How to Use Bivariate Regression Analysis in Data Analysis
Introduction
Bivariate regression analysis is an important statistical technique used to examine the relationship between two variables. It is particularly useful when a researcher wants to determine whether changes in one variable are associated with changes in another variable.
In quantitative research, bivariate regression can help answer questions such as:
- Does access to credit influence business performance?
- Does education level predict household income?
- Does farm size influence agricultural income?
- Does employee training affect productivity?
- Does advertising expenditure predict sales?
- Does access to markets influence farmers’ income?
Bivariate regression is relatively simple to conduct using statistical software such as SPSS, Stata, R and Python, making it particularly useful for undergraduate, master’s and PhD research.
1. What Is Bivariate Regression Analysis?
Bivariate regression analysis examines the relationship between one dependent variable (Y) and one independent variable (X).
It is therefore sometimes referred to as simple linear regression.
The basic regression model is:
Y = β₀ + β₁X + ε
Where:
- Y = dependent variable
- X = independent variable
- β₀ = constant/intercept
- β₁ = regression coefficient
- ε = error term
The purpose is to estimate how much the dependent variable changes when the independent variable changes by one unit.
Example
Suppose a researcher wants to investigate whether access to credit influences SME performance.
The variables could be:
- X = Access to credit
- Y = SME performance
The regression model would therefore be:
SME performance = β₀ + β₁(access to credit) + ε
Because there is only one independent variable, this is a bivariate/simple regression model.
2. Difference Between Bivariate Correlation and Bivariate Regression
These two techniques are related but they answer somewhat different questions.
Bivariate correlation
Correlation examines the strength and direction of association between two variables.
For example:
Access to credit and business performance: r = 0.62, p < 0.001
This tells the researcher that the two variables have a positive association.
Bivariate regression
Regression goes further by estimating the expected change in the dependent variable associated with a one-unit change in the independent variable.
For example:
β = 0.62, p < 0.001
The coefficient provides information about the estimated relationship between X and Y.
Therefore, researchers should select the technique according to their research question.
3. When Should You Use Bivariate Regression?
Bivariate regression is appropriate when your research objective focuses on the relationship between one predictor and one outcome.
For example:
Objective
To examine the effect of access to credit on SME performance.
Hypothesis
H₀: Access to credit has no significant relationship with SME performance.
H₁: Access to credit has a significant relationship with SME performance.
The researcher can use bivariate regression to test the hypothesis.
Other examples include:
| Independent variable | Dependent variable |
|---|---|
| Farm size | Farm income |
| Education | Household income |
| Training | Employee productivity |
| Advertising | Sales |
| Access to credit | Business performance |
| Market access | Agricultural income |
| Experience | Business profitability |
4. Preparing Your Data
Before conducting regression analysis, the researcher needs to prepare the dataset.
For example, suppose 100 farmers were surveyed.
The dataset could contain:
| Farmer | Farm size (acres) | Annual income (UGX) |
|---|---|---|
| 1 | 2 | 3,500,000 |
| 2 | 4 | 6,200,000 |
| 3 | 1 | 2,100,000 |
| 4 | 5 | 8,400,000 |
| 5 | 3 | 4,900,000 |
Here:
Independent variable (X): Farm size
Dependent variable (Y): Annual farm income
The research question could be:
Does farm size predict annual farm income?
5. Develop the Regression Model
The regression equation would be:
Farm income = β₀ + β₁(Farm size) + ε
Suppose the analysis produces:
β₀ = 1,200,000
β₁ = 1,400,000
The estimated equation becomes:
Farm income = 1,200,000 + 1,400,000(Farm size)
The coefficient of 1,400,000 means that a one-acre increase in farm size is associated with an estimated increase of approximately UGX 1.4 million in annual farm income, assuming the model specification and measurement are appropriate.
6. How to Conduct Bivariate Regression in SPSS
SPSS provides a straightforward procedure for conducting simple linear regression.
Step 1: Enter your data
Enter each observation as a row and each variable as a column.
For example:
- Farm_size
- Farm_income
Step 2: Open the regression procedure
Go to:
Analyze → Regression → Linear
Step 3: Select the variables
Place:
- Farm_income in the Dependent box.
- Farm_size in the Independent(s) box.
Step 4: Select statistics
Depending on the research requirements, you may request:
- Estimates
- Confidence intervals
- Model fit
- Descriptives
- Collinearity diagnostics where relevant
For a single predictor, multicollinearity is generally not an issue because there is only one independent variable.
Step 5: Examine the output
SPSS will produce several tables, including:
- Model Summary
- ANOVA
- Coefficients
These tables provide the information needed to interpret the regression.
7. Understanding the SPSS Model Summary
Suppose SPSS produces:
| Model | R | R Square | Adjusted R Square | Std. Error |
|---|---|---|---|---|
| 1 | 0.680 | 0.462 | 0.456 | 1.25 |
R
The R value represents the correlation between the observed dependent variable and the values predicted by the regression model.
R Square
The R² value indicates the proportion of variation in the dependent variable explained by the independent variable.
In this example:
R² = 0.462
This means that approximately 46.2% of the variation in farm income is explained by farm size in this model.
The remaining variation is associated with other factors not included in the model and random variation.
8. Understanding the ANOVA Table
A typical SPSS ANOVA table might look like this:
| Model | Sum of Squares | df | Mean Square | F | Sig. |
|---|---|---|---|---|---|
| Regression | 125.40 | 1 | 125.40 | 80.26 | .000 |
| Residual | 145.60 | 98 | 1.49 | ||
| Total | 271.00 | 99 |
The F-statistic tests whether the regression model provides evidence of a relationship between the predictor and the outcome.
The Sig. column gives the p-value.
If:
p < 0.05
the overall regression model is statistically significant at the 5% level.
A reported value of .000 in SPSS should normally be written as:
p < .001
rather than p = .000.
9. Understanding the Coefficients Table
The coefficients table is one of the most important parts of the regression output.
For example:
| Variable | B | Std. Error | Beta | t | Sig. |
|---|---|---|---|---|---|
| Constant | 1.20 | 0.31 | — | 3.87 | <.001 |
| Farm size | 1.40 | 0.16 | 0.68 | 8.96 | <.001 |
Constant
The constant is 1.20 million in this example.
It represents the estimated value of farm income when farm size is zero. Whether this has meaningful practical interpretation depends on the context and whether zero is within the relevant range of the data.
B coefficient
The unstandardized coefficient for farm size is:
B = 1.40
If income is measured in millions of UGX, this means that each additional acre is associated with an estimated UGX 1.4 million increase in annual income.
Beta
The standardized Beta is useful when comparing predictors measured on different scales. In a strictly bivariate regression with one predictor, it is closely related to the correlation coefficient.
t-statistic
The t-statistic is used to test whether the regression coefficient differs from zero.
Sig.
This is the p-value associated with the coefficient.
10. How to Interpret the Regression Result
Suppose your results are:
β = 0.68, p < .001, R² = 0.462
A suitable interpretation would be:
