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Predicting Readmission Rates: Logistic Regression and a Shiny App for Actionable Hospital Decision-Making
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    This report provides insight on patient readmissions by answering three questions:

    1. What is the most common primary diagnosis by age group?
    2. Some doctors believe diabetes might play a central role in readmission. What is the effect of a diabetes diagnosis on readmission rates?
    3. On what groups of patients should the hospital focus their follow-up efforts to better monitor patients with a high probability of readmission?

    To summarize the results of this report:

    1. I find that circulatory ailments are the most common diagnosis for most age groups, making up more than 30% of the primary diagnoses of those 50 and older.

    2. To estimate the effect of diabetes on readmits, I use a logistic regression model and control for patient characteristics, medical provision, and, secondary and tertiary diagnoses. I find that a patient with diabetes as a primary diagnosis has a 55.7% higher probability of being readmitted compared to someone whose primary diagnosis was not diabetes, holding the above groups constant.

    3. I identify the groups most at-risk of readmit. I find that the groups with the highest probability of readmittance are those who have a high number of inpatient and/or outpatient visits in the previous year, those aged between 60 and 90, and those who are diabetic. To put this information into practice, I made a Shiny App that gives the hospital the tools to optimize their follow-ups on these high readmit-probability patients. This app would allow the hospital to have a higher return on reducing readmit by targetting these high probability patients.

    The Data

    Let's load in our data

    library(tidyverse, quietly = TRUE)
    library(rmarkdown, quietly = TRUE)
    # read in data
    df <- readr::read_csv('data/hospital_readmissions.csv', show_col_types = FALSE)
    Hidden output

    Have a look at the data. There are some 'Missing' values in the diagnoses columns. Let's get rid of those for our analysis.

    I feel good about this. Let's answer the first question...

    Q1: What is the most common primary diagnosis by age group?

    Below is a table that gives the top primary diagnosis and its share for by age group

    df %>%
      filter(diag_1 != 'Missing',
             diag_2 != 'Missing',
             diag_3 != 'Missing') %>% 
      select(age,diag_1) %>%
      group_by(age,diag_1) %>% 
      summarise(diag_count = length(diag_1)) %>% 
      ungroup() %>% 
      group_by(age) %>% 
      mutate(diag_share = round(diag_count/sum(diag_count),3 ) ) %>% 
      rename(Age = age,
             `Primary Diagnosis` = diag_1,
             `Diagnosis Share` = diag_share) %>% 
      group_by(Age) %>% 
      filter(`Diagnosis Share` == max(`Diagnosis Share`)) %>% 
      select(-diag_count) %>% 
      mutate(`Diagnosis Share` = paste(round(`Diagnosis Share`*100),'%')) %>% 

    Clearly, Circulatory diagnoses are the most common -- making up about a third of each age group's diagnoses. To give an idea of the breakdown of primary diagnoses, I share a plot that gives, by age group, the share of each primary diagnosis.

    # plot 
    p<- df %>%
      filter(diag_1 != 'Missing') %>% 
      select(age,diag_1) %>%
      group_by(age,diag_1) %>% 
      summarise(diag_count = length(diag_1)) %>% 
      ungroup() %>% 
      group_by(age) %>% 
      mutate(diag_share = round(diag_count/sum(diag_count),3 ) ) %>% 
      rename(Age = age,
             `Primary Diagnosis` = diag_1,
             `Diagnosis Share` = diag_share) %>% 
      ggplot(aes(Age,`Diagnosis Share`, fill = `Primary Diagnosis`, text = paste("Diagnosis Share:",paste(`Diagnosis Share`*100,'%'))))+geom_bar(stat = 'identity')+theme_minimal()+
      scale_y_continuous(labels = scales::percent)+ ggtitle('Shares of Primary Diagnoses by Age')
    plotly::ggplotly(p, tooltip= c('x','text','fill')) 

    Clearly Circulatory and Other diagnoses are the most common diagnosis for all age groups -- making up about 50% in total. Let's move to the second question

    Q2: Some doctors believe diabetes might play a central role in readmission. Explore the effect of a diabetes diagnosis on readmission rates

    To answer this question, I use several specifications of a logistic regression model to estimate the effect of a diabetes diagnosis on readmission rates. The logistic regression model suits the use-case as it provides a clear interpretation of the propensity for readmit by each covariety and is simple to implement in practice.

    Below is a table of the different estimators, each column is a different specification and each row is a covariate. If the covariate is a categorical variable, we can report the log-odds as that compared to the reference group. Here is a list of the categorical variables and their reference groups.

    List of Reference Groups for each Categorical Variable:

    • diag_1 : non-Diabetic
    • diag_2 : non-Diabetic
    • diag_3 : non-Diabetic
    • Age : [40-50)
    • Medical Speciality : Family/GeneralPractice
    • glucose_test : no
    • A1Ctest: no
    • change: no
    • diabetes_med : no

    To further simplify the interpretation, I convert the diagnoses variables to be either Diabetic or non-Diabetic. In this case, doctors can compare the average non-diabetic patient with the average diabetic patient making it a simpler calcuation (instead of, for example, comparing the diabetic patient to a Circulatory patient).

    Let's check out the estimators:

    # prep the data for logistic regressions
    df_features <- df %>% 
      filter(diag_1 != 'Missing',
             diag_2 != 'Missing',
             diag_3 != 'Missing',
             medical_specialty != 'Missing') %>% 
      mutate(readmitted = ifelse(readmitted == 'yes',1,0),
             time_in_hospital_sq = time_in_hospital^2,
             n_medications_dummy = ifelse(n_medications < 50,1,0),
             n_medications_sq = n_medications^2,
             n_medications_cube = n_medications^3,
             n_inpatient_sq = n_inpatient^2
             ) %>% 
      mutate(diag_1 = ifelse(diag_1 == 'Diabetes',1,0), # to make the contrast diabetic diagnosis vs all others
             diag_2 = ifelse(diag_2 == 'Diabetes',1,0),
             diag_3 = ifelse(diag_3 == 'Diabetes',1,0)) %>% 
      mutate(A1Ctest = factor(A1Ctest, levels = c("no", "normal", "high")), # refactoring for clear interpreation of reference 
             glucose_test =factor(glucose_test, levels = c("no", "normal", "high")),
            age = factor(age, levels = c("[40-50)","[50-60)" ,"[60-70)","[70-80)" ,"[80-90)","[90-100)" )),
            medical_specialty = factor(medical_specialty, levels = c('Family/GeneralPractice', 'Cardiology','Emergency/Trauma','InternalMedicine', 'Surgery', 'Other' )))
    # add series of specifications and print out with stargazer
    # logistic regression models
    # solo
    fit1 <- glm(data = df_features,readmitted ~ diag_1, family='binomial') 
    # other diagnoses
    fit2 <- glm(data = df_features,readmitted ~ diag_1+ diag_2 + diag_3, family='binomial') 
    # with patient characteristics
    fit3 <-  glm(data = df_features,readmitted ~ diag_1+ diag_2 + diag_3+age +n_outpatient + 
                   n_inpatient+n_emergency, family='binomial')
    # with hospital attendance characteristics
    fit4 <- glm(data = df_features,readmitted ~ diag_1+ diag_2 + diag_3+age +n_outpatient + 
                  n_inpatient+n_emergency+n_procedures+ n_lab_procedures+ medical_specialty+diabetes_med+glucose_test+A1Ctest+change, family='binomial')
    # all variables with interactions for non-linearities
    fit5 <- glm(data = df_features,readmitted ~ diag_1+ diag_2 + diag_3+  age + n_procedures+n_outpatient + n_inpatient +n_emergency+ n_inpatient_sq +n_lab_procedures+medical_specialty+diabetes_med+glucose_test+A1Ctest+change, 
    # make model list to input into se option in stargazer
    model.lst = list(fit1,fit2,fit3,fit4,fit5)
            digits = 2,
            se=lapply(model.lst, function(x) sqrt(diag(vcovHC(x, type = "HC1")))),# heteroskedastic-consistent standard errors 
            type = 'text')

    Model Specifications and Initial Results

    There are five specifications that include groups of controls. The first specification is only interested in whether or not the patient's primary diagnosis was Diabetes. The second specification adds in the secondary and tertiary diagnoses. The third specification includes patient characteristics outside of their visit. This includes patient age, number of inpatient, outpatient, and emergency visits in the past year. The fourth specification includes medical goods and services consumed during the patient's visit. This includes the number of procedures, lab procedures, the type of specialty of the overseeing physician, medication changes or administration, and other tests for the severity of diabetes. The fifth specification adds some non-linearity that were observed in the data exploring process.

    We can see that having a primary diagnosis of diabetes has a positive effect on being readmitted to the hospital throughout each of the specifications. The severity of a readmit is diminished when controlling for patient characteristics. It is further reduced when medical provisions are given to the patient. Regardless, the readmit effect is still strong with a log odds ratio of 23% (55.7% probability) in the fourth and fifth specifications. Each model's standard errors are consistent against heteroskedasticity.

    Interpreting the Results

    Regardless of what is done in the hospital, the patient's characteristics, and other diagnoses, primary diagnosis diabetic patients are still more likely to be readmitted than non-diabetic patients. We see that after controlling for patient characteristics, secondary and tertiary diagnoses have, on average, no effect on readmission rates. This result allows the hospital to focus squarely on the primary diagnosis as an indicator for readmittance.

    We can also see that the patient's age plays a significant role in readmits. As age increases, we see that the readmit rate is higher than those in the reference group (Ages 40-49). Note, that the relative probability of readmit has an upside-down U-shape. This upside-down U-shape can be attributed to selection bias in the data. The story goes like this: those who take worse care of themselves will likely die at a younger age than those who do. Readmittance is a result, in part, of not maintaining a sufficient stock of health capital. As the hospital observes more older patients, these patients tend to be those who can better maintain themselves through investments in their health capital. This is because those who would readmit, have died off. Further analysis requires a model that explains patient behaviour and readmit. In sum, the upside-down U-shape is driven by this selection bias.

    We can also see that a patient with more previous encounters will have a higher probabiliity of readmit. Inpatient visits have a higher propensity than both outpatient and emergency visits. This may be because inpatient visits treat chronic care while outpatient and emergency visits are for more acute care. Chronic care may just be harder to manage for patients than acute care which could be a reason why inpatient encounters give a higher propensity to readmit. Further, another reason for the difference could be that acute encounters "scares" patients into making better investments in there health while Chronic encounters do not force these patients to change their health investment.

    The attending physician's medical specialty also plays a large role in if a patient is readmitted. More severe encounters (i.e Surgery or Internal Medicine) reduce the probability of readmit. This could be another case where the type of doctor or encounter "scares" the patient into making better health investments. To argue this case: patients often substitute emergency visits for primary care (Family Medicine). Therefore, if the severity is the same for both type of visits, then there is no difference in readmit probability. Therefore, the "scare" interpretation may fit the data well. Further, it could also be that the differences in medical speciality is due to propensities for follow-up. An ER\Family Medicine physician may not care much about follow up than a Surgeon, for example.

    Last to note, the medical provisions related to diabetes have strong effects on readmits. A normal glucose test positively increases the probability of readmit. We can also see that if someone was prescribed diabetes medication, then they will likely readmit. This signals that the severity of the diabetes is so high that it requires more chronic care. Maintaining chronic care is difficult for diabetic patients.

    Other specifications and models considered: Shifting from to

    The main objective in this report is to maximize interpretability. That being said, there are other models and specifications that could be used to better predict the probability distribution of readmits. For example, a tree-based method could acccount for other non-linearities not captured by the logistic regression.

    I test this hypothesis using an out-of-the-box Random Forest model and a more sophisticated stacked ensemble model using H2o's autoML function. I find that the Random Forest model produces a mean squared error of 24% and the stacked ensemble gave an AUC of 65%. These poor results demonstrate that we cannot predict readmissions with the covariates available. However, there is still hope. Since the logistic regression provides parameters, we can focus on estimating unbiased and consistent parameters. We therefore can shift attention away from and focus instead on .

    This is what makes the logistic regression the better model: if our parameters are unbiased, we can at least explain the world we observe and everything outside our world has an average effect of 0. In this case, we do not care how much of our world we can explain given our observations if what we do observe we know to be true. The parameters of our model give us a good idea of the magnitude and direction of the covariates on readmissions and we believe that all else equal our error term, on average, has no effect on readmissions. Also, more practically, the parameters provide simple interpretation using log-odds which allows the hospital to hueristically assess a patients chance of admissions.

    Summary and the Probability Distribution

    To summarise the models presented above, clearly a primary diagnosis of diabetes has a strong, postive effect on readmits per the doctor's hypothesis. We test this hypothesis using a logistic regression model and estimate using several, robust specifications. The results also yield interesting interpretations for the other covariates like Age and Medical Speciality. These interpretations hopefully provide more insight for the doctors to not just think about diabetic readmits, but readmits more generally.

    Now that we have a model that estimates the probability distribution (or propensity scores) of a patient being readmitted, we can observe this distribution. See below a plot of the probability distrbutions comparing those who had a primary diagnosis of Diabetes and those who did not. We can clearly see that there is a log-normal probability distribution for each and where the median diabetic patient has a higher probability of readmit (47.6% compared to 41.2%) (see table below the plot)

    Check it out:

    cbind(df_features, tibble(fit5$fitted.values)) %>% 
      rename(fitted_prob = `fit5$fitted.values`) %>%
    mutate(diag_1 = ifelse(diag_1 == 1,'Diabetes','Non-Diabetes')) %>%
    ggplot(aes(x=fitted_prob, color = as.factor(diag_1), fill = as.factor(diag_1)))+
    geom_density(alpha = 0.25) +
    scale_fill_discrete(name = "Type of Diagnosis")+
    xlab('Predicted Probabilities of Readmit')+
    ggtitle('Densities of Probability Distributions by Type of Diagnosis')