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Logistic Regression

Odds ratios

Prof. Maria Tackett

1

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Topics

  • Use the odds ratio to compare the odds of two groups

  • Interpret the coefficients of a logistic regression model with

    • a single categorical predictor
    • a single quantitative predictor
    • multiple predictors
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Risk of coronary heart disease

This dataset is from an ongoing cardiovascular study on residents of the town of Framingham, Massachusetts. We want to examine the relationship between various health characteristics and the risk of having heart disease.

high_risk:

  • 1: High risk of having heart disease in next 10 years
  • 0: Not high risk of having heart disease in next 10 years

age: Age at exam time (in years)

education: 1 = Some High School; 2 = High School or GED; 3 = Some College or Vocational School; 4 = College

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High risk vs. education

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403
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Compare the odds for two groups

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403

  • We want to compare the risk of heart disease for those with a High School diploma/GED and those with a college degree.

  • We'll use the odds to compare the two groups

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Compare the odds for two groups

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403


odds=P(success)P(failure)=# of successes# of failures

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Odds of being high risk for the High school or GED group

1471106=0.133

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Odds of being high risk for the High school or GED group

1471106=0.133

Odds of being high risk for the College group

70403=0.174

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Odds of being high risk for the High school or GED group

1471106=0.133

Odds of being high risk for the College group

70403=0.174

Based on this, we see those with a college degree had higher odds of being high risk of heart disease than those with a high school diploma or GED.

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Odds ratio

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403

Let's summarize the relationship between the two groups. To do so, we'll use the odds ratio (OR).

OR=odds1odds2=ω1ω2

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Odds ratio: College vs. High school or GED

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403

OR=oddsCollegeoddsHS=0.1740.133=1.308

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Odds ratio: College vs. High school or GED

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403

OR=oddsCollegeoddsHS=0.1740.133=1.308

The odds of being high risk of heart disease are 1.30 times higher for those with a college degree than those with a high school diploma or GED.

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Odds ratio: College vs. Some high school

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403

OR=oddsCollegeoddsSomeHS=70/403323/1397=0.751

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Odds ratio: College vs. Some high school

High Risk Not High Risk
Some high school 323 1397
High school or GED 147 1106
Some college or vocational school 88 601
College 70 403

OR=oddsCollegeoddsSomeHS=70/403323/1397=0.751

The odds of being high risk of having heart disease for those with a college degree are 0.751 times the odds of being high risk for heart disease for those with some high school.

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More natural interpretation

It's more natural to interpret the odds ratio with a statement with the odds ratio greater than 1.

12

More natural interpretation

It's more natural to interpret the odds ratio with a statement with the odds ratio greater than 1.

The odds of being high risk for heart disease are 1.33 times higher for those with some high school than those with a college degree.

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Logistic regression: categorical predictor

Recall: Education - 1 = Some High School; 2 = High School or GED; 3 = Some College or Vocational School; 4 = College

risk_model <- glm(high_risk ~ education, 
                  data  = heart, family = "binomial")
term estimate std.error statistic p.value
(Intercept) -1.464 0.062 -23.719 0.000
education2 -0.554 0.107 -5.159 0.000
education3 -0.457 0.130 -3.520 0.000
education4 -0.286 0.143 -1.994 0.046
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Interpreting education4 - log-odds

term estimate std.error statistic p.value
(Intercept) -1.464 0.062 -23.719 0.000
education2 -0.554 0.107 -5.159 0.000
education3 -0.457 0.130 -3.520 0.000
education4 -0.286 0.143 -1.994 0.046

The log-odds of being high risk of heart disease are expected to be 0.286 less for those with a college degree compared to those with some high school (the baseline group).

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Interpreting education4 - odds

term estimate std.error statistic p.value
(Intercept) -1.464 0.062 -23.719 0.000
education2 -0.554 0.107 -5.159 0.000
education3 -0.457 0.130 -3.520 0.000
education4 -0.286 0.143 -1.994 0.046

The odds of being high risk of heart disease for those with a college degree are expected to be 0.751 (exp(-0.286)) times the odds for those with some high school.

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Coefficients + odds ratios

The model coefficient, -0.286, is the expected change in the log-odds when going from the Some high school group to the College group.

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Coefficients + odds ratios

The model coefficient, -0.286, is the expected change in the log-odds when going from the Some high school group to the College group.

Therefore, exp{−0.286} = 0.751 is the expected change in the odds when going from the Some high school group to the College group.

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Coefficients + odds ratios

The model coefficient, -0.286, is the expected change in the log-odds when going from the Some high school group to the College group.

Therefore, exp{−0.286} = 0.751 is the expected change in the odds when going from the Some high school group to the College group.

OR=exp{ˆβj}=eˆβj

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Logistic regression: quantitative predictor

term estimate std.error statistic p.value
(Intercept) -5.619 0.288 -19.498 0
age 0.076 0.005 14.174 0
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Interpreting age: log-odds

term estimate std.error statistic p.value
(Intercept) -5.619 0.288 -19.498 0
age 0.076 0.005 14.174 0

For each additional year in age, the log-odds of being high risk of heart disease are expected to increase by 0.076.

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Interpretating age: odds

term estimate std.error statistic p.value
(Intercept) -5.619 0.288 -19.498 0
age 0.076 0.005 14.174 0

For each additional year in age, the odds of being high risk of heart disease are expected to multiply by a factor of 1.08 (exp(0.076)).

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Interpretating age: odds

term estimate std.error statistic p.value
(Intercept) -5.619 0.288 -19.498 0
age 0.076 0.005 14.174 0

For each additional year in age, the odds of being high risk of heart disease are expected to multiply by a factor of 1.08 (exp(0.076)).

Alternate interpretation

For each additional year in age, the odds of being high risk for heart disease are expected to increase by 8%.

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Logistic regression: multiple predictors

risk_model_3 <- glm(high_risk ~ education + age, 
                    data = heart, family = "binomial")
term estimate std.error statistic p.value
(Intercept) -5.385 0.308 -17.507 0.000
education2 -0.242 0.112 -2.162 0.031
education3 -0.235 0.134 -1.761 0.078
education4 -0.020 0.148 -0.136 0.892
age 0.073 0.005 13.385 0.000
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Interpretation in terms of the log-odds

term estimate std.error statistic p.value
(Intercept) -5.385 0.308 -17.507 0.000
education2 -0.242 0.112 -2.162 0.031
education3 -0.235 0.134 -1.761 0.078
education4 -0.020 0.148 -0.136 0.892
age 0.073 0.005 13.385 0.000

education4: The log-odds of being high risk of heart disease are expected to be 0.020 less for those with a college degree compared to those with some high school, holding age constant.

21

Interpretation in terms of the log-odds

term estimate std.error statistic p.value
(Intercept) -5.385 0.308 -17.507 0.000
education2 -0.242 0.112 -2.162 0.031
education3 -0.235 0.134 -1.761 0.078
education4 -0.020 0.148 -0.136 0.892
age 0.073 0.005 13.385 0.000

age: For each additional year in age, the log-odds of being high risk of heart disease are expected to increase by 0.073, holding education level constant.

22

Interpretation in terms of the odds

term estimate std.error statistic p.value
(Intercept) -5.385 0.308 -17.507 0.000
education2 -0.242 0.112 -2.162 0.031
education3 -0.235 0.134 -1.761 0.078
education4 -0.020 0.148 -0.136 0.892
age 0.073 0.005 13.385 0.000

education4: The odds of being high risk of heart disease for those with a college degree are expected to be 0.98 (exp(-0.020)) times the odds for those with some high school, holding age constant.

23

Interpretation in terms of the odds

term estimate std.error statistic p.value
(Intercept) -5.385 0.308 -17.507 0.000
education2 -0.242 0.112 -2.162 0.031
education3 -0.235 0.134 -1.761 0.078
education4 -0.020 0.148 -0.136 0.892
age 0.073 0.005 13.385 0.000

age: For each additional year in age, the odds being high risk of heart disease are expected to multiply by a factor of 1.08 (exp(0.073)), holding education level constant.

24

Recap

  • Use the odds ratio to compare the odds of two groups

  • Interpret the coefficients of a logistic regression model with

    • a single categorical predictor
    • a single quantitative predictor
    • multiple predictors
25

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