measure the timing of an event
  
• calendar time since the baseline survey
• age at death (time since birth)
• first day of treatment
• day of randomization

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• Entry period
• Must be clearly defined based on study objectives
• Date of diagnosis
• Date of treatment/Date of Randomization
• Date of recurrence
• Date of beginning/detection of a exposure
• Date of birth

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• End-point
• Based on study objectives
• Deaths from any causes
• Death from disease
• Recurrence of disease
• Other end-points
• Development of disease
• Detection of intermediate end-point
• Detection of biomaker
• Persistence or regression of a disease process

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Time (months)
0
6
12
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Patient Accrual
Observation Period
Diagram showing patients entering the study at
different times and the observation of known (?)
and censored (?) survival times
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Time (months)
0
6
12
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Patient Accrual
Observation Period
Diagram showing patients reorganized by survival
time
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• Survival probability
• Proportion of the population that survives a
  given length of time in the same circumstances
• Assumption in survival analysis
• In a sample of N individuals observations are
  independent
• The random variables survival time and
  censoring time are independent
• Special feature of survival analysis
• Covariates may be time dependent (e.g. age,
  calendar period of observation)

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• Methods of survival analysis
• Accommodate censoring
• Account for different periods of observation on
  each experimental unit
• Account for time at which event occur

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Planning, Reporting, and Interpreting Survival
Analysis

• Sample size
• Follow-up
• Date of entry or beginning of F-U
• Date of end of F-U
• Cut-off date for analysis
• Summary of F-U (median F-U)
• Percent of censored data

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Planning, Reporting and Interpreting Survival
Analysis

• Clear Definition
• Entry point
• end-points
• Losses to F-U
• Competing events (e.g. deaths from other causes)
• Relapse-free survival, disease-free survival,
  remission duration (only patients responding to
  treatment are analyzed)
• Progression-free survival (all patients are
  analyzed)

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Planning, Reporting, and Interpreting Survival
Analysis

• Explanatory variables/Prognostic factors
• Number and definition variable
• Coding of variables
• Presence of missing values
• Cut-off for categorization of continuous
  variables

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Planning, Reporting, and Interpreting Survival
Analysis

• Graphical Representations
• Quality of graphs
• Scales
• Marks for censoring times
• Clear identification of groups
• Avoid over-interpretation of the right hand of
  the curve
• Survival curve shows the pattern of mortality in
  time, not the details

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Planning, Reporting, and Interpreting Survival
Analysis

• Type of analysis
• Univariate analysis
• Only one explanatory variable at a time
• Method of analysis
• Test for comparison of groups
• Median survival time should be reported when
  possible
• Estimates the time period beyond which 50 of
  patients are expected to survived in the study
  population

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Planning, Reporting, and Interpreting Survival
Analysis

• Type of analysis (continue)
• Multivariate analysis
• At least two explanatory variables
• Method of analysis
• Statistical software used

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Methods of Estimation of survival Probability

• Product Limit Method (Kaplan-Meier method)
• Time variable continuous
• Life-Table Method (Actuarial method)
• Time variable grouped

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Methods for Comparing Survival Curves

• Kaplan-Meier method (1958)
• non-parametric method
• very often is the primary comparison between
  treatment and control groups in cancer clinical
  trials
• Kaplan-Meier method produces survival curves

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Methods of Estimation of survival Probability

• Log Rank test
• most common method to compared independent groups
  of survival times
• test the H0 that two survival curves are
  identical
• based on comparison of observed and expected
  death
• expected number calculated under the assumption
  of no difference in survival between groups
• appropriate when relative mortality does not
  change over time

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Methods of Estimation of survival Probability

• Hazard Ratio
• Provides information on the magnitude of the
  difference between groups
• Measures relative survival comparing the observed
  number of events with the expected number of
  events

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Cox Proportional Hazard Model
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Cox Proportional Hazard Model

• Most used regression method for analysis of
  censored survival data
• Introduced by Cox in 1972
• Used for
• Identification of differences in survival due to
  treatment effect or prognostic (risk,
  explanatory, or predictor) factors
• Control of confounding in cohort studies

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Cox Proportional Hazard Model

• Given a set of p covariates (explanatory
  variables) xi (x1i, x2i,xpi)
• The hazard function for a given individual i is
  modeled by
• hi(t, Xi) h0(t) e?ßiXi

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Cox Proportional Hazard Model

• The hazard is the product of two quantities
• An arbitrary nonnegative baseline hazard h0(t)
• An exponential linear function of the p
  covariates (explanatory/predictor/prognostic
  variables)
• The model is nonparametric because h0(t) is
  unspecified
• Baseline hazard is a function of t but does not
  involve the Xs

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Cox Proportional Hazard Model

• The linear function
• ?ßiXi ß1x1i ß2x2i . ßpxpi
• also known as prognostic index for the ith
  individual
• The linear function is exponentiated to insure a
  nonnegative hazard
• Linear function involves Xs but does not involve
  t
• time independent variables
• If Xs involve t, Xs are called time-dependent
  variables
• Extended Cox Model

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Cox Proportional Hazard Model

• Time independent variables
• Values do not change over time
• Gender
• Values are only measured once
• Values are assumed not to change once they are
  measured
• Age
• Smoking status
• weight

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Cox Proportional Hazard Model

• Properties of Cox PH Model
• If all Xs 0 formula reduces to the baseline
  hazard function h0(t)
• x1 x2 . xp 0
• h(t, Xi) h0(t) e?ßiXi
• h0(t) e0
• h0(t)

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Cox Proportional Hazard Model

• Properties of Cox PH Model
• The baseline hazard function h0(t)
• Unspecified function
• nonparametric method

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Cox Proportional Hazard Model

• Properties of Cox PH Model
• Even when the h0(t) is unspecified
• Still possible to estimate ßs to assess effect
  of predictor/explanatory variables
• hazard ratios calculated without estimates of
  h0(t)

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Cox Proportional Hazard Model

• Properties of Cox PH Model
• Form of the model
• exponential part of the model (e?ßiXi) ensures
  hazard estimates that are not negative
• Hazard function ranges between 0 and plus 8

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Cox Proportional Hazard Model

• Properties of Cox PH Model
• Hazard Function h(t, X) and corresponding
  survival curve S(t, X) can be estimated using
  minimum assumptions
• even if the baseline hazard baseline function
  h0(t) is not specified

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Cox Proportional Hazard Model

• Properties of Cox PH Model
• Prefer over logistic regression model when
  survival time is available and there is censoring
• Cox model uses more information (survival time)
• Logistic regression uses (1,0) outcome and
  ignores survival time and censoring

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Cox Proportional Hazard Model

• Estimation of Cox PH Model Parameters (ß coeff)
• Parameters are called ML estimates
• ßi hat
• Derived by maximizing the likelihood function L
• L describes the joint probability of obtaining
  the data observed in the study subjects as a
  function of the unknown parameters (ßs) in the
  model
• L Joint probability of observed data
• L sometimes written as L (ß)

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Cox Proportional Hazard Model

• Estimation of Cox PH Model Parameters (ß coeff)
• Cox models L is a partial likelihood function
• L does not consider probabilities for all study
  subjects
• only probabilities for those subjects who fail
• but not for those who are censored

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Cox Proportional Hazard Model

• Estimation of Cox PH Model Parameters (ß coeff)
• L is the product of several likelihoods, one for
  each k failure times
• L L1 x L2 x L3 x x Lk ? Lj
• Lj denotes likelihood of failing at jth failure
  time, given survival up to this time, risk set
  R(t(j))
• R(t(j)) over time as failure time

K
j 1
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Cox Proportional Hazard Model

• Estimation of Cox PH Model Parameters (ß coeff)
• L focus on subjects who fail, but the survival
  time of censored subjects is use
• L is then maximized by maximizing the natural log
  of L
• Taking partial derivatives of L with respect to
  each parameter in the model
• ?L / ?ßi 0, i 1.p ( of parameters)

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Cox Proportional Hazard Model

• Estimation of Cox PH Model Parameters (ß coeff)
• Solution is obtained by iteration- in stepwise
  manner
• Guess a value for the solution
• Modifies guess value in successive steps
• Stops when solution is obtained

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Cox Proportional Hazard Model

• Estimation of Cox PH Model Parameters (ß coeff)
• Once ML estimates obtained
• Statistical inference about the hazard ratio
• Wild test
• LR test
• 95 CI
• Estimated HR computed by exponentiating the ß
  coefficient (0,1) of a explanatory variable of
  interest
• HR eß

?
?
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Cox Proportional Hazard Model

• Computing the Hazard Ratio
• Hazard for one individual divided by the hazard
  of another individual
• Individuals are distinguished by their values of
  the predictor variables of interest (Xs)
• HR h(t,X) / h(t,X)
• where, X (X1, X2,. Xp)
• and, X (X1, X2,. Xp)
• denote the set of predictors for two individuals

?
?
?
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Cox Proportional Hazard Model

• Computing the Hazard Ratio
• Easier to interpret a HR that exceeds 1
• HP Formula in terms of regression coefficients
• X (X1, X2,. Xp), Unexposed group (X1
  0)
• X (X1, X2,. Xp), Exposed group (X1 1)
• HR h(t,X) / h(t,X)
• h0(t) e?ßiXi / h0(t) e?ßiXi
• e?ßi(Xi - Xi)

?
?
?
We Obtain HR formula substituting the Cox model
formula with the regression coefficients in the
numerator and denominator.
?
p
p
?
?
?
I1
I1
p
?
I1
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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example

Mitchell MF, Tortolero-Luna G, et al, Ob Gyn, 1998
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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Background
• In patients with biopsy-proven squamous
  intraepithelial lesions (SIL) of the cervix who
  have negative findings on endocervical curettage,
  a satisfactory colposcopy examination, and
  concordant Papanicolaou smear and biopsy results,
  ablation of the transformation zone has been the
  standard of care for several decades. Several
  techniques for ablation are available
  cryotherapy, laser ablation, and loop excision of
  the transformation zone, also called the loop
  electrosurgical excision procedure

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Cryotherapy
• Introduced in 1972
• Advantages reliability, ease of use, and low
  cost
• Disadvantages lack of ability to tailor treatment
  to the size of the lesion and lack of a tissue
  specimen

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Laser vaporization
• Introduced in 1977
• Advantage easily tailored to lesion size
• Disadvantages cost and lack of tissue specimen

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Loop electrical excision procedure (LEEP)
• Introduced in 1989
• Advantages reliable, easy to use, can be
  tailored to lesion size, and provides a tissue
  specimen
• Disadvantage of increased risk of bleeding and
  infection, and an increased cost
• Bleeding after treatment with LEEP have been
  reported in 2 to 7 of cases

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Rationale for Study
• No previous randomized clinical trial has
  assessed the effectiveness all three-treatment
  modalities
• Objectives of the Study
• to compare differences in the rates of persistent
  and recurrent disease among treatment modalities
• to identify predictor factors for persistent and
  recurrent disease
• to assess differences in complications rate among
  treatment groups

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Study Population
• Women referred to the University of Texas M. D.
  Anderson Cancer Center Colposcopy clinic with a
  diagnosis of cervical intraepithelial neoplasia
  (CIN), between March 1992 and April 1994

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Eligibility Criteria
• non-pregnant
• 18 years of age or older
• using a contraceptive methods
• biopsy-proven CIN lesion
• negative endocervical curettage (ECC)
• satisfactory colposcopic examination
  (visualization of squamocolumnar junction and
  entire of the lesion
• consistent Pap smear and biopsy result

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Exclusion Criteria
• suspicion or evidence of invasive cervical
  lesions on Pap smear, biopsy, or colposcopic
  examination
• suspicion of pregnancy
• current pelvic inflammatory disease, cervicitis
  or other gynecological infection

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Data Collection
• Complete medical history
• Physical examination
• Pan-colposcopy of the vulva, perineum, vagina,
  and cervix
• Colposcopically directed biopsies
• human papillomavirus (HPV) testing by
  Virapap/Viratype assay (Digene Diagnostics, Inc.,
  Silverspring, MD)

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Randomization
• Eligibility patients randomly assigned to one of
  the treatment groups following a stratified
  assignment schedule
• grade of CIN (1, 2, or 3)
• endocervical gland involvement
• lesion size (less than one-third, one- to
  two-thirds, or greater than two-thirds of the
  surface area of the cervix)
• Patients were scheduled for treatment within 7-15
  days from initial evaluation

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Follow-up Schedule
• 1 month after treatment
• Every 4 months for 2 years (4, 8, 12, 16, 20, and
  24 month post-treatment)
• Follow-up Evaluation
• complete physical and pelvic examination
• colposcopic exam
• Pap smear and colposcopic directed biopsies as
  needed
• Complications
• bleeding, infection (fever, vaginal discharge),
  visits to the emergency room, and use of pain
  medication within 24 hours post-treatment or
  latter
• evaluated for cervical stenosis

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Outcomes
• persistent and recurrent disease
• assessed on the bases of cytology and histology
• cytologist and pathologist were blinded to
  treatment assignment
• Definitions
• Persistent disease cytological or histological
  presence CIN at the time of their second
  follow-up visit (within 6 months after treatment)
• Recurrent disease cytological or histological
  presence CIN diagnosed at a subsequent follow-up
  visit (6 months after treatment) in a patients
  who had at least one negative cytological smear
  after treatment

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• to determine the association between treatment
  modality and other covariates of interest and
  disease-free survival
• age
• HPV status
• smoking habits
• grade of disease
• glandular involvement
• size and location of lesion
• history of prior treatment for CIN

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Cox Proportional Hazard Model

• Computing the Hazard Ratio Example
• Differences in disease-free survival
• (date of treatment date of recurrence)
• K-M method and the log-rank test
• Univariate and multivariate Cox proportional
  hazard models

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Mitchell MF, Tortolero-Luna G, et al, Ob Gyn, 1998
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• For the purpose of this presentation we will
  focus on the role of HPV status on recurrence of
  CIN
• Analysis was conducted using SPSS
• For previous model and each of the following
• 1st column variable(s) included
• 2nd column regression coefficients (B)
• 3rd column standard error of the coefficient
  (SE)
• 4th column Wald statistic (B/SE)
• 5th column degrees of freedom (df)
• 6th column p value for the Wald test (Z
  statistic)
• 7th column exp(B) (HR)
• 8th column 95 CI for the exp(B)

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When There is only one variable in the model the
estimated hazard ratio can simplify to
HR e?ß1(Xi - Xi)
e?ß1(1 - 0) eß1 e(.544) 1.722




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Control of Confounding
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Crude Model
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Control of Confounding

• Comparison of the HR of the crude model with the
  HR of the adjusted model
• If the estimated HR of the of the crude model
  meanignfully differs from the estimated HR of
  the adjusted model then there is confounding
• If confounding is present, we must control for
  the confounding variable to obtain a valid
  estimate of effect
• However, if a variable is not found to be a
  confounder of the effect, we might decide to
  include it in the model for other reason
  (biological meaning, previous studies, or to
  increase precision)

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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of Confounding
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Control of ConfoundingMultivariate Model
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Control of ConfoundingMultivariate Model
Crude Model -2 LL 650.283 HPV 16/18 1.973
(1.111-3.504) HPV Other 1.302 (0.623-2.717)
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Adjusted Overall Survival Function
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Adjusted Survival Function
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Model Selection in Cox Regression Analysis

• Similar to other modeling procedures
• Initial selection of variables based on
• Biological plausibility
• Literature
• Previous research
• Pilot data

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Model Selection in Cox Regression Analysis

• Step 1
• Assess the influence of each covariate of
  interest (univariate or bivariate analysis)
• Likelihood ratio test
• Wald test
• Using a significant level of 0.1 to .25

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Model Selection in Cox Regression Analysis

• Step 2
• All significant variables fitted in a
  multivariable model
• If we are interested on a particular variable the
  other covariates are regarded as confounders
• If some variables become no longer significant,
  assess the effect on the model of removing each
  variable at the time
• LRT and Wald test
• If a substantial changed occurred in the
  remaining coefficients or variable of interest
  (e.g., 20 or greater) add variable back to the
  model

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Model Selection in Cox Regression Analysis

• Step 3
• Add all variables found to be not significant in
  step1
• All variables found to be significant in this
  step, would be added to a new multivariable model
• If variables included in step 2 lose their
  significance, test them fro deletion
• Step 4
• Check that not covariate can be removed or
  removed to the model
• Stepwise procedures
• Backward
• Forward

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Selecting the Best Model
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Selecting the Best Model Backward Selection
Crude Model -2 LL 632.495 HPV 16/18 2.011
(1.103-3.666) HPV Other 1.212 (0.573-2.565)
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Selecting the Best Model Backward Selection
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Selecting the Best Model Backward Selection
Crude Model -2 LL 632.495 HPV 16/18 2.011
(1.103-3.666) HPV Other 1.212 (0.573-2.565)
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Selecting the Best Model Backward Selection
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Selecting the Best Model Backward Selection
Crude Model -2 LL 632.495 HPV 16/18 2.011
(1.103-3.666) HPV Other 1.212 (0.573-2.565)
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Selecting the Best Model Backward
StepwiseProcedure
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Selecting the Best Model Backward
StepwiseProcedure
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Selecting the Best Model Backward
StepwiseProcedure
Block 1 Method Backward Stepwise (Likelihood
Ratio)
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Selecting the Best Model Backward
StepwiseProcedure Step 1
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Selecting the Best Model Backward
StepwiseProcedure Step 2
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Selecting the Best Model Backward
StepwiseProcedure Step 3
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Selecting the Best Model Backward
StepwiseProcedure Step 4
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Selecting the Best Model
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Selecting the Best Model
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Adjusted Survival Curve
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Adjusted Survival Function