Simplifying Linear Regression for Clinical Data Managers

1 Linear Regression

1.1 Introduction

Linear regression is a simple yet powerful statistical technique used to understand the relationship between variables in clinical research. It helps us predict or estimate a continuous outcome variable based on one or more input variables.

1.2 Assumptions of Linear Regression

It is important to note the data related assumptions for Linear Regression model. It will help to know if linear regression is the right tool for the analysis. If the assumptions are violated, the model's results may be misleading or invalid. Know the data assumptions also helps in identifying and remediating issues in data that are leading to violation of these assumptions and therefore help avoid misleading or invalid results.

1.2.1 Linearity

The relationship between the input variables and the outcome variable should be approximately linear. In clinical research, this means that changes in the input variables should be associated with proportional changes in the outcome variable.

1.2.2 Independence

The observations should be independent of each other. In clinical research, this assumes that each patient's data is independent of other patients.

1.2.3 Homoscedasticity

Homoscedasticity refers to the assumption that the variability of the outcome variable is constant across different levels of the input variables. Lets consider If there is homoscedasticity, the spread of test scores would be similar for students who study 2 hours, 4 hours, 6 hours, and so on. This means that the variability (how much the scores differ from each other) in test scores is about the same regardless of the number of study hours.

For example:

Students who study 2 hours might have scores ranging from 70 to 80.

Students who study 4 hours might have scores ranging from 85 to 95.

Students who study 6 hours might have scores ranging from 90 to 100.

In this case, the range of scores is consistent (around 10 points) across different study hours, showing homoscedasticity.

If the variability was not the same (e.g., students who study 2 hours have scores ranging from 60 to 80, while students who study 6 hours have scores ranging from 85 to 100), it would show heteroscedasticity, not homoscedasticity.?

In clinical research, this means that the spread or range of the outcome variable should be consistent for all values of the input variables.?

1.2.4 Normality

The residuals (the differences between the predicted and observed values) should follow a normal distribution. In clinical research, this assumes that the errors are normally distributed around the regression line.

1.3 Model Fitting, Interpretation, and Evaluation

1.3.1 Model Fitting

Model fitting in linear regression involves finding the best-fit line that represents the relationship between the input variables and the outcome variable. This line is calculated by estimating two main components:

Intercept:?This is the value of the outcome variable when all input variables are zero. It's where the line crosses the y-axis.

Slope coefficients:?These represent the change in the outcome variable for a one-unit change in the corresponding input variable.

The best-fit line is determined by minimizing the sum of the squared differences between the observed values (actual data points) and the predicted values (points on the regression line). This method is called "least squares."

1.3.2 Interpretation

Once the model is fitted, we need to interpret the coefficients:

Intercept:?The intercept tells us the starting point of the outcome variable when all input variables are zero. For example, if we are predicting blood pressure and the intercept is 70, it means that if all other factors are zero, the predicted blood pressure is 70 mmHg.

Slope coefficients:?Each slope coefficient shows how much the outcome variable is expected to increase or decrease with a one-unit change in the input variable. For example, if the slope for age is 1.5, it means that for every additional year of age, the blood pressure is expected to increase by 1.5 mmHg.

1.3.3 Evaluation

Evaluating the performance of a linear regression model involves several metrics:

R-squared (R2): This metric indicates how well the model explains the variability of the outcome variable. An R2 value of 1 means the model explains all the variability, while an R2 of 0 means it explains none. For example, if R2 is 0.8, it means 80% of the variability in the outcome variable is explained by the model.

Root Mean Squared Error (RMSE):?RMSE measures the average difference between the predicted values and the actual values. A lower RMSE indicates a better fit. For example, if the RMSE is 5, it means that, on average, the predicted values are within 5 units of the actual values.

1.4 Example Application in Clinical Research

In a clinical study on hypertension, researchers investigate the relationship between blood pressure (outcome variable) and age and body mass index (input variables). They collect data from 100 patients and perform a linear regression analysis. The model shows that for every one-year increase in age, blood pressure increases by 1.5 mmHg, and for every one-unit increase in BMI, blood pressure increases by 0.8 mmHg.

1.5 Key take away

Linear regression is a valuable tool in clinical research for understanding the relationship between variables and predicting continuous outcomes. It relies on assumptions of linearity, independence, homoscedasticity, and normality. Model fitting, interpretation, and evaluation help us understand and evaluate the predictive performance of the linear regression model.