lecture 2lecturecontent.s3.amazonaws.com/pdf/14993.pdf · Lecture 3, part A: St atistical Reasoning...

Lecture 3, part A: Statistical Reasoning 2

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Simple Logistic Regression

Lecture 2


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Learning Objectives

In this set of lectures we will develop a framework for simple logistic regression, a method for relating a binary outcome to a single predictor that can be binary, categorical or continuous


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Section A

Simple Logistic Regression With a Binary (or Categorical) Predictor


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Learning Objectives

Understand how logistic regression relates a function of the probability (proportion) of a binary outcome to a predictor via a linear equation

Interpret the resulting intercept and slope(s) from a logistic regression model in which the predictor of interest is binary or categorical


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The “Left Hand Side”

For logistic regression, the equation is a bit more convoluted than with linear regression : the regression models the log odds of a binary outcome(y) as a function of the predictor x

Where p = proportion (probability) of y=1

As noted in the previous section, x can be binary, nominal categorical or continuous

11o1ln x

pp


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As with everything else we have done thus far, we will only be able to estimate the regression equation from a sample of data: to indicate the estimates, can write as:

In the next section, the reason for this choice of scaling will be explained

11oˆˆ

1ln x

pp


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For a given value of x1, we can estimate the ln(odds) via the equation

The slope compared the ln(odds of y=1) for two groups who differ by one unit of x1, and hence is interpretable as a difference in ln(odds) between two groups

11oˆˆ

1ln x

pp


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Difference in ln(odds)?


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Example 1: Breast Feeding and Sex

Data on anthropometric measures from a random sample of 236 Nepali children [0, 36) months old

Question: what is the relationship between breast feeding and sex of a child?

Data: Breast fed: 75% Sex: 52% female (1= male, 0 = female)


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Notice: this equation is only estimating two values: ln(odds of being breast fed for males) and ln(odds of being breast fed for females)

For male children:

For female children

10101ˆˆ1ˆˆ1x :ln Odds

0101ˆ0ˆˆ0x :ln Odds


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Interpretation: 1


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Interpretation: o


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The resulting equation

: the ln(odds ratio) of being breast fed for males to females is 0.002

: the ln(odds) of being breast fed for female children is 1.12

002.0ˆ1

12.1ˆ o

11oˆˆ

1ln x

pp


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Results, antilogged


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Question

The coding choice for a binary predictor is completely arbitrary. For this breast feeding arm circumference and sex analysis, what would the values of and be if sex was coded as a 1 for females, and 0 for males?

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o1


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Example 2: Respiratory Failure

Respiratory Failure and Gestational Age1

1 Respiratory Morbidity in Late Preterm Births: The Consortium on Safe Labor, JAMA. 2010;304(4):419-425


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Respiratory failure and gestational Age1

1 Respiratory Morbidity in Late Preterm Births: The Consortium on Safe Labor, JAMA. 2010;304(4):419-425

Gestational Age Percentage Total34 weeks 0.02 3,70035 weeks 0.03 5,47736 weeks 0.05 10,157

37‐40 weeks 0.90 165,993


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Even though the gestational age categories are ordinal, authors did not want to assume linearity of ln(odds) of respiratory failure and gestational age category

There are four categories: make one category the reference, and make binary xs indicators for the other 3. The authors used 37-40 weeks as the reference.

x1 = 1 if gestational age =34 weeksx2 = 1 if gestational age =35 weeksx3 = 1 if gestational age =36 weeks


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Respiratory failure and gestational age

332211oˆˆˆˆ

1ln xxx

pp


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In this model

40-37 agegest failure,y respirator of odds

34 agegest failure,y respirator of oddsln1


35 agegest failure,y respirator of oddslnˆ2


36 agegest failure,y respirator of oddslnˆ3

40-37 agegest failure,y respirator of oddslnˆ0


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321 0.28.24.35.51

ln xxxp

p


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321 0.28.24.35.51

ln xxxp

p


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Example 2: Summary

Logistic regression is a method for relating a binary outcome to a predictor x via a linear equation The predictor can be binary, categorical or continuous

The resulting linear equation relates the ln(odds) of the binary outcome to the predictor x

Slopes from logistic regression have ln(odds ratio) interpretation and can be exponentiated to estimate odds ratios

The intercept estimates the ln(odds) for the groups with x = 0


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Section B

Simple Logistic Regression With a Continuous Predictor


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Learning Objectives

Understand why transforming the estimated probability (proportion) that y=1 where y is a binary outcome is necessary to be able to properly estimate logistic regression equations

Use a lowess plot to get a snapshot of the relationship between the ln(odds of y=1) and the continuous predictor x1

Interpret the slope and intercept from simple logistic regression models

Translate the estimated slope into an estimated odds ratio


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Background: Underlying Model

Why model the ln(odds) as a linear function of x1?


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Background: Underlying Model

Equation for Pr(y = 1) – the proportion of subjects with y =1

e is the “natural constant” 2.718

p = probability (proportion) of y=1

.ˆˆ

ˆˆ

110

110

1 x

x

eep


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Example 1: Risk of Obesity and HDL

Data from 2009-10 NHANES1

Sample of over 6,400 US residents, 16-80 years old

HDL levels: mean 52.4 mg/dl, sd = 16, range 11-1415% of sample is obese by (BMI)

1 Data obtained via Hosmer, D.W., Lemeshow, S. and Sturdivant, R.X. (2013) Applied Logistic Regression: Third Edition.


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Question : does a line reasonably describe the general shape of the relationship between obesity and HDL?

We can estimate a line, using the computer

The line we estimate will be of the form:

Here: p is probability of being obese (proportion of individuals who are obese), for a given value of HDL cholesterol, x1

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11ˆˆ

1ln x

pp

o


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This formulation makes a strong assumption about the nature of the relationship between the ln(odds) of obesity and HDL cholesterol

How to investigate this assumption?

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11ˆˆ

1ln x

pp

o


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A smoothed scatterplot of estimated ln(odds) versus HDL

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-8-6

-4-2

02

Estim

ated

ln(o

dds)

Of O

besi

ty

0 50 100 150HDL Cholesterol (mg/dL)

bandwidth = .5

Data from 2009-10 NHANESEstimated ln(odds) of Obesity By HDL Cholesterol Level


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Equation of regression line relating ln(odds) of obesity to HDL : from computer

Here, and

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1033.00.05 obesity) of oddsln(1

ln xp

p

05.0ˆ o 033.0ˆ1


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Interpretation of

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So the odds ratio estimate is 0.967, or ≈ 0.97. The odds ratio of being obese for two groups of persons who differ by one mg/dL in HDL levels is 0.97, higher HDL to lower LDL In other words, higher HDL subjects (by one mg/dL) have 3%

lower odds of being obese when compared to the lower HDL subjects

This estimate is for any two groups who differ by one mg/dL in HDL in our the population from which the samples was taken 60 mg/dL to 59 mg/dL 44 mg/dL to 43 mg/dL Etc..

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Interpretation of

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o


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What is odds ratio of being obese for persons with HDL of 100 mg/dLversus persons with HDL of 80 mg/dL?

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Data on a random sample of 192 Nepali Children between 1 and 3 years old (12-36 months) . Information includes breast feeding status at time of study (1 = yes, 0 = no), and age of the child in months

The following model can be used to estimate this breast feeding/age association

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Example 2: Breast Feeding and Age

11o xββ)p1

pln(


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A smoothed scatterplot of estimated ln(odds) versus age (months)

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-20

24

6ln

(odd

s) o

f Bei

ng B

reas

tfed

10 15 20 25 30 35 Age of Child (months)

bandwidth = .8

192 Nepalese Children 12-36 MonthsEstimated ln(odds) of Being Breastfed by Age


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Equation of regression line relating ln(odds) of being breast fed to age (months): from computer

Here, and

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124.030.7 breastfed) being of oddsln(1

ln xp

p

30.7ˆ o 24.0ˆ1


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Interpretation of

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1


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So the odds ratio estimate is 0.79: The odds ratio of being breastfed for two groups of children who differ by one month in age is 0.79, older to younger In other words, older children (by one month of age) have 21%

lower odds of being breast fed when compared to younger children

This estimate is for any two groups who differ by one month of age in the population of Nepalese children 12-36 months 15 months to 14 months 27 months to 26 months, etc..

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Question: what is the estimated relative odds (odds ratio) of being breast fed for children who are 30 months old compared to children who are 24 months old?

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Interpretation of

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o


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Example 3: To Categorize or Not?


321 4.38.20.25.51

ln xxxp

p


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Simple logistic regression can be done with binary, categorical and continuous predictors

When the predictor x1 is continuous, the model estimates a linear relationship between the ln(odds y=1) and x1

The resulting estimated slope from logistic regression with a continuous predictor still has a ln(odds ratio) interpretation, and the intercept a ln(odds when x1=0) interpretation

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Summary


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Section C

Simple Logistic Regression : Accounting for Uncertainty in the Estimates


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Learning Objectives

Create 95% CIs for the intercept and slopes from simple logistic regression and convert these to 95% CIs for odds and odds ratios

Estimate p-values for testing the null Ho: β1=0 (and hence the OR =1)


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So in the last two sections, we showed the results from several simple logistic regression models

For example, the relationship between breast feeding and child sex estimated from a random sample of 236 Nepali children [0, 36) months old was given by the following equation: (x1=1 for males)

I told you this came from a computer package: but what is the algorithm to estimate this equation?


1002.012.11

ln xp

p


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There must be some algorithm that will always yield the same results for the same data set

For logistic regression, this approach is called “maximum likelihood”: the estimates for the intercept ( ) and the slope ( ) are the values that make the observed data “most” likely among all choices for and .

This must be done via the computer

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o 1

o 1


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The values chosen for are just estimates based on a single sample. For a different random sample of 236 children from the same population of [0,36) month olds, the resulting estimates would likely be different:

As such, all regression coefficients have an associated standard error that can be used to make statements about the true relationship between ln(odds y=1) and x1 (for example, the true slope ) based on a single sample


1ˆˆ and o

1


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The estimated regression equation for the relationship between breast feeding and child sex estimated from a random sample of 236 Nepali children [0, 36) months old was given by the following equation


210)ˆ(ES and12.1ˆ30.0)ˆ(ES and 002.0ˆ

o

11

. o

1002.012.11

ln xp

p


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Random sampling behavior of estimated regression coefficients is normal

As such, we can use same ideas to create 95% CIs and get p-values



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The estimated regression equation relating arm circumference to sex was:

95% CI for β1


)602.0,598.0(30.02002.0)ˆ(ˆ2ˆ11 ES

1002.012.11

ln xp

p

30.0)ˆ(ES and 002.0ˆ11


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95% CI for eβ1



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p-value for testing:Ho: β1 =0 (eβ1 =1) HA: β1 ≠0 (eβ1 ≠ 1)

Assume null true, and calculate standardized “distance “ of from 0

The p-value is probability of being 0.01 or more standard errors away from mean of 0 on a normal curve is very large: in this example, p = .997


1

01.03.0

002.0)(ˆ

ˆ

)(ˆ0ˆ

1

1

1

1

ESES

z


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Logistic regression was used to estimate the relationship between breast feeding and child sex using data from a random samples of 236 Nepalese children 0-36 months old. The results showed no substantive or statistically significant association between breast feeding status and sex (odds ratio = 1.00, 95% CI 0.55 to 1.83)

Summarizing findings: Breast Feeding and Sex


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ln xp

p

13.0)ˆ(;05.0ˆ oo SE

003.0)ˆ(ˆ ;033.0ˆ11 ES


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95% CI for β1

95% CI for eβ1

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003.0)ˆ(ˆ ;033.0ˆ11 ES


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What is a 95% CI for the odds ratio of being obese for persons with HDL of 100 mg/dL versus persons with HDL of 80 mg/dL?

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Example 3: Respiratory Failure and Gestational Age


321 0.28.24.35.51

ln xxxp

p

066.0)ˆ(ˆ ;4.3ˆ11 ES039.0)ˆ(ˆ ;5.5ˆ

00 ES


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Example 3: Respiratory Failure and Gestational Age

Respiratory failure and gestational age; p-value for slope

321 0.28.24.35.51

ln xxxp

p

066.0)ˆ(ˆ ;4.3ˆ11 ES


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Summary


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Section D

Estimating Risk and Functions of Risk from Logistic Regression Results


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Learning Objectives

While the results from logistic regression can be interpreted in terms of odds and odds ratios (after exponentiation), for prospective cohort studies, risks can be estimated

With a little bit of work, the results from logistic regression can be converted to probabilities (proportions, risks) and presented on this scale


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In the last several sections we have explored how to relate a binary outcome to a predictor (binary, ordinal and nominal categorical , continuous) via simple logistic regression

We have shown how translate the results into estimates of odds and odds ratio

The results from logistic regression can also be used to get estimated risks and functions of risk (if the study design allows for risk estimates)

Risk Estimates From Logistic Regression


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Recall, the estimated odds of a binary outcome is given by:

where is the estimated proportion of sample (probability of, risk of ) having event

This expression can be solved in terms of :

Risk Estimates From Logistic Regression

ppSDOD

ˆ1ˆˆ

p

p


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There are four categories, and 37-40 weeks is the reference category.

x1 = 1 if gestational age =34 weeksx2 = 1 if gestational age =35 weeksx3 = 1 if gestational age =36 weeks

321 0.28.24.35.51

ln xxxp

p


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To compute estimate risk (probability, proportion) of respiratory failure for reference group: (37-40 weeks)

To compute estimate risk (probability, proportion) of respiratory failure for gestational age=34 weeks:



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What is the estimate proportion of obese persons with HDL measurements of 75? (ie: estimated risk of obesity for persons with HDL of 75)

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ln xp

p


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What is estimated risk difference and relative risk of being obese for persons with HDL of 100 mg/dL versus persons with HDL of 75 mg/dL?

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The following graphic shows the estimated risk of obesity as function of HDL level

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0.1

.2.3

.4E

stim

ated

Pro

porti

on (P

roba

bilit

y) o

f Obe

se

0 50 100 150HDL (mg/dl)

Simple Logistic Regession Using NHANESEstimated Probability (Proportion) of Obesity


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Example 3: Breast Feeding Status and Age

Recall the resulting equation from the same of 192 Nepali children 12-36 months, relating breast feeding status to age in months

What does the above estimate for 24 month old children?

124.030.7 breastfed) being of oddsln(1

ln xp

p


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This is the estimated ln odds of being breast fed of 24 month olds in the sample

To get the corresponding odds, exponentiate the ln(odds)

To estimate from the odds:

So the above results translate into an estimated probability of .82 (82%)

1.5424.24.7.3024Age;p1

pln

66.41.54 e

82.066.566.4

ˆ1

ˆˆ

sdodsdodpp


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So an estimated 82% of 24 month olds are breast fed in this sample of Nepali children

What about the estimated proportion of 16 month olds?

the corresponding odds is

the corresponding estimated probability is

46.361.24.7.361Age;p1

pln


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So an estimated 82% of 24 month olds are breast fed in this sample of Nepali children; and an estimated 95% of 16 month olds are breast fed

What about the estimated relative risk of being breast fed for 24 month olds to 16 month olds?

The estimate risk difference for the same age comparison:

0.850.970.82

pp

RR61age

24age

0.1597.0.82pp 61age24age


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The following graphic shows the estimated risk (probability, proportion) of being breast fed as function of child age.

.2.4

.6.8

1P

roba

bilit

y (P

ropr

ortio

n, R

isk)

of B

reas

t Fed

10 15 20 25 30 35Age of Child (months)

192 Nepali Children 12-36 MonthsProbability of Being Breast Fed by Age


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Summary

For most types of studies (case control studies excepted), the results from logistic regression can be used to estimate risk (probability, proportion) and hence risk differences and relative risks

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