Question

1. Explain Simple Regression analysis

2. Give an example of a simple regression equation in any STEM subject

3. How will you implement such the regression model?

4. List some examples of how regression can be used in research and practice in STEM.

Answer 1

1). Simple linear regression analysis is a statistical tool for quantifying the relationship between just one independent variable (hence "simple") and one dependent variable based on past experience (observations). For example, simple linear regression analysis can be used to express how a company's electricity cost (the dependent variable) changes as the company's production machine hours (the independent variable) change.

Prior to using simple linear regression analysis it is important to follow these preliminary steps:-

• seek an independent variable that is likely to cause or drive the change in the dependent variable
• make certain that the past amounts for the independent variable occur in the exact same period as the amount of the dependent variable
• plot the past observations on a graph using the y-axis for the cost (monthly electricity bill) and the x-axis for the activity (machine hours used during the exact period of the electricity bill)
• review the plotted observations for a linear pattern and for any outliers
• keep in mind that there can be correlation without cause and effect

2). Example of Simple linear equation in Maths:-

Parallel lines have equal slopes. if y = mx + b, then m is the slope and b is the y-intercept (i.e., the value of y when x = 0). Often linear equations are written in standard form with integer coefficients (Ax + By = C). Such relationships must be converted into slope-intercept form (y = mx + b) for easy use on the graphing calculator. One other form of an equation for a line is called the point-slope form and is as follows: y - y₁ = m(x - x₁). The slope, m, is as defined above, x and y are our variables, and (x₁, y₁) is a point on the line.

3). When implementing simple linear regression, you typically start with a given set of input-output (?-?) pairs (green circles). These pairs are your observations. For example, the leftmost observation (green circle) has the input ? = 5 and the actual output (response) ? = 5. The next one has ? = 15 and ? = 20, and so on. The estimated regression function (black line) has the equation ?(?) = ?₀ + ?₁?. Your goal is to calculate the optimal values of the predicted weights ?₀ and ?₁ that minimize SSR and determine the estimated regression function. The value of ?₀, also called the intercept, shows the point where the estimated regression line crosses the ? axis. It is the value of the estimated response ?(?) for ? = 0.

The value of ?₁ determines the slope of the estimated regression line. The predicted responses (red squares) are the points on the regression line that correspond to the input values. For example, for the input ? = 5, the predicted response is ?(5) = 8.33 (represented with the leftmost red square). The residuals (vertical dashed gray lines) can be calculated as ?ᵢ - ?(?ᵢ) = ?ᵢ - ?₀ - ?₁?ᵢ for ? = 1, …, ?. They are the distances between the green circles and red squares. When you implement linear regression, you are actually trying to minimize these distances and make the red squares as close to the predefined green circles as possible.

4). Regression analysis can handle many things. For example, you can use regression analysis to do the following:-

• Model multiple independent variables
• Include continuous and categorical variables
• Use polynomial terms to model curvature
• Assess interaction terms to determine whether the effect of one independent variable depends on the value of another variable

These capabilities are all cool, but they don’t include an almost magical ability. Regression analysis can unscramble very intricate problems where the variables are entangled like spaghetti. For example, imagine you’re a researcher studying any of the following:-

• Do socio-economic status and race affect educational achievement?
• Do education and IQ affect earnings?
• Do exercise habits and diet effect weight?
• Are drinking coffee and smoking cigarettes related to mortality risk?
• Does a particular exercise intervention have an impact on bone density that is a distinct effect from other physical activities?

All these research questions have entwined independent variables that can influence the dependent variables. How do you untangle a web of related variables? Which variables are statistically significant and what role does each one play? Regression comes to the rescue because you can use it for all of these scenarios.