01/20151 EPI 5344: Survival Analysis in Epidemiology Interpretation of Models March 17, 2015 Dr. N. Birkett, School of Epidemiology, Public Health & Preventive.

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EPI 5344:Survival Analysis in

EpidemiologyInterpretation of Models

March 17, 2015

Dr. N. Birkett,School of Epidemiology, Public Health &

Preventive Medicine,University of Ottawa

2

Objectives

• Epidemiology often tries to determine risk associated

with ‘exposures’ or like events

• Various types of exposures– nominal

– ordinal

– count

– continuous

• Session looks at how to use these different types of

exposures in Cox models

• And, how to interpret the results01/2015

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Nominal variables (1)

• Categories– Male/female– Occupation– City of residence

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Nominal variables (2)

• Can not be rank ordered– If you ‘impose’ a ranking, then you are defining a new

variable.– No intrinsic numerical value

• Even if they are commonly coded as 1/2, they are NOT NUMERIC

• Use of 1/2/3 coding dates back to early days of computers• Can assign letters or words to categories (e.g.

Male/Female)

– Usually analyzed using dummy variables• Indicator variables is a better term.

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Nominal variables (3)

• SAS must be aware that these variables are categories, not numbers– CLASS statement

• Indicator variables– ‘n’ response levels ‘n-1’ indicator variables– Why?

• The MLE equations are indeterminate if you include ‘n’ indicator variables.

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• Consider a two-level nominal variable– e.g. M/F

• Some possible IV codings:

• All (and many more) are possible.• #1 is the most commonly used

Nominal variables (4)

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Level #1 #2 #3 #4

1 (M)

2 (F)

Level #1 #2 #3 #4

1 (M) 1

2 (F) 0

Level #1 #2 #3 #4

1 (M) 1 1

2 (F) 0 -1

Level #1 #2 #3 #4

1 (M) 1 1 2

2 (F) 0 -1 1

Level #1 #2 #3 #4

1 (M) 1 1 2 10

2 (F) 0 -1 1 -18

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• Suppose we have 3 levels (blue/green/red)

• Some possible Indicator Variable codings:

Nominal variables (5)

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Reference Effect OrthogonalPolynomial

Level #1X1 X2

#2X1 X2

#3X1 X2

1 (blue)

2 (green)

3 (red)

Level #1X1 X2

#2X1 X2

#3X1 X2

1 (blue) 1 0

2 (green) 0 1

3 (red) 0 0

Level #1X1 X2

#2X1 X2

#3X1 X2

1 (blue) 1 0 1 0

2 (green) 0 1 0 1

3 (red) 0 0 -1 -1

Level #1X1 X2

#2X1 X2

#3X1 X2

1 (blue) 1 0 1 0 -1.225 0.707

2 (green) 0 1 0 1 0 -1.414

3 (red) 0 0 -1 -1 1.225 0.707

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Ordinal variables

• Still categorical (name) type data.

• But, there is an implicit ordering with the levels.– Disease severity

• mild, moderate, severe

– Rating scale• Agree/don’t care/disagree

• Interval data which has been categorized is really ‘ordinal’, not

interval.

• Commonly treat as nominal variables and ignore rank order

• Can use ordinal variables for trend or dose response analyses– Key issue: determining the spacing/numerical values to use

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Interval variables

• Has a meaningful ‘0’

• Can assume any value (perhaps within a range)– Infinite precision is possible.

• Examples– Blood pressure

– Cholesterol

– Income

• Usually treated as a continuous number

• Can categorize and then use ordinal methods– Can reduce measurement errors

– Useful for EDA to look for non-linear dose-response.

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Count variables

• The number of times an event has happened– # pregnancies

– # years of education

• Very common type of data in medical research

• Often regarded as continuous but isn’t really

• Can affect choice of model– Poisson distribution

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Parameter interpretation (1)

• Cox models Hazard Ratios not the hazard– Can not provide a direct estimates of S(t) or

h(t)– Needs additional assumptions or methods– Next week

• Cox model:• Contains no explicit intercept term

– That is the h0(t)

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Parameter interpretation (2)

• To explore Betas, we will use this form of

the Cox model:

• Start with 2-level nominal variable

• Consider 3 different coding approaches

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Parameter interpretation (3)

• We will have only one parameter.

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xi= 1 if exposed

= 0 otherwise

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H0: β1=0 Wald Score Likelihood ratio

95% CI

or

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xi= 1 if exposed

= -1 otherwise

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• Changing the coding has changed the interpretation of

the Beta value

• ‘Effect’ coding compares each level to the mean effect

• ‘sort of’ since it is dependent on the number of subjects in

each level.

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xi= 2 if exposed

= 1 otherwise

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• Changing the coding doesn’t always change the

interpretation of the Beta value

• Need to be aware of the coding used in order to correctly

interpret the model

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Xi is continuous (interval)

Let’s consider the effect of changing the exposure by ‘1’ unit

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So,β1 relates to the HR for a 1 unit change in x1.What about an ‘s’ unit change?

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Xi is continuous (interval)

Let’s consider the effect of changing the exposure by ‘s’ unit

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• SAS has an option to do this automatically.• Important to consider the level for a

‘meaningful’ change

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Level #1X1 X2

#2X1 X2

#3X1 X2

1 (blue) 1 0 1 0 -1.225 0.707

2 (green) 0 1 0 1 0 -1.414

3 (red) 0 0 -1 -1 1.225 0.707

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‘reference’ coding; level 3=ref

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‘effect’ coding

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• Effect coding– βi does NOT let you compare two levels

directly– Use ‘Reference Coding’ to do that

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‘orthogonal polynomial’ coding

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• Doesn’t look that useful or interesting • This coding can be used to test for linear

and quadratic effects in the dose-response relationship• Test β1 to look for a linear effect

• Test β2 to look for a quadratic effect

• Can be extended to higher orders if there are more levels.

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Hypothesis testing (1)

• Does adding new variable(s) to a model lead to a

better model?– Test the statistical significance of the additional variables.

• Let’s start with 1 variable

• The likelihood is a function of the variable

– L = f(β1)

• To test β1=0, compare the likelihood of the model with

the MLE estimate to the ‘β1=0’ model

– f(0) vs. f(βMLE)01/2015

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Hypothesis testing (2)

• As is common in statistics, this gives the best results if you:– Take logarithms– Multiple by -2

• Likelihood ratio test of H0: β1=0

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Hypothesis testing (3)

• Can extend to test the null hypothesis that ‘k’ parameters are simultaneously ‘0’.

• Likelihood ratio test of H0: β1=β2=….. βk=0

• Can be used to compare any two models– BUT, Models must be Nested or hierarchical

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Hypothesis testing (4)

• Nested– All of the variables in one model must be

contained in the other model• #1: x1, x2, x3, x4

• #2: x1, x2

– AND: Analysis samples must be identical• Watch for issues with missing data in extra variables

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Hypothesis testing (5)

• You can also do many of these tests using the Wald and Score approximations– Hopefully, you get the same answer

• Usually do (at least, in the same ballpark)

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Confidence Intervals (1)

• Always based on the Wald Approximation• First, determine 95% CI’s for the Betas

• Second, convert to 95% CI’s for the HR’s

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