lecture 2lecturecontent.s3.amazonaws.com/pdf/14993.pdf · Lecture 3, part A: St atistical Reasoning...
Transcript of lecture 2lecturecontent.s3.amazonaws.com/pdf/14993.pdf · Lecture 3, part A: St atistical Reasoning...
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Lecture 3, part A: Statistical Reasoning 2
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Simple Logistic Regression
Lecture 2
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Lecture 3, part A: Statistical Reasoning 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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Lecture 3, part A: Statistical Reasoning 2
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Section A
Simple Logistic Regression With a Binary (or Categorical) Predictor
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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The “Left Hand Side”
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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Lecture 3, part A: Statistical Reasoning 2
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The “Left Hand Side”
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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Lecture 3, part A: Statistical Reasoning 2
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The “Left Hand Side”
Difference in ln(odds)?
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Breast Feeding and Sex
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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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Breast Feeding and Sex
Interpretation: 1
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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Breast Feeding and Sex
Interpretation: o
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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Breast Feeding and Sex
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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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Breast Feeding and Sex
Results, antilogged
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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
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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Lecture 3, part A: Statistical Reasoning 2
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Example 2: Respiratory Failure
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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Lecture 3, part A: Statistical Reasoning 2
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Example 2: Respiratory Failure
Respiratory failure and gestational age
332211oˆˆˆˆ
1ln xxx
pp
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Lecture 3, part A: Statistical Reasoning 2
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Example 2: Respiratory Failure
In this model
40-37 agegest failure,y respirator of odds
34 agegest failure,y respirator of oddsln1
40-37 agegest failure,y respirator of odds
35 agegest failure,y respirator of oddslnˆ2
40-37 agegest failure,y respirator of odds
36 agegest failure,y respirator of oddslnˆ3
40-37 agegest failure,y respirator of oddslnˆ0
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Lecture 3, part A: Statistical Reasoning 2
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Example 2: Respiratory Failure
Respiratory failure and gestational age
321 0.28.24.35.51
ln xxxp
p
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Lecture 3, part A: Statistical Reasoning 2
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Example 2: Respiratory Failure
Respiratory failure and gestational age
321 0.28.24.35.51
ln xxxp
p
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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Section B
Simple Logistic Regression With a Continuous Predictor
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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Background: Underlying Model
Why model the ln(odds) as a linear function of x1?
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Risk of Obesity and HDL
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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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Risk of Obesity and HDL
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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Lecture 3, part A: Statistical Reasoning 2
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Example 1: Risk of Obesity and HDL
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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Lecture 3, part A: Statistical Reasoning 2
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Equation of regression line relating ln(odds) of obesity to HDL : from computer
Here, and
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Example 1: Risk of Obesity and HDL
1033.00.05 obesity) of oddsln(1
ln xp
p
05.0ˆ o 033.0ˆ1
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Lecture 3, part A: Statistical Reasoning 2
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Interpretation of
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Example 1: Risk of Obesity and HDL
1
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Lecture 3, part A: Statistical Reasoning 2
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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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Example 1: Risk of Obesity and HDL
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Lecture 3, part A: Statistical Reasoning 2
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Interpretation of
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Example 1: Risk of Obesity and HDL
o
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Lecture 3, part A: Statistical Reasoning 2
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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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Example 1: Risk of Obesity and HDL
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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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Example 2: Breast Feeding and Age
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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Lecture 3, part A: Statistical Reasoning 2
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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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Example 2: Breast Feeding and Age
124.030.7 breastfed) being of oddsln(1
ln xp
p
30.7ˆ o 24.0ˆ1
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Lecture 3, part A: Statistical Reasoning 2
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Interpretation of
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Example 2: Breast Feeding and Age
1
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Lecture 3, part A: Statistical Reasoning 2
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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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Example 2: Breast Feeding and Age
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Lecture 3, part A: Statistical Reasoning 2
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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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Example 2: Breast Feeding and Age
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Interpretation of
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Example 2: Breast Feeding and Age
o
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Example 3: To Categorize or Not?
Respiratory failure and gestational age
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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Lecture 3, part A: Statistical Reasoning 2
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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?
Example 1: Breast Feeding and Sex
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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Example 1: Breast Feeding and Sex
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
Example 1: Breast Feeding and Sex
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
Example 1: Breast Feeding and Sex
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
Example 1: Breast Feeding and Sex
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The estimated regression equation relating arm circumference to sex was:
95% CI for β1
Example 1: Breast Feeding and Sex
)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
Example 1: Breast Feeding and Sex
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Lecture 3, part A: Statistical Reasoning 2
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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
Example 1: Breast Feeding and Sex
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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Lecture 3, part A: Statistical Reasoning 2
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Equation of regression line relating ln(odds) of obesity to HDL : from computer
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Example 2: Risk of Obesity and HDL
1033.00.05 obesity) of oddsln(1
ln xp
p
13.0)ˆ(;05.0ˆ oo SE
003.0)ˆ(ˆ ;033.0ˆ11 ES
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Lecture 3, part A: Statistical Reasoning 2
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95% CI for β1
95% CI for eβ1
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Example 2: Risk of Obesity and HDL
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 2: Risk of Obesity and HDL
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Example 3: Respiratory Failure and Gestational Age
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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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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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Example 2: Respiratory Failure
Respiratory failure and gestational age
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:
Example 1: Breast Feeding and Sex
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Equation of regression line relating ln(odds) of obesity to HDL : from computer
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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Example 2: Risk of Obesity and HDL
1033.00.05 obesity) of oddsln(1
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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Example 2: Risk of Obesity and HDL
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The following graphic shows the estimated risk of obesity as function of HDL level
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Example 2: Risk of Obesity and HDL
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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Lecture 3, part A: Statistical Reasoning 2
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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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Lecture 3, part A: Statistical Reasoning 2
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Example 3: Breast Feeding Status and Age
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
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sdodsdodpp
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Example 3: Breast Feeding Status and Age
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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Example 3: Breast Feeding Status and Age
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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Lecture 3, part A: Statistical Reasoning 2
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Example 3: Breast Feeding Status and Age
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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Lecture 3, part A: Statistical Reasoning 2
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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