Type of Study

This cross-sectional observational study based on gathered, non-experimental data. We might be able to make meaningful associations based on it, but due to the limited ethnic diversity and the gender bias within the set, it is definitely not going to provide generalizable results for humanity at large. However, it can provide insight for future controlled experiments or observational studies which take diversity across ages, genders and races into consideration.

Data Source

The set was gathered from the patient records in area north-east of Andhra Pradesh, India and donated to to University of California Irvine by:

1. Bendi Venkata Ramana ramana.bendi ‘@’ gmail.com Associate Professor, Department of Information Technology, Aditya Instutute of Technology and Management, Tekkali - 532201, Andhra Pradesh, India.

2. Prof. M. Surendra Prasad Babu drmsprasadbabu ‘@’ yahoo.co.in Deptartment of Computer Science & Systems Engineering, Andhra University College of Engineering, Visakhapatnam-530 003 Andhra Pradesh, India.

3.Prof. N. B. Venkateswarlu venkat_ritch ‘@’ yahoo.com Department of Computer Science and Engineering, Aditya Instutute of Technology and Management, Tekkali - 532201, Andhra Pradesh, India.

About the

There are 583 observations, 416 represent subjects with diseased livers, 167 represent subjects without diseased livers.

The data represent 441 male subjects (of whom 324 have liver disease) and 142 female subjects (of whom 92 have liver disease).

Data Dictionary

1. Age = Age of the patient (all subjects greater than 89 are labelled 90)
2. Sex = Gender of the patient Female Male
3. Tot_Bil = Total Bilirubin
4. Dir_Bil = Direct Bilirubin
5. Alk_Phos = Alkaline Phosphotase
6. Alamine = Alamine Aminotransferase
7. Aspartate = Aspartate Aminotransferase
8. Tot_Prot = Total Protiens
9. Albumin = Albumin 10.A_G_Ration = Albumin and Globulin Ratio 11.Disease = Disease State (classified labeled by the medical experts ) 0 - not diseased 1- diseased

Predictor Variables are columns 1 through 10 and the outcome variable is column 11, Disease.

Age

Gender

Total Bilirubin

Bilirubin is a byproduct of hemolytic catabolism, it is one of many substances the liver filters from the body. Heighten presence of either or both can be indicative of liver disease and is the cause of skin yellowing associated with jaundice. (medscape)

Direct Bilirubin

Alkaline Phosphotase

This is one of the enzymes included in a normal liver panel, frequently used to estimate overall liver health.

Alamine Aminotransferase

Alamine Aminotransferase is a natural art of the liver ecosystem tested for in a liver panel.

Aspartate Aminotransferase

Aspartate Aminotransferase is also a natural part of the liver ecosystem tested for in a liver panel.

Total Proteins

Total protein is a measure of both albumin and globulin combined.

Albumin

Albumin is a blood protein which adds structure to the vascular system keeping preventing blood from seeping out through the vessel walls.

Albumin-Globulin Ratio

The Albumin-Globulin ratio is considered an index of systemic diseases in general.

The above graphs suggest that the enzymes and proteins in question vary relative to the diseased state which we may be able to understand more completely after the analysis.

Training Summary

Initial Logistic Regression Model

fit <- glm(Disease ~ Age + Sex + Tot_Bil + Dir_Bil + Alkphos + Alamine + Aspartate + 
    Tot_Prot + Albumin + A_G_Ratio, data = train, family = binomial(link = "logit"))

Coefficients:

Model Pseudo R-Square and Log-Likelihoods (from pscl package)

Based on the above table using the MacFadden \(R_2\) as a guide, it appears that this model explains approximately 22.5% of the disease classification.

Another estimate of utility is the Coefficient of Discrimination is test metric which subtracts mean of all No-disease probabilities from the mean of all disease-probabilities based on the predicted outcomes of the test data.

# Start by making a data frame of predictions
Test_Predictions <- data.frame(Probability = predict(fit, test, type = "response"))
# add then add predicted class using 50% probability to round up and down
Test_Predictions$Prediction <- ifelse(Test_Predictions > 0.5, 1, 0)
# added the actual diagnosed disease/ non classification to the set
Test_Predictions$Disease <- test$Disease

# accuracy is simply the mean of all trues (1) where prediction = reality
accuracy <- mean(Test_Predictions$Disease == Test_Predictions$Prediction, na.rm = TRUE)

# For the Coefficient of Discrimination create arrays of predicted
# #probabilities where the true outcome is disease and one where it is not
# taking an average of the two give a sense of the overall predictive power
disease <- Test_Predictions$Probability[which(Test_Predictions$Disease == 1)]
non <- Test_Predictions$Probability[which(Test_Predictions$Disease == 0)]
Coef_Desc <- mean(disease, na.rm = TRUE) - mean(non, na.rm = TRUE)

Coefficient of Discrimination : 0.1420897

Accuracy: 0.6889952

The accuracy is simply the number of times the model was right…which seems considerably higher than the Model Fit Estimates would suggest, at around 69%.

plot(Test_Predictions$Probability, Test_Predictions$Disease, xlim = c(0.1, 1.1), 
    xlab = "Probability", ylim = c(-0.1, 1.1), ylab = "Disease", col = "blue", 
    pch = 18)

Scope of Analysis

Improvements

Although there were variables which were not in and of themselves statistically significant (meaning we cannot tell what their precise contribution to the diseased diagnosis is reliable) removing them in any order did nothing but drop the accuracy, the Coefficient of Discrimination and the Pseudo R-square scores, indicating that for shear ability to predict the likelihood of a patient having liver disease based on the included variables, the whole model is more accurate than an abridged logistic model.

Model Conditions

The two conditions which we need to meet are:

1. That each predictor Xi is linearly related to the logit( \(p_i\) )when all other predictors are held constant

Based on the following samples of x plotted against the predicted probabilities of the logistic regression, it seems as though there may be some concerns with using these particular regressors in a logistic model, even transformed.

train$predictions <- predict(fit, train, type = "response")
require(ggplot2)
ggplot(train, aes(x = Tot_Bil, y = predictions)) + geom_point() + geom_smooth(method = "lm", 
    se = FALSE)

ggplot(train, aes(x = Dir_Bil, y = predictions)) + geom_point() + geom_smooth(method = "lm", 
    se = FALSE)

ggplot(train, aes(x = Alkphos, y = predictions)) + geom_point() + geom_smooth(method = "lm", 
    se = FALSE)

ggplot(train, aes(x = Albumin, y = predictions)) + geom_point() + geom_smooth(method = "lm", 
    se = FALSE)

2. Each \(Y-i\) is independent of other outcomes

no mention of relations between family members was included in the metadata for this set, so the assumption is that all subjects are independent of each other