Methodology: 824625

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Answers:

Research Question

Is it possible that the levels of creatine kinase (CK) and hemopexin (H) be used to develop a predictive model to determine if a woman is a carrier or not?

Data

Data for this project is obtained from the Sleuth3 library in R statistical programming language under the dataset “ex2012”. It contains 3 variables, i.e. Group which has 2 levels that is Control and cases, the group variable is as well the response variable while creatine kinase (CK) and hemopexin (H) are the predictor variables. Moreover, there are 120 entries in the dataset.

The control level indicates the test subjects that are not diagnosed with DMD while the case level indicate the test subjects diagnosed with DMD.

Logistic regression

Logistic regression is often categorized under a classification algorithm in machine learning. It is generally applied in prediction of binary outcomes i.e. when the expected outcome falls into two distinct categories. Therefore, it predicts the probability of an event occurring through data fitting using a “logit” function.

Generalized Linear Model

The logistic regression model is one of the wider range of Generalized Line Models (glm’s) with a general equation of:

g(E(y)) = α + βx1 + γx2

where g() is link function, E(Y) expectation of target variable and “α + βx1 + γx2” as the linear predictor.

Assumptions

1. There is no linear relationship between predictor and response variables

2. There is no strict need for the dependent variable to be normally distributed

3. Errors ought to be independent but not necessarily having a normal distribution

Evaluation of Logistic Regression Model

The logistic regression model is evaluated using:

1- The AIC (Akaike Information Criterion)-where the best model has the least AIC

2- Null Deviance and Residual Deviance– implying the predicted response through use of intercept. Low residual deviance indicates good models

3- Null Deviance and Residual Deviance-Enables reduction of overfitting and determination of model accuracy

4- ROC Curve-The ROC curve summarizes the logistic model performance in predicting the response variable where the area under the curve should be large enough for the model to be a good predictive model.

Part 1:

1. a) Graph

From the graph above, the control data form the train data i.e. they form approximately 30% of the data while the case data are spread wider. Therefore, it suggests that the two variables CK and H can be used as good predictors for the response variable.

Mean by groups

Mean of CK and H

Under the Control level, CK has a mean of 40.48 while H has a mean of 82.64.

In the Case level, CK has a mean of 175.87 while H has a mean of 93.99.

Part 2

Constructing a Logistic Model

1. Logistic model

1. Logistic model Diagnostics

2. Significant Variables

At a significance level of 0.05, both CK and H are significant in predicting DMD, i.e. the p-value of both CK and H are approximately 0.000 which is less than 0.05 hence we reject the null hypothesis which assumes zero difference between the predictor and response variables and conclude that there is a relationship between the predictor and response variables.

3. Evidence of interaction effect

Taking into consideration the logistic regression results:

Y=16.16695 -0.06838CK – 0.12732H

where Y is the DMD, we realize that for every 1 positive CK and H, the effect on DMD is 15.97125. Therefore, there is evidence of an interaction effect between the two predictor variables and DMD

4. Goodness-of-fit test of the Model

Mathew (2015) argues that a McFadden value closer to 0 indicates a model with no predictive power hence low goodness of fit. From the McFadden results of our model, i.e. 0.5847283 is well above 0 hence the model passes the goodness of fit test.

Part 3

1. Using the model to give a 95% Multiplicative effect on the odds of increasing either H or CK

The results above indicate that in the multiplicative effect on the odds of CK when increasing H is that the odds of CK increase while a decrease in H leads to a decrease in the odds of CK. For instance, when H is increased by 0.92%, CK increases by 0.95 with odds of between 0.93 and

0.97 compared to when H is decreased by 0.88% where the odds of CK reduce to between 0.91 and 0.96.

2. How good a job the model does, i.e. Predicting performance of the model

To explore the performance of the model we use:

1. Confusion matrix

1. ROC curve

Confusion matrix

The model predicts 79 control cases correctly with only 3 cases falsely while most DMD cases are falsely predicted i.e. 28 with 10 being true

ROC Curve

From the ROC graph above, it is clear that there is a large area under the curve indicating that logistic model performs well in predicting the response variable i.e. where the area under the curve should be large enough for the model to be a good predictive model.

Conclusion

In conclusion, from our logistic regression, creatine kinase (CK) and hemopexin (H) are both significant in predicting DMD, i.e. through the low Akaike Information criterion, large ROC curve area as well as high R-squared statistic indicating that the model used in predicting the response variable is a good fit hence the regression results can be used to predict the outcome of the presence of both CK and H in a woman’s hormone to determine whether they are carriers or not.

References

Michael, A. (2015). Evaluating Logistic Regression Models. Available from:

https://www.r-bloggers.com/evaluating-logistic-regression-models/