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Regression Analysis

What is Regression Analysis?

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. 

What does a regression analysis tell you?

Regression analysis is a reliable method of identifying which variables have impact on a topic of interest. The process of performing a regression allows you to confidently determine which factors matter most, which factors can be ignored, and how these factors influence each other.

In order to understand regression analysis fully, it’s essential to comprehend the following terms:

  • Dependent Variable: This is the main factor that you’re trying to understand or predict. 
  • Independent Variables: These are the factors that you hypothesize have an impact on your dependent variable.

In our application training example above, attendees’ satisfaction with the event is our dependent variable. The topics covered, length of sessions, food provided, and the cost of a ticket are our independent variables.

How does regression analysis work?

In order to conduct a regression analysis, you’ll need to define a dependent variable that you hypothesize is being influenced by one or several independent variables.

You’ll then need to establish a comprehensive dataset to work with. Administering surveys to your audiences of interest is a terrific way to establish this dataset. Your survey should include questions addressing all of the independent variables that you are interested in.

To begin investigating whether or not there is a relationship between these two variables, we would begin by plotting these data points on a chart.

Our dependent variable should be plotted on the y-axis, while our independent variable should be plotted on the x-axis.

Once your data is plotted, you may begin to see correlations. Draw a line through the middle of all of the data points on the chart. This line is referred to as your regression line, and it can be precisely calculated using a standard statistics program like Excel.

The regression line represents the relationship between your independent variable and your dependent variable. 

Excel will even provide a formula for the slope of the line, which adds further context to the relationship between your independent and dependent variables. 

The formula for a regression line might look something like Y = 100 + 7X + error term.

This tells you that if there is no “X”, then Y = 100. If X is our increase in ticket price, this informs us that if there is no increase in ticket price, event satisfaction will still increase by 100 points. 

You’ll notice that the slope formula calculated by Excel includes an error term. Regression lines always consider an error term because in reality, independent variables are never precisely perfect predictors of dependent variables. This makes sense while looking at the impact of  ticket prices on event satisfaction — there are clearly other variables that are contributing to event satisfaction outside of price.

Your regression line is simply an estimate based on the data available to you. So, the larger your error term, the less definitively certain your regression line is.

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What is regression analysis used for?

Regression analysis is helpful statistical method that can be leveraged across an organization to determine the degree to which particular independent variables are influencing dependent variables. 

Use regression analysis to describe the relationships between a set of independent variables and the dependent variable. Regression analysis produces a regression equation where the coefficients represent the relationship between each independent variable and the dependent variable. You can also use the equation to make predictions.

The possible scenarios for conducting regression analysis to yield valuable, actionable business insights are endless.

The next time someone in your business is proposing a hypothesis that states that one factor, whether you can control that factor or not, is impacting a portion of the business, suggest performing a regression analysis to determine just how confident you should be in that hypothesis! This will allow you to make more informed business decisions, allocate resources more efficiently, and ultimately boost your bottom line.

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

Which regression model is best?

There are numerous types of regression models that you can use. This choice often depends on the kind of data you have for the dependent variable and the type of model that provides the best fit.

1. Linear regression

Linear regression, also known as ordinary least squares (OLS) and linear least squares, is the real workhorse of the regression world. Use linear regression to understand the mean change in a dependent variable given a one-unit change in each independent variable. You can also use polynomials to model curvature and include interaction effects. Despite the term “linear model,” this type can model curvature.

Linear models are the oldest type of regression. It was designed so that statisticians can do the calculations by hand. However, OLS has several weaknesses, including a sensitivity to both outliers and multicollinearity, and it is prone to overfitting. 

To address these problems, statisticians have developed several advanced variants:

  • Ridge regression: Ridge regression allows you to analyze data even when severe multicollinearity is present and helps prevent overfitting. This type of model reduces the large, problematic variance that multicollinearity causes by introducing a slight bias in the estimates. The procedure trades away much of the variance in exchange for a little bias, which produces more useful coefficient estimates when multicollinearity is present.
  • Lasso regression (least absolute shrinkage and selection operator): Lasso regression performs variable selection that aims to increase prediction accuracy by identifying a simpler model. It is similar to Ridge regression but with variable selection.
  • Partial least squares (PLS) regression: PLS Regression is useful when you have very few observations compared to the number of independent variables or when your independent variables are highly correlated. PLS decreases the independent variables down to a smaller number of uncorrelated components, similar to Principal Components Analysis. 

2. Non-linear regression

Nonlinear regression also requires a continuous dependent variable, but it provides a greater flexibility to fit curves than linear regression.

Like OLS, nonlinear regression estimates the parameters by minimizing the SSE. However, nonlinear models use an iterative algorithm rather than the linear approach of solving them directly with matrix equations. What this means for you is that you need to worry about which algorithm to use, specifying good starting values, and the possibility of either not converging on a solution or converging on a local minimum rather than a global minimum SSE. And, that’s in addition to specifying the correct functional form!

Most nonlinear models have one continuous independent variable, but it is possible to have more than one. When you have one independent variable, you can graph the results using a fitted line plot.

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Regression Analysis

October 14, 2020

Table of contents

Key takeawaysCollaboration platforms are essential to the new way of workingEmployees prefer engati over emailEmployees play a growing part in software purchasing decisionsThe future of work is collaborativeMethodology

What is Regression Analysis?

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. 

What does a regression analysis tell you?

Regression analysis is a reliable method of identifying which variables have impact on a topic of interest. The process of performing a regression allows you to confidently determine which factors matter most, which factors can be ignored, and how these factors influence each other.

In order to understand regression analysis fully, it’s essential to comprehend the following terms:

  • Dependent Variable: This is the main factor that you’re trying to understand or predict. 
  • Independent Variables: These are the factors that you hypothesize have an impact on your dependent variable.

In our application training example above, attendees’ satisfaction with the event is our dependent variable. The topics covered, length of sessions, food provided, and the cost of a ticket are our independent variables.

How does regression analysis work?

In order to conduct a regression analysis, you’ll need to define a dependent variable that you hypothesize is being influenced by one or several independent variables.

You’ll then need to establish a comprehensive dataset to work with. Administering surveys to your audiences of interest is a terrific way to establish this dataset. Your survey should include questions addressing all of the independent variables that you are interested in.

To begin investigating whether or not there is a relationship between these two variables, we would begin by plotting these data points on a chart.

Our dependent variable should be plotted on the y-axis, while our independent variable should be plotted on the x-axis.

Once your data is plotted, you may begin to see correlations. Draw a line through the middle of all of the data points on the chart. This line is referred to as your regression line, and it can be precisely calculated using a standard statistics program like Excel.

The regression line represents the relationship between your independent variable and your dependent variable. 

Excel will even provide a formula for the slope of the line, which adds further context to the relationship between your independent and dependent variables. 

The formula for a regression line might look something like Y = 100 + 7X + error term.

This tells you that if there is no “X”, then Y = 100. If X is our increase in ticket price, this informs us that if there is no increase in ticket price, event satisfaction will still increase by 100 points. 

You’ll notice that the slope formula calculated by Excel includes an error term. Regression lines always consider an error term because in reality, independent variables are never precisely perfect predictors of dependent variables. This makes sense while looking at the impact of  ticket prices on event satisfaction — there are clearly other variables that are contributing to event satisfaction outside of price.

Your regression line is simply an estimate based on the data available to you. So, the larger your error term, the less definitively certain your regression line is.

Build an AI chatbot to engage your always-on customers

What is regression analysis used for?

Regression analysis is helpful statistical method that can be leveraged across an organization to determine the degree to which particular independent variables are influencing dependent variables. 

Use regression analysis to describe the relationships between a set of independent variables and the dependent variable. Regression analysis produces a regression equation where the coefficients represent the relationship between each independent variable and the dependent variable. You can also use the equation to make predictions.

The possible scenarios for conducting regression analysis to yield valuable, actionable business insights are endless.

The next time someone in your business is proposing a hypothesis that states that one factor, whether you can control that factor or not, is impacting a portion of the business, suggest performing a regression analysis to determine just how confident you should be in that hypothesis! This will allow you to make more informed business decisions, allocate resources more efficiently, and ultimately boost your bottom line.

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

Which regression model is best?

There are numerous types of regression models that you can use. This choice often depends on the kind of data you have for the dependent variable and the type of model that provides the best fit.

1. Linear regression

Linear regression, also known as ordinary least squares (OLS) and linear least squares, is the real workhorse of the regression world. Use linear regression to understand the mean change in a dependent variable given a one-unit change in each independent variable. You can also use polynomials to model curvature and include interaction effects. Despite the term “linear model,” this type can model curvature.

Linear models are the oldest type of regression. It was designed so that statisticians can do the calculations by hand. However, OLS has several weaknesses, including a sensitivity to both outliers and multicollinearity, and it is prone to overfitting. 

To address these problems, statisticians have developed several advanced variants:

  • Ridge regression: Ridge regression allows you to analyze data even when severe multicollinearity is present and helps prevent overfitting. This type of model reduces the large, problematic variance that multicollinearity causes by introducing a slight bias in the estimates. The procedure trades away much of the variance in exchange for a little bias, which produces more useful coefficient estimates when multicollinearity is present.
  • Lasso regression (least absolute shrinkage and selection operator): Lasso regression performs variable selection that aims to increase prediction accuracy by identifying a simpler model. It is similar to Ridge regression but with variable selection.
  • Partial least squares (PLS) regression: PLS Regression is useful when you have very few observations compared to the number of independent variables or when your independent variables are highly correlated. PLS decreases the independent variables down to a smaller number of uncorrelated components, similar to Principal Components Analysis. 

2. Non-linear regression

Nonlinear regression also requires a continuous dependent variable, but it provides a greater flexibility to fit curves than linear regression.

Like OLS, nonlinear regression estimates the parameters by minimizing the SSE. However, nonlinear models use an iterative algorithm rather than the linear approach of solving them directly with matrix equations. What this means for you is that you need to worry about which algorithm to use, specifying good starting values, and the possibility of either not converging on a solution or converging on a local minimum rather than a global minimum SSE. And, that’s in addition to specifying the correct functional form!

Most nonlinear models have one continuous independent variable, but it is possible to have more than one. When you have one independent variable, you can graph the results using a fitted line plot.

Let's build your first AI Chatbot today!

 

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