Hierarchical regression analysis

Hierarchical regression analysis is a type of multivariate regression analysis that is used to assess the relationship between a dependent variable and multiple independent variables. In this type of analysis, the independent variables are entered into the model in a specific order, with each step adding additional variables to the model. This type of analysis allows researchers to identify which variables are most influential in predicting the dependent variable, and to identify any potential interactions between the variables in the model. It is useful for managers and researchers to identify the most important factors in predicting a given outcome or behavior.

Example of hierarchical regression analysis

• For example, a researcher may use hierarchical regression analysis to examine the relationship between student academic achievement and several independent variables, such as family income, parent involvement, and teacher quality. The researcher would first enter family income into the model as the first step of the analysis. The researcher would then add the remaining independent variables to the model in order, examining the effects of each on the dependent variable. This type of analysis can be used to identify which of the independent variables are most influential in predicting student academic achievement, as well as any potential interactions between the variables.

Formula of hierarchical regression analysis

Hierarchical regression analysis is a form of multiple regression analysis where the independent variables are entered into the model in a predetermined hierarchical order. The basic formula for hierarchical regression is:

$$Y = β_0 + β_1X_1 + β_2X_2 + ... + β_kX_k + ɛ$$

where Y is the dependent variable, $$X_1, X_2, ..., X_k$$ are the independent variables, $$β_0$$ is the intercept, $$β_1, β_2, ..., β_k$$ are the regression coefficients for the independent variables, and ɛ is the error term.

The hierarchical regression model can be further extended by adding interaction terms between the independent variables $$(X_1, X_2, ..., X_k)$$. The formula for the extended hierarchical regression model is:

$$Y = β_0 + β_1X_1 + β_2X_2 + ... + β_kX_k + β_ijX_iX_j + ɛ$$

where $$β_ij$$ is the regression coefficient for the interaction term between $$X_i$$ and $$X_j$$.

The hierarchical regression model can also be extended by adding polynomial terms to the model (e.g., $$X_1^2, X_2^3$$, etc.). The formula for the extended hierarchical regression model with polynomial terms is:

$$Y = β_0 + β_1X_1 + β_2X_2 + ... + β_kX_k + β_iX_i^n + ɛ$$

where β_i is the regression coefficient for the polynomial term of order n for $$X_i$$.

When to use hierarchical regression analysis

Hierarchical regression analysis can be used in a variety of settings to assess the relationship between a dependent variable and multiple independent variables. It can be used to:

• Investigate the effects of different variables on a single outcome, such as a company’s profit or a consumer’s purchasing behavior.
• Identify which independent variables have the largest effect on a dependent variable, allowing for more precise targeting of interventions or marketing initiatives.
• Examine the interactions between the independent variables and the dependent variable, such as how one variable can modify the effect of another variable.
• Understand how different independent variables are related to one another.
• Explore how changes in one variable affect the others, allowing for more precise predictions of a dependent variable’s behavior.

Types of hierarchical regression analysis

Hierarchical regression analysis is a type of multivariate regression analysis that is used to assess the relationship between a dependent variable and multiple independent variables. Types of hierarchical regression analysis include:

• Stepwise Regression: This type of hierarchical regression analysis involves the selection of independent variables in a stepwise manner, with each step adding additional variables to the model. This type of analysis allows the researcher to identify which variables are most important in predicting the dependent variable.
• Hierarchical Multiple Regression: This type of hierarchical regression analysis involves the assessment of multiple independent variables at once, as opposed to the stepwise approach. This type of analysis is useful in situations where the researcher is interested in assessing the individual impact of each variable on the dependent variable, as well as the influence of interactions between the independent variables.
• Interaction Effects: This type of analysis is used to assess the potential interactions between the independent variables in the model. This type of analysis is useful in identifying any potential interactions between the variables in the model and can help to identify any potential confounding factors that may be influencing the results.
• Multicollinearity: This type of analysis is used to identify any potential multicollinearity between the independent variables in the model. This type of analysis helps to identify any potential problems with the model and can help to improve the accuracy of the results.

Advantages of hierarchical regression analysis

Hierarchical regression analysis is a powerful tool for assessing the relationship between a dependent variable and multiple independent variables. It allows researchers to identify the most influential predictors of a given outcome, and to identify any interactions between the variables in the model. The advantages of hierarchical regression analysis include:

• It allows researchers to identify the most important predictors of a given outcome in an effective and efficient manner.
• It provides a way to compare different models and identify the best model for predicting a given outcome.
• It can be used to identify the relative importance of each predictor in the model.
• It allows researchers to identify any interactions between the variables in the model.
• It can be used to account for any nonlinear relationships between the variables in the model.
• It is relatively easy to interpret the results of a hierarchical regression analysis.

Limitations of hierarchical regression analysis

Hierarchical regression analysis is a valuable tool for understanding the influence of multiple independent variables on a dependent variable. However, it is subject to several limitations, including:

• Data Assumptions: The model assumes that the data is linear, normally distributed, and homoscedastic. If any of these assumptions are violated, the results of the analysis may be unreliable.
• Multi-Collinearity: Hierarchical regression analysis is sensitive to multicollinearity, which is when two or more independent variables are highly correlated. This can lead to inflated standard errors, which can make the results of the analysis difficult to interpret.
• Nonlinear Relationships: This type of analysis is based on linear relationships between the independent and dependent variables. If the relationship between the variables is nonlinear, the results of the analysis may be inaccurate.
• Inappropriate Hierarchy: The order in which the independent variables are entered into the model can have a significant effect on the results of the analysis. If the order is inappropriate, the results may be inaccurate.

Other approaches related to hierarchical regression analysis

Hierarchical regression analysis is a type of multivariate regression analysis that is used to assess the relationship between a dependent variable and multiple independent variables. Other approaches related to hierarchical regression analysis include:

• Multiple Linear Regression Analysis. This technique involves fitting a line of best fit through a set of data points to determine the relationship between a dependent variable and multiple independent variables.
• Logistic Regression Analysis. This technique is used to create a model of a categorical dependent variable, such as whether a person is likely to purchase a product or not. It is used to predict the likelihood of an event occurring based on a set of independent variables.
• Discriminant Analysis. This technique is used to distinguish between two or more groups based on a set of independent variables. It can be used for classification, prediction, or decision making.

In summary, hierarchical regression analysis is a type of multivariate regression analysis that can be used to assess the relationship between a dependent variable and multiple independent variables. Other related approaches include multiple linear regression analysis, logistic regression analysis, and discriminant analysis.

Hierarchical regression analysis — recommended articles

Principal component analysis — Logistic regression model — Exploratory factor analysis — Statistical methods — Multivariate data analysis — Analysis of covariance — Standardized regression coefficients — Logistic regression analysis — Latent class analysis

References

• de Jong, P. F. (1999). Hierarchical regression analysis in structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 6(2), 198-211.
• Lewis, M. (2007). Stepwise versus Hierarchical Regression: Pros and Cons. Online Submission.