Use the odds ratio to compare the odds of two groups
Interpret the coefficients of a logistic regression model with
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:
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
| 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 |
| 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
| 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 |
$$\text{odds} = \frac{P(\text{success})}{P(\text{failure})} = \frac{\text{# of successes}}{\text{# of failures}}$$
Odds of being high risk for the High school or GED group
$$\frac{147}{1106} = 0.133$$
Odds of being high risk for the High school or GED group
$$\frac{147}{1106} = 0.133$$
Odds of being high risk for the College group
$$\frac{70}{403} = 0.174$$
Odds of being high risk for the High school or GED group
$$\frac{147}{1106} = 0.133$$
Odds of being high risk for the College group
$$\frac{70}{403} = 0.174$$
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.
| 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).
$$OR = \frac{\text{odds}_1}{\text{odds}_2} = \frac{\omega_1}{\omega_2}$$
| 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 |
$$OR = \frac{\text{odds}_{College}}{\text{odds}_{HS}} = \frac{0.174}{0.133} = \mathbf{1.308}$$
| 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 |
$$OR = \frac{\text{odds}_{College}}{\text{odds}_{HS}} = \frac{0.174}{0.133} = \mathbf{1.308}$$
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.
| 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 |
$$OR = \frac{\text{odds}_{College}}{\text{odds}_{Some HS}} = \frac{70/403}{323/1397} = 0.751$$
| 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 |
$$OR = \frac{\text{odds}_{College}}{\text{odds}_{Some HS}} = \frac{70/403}{323/1397} = 0.751$$ 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.
It's more natural to interpret the odds ratio with a statement with the odds ratio greater than 1.
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.
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 |
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).
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.
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.
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.
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.
$$OR = \exp\{\hat{\beta}_j\} = e^{\hat{\beta}_j}$$
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | -5.619 | 0.288 | -19.498 | 0 |
| age | 0.076 | 0.005 | 14.174 | 0 |
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.
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)).
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%.
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 |
| 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.
| 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.
| 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.
| 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.
Use the odds ratio to compare the odds of two groups
Interpret the coefficients of a logistic regression model with
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