• My presentation will focus on the concept of simple linear regression and how it can be applied in a real working environment.

• Specifically, I want to explore how we can use this math concept to predict or explain how one variable may affect another. I will be using the medical topic of body mass index versus blood pressure for this presentation.

• The basic formula looks something like this: \[ y = \beta_0 + \beta_1 x + \epsilon \]

Where: \(y\) = Blood Pressure, \(x\) = Body Mass Index, \(\beta_0\) = Intercept, \(\beta_1\) = Slope

Simple Linear Regression is a strong math tool that is a major component of statistical analysis.

• It can allow us to explore correlations between two variables.

• This can then allow us to predict certain outcomes related to the variables, which gives us a a better chance of making more informed clinical decisions.

• This presentation will focus on a medical application of such concepts, specifically with the relationship between blood pressure and body mass index.

BMI (predictor), Blood Pressure (outcome)

We will use a randomly generated dataset of patients as well as their BMI and their corresponding blood pressure levels.

• Although random, these numbers are set in a range that makes reasonable sense.

#example dataset
set.seed(123)
BMI <- rnorm(100, mean = 25, sd = 5)
bloodPressure <- 80 + 2.5 * BMI + rnorm(100, sd = 10)
medicalData <- data.frame(BMI, bloodPressure)

• This code creates a fully randomized selection of patient data and serves as the baseline foundation for the data analysis we will perform later on.

• set.seed(123) ensures that the same sequence of random numbers are generated by rnorm every time.

• BMI <- norm(100, mean = 25, sd = 5) generates vector of \(100\) random BMI values from a normal distribution, with an average value at \(25\) and standard deviation at \(5\).

• bloodPressure <- 80 + 2.5 * BMI generates vector of assumed blood pressure levels based on BMI values. \(80\) is considered the base blood pressure, while \(2.5 * BMI\) represents the generally accepted relationship between both variables.

• The added rnorm(100, sd = 10) is to add variability to the data to best simulate the situation and gain more credible information.

set.seed(123)
BMI <- rnorm(100, mean = 25, sd = 5)
bloodPressure <- 80 + 2.5 * BMI + rnorm(100, sd = 10)
medicalData <- data.frame(BMI, bloodPressure)
print(head(medicalData))

##        BMI bloodPressure
## 1 22.19762      128.3900
## 2 23.84911      142.1916
## 3 32.79354      159.5169
## 4 25.35254      139.9059
## 5 25.64644      134.5999
## 6 33.57532      163.4880

• Using this data set, the relationship between the two variables can be visualized in a more effective manner.

• The next few slides have various models that better depict the nuances of the relationship between BMI and blood pressure.

• This information reveals telling insights into how we can start to make more informed clinical decisions based on this correlation.

## 
## Call:
## lm(formula = bloodPressure ~ BMI, data = medicalData)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -19.073  -6.835  -0.875   5.806  32.904 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  81.5956     5.5265   14.76   <2e-16 ***
## BMI           2.3951     0.2138   11.21   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.707 on 98 degrees of freedom
## Multiple R-squared:  0.5616, Adjusted R-squared:  0.5571 
## F-statistic: 125.5 on 1 and 98 DF,  p-value: < 2.2e-16

Therefore, the final equation for predicting the blood pressure based on the units of BMI is:

\[ \text{Blood Pressure} = 81.5956 + 2.3951 \times \text{BMI} \]

Where:

• \(81.5956\) is the intercept that represents the baseline blood pressure when BMI is at zero.

• \(2.3951\) is the slope that represents the change in blood pressure for each added unit of BMI.

• The purpose of the residual plot is to showcase how the regression model fits into the relationship between the two variables.

• The residual plot can help researchers assess whether the relationship is linear, how the variance of residuals are across all BMI levels which can reveal issues with data collection, as well as any outliers.

• The dashed line in the center represents no variance. Ideally the plot should contain a random distribution of points around the line all throughout.

• The residual plot from the previous slide has a random distribution which indicates that the data selection was truly random and that the results from the linear regression model are credible.