6.3.4.2 Regression and statistical methods

Achievable CMA Part 1

6. Technology and analytics

6.3. Data analytics

6.3.4. Types of data analytics

Our CMA Part 1 course is currently in development and is a work-in-progress.

Regression and statistical methods

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Learning outcome statements

The learning outcome statements relevant for this section are:

demonstrate an understanding of multiple regression and logistic regression and recognize when these techniques are appropriate
calculate the result of multiple regression equations as applied to a specific situation
demonstrate an understanding of the coefficient of determination (R squared) and the correlation coefficient $(r)$
identify and explain the benefits and limitations of regression analysis
define standard error of the estimate, goodness of fit, and confidence interval
explain how to use predictive analytics techniques to draw insights and make recommendations

Defining a multiple regression equation

The topic of multiple regression has already been discussed in CMA Part 1, Section B: Forecasting. However, additional discussions are explored here in CMA Part 1, Section F to address the expanded learning outcome statements related to its applications in data analytics and decision-making.

Multiple regression involves more than one independent variable. The equation for multiple regression is:

$Y = a + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{n} X_{n} + ε$

Where:

Y = dependent variable (the outcome being explained)
a = intercept (value of Y when all X’s are zero)
b1, b2, … bn = coefficients (slopes showing the impact of each independent variable on Y)
X1, X2, … Xn = independent variables (factors that influence Y)
ε = error term (variation not explained by the model)

When to use multiple regression in data analytics

Multiple regression is used when a dependent variable is influenced by multiple factors, allowing analysts to measure relationships and make predictions. Below are common applications of multiple regression:

Multiple regression is appropriate when multiple independent variables influence an outcome, helping businesses and analysts improve predictions, optimize strategies, and make data-driven decisions.

Benefits and shortcomings of regression analysis

Benefits

Predictive power: Regression analysis allows companies to forecast outcomes, such as sales or costs, based on different levels of input (independent) variables.
Decision-making: By understanding the relationship between variables, businesses can make informed decisions about how changes in one factor (e.g., production volume) will impact others (e.g., costs or revenue).
Identifying trends: Regression can help identify underlying trends in historical data, allowing organizations to project future performance.

Shortcomings

Assumption of linearity: Simple regression assumes a linear relationship between variables, which may not always be the case in real-world situations.
Data limitations: Regression analysis depends on the quality of data. Inaccurate or incomplete data can lead to unreliable predictions.
Overfitting: Using too many variables in a regression model can lead to overfitting, where the model becomes too tailored to historical data and fails to predict future outcomes accurately.

Coefficient of determination (R²) and correlation coefficient $(r)$

In data analytics, understanding relationships between variables is critical for making informed predictions and decisions. Two key statistical measures used for this purpose are:

coefficient of determination (R²) and
correlation coefficient $(r)$

Correlation coefficient $(r)$

The $correlation coefficient (r)$ measures the strength and direction of a linear relationship between two variables. It ranges from $-1 to 1$

$r = 1$ indicates a perfect positive correlation, meaning as one variable increases, the other also increases.
$r = -1$ indicates a perfect negative correlation, meaning as one variable increases, the other decreases.
$r = 0$ suggests no linear correlation between the variables.

In data analytics, correlation helps determine whether a relationship exists between two variables, such as marketing spend and revenue.

Coefficient of determination (R²)

The $R² value$ tells us how well the independent variable(s) explain the variability of the dependent variable. It ranges from $0 to 1$ :

$R² = 0$ means the independent variable explains none of the variation in the dependent variable.
$R² = 1$ means the independent variable explains 100% of the variation.

In data analytics, a higher $R²$ value indicates a better model fit, meaning the independent variables are strong predictors of the dependent variable.

For example, in sales forecasting, if an R² value is 0.85, it means 85% of the variation in sales can be explained by the independent variables (such as advertising spend, promotions, and seasonality). This helps businesses evaluate the reliability of their predictive models.

Standard error of the estimate (SEE), goodness of fit, and confidence interval

In data analytics, these statistical concepts help evaluate how well a model fits data and how reliable predictions are. These concepts play a crucial role in data analytics by ensuring models are statistically valid, reducing uncertainty, and improving decision-making based on data-driven insights.

Standard error of the estimate (SEE)

The SEE measures the accuracy of predictions in a regression model. A lower SEE means that the predicted values are closer to the actual values, indicating a better model fit. It is useful in data analytics to assess the reliability of predictive models, such as forecasting customer demand based on past sales data.

Goodness of fit

This term refers to how well a statistical model matches the observed data. $R²$ is a commonly used measure of goodness of fit. A model with poor goodness of fit may lead to inaccurate insights and poor decision-making. In financial risk modeling, analysts assess the goodness of fit to determine if a model accurately predicts stock price movements.

Confidence interval (CI)

A confidence interval (CI) is a statistical range that estimates where a true population parameter (such as a mean or regression coefficient) is likely to fall, given a specified level of confidence (e.g., 95%). It reflects the degree of uncertainty around an estimate rather than providing a single point value.

For example, if revenue is forecasted to increase by $10 million with a 95% CI of ±$1 million, it means we can be 95% confident that the actual increase will fall between $9 million and $11 million.

Multiple regression equation

Predicts dependent variable (Y) using multiple independent variables (X₁, X₂, …, Xₙ)
Equation: Y = a + b₁X₁ + b₂X₂ + … + bₙXₙ + ε
Coefficients (b₁, b₂, …) show impact of each X on Y

When to use multiple regression

Use when outcome is influenced by multiple factors
Measures relationships and predicts outcomes
Supports data-driven business decisions

Benefits and shortcomings of regression analysis

Benefits:
- Forecasting and predictive power
- Informs decision-making by quantifying relationships
- Identifies trends in historical data
Shortcomings:
- Assumes linearity between variables
- Sensitive to data quality and completeness
- Risk of overfitting with too many variables

Coefficient of Determination (R²) and Correlation Coefficient ®

Correlation coefficient ®:
- Measures strength/direction of linear relationship (range: -1 to 1)
- r = 1: perfect positive, r = -1: perfect negative, r = 0: no linear correlation
Coefficient of determination (R²):
- Proportion of variance in Y explained by X’s (range: 0 to 1)
- Higher R² = better model fit

Standard Error of the Estimate (SEE), goodness of fit, and confidence interval

Standard Error of the Estimate (SEE):
- Measures prediction accuracy of regression model
- Lower SEE = predictions closer to actual values
Goodness of fit:
- Indicates how well model matches observed data
- Commonly measured by R²
Confidence interval (CI):
- Range estimating where true parameter likely falls
- Reflects uncertainty; e.g., 95% CI gives interval with 95% confidence

Next | 6.3.4.3 Time series analysis

Regression and statistical methods

Learning outcome statements

The learning outcome statements relevant for this section are:

demonstrate an understanding of multiple regression and logistic regression and recognize when these techniques are appropriate
calculate the result of multiple regression equations as applied to a specific situation
demonstrate an understanding of the coefficient of determination (R squared) and the correlation coefficient $(r)$
identify and explain the benefits and limitations of regression analysis
define standard error of the estimate, goodness of fit, and confidence interval
explain how to use predictive analytics techniques to draw insights and make recommendations

Defining a multiple regression equation

Multiple regression involves more than one independent variable. The equation for multiple regression is:

$Y = a + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{n} X_{n} + ε$

Where:

Y = dependent variable (the outcome being explained)
a = intercept (value of Y when all X’s are zero)
b1, b2, … bn = coefficients (slopes showing the impact of each independent variable on Y)
X1, X2, … Xn = independent variables (factors that influence Y)
ε = error term (variation not explained by the model)

When to use multiple regression in data analytics

Multiple regression is appropriate when multiple independent variables influence an outcome, helping businesses and analysts improve predictions, optimize strategies, and make data-driven decisions.

Benefits and shortcomings of regression analysis

Benefits

Predictive power: Regression analysis allows companies to forecast outcomes, such as sales or costs, based on different levels of input (independent) variables.
Decision-making: By understanding the relationship between variables, businesses can make informed decisions about how changes in one factor (e.g., production volume) will impact others (e.g., costs or revenue).
Identifying trends: Regression can help identify underlying trends in historical data, allowing organizations to project future performance.

Shortcomings

Assumption of linearity: Simple regression assumes a linear relationship between variables, which may not always be the case in real-world situations.
Data limitations: Regression analysis depends on the quality of data. Inaccurate or incomplete data can lead to unreliable predictions.
Overfitting: Using too many variables in a regression model can lead to overfitting, where the model becomes too tailored to historical data and fails to predict future outcomes accurately.

Coefficient of determination (R²) and correlation coefficient $(r)$

In data analytics, understanding relationships between variables is critical for making informed predictions and decisions. Two key statistical measures used for this purpose are:

coefficient of determination (R²) and
correlation coefficient $(r)$

Correlation coefficient $(r)$

The $correlation coefficient (r)$ measures the strength and direction of a linear relationship between two variables. It ranges from $-1 to 1$

$r = 1$ indicates a perfect positive correlation, meaning as one variable increases, the other also increases.
$r = -1$ indicates a perfect negative correlation, meaning as one variable increases, the other decreases.
$r = 0$ suggests no linear correlation between the variables.

In data analytics, correlation helps determine whether a relationship exists between two variables, such as marketing spend and revenue.

Coefficient of determination (R²)

The $R² value$ tells us how well the independent variable(s) explain the variability of the dependent variable. It ranges from $0 to 1$ :

$R² = 0$ means the independent variable explains none of the variation in the dependent variable.
$R² = 1$ means the independent variable explains 100% of the variation.

In data analytics, a higher $R²$ value indicates a better model fit, meaning the independent variables are strong predictors of the dependent variable.

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

For example, if revenue is forecasted to increase by $10 million with a 95% CI of ±$1 million, it means we can be 95% confident that the actual increase will fall between $9 million and $11 million.

Achievable CMA Part 1

6. Technology and analytics

6.3. Data analytics

6.3.4. Types of data analytics

Our CMA Part 1 course is currently in development and is a work-in-progress.

Regression and statistical methods

6 min read

Font

Discuss

Feedback

Learning outcome statements

The learning outcome statements relevant for this section are:

demonstrate an understanding of multiple regression and logistic regression and recognize when these techniques are appropriate
calculate the result of multiple regression equations as applied to a specific situation
demonstrate an understanding of the coefficient of determination (R squared) and the correlation coefficient $(r)$
identify and explain the benefits and limitations of regression analysis
define standard error of the estimate, goodness of fit, and confidence interval
explain how to use predictive analytics techniques to draw insights and make recommendations

Defining a multiple regression equation

Multiple regression involves more than one independent variable. The equation for multiple regression is:

$Y = a + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{n} X_{n} + ε$

Where:

Y = dependent variable (the outcome being explained)
a = intercept (value of Y when all X’s are zero)
b1, b2, … bn = coefficients (slopes showing the impact of each independent variable on Y)
X1, X2, … Xn = independent variables (factors that influence Y)
ε = error term (variation not explained by the model)

When to use multiple regression in data analytics

Multiple regression is appropriate when multiple independent variables influence an outcome, helping businesses and analysts improve predictions, optimize strategies, and make data-driven decisions.

Benefits and shortcomings of regression analysis

Benefits

Predictive power: Regression analysis allows companies to forecast outcomes, such as sales or costs, based on different levels of input (independent) variables.
Decision-making: By understanding the relationship between variables, businesses can make informed decisions about how changes in one factor (e.g., production volume) will impact others (e.g., costs or revenue).
Identifying trends: Regression can help identify underlying trends in historical data, allowing organizations to project future performance.

Shortcomings

Assumption of linearity: Simple regression assumes a linear relationship between variables, which may not always be the case in real-world situations.
Data limitations: Regression analysis depends on the quality of data. Inaccurate or incomplete data can lead to unreliable predictions.
Overfitting: Using too many variables in a regression model can lead to overfitting, where the model becomes too tailored to historical data and fails to predict future outcomes accurately.

Coefficient of determination (R²) and correlation coefficient $(r)$

In data analytics, understanding relationships between variables is critical for making informed predictions and decisions. Two key statistical measures used for this purpose are:

coefficient of determination (R²) and
correlation coefficient $(r)$

Correlation coefficient $(r)$

The $correlation coefficient (r)$ measures the strength and direction of a linear relationship between two variables. It ranges from $-1 to 1$

$r = 1$ indicates a perfect positive correlation, meaning as one variable increases, the other also increases.
$r = -1$ indicates a perfect negative correlation, meaning as one variable increases, the other decreases.
$r = 0$ suggests no linear correlation between the variables.

In data analytics, correlation helps determine whether a relationship exists between two variables, such as marketing spend and revenue.

Coefficient of determination (R²)

The $R² value$ tells us how well the independent variable(s) explain the variability of the dependent variable. It ranges from $0 to 1$ :

$R² = 0$ means the independent variable explains none of the variation in the dependent variable.
$R² = 1$ means the independent variable explains 100% of the variation.

In data analytics, a higher $R²$ value indicates a better model fit, meaning the independent variables are strong predictors of the dependent variable.

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

For example, if revenue is forecasted to increase by $10 million with a 95% CI of ±$1 million, it means we can be 95% confident that the actual increase will fall between $9 million and $11 million.

Multiple regression equation

Predicts dependent variable (Y) using multiple independent variables (X₁, X₂, …, Xₙ)
Equation: Y = a + b₁X₁ + b₂X₂ + … + bₙXₙ + ε
Coefficients (b₁, b₂, …) show impact of each X on Y

When to use multiple regression

Use when outcome is influenced by multiple factors
Measures relationships and predicts outcomes
Supports data-driven business decisions

Benefits and shortcomings of regression analysis

Benefits:
- Forecasting and predictive power
- Informs decision-making by quantifying relationships
- Identifies trends in historical data
Shortcomings:
- Assumes linearity between variables
- Sensitive to data quality and completeness
- Risk of overfitting with too many variables

Coefficient of Determination (R²) and Correlation Coefficient ®

Correlation coefficient ®:
- Measures strength/direction of linear relationship (range: -1 to 1)
- r = 1: perfect positive, r = -1: perfect negative, r = 0: no linear correlation
Coefficient of determination (R²):
- Proportion of variance in Y explained by X’s (range: 0 to 1)
- Higher R² = better model fit

Standard Error of the Estimate (SEE), goodness of fit, and confidence interval

Standard Error of the Estimate (SEE):
- Measures prediction accuracy of regression model
- Lower SEE = predictions closer to actual values
Goodness of fit:
- Indicates how well model matches observed data
- Commonly measured by R²
Confidence interval (CI):
- Range estimating where true parameter likely falls
- Reflects uncertainty; e.g., 95% CI gives interval with 95% confidence

Next | 6.3.4.3 Time series analysis

Regression and statistical methods

Learning outcome statements

The learning outcome statements relevant for this section are:

demonstrate an understanding of multiple regression and logistic regression and recognize when these techniques are appropriate
calculate the result of multiple regression equations as applied to a specific situation
demonstrate an understanding of the coefficient of determination (R squared) and the correlation coefficient $(r)$
identify and explain the benefits and limitations of regression analysis
define standard error of the estimate, goodness of fit, and confidence interval
explain how to use predictive analytics techniques to draw insights and make recommendations

Defining a multiple regression equation

Multiple regression involves more than one independent variable. The equation for multiple regression is:

$Y = a + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{n} X_{n} + ε$

Where:

Y = dependent variable (the outcome being explained)
a = intercept (value of Y when all X’s are zero)
b1, b2, … bn = coefficients (slopes showing the impact of each independent variable on Y)
X1, X2, … Xn = independent variables (factors that influence Y)
ε = error term (variation not explained by the model)

When to use multiple regression in data analytics

Multiple regression is appropriate when multiple independent variables influence an outcome, helping businesses and analysts improve predictions, optimize strategies, and make data-driven decisions.

Benefits and shortcomings of regression analysis

Benefits

Predictive power: Regression analysis allows companies to forecast outcomes, such as sales or costs, based on different levels of input (independent) variables.
Decision-making: By understanding the relationship between variables, businesses can make informed decisions about how changes in one factor (e.g., production volume) will impact others (e.g., costs or revenue).
Identifying trends: Regression can help identify underlying trends in historical data, allowing organizations to project future performance.

Shortcomings

Assumption of linearity: Simple regression assumes a linear relationship between variables, which may not always be the case in real-world situations.
Data limitations: Regression analysis depends on the quality of data. Inaccurate or incomplete data can lead to unreliable predictions.
Overfitting: Using too many variables in a regression model can lead to overfitting, where the model becomes too tailored to historical data and fails to predict future outcomes accurately.

Coefficient of determination (R²) and correlation coefficient $(r)$

In data analytics, understanding relationships between variables is critical for making informed predictions and decisions. Two key statistical measures used for this purpose are:

coefficient of determination (R²) and
correlation coefficient $(r)$

Correlation coefficient $(r)$

The $correlation coefficient (r)$ measures the strength and direction of a linear relationship between two variables. It ranges from $-1 to 1$

$r = 1$ indicates a perfect positive correlation, meaning as one variable increases, the other also increases.
$r = -1$ indicates a perfect negative correlation, meaning as one variable increases, the other decreases.
$r = 0$ suggests no linear correlation between the variables.

In data analytics, correlation helps determine whether a relationship exists between two variables, such as marketing spend and revenue.

Coefficient of determination (R²)

The $R² value$ tells us how well the independent variable(s) explain the variability of the dependent variable. It ranges from $0 to 1$ :

$R² = 0$ means the independent variable explains none of the variation in the dependent variable.
$R² = 1$ means the independent variable explains 100% of the variation.

In data analytics, a higher $R²$ value indicates a better model fit, meaning the independent variables are strong predictors of the dependent variable.

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

For example, if revenue is forecasted to increase by $10 million with a 95% CI of ±$1 million, it means we can be 95% confident that the actual increase will fall between $9 million and $11 million.

Regression and statistical methods

Learning outcome statements

Defining a multiple regression equation

When to use multiple regression in data analytics

Benefits and shortcomings of regression analysis

Benefits

Shortcomings

Coefficient of determination (R²) and correlation coefficient (r)

Correlation coefficient (r)

Coefficient of determination (R²)

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

Regression and statistical methods

Learning outcome statements

Defining a multiple regression equation

When to use multiple regression in data analytics

Benefits and shortcomings of regression analysis

Benefits

Shortcomings

Coefficient of determination (R²) and correlation coefficient (r)

Correlation coefficient (r)

Coefficient of determination (R²)

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

Regression and statistical methods

Learning outcome statements

Defining a multiple regression equation

When to use multiple regression in data analytics

Benefits and shortcomings of regression analysis

Benefits

Shortcomings

Coefficient of determination (R²) and correlation coefficient (r)

Correlation coefficient (r)

Coefficient of determination (R²)

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

Regression and statistical methods

Learning outcome statements

Defining a multiple regression equation

When to use multiple regression in data analytics

Benefits and shortcomings of regression analysis

Benefits

Shortcomings

Coefficient of determination (R²) and correlation coefficient (r)

Correlation coefficient (r)

Coefficient of determination (R²)

Standard error of the estimate (SEE), goodness of fit, and confidence interval

Standard error of the estimate (SEE)

Goodness of fit

Confidence interval (CI)

Coefficient of determination (R²) and correlation coefficient $(r)$

Correlation coefficient $(r)$

Coefficient of determination (R²) and correlation coefficient $(r)$

Correlation coefficient $(r)$

Coefficient of determination (R²) and correlation coefficient $(r)$

Correlation coefficient $(r)$

Coefficient of determination (R²) and correlation coefficient $(r)$

Correlation coefficient $(r)$