2024-04-24
Used the Wald test and Wald 95% confidence interval to interpret coefficients in a fitted model
This time: Interpret using odds ratio
Interpret odds ratios from fitted simple logistic regression model for a continuous explanatory variable.
Interpret odds ratios from fitted simple logistic regression model for a binary explanatory variable.
Interpret odds ratios from fitted simple logistic regression model for a multi-level categorical explanatory variable.
Report the odds ratio using a table and/or a forest plot.
For a population simple logistic regression model with a continuous predictor \[\text{logit}(\pi(X)) = \beta_0 + \beta_1 \cdot X\]
For our fitted simple logistic regression model with a continuous predictor \[\text{logit}(\widehat{\pi}(X)) = \widehat{\beta}_0 + \widehat{\beta}_1 \cdot X\]
bc_reg = glm(Late_stage_diag ~ Age_c, data = bc, family = binomial)
tidy(bc_reg, conf.int=T) %>% gt() %>% tab_options(table.font.size = 35) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | −0.989 | 0.023 | −42.637 | 0.000 | −1.035 | −0.944 |
Age_c | 0.057 | 0.003 | 17.780 | 0.000 | 0.051 | 0.063 |
For our fitted simple logistic regression model with age as a predictor \[\text{logit}(\widehat{\pi}(Age^c)) = −0.989 + 0.057 \cdot Age^c\]
\(\widehat{\beta}_0\): The estimated log-odds is -0.989 when age is 61.71 years (95% CI: -1.035, -0.944)
\(\widehat{\beta}_1\): The estimated increase in log-odds is 0.057 for every 1 year increase in age (95% CI: 0.051, 0.063).
For our fitted simple logistic regression model with a continuous predictor \[\text{logit}(\widehat{\pi}(X)) = \widehat{\beta}_0 + \widehat{\beta}_1 \cdot X\]
How do we go from interpretations of \(\widehat{\beta}_0\) and \(\widehat{\beta}_1\) using log odds to odds ratios?
We will need to take the exponential of our model:
Important distinction:
Interpret coefficients from fitted logistic regression model
The interpretation of the coefficients involves two issues:
We will learn the interpretation for
Binary independent variable
Categorical independent variable with multiple groups
Continuous independent variable
Interpret odds ratios from fitted simple logistic regression model for a binary explanatory variable.
Interpret odds ratios from fitted simple logistic regression model for a multi-level categorical explanatory variable.
Report the odds ratio using a table and/or a forest plot.
For simplicity, we assume the linear relationship between logit and continuous variable 𝑥
Again using simple logistic regression model to illustrate the interpretation of \(\widehat{\beta}\) for a continuous variable \(x\) \[\text{logit}(\widehat{\pi}(X)) = \widehat{\beta}_0 + \widehat{\beta}_1 \cdot X\]
The estimated slope coefficient, \(\widehat{\beta}_1\), is the expected change in the log odds for 1 unit increase in \(x\)
For \(\text{exp}(\widehat{\beta}_1)\)
\(\widehat{\beta}_1\) is 0.057, suggesting that one year increase in age is associated with 0.057 increase in log odds of receiving a late stage breast cancer diagnosis
\(\exp\left({\widehat{\beta}}_1\right)\) is 1.06, suggesting that one year increase in age is associated with 1.06 times the odds of receiving a late stage breast cancer diagnosis
For continuous covariates in logistic regression model, it is helpful to subtract 1 from the odds ratio and multiply by 100 to obtain the percentage change in odds for 1-unit increase.
Odds ratio from logistic regression
Compute the estimate and 95% confidence interval for odds ratio for late stage breast cancer diagnosis for every 1 year increase in age.
Needed steps:
Odds ratio from logistic regression
Compute the estimate and 95% confidence interval for odds ratio for late stage breast cancer diagnosis for every 1 year increase in age.
Call:
glm(formula = Late_stage_diag ~ Age_c, family = binomial, data = bc)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.989422 0.023205 -42.64 <2e-16 ***
Age_c 0.056965 0.003204 17.78 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 11861 on 9999 degrees of freedom
Residual deviance: 11510 on 9998 degrees of freedom
AIC: 11514
Number of Fisher Scoring iterations: 4
Odds ratio from logistic regression
Compute the estimate and 95% confidence interval for odds ratio for late stage breast cancer diagnosis for every 1 year increase in age.
tidy()
tidy_bc_reg = tidy(bc_reg, conf.int=T, exponentiate = T)
tidy_bc_reg %>% gt() %>% tab_options(table.font.size = 35) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | 0.372 | 0.023 | −42.637 | 0.000 | 0.355 | 0.389 |
Age_c | 1.059 | 0.003 | 17.780 | 0.000 | 1.052 | 1.065 |
[1] 0.3551931 1.0520321
Odds ratio from logistic regression
Compute the estimate and 95% confidence interval for odds ratio for late stage breast cancer diagnosis for every 1 year increase in age.
logistic.display()
Odds ratio from logistic regression
Compute the estimate and 95% confidence interval for odds ratio for late stage breast cancer diagnosis for every 1 year increase in age.
For every one year increase in age, there is an estimated 5.86% increase in the estimated odds of late stage breast cancer diagnosis (95% CI: 5.2%, 6.53%).
Sometimes a change in “1” unit may not be considered clinically interesting
For example, a 1 year increase in age or a 1 mm Hg increase in systolic blood pressure may be too small for a meaningful change in log odds
Instead, we may be interested to find out the log odds change for a increase of 10 years in age or 10 mm Hg in systolic blood pressure
On the other hand, if the range of x is small (say 0-1), than a change in 1 unit of 𝑥 is too large to be meaningful
We should be able to compute and interpret coefficients for a continuous independent covariate \(x\) for an arbitrary change of “c” units in \(x\)
The estimated log odds ratio for a change of c units in x can be obtained from \[\hat{g}\left(x+c\right)-\hat{g}\left(x\right)=c{\hat{\beta}}_1\]
The 95% CI for \(\widehat{OR}(c)\) is: \[\exp \left( c \hat{\beta}_1 \pm 1.96 \cdot c \cdot SE_{\hat{\beta}_1} \right)\]
The \(c\) is chosen to be a clinically meaningful unit change in \(x\)
The value of 𝑐 should be clearly specified in all tables and calculations
bc_reg = glm(Late_stage_diag ~ Age_c, data = bc, family = binomial)
tidy(bc_reg, conf.int=T) %>% gt() %>% tab_options(table.font.size = 35) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | −0.989 | 0.023 | −42.637 | 0.000 | −1.035 | −0.944 |
Age_c | 0.057 | 0.003 | 17.780 | 0.000 | 0.051 | 0.063 |
\[ \widehat{OR}\left(10\right)=\exp{\left(10\cdot{\hat{\beta}}_1\right)}=\exp{\left(0.56965\right)}=\mathrm{\mathrm{1.767}}\]
bc2 = bc %>% mutate(Age_c_10 = Age_c/10)
bc_reg_10 = glm(Late_stage_diag ~ Age_c_10, data = bc2, family = binomial)
tidy(bc_reg_10, conf.int=T, exponentiate = T) %>% gt() %>% tab_options(table.font.size = 35) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | 0.372 | 0.023 | −42.637 | 0.000 | 0.355 | 0.389 |
Age_c_10 | 1.768 | 0.032 | 17.780 | 0.000 | 1.661 | 1.883 |
Notice that the logistic regression model suggests that logit is linear in the covariate
The model implies the additional risk of late stage breast cancer diagnosis for a 40 year-old compared to a 30 year-old is the same as the additional risk of late stage breast cancer diagnosis for a 60 year-old compared to a 50-year-old
This assumption may not be realistic
To address this, we may consider using higher order terms (e.g., \(x^2\), \(x^3\),…) or other nonlinear transformation(e.g., \(log(x)\))
Categorize the continuous variable may be another option
For \(\text{exp}(\widehat{\beta}_0)\)
When \(X=0\), we have \[\text{logit}(\widehat{\pi}(X=0)) = \widehat{\beta}_0\]
Thus, \[\begin{aligned} \widehat{\beta}_0 & = \text{logit}(\widehat{\pi}(X)) \\ \text{exp}\big[\widehat{\beta}_0\big] & = \text{exp}\big[\text{logit}(\widehat{\pi}(X))\big] \\ \text{exp}\big[\widehat{\beta}_0\big] & = \text{exp}\Bigg[\text{log}\Bigg(\dfrac{\widehat{\pi}(X)}{1-\widehat{\pi}(X)}\Bigg)\Bigg] \\ \text{exp}\big[\widehat{\beta}_0\big] & = \dfrac{\widehat{\pi}(X)}{1-\widehat{\pi}(X)} \\ \end{aligned}\]
Interpret odds ratios from fitted simple logistic regression model for a multi-level categorical explanatory variable.
Report the odds ratio using a table and/or a forest plot.
Independent variable \(x\) is a binary variable (\(x\) can take values: 0 or 1)
We are fitting the simple logistic regression model: \[\text{logit}\left(\pi(X) \right) = \beta_0 + \beta_1 \cdot I(X=1)\]
The logit difference is \(\beta_1\) for binary independent variable
It will be much easier to understand if we can interpret the coefficient using odds ratio (OR)
For individuals with \(X=0\): \[\text{logit}\left(\pi(X=0)\right)=\beta_0+\beta_1\times\left(0\right)=\beta_0\]
For individuals with \(X=1\): \[\text{logit}\left(\pi(X=1)\right)=\beta_0+\beta_1\times\left(1\right)=\beta_0 + \beta_1\]
To solve for \(\beta_1\), we take the difference of the logits: \[ \text{logit}\left(\pi(X=1)\right) - \text{logit}\left(\pi(X=0)\right) = \left( \beta_0 + \beta_1 \right) - \left( \beta_0 \right) = \beta_1\]
\[ \text{logit}\left(\pi(X=1)\right) - \text{logit}\left(\pi(X=0)\right) = \left( \beta_0 + \beta_1 \right) - \left( \beta_0 \right) = \beta_1\]
\[\begin{aligned} \beta_1&=l\mathrm{ogit}\left(\pi(X=1)\right)\ -l\mathrm{ogit}\left(\pi\left(X=0\right)\right) \\ \beta_1&=l\mathrm{og}\left(\dfrac{\pi(X=1)}{1-\pi(X=1)}\right)-l\mathrm{og}\left(\dfrac{\pi\left(X=0\right)}{1-\pi\left(X=0\right)}\right) \\ \beta_1&=\log{\left(\dfrac{\dfrac{\pi(X=1)}{1-\pi(X=1)}}{\dfrac{\pi(X=0)}{1-\pi(X=0)}}\right)} \\ \exp{\left(\beta_1\right)}&=\dfrac{\dfrac{\pi(X=1)}{1-\pi(X=1)}}{\dfrac{\pi(X=0)}{1-\pi(X=0)}} \end{aligned}\]
Odds for a subject with \(X=1\): \[\text{odds}_1 = \dfrac{\pi(X=1)}{1-\pi(X=1)}\]
Odds for a subject with \(X=0\): \[\text{odds}_0 = \dfrac{\pi(X=0)}{1-\pi(X=0)}\]
Odds Ratio for \(X=1\) vs. \(X=0\): \[OR = \dfrac{\dfrac{\pi(X=1)}{1-\pi(X=1)}}{\dfrac{\pi(X=0)}{1-\pi(X=0)}}\]
\[ OR=\dfrac{a/c}{b/d}=\dfrac{\dfrac{\left(\dfrac{\exp{\left(\beta_0+\beta_1\right)}}{1+\exp{\left(\beta_0+\beta_1\right)}}\right)}{\left(\dfrac{1}{1+\exp(\beta_0+\beta_1)}\right)}}{\dfrac{\left(\dfrac{\exp(\beta_0)}{1+\exp(\beta_0)}\right)}{\left(\dfrac{1}{1+\exp{\left(\beta_0\right)}}\right)}}=\dfrac{\exp(\beta_0+\beta_1)}{\exp(\beta_0)}=e^{\beta_1} \]
Simple relationship between coefficient and odds ratio is a primary reason why we report OR for categorical data analysis.
For binary independent variable x, OR computed in logistic regression model is the same as OR computed using contingency table
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for older individuals (>65 years old) compared to younger individuals (≤65 years old)?
Two options to calculate this value:
Option 1: Calculate \(\widehat{OR}\) from 2x2 contingency table
Option 2: Calculate \(\widehat{OR}\) from logistic regression
Needed steps for Option 2:
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for older individuals (>65 years old) compared to younger individuals (≤65 years old)?
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for older individuals (>65 years old) compared to younger individuals (≤65 years old)?
age_bin_tidy = tidy(age_bin_glm, conf.int=T, exponentiate = T)
age_bin_tidy %>% gt() %>%
tab_options(table.font.size = 35) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | 0.297 | 0.031 | −39.608 | 0.000 | 0.280 | 0.315 |
Age_binary | 1.875 | 0.045 | 13.928 | 0.000 | 1.716 | 2.048 |
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for older individuals (>65 years old) compared to younger individuals (≤65 years old)?
The estimated odds of late stage breast cancer among individuals over 65 years old is 1.87 (95% CI: (1.72, 2.05)) times that of individuals 65 years or younger.
Interpret odds ratios from fitted simple logistic regression model for a continuous explanatory variable.
Interpret odds ratios from fitted simple logistic regression model for a binary explanatory variable.
Independent variable \(x\) is a multi-level categorical variable
Let’s say \(X\) takes values: a, b, c, or d
We are fitting the simple logistic regression model: \[\text{logit}\left(\pi(X) \right) = \beta_0 + \beta_1 \cdot I(X=b) + \beta_2 \cdot I(X=c) + \beta_3 \cdot I(X=d)\]
The logit difference is \(\beta_1\) for binary independent variable
It will be much easier to understand if we can interpret the coefficient using odds ratio (OR)
We are fitting the simple logistic regression model with reference group \(a\): \[\text{logit}\left(\pi(X) \right) = \beta_0 + \beta_1 \cdot I(X=b) + \beta_2 \cdot I(X=c) + \beta_3 \cdot I(X=d)\]
\[ \text{logit}\left(\pi(X=c)\right) - \text{logit}\left(\pi(X=a)\right) = \left( \beta_0 + \beta_1\cdot 0 + \beta_2\cdot 1 + \beta_3\cdot 0 \right) - \left( \beta_0 + \beta_1\cdot 0 + \beta_2\cdot 0 + \beta_3\cdot 0 \right) = \beta_2\]
\[\begin{aligned} \beta_2&=l\mathrm{ogit}\left(\pi(X=c)\right)\ -l\mathrm{ogit}\left(\pi\left(X=a\right)\right) \\ \beta_2&=l\mathrm{og}\left(\dfrac{\pi(X=c)}{1-\pi(X=c)}\right)-l\mathrm{og}\left(\dfrac{\pi\left(X=a\right)}{1-\pi\left(X=a\right)}\right) \\ \beta_2&=\log{\left(\dfrac{\dfrac{\pi(X=c)}{1-\pi(X=c)}}{\dfrac{\pi(X=a)}{1-\pi(X=a)}}\right)} \\ \exp{\left(\beta_2\right)}&=\dfrac{\dfrac{\pi(X=c)}{1-\pi(X=c)}}{\dfrac{\pi(X=a)}{1-\pi(X=a)}} \end{aligned}\]
We are fitting the simple logistic regression model with reference group \(a\): \[\text{logit}\left(\pi(X) \right) = \beta_0 + \beta_1 \cdot I(X=b) + \beta_2 \cdot I(X=c) + \beta_3 \cdot I(X=d)\]
The choice can be more apparent for multi-group categorical independent variables within studies
For example, if we want to evaluate the association between clinical response and four treatments.
The treatment variable has 4 categories: “active treatment A”, “active treatment B”, “active treatment C” and “Placebo treatment”
The investigator is interested in comparing each of the three active treatment with the placebo treatment
Then the placebo treatment should be picked as the reference group
Chose Non-Hispanic White individuals as reference group
Underlying health disparities linked to racism in healthcare and in clinical studies
There is evidence that white individuals receive a certain standard of care that is not paralleled for POC Mateo and Williams (2021)
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for Non-Hispanic Asian/Pacific Islander individuals compared to Non-Hispanic White individuals?
Needed steps:
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for Non-Hispanic Asian/Pacific Islander individuals compared to Non-Hispanic White individuals?
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for Non-Hispanic Asian/Pacific Islander individuals compared to Non-Hispanic White individuals?
RE_tidy = tidy(RE_glm, conf.int=T, exponentiate = T)
RE_tidy %>% gt() %>%
tab_options(table.font.size = 35) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | 0.372 | 0.026 | −37.553 | 0.000 | 0.353 | 0.392 |
Race_EthnicityHispanic-Latino | 0.968 | 0.082 | −0.398 | 0.691 | 0.822 | 1.135 |
Race_EthnicityNH American Indian/Alaskan Native | 0.948 | 0.476 | −0.111 | 0.911 | 0.342 | 2.287 |
Race_EthnicityNH Asian/Pacific Islander | 1.131 | 0.082 | 1.497 | 0.134 | 0.961 | 1.327 |
Race_EthnicityNH Black | 1.405 | 0.070 | 4.826 | 0.000 | 1.223 | 1.611 |
Odds ratio from logistic regression
What is the odds ratio of late stage breast cancer diagnosis for Non-Hispanic Asian/Pacific Islander individuals compared to Non-Hispanic White individuals?
The estimated odds of late stage breast cancer among Non-Hispanic Asian/Pacific Islander individuals is 1.13 (95% CI: (0.96, 1.33)) times that of Non-Hispanic White individuals.
What if we want to estimate OR comparing Non-Hispanic Asian Pacific Islander to Non-Hispanic Black individuals?
Option 1: Change reference group and refit the model (maybe the easiest option)
Option 2: Estimate OR using fitted coefficients (\(\widehat{\beta}\)’s) in the current model: \[\begin{aligned} \text{log}\left( OR (\text{NH API}, \text{NH B}) \right) &= \text{logit}\left(\pi \left(X = \text{NH API}\right)\right) - \text{logit}\left(\pi \left(X = \text{NH B}\right)\right) \\ & = \left[\beta_0 + \beta_3 \cdot 1\right] - \left[\beta_0 + \beta_4 \cdot 1 \right] \\ \text{log}\left( \widehat{OR} (\text{NH API}, \text{NH B}) \right) &= \widehat{\beta}_3 - \widehat{\beta}_4 \\ \widehat{OR} (\text{NH API}, \text{NH B}) &= \exp \left( \widehat{\beta}_3 - \widehat{\beta}_4 \right) \end{aligned}\]
bc3 = bc %>%
mutate(Race_Ethnicity = relevel(Race_Ethnicity, ref = "NH Black"))
RE_glm2 = glm(Late_stage_diag ~ Race_Ethnicity, data = bc3,
family = binomial)
tidy(RE_glm2, conf.int=T, exponentiate = T) %>% gt() %>%
tab_options(table.font.size = 38) %>%
fmt_number(decimals = 3)
term | estimate | std.error | statistic | p.value | conf.low | conf.high |
---|---|---|---|---|---|---|
(Intercept) | 0.523 | 0.065 | −9.934 | 0.000 | 0.459 | 0.594 |
Race_EthnicityNH White | 0.712 | 0.070 | −4.826 | 0.000 | 0.621 | 0.818 |
Race_EthnicityHispanic-Latino | 0.689 | 0.102 | −3.664 | 0.000 | 0.564 | 0.840 |
Race_EthnicityNH American Indian/Alaskan Native | 0.675 | 0.479 | −0.819 | 0.413 | 0.242 | 1.641 |
Race_EthnicityNH Asian/Pacific Islander | 0.805 | 0.102 | −2.131 | 0.033 | 0.659 | 0.982 |
Interpret odds ratios from fitted simple logistic regression model for a continuous explanatory variable.
Interpret odds ratios from fitted simple logistic regression model for a binary explanatory variable.
Interpret odds ratios from fitted simple logistic regression model for a multi-level categorical explanatory variable.
tbl_regression()
in the gtsummary
package is helpful for presenting the odds ratios in a clean waylibrary(gtsummary)
tbl_regression(RE_glm, exponentiate = TRUE) %>%
as_gt() %>% # allows us to use tab_options()
tab_options(table.font.size = 38)
Characteristic | OR1 | 95% CI1 | p-value |
---|---|---|---|
Race_Ethnicity | |||
NH White | — | — | |
Hispanic-Latino | 0.97 | 0.82, 1.14 | 0.7 |
NH American Indian/Alaskan Native | 0.95 | 0.34, 2.29 | >0.9 |
NH Asian/Pacific Islander | 1.13 | 0.96, 1.33 | 0.13 |
NH Black | 1.40 | 1.22, 1.61 | <0.001 |
1 OR = Odds Ratio, CI = Confidence Interval |
library(broom.helpers)
RE_tidy = tidy_and_attach(RE_glm, conf.int=T, exponentiate = T) %>%
tidy_remove_intercept() %>%
tidy_add_reference_rows() %>% tidy_add_estimate_to_reference_rows() %>%
tidy_add_term_labels()
glimpse(RE_tidy)
Rows: 5
Columns: 16
$ term <chr> "Race_EthnicityNH White", "Race_EthnicityHispanic-Latin…
$ variable <chr> "Race_Ethnicity", "Race_Ethnicity", "Race_Ethnicity", "…
$ var_label <chr> "Race_Ethnicity", "Race_Ethnicity", "Race_Ethnicity", "…
$ var_class <chr> "factor", "factor", "factor", "factor", "factor"
$ var_type <chr> "categorical", "categorical", "categorical", "categoric…
$ var_nlevels <int> 5, 5, 5, 5, 5
$ contrasts <chr> "contr.treatment", "contr.treatment", "contr.treatment"…
$ contrasts_type <chr> "treatment", "treatment", "treatment", "treatment", "tr…
$ reference_row <lgl> TRUE, FALSE, FALSE, FALSE, FALSE
$ label <chr> "NH White", "Hispanic-Latino", "NH American Indian/Alas…
$ estimate <dbl> 1.0000000, 0.9678002, 0.9484848, 1.1310170, 1.4046741
$ std.error <dbl> NA, 0.08224948, 0.47558680, 0.08224988, 0.07041472
$ statistic <dbl> NA, -0.3979312, -0.1112089, 1.4968682, 4.8257715
$ p.value <dbl> NA, 6.906809e-01, 9.114507e-01, 1.344276e-01, 1.394623e…
$ conf.low <dbl> NA, 0.8223138, 0.3417844, 0.9612074, 1.2226824
$ conf.high <dbl> NA, 1.135332, 2.286596, 1.327092, 1.611466
ggplot(data=RE_tidy, aes(y=label, x=estimate, xmin=conf.low, xmax=conf.high)) +
geom_point(size = 3) + geom_errorbarh(height=.2) +
geom_vline(xintercept=1, color='#C2352F', linetype='dashed', alpha=1) +
theme_classic() +
labs(x = "OR (95% CI)", y = "Race and ethnicity",
title = "Odds ratios of Late Stage \n Breast Cancer Diagnosis") +
theme(axis.title = element_text(size = 25), axis.text = element_text(size = 25), title = element_text(size = 25))
Lesson 8: Interpretations and Visualizations of Odds Ratios