What Is a Regression Equation? How to Read and Write It in SPSS
What is a regression equation
You open your SPSS output and see the Coefficients table, but you are not sure which numbers belong in the equation. A regression equation represents the estimated relationship between one dependent variable and one or more independent variables. In a quantitative thesis, the equation helps answer a practical question: when one factor changes, in which direction and by how much does the outcome variable change, while the other factors remain in the model.
For simple linear regression, the model has one independent variable:
Y = β0 + β1X1 + ε
For multiple linear regression, the model has several independent variables:
Y = β0 + β1X1 + β2X2 + ... + βkXk + ε
Here, Y is the dependent variable, X1 to Xk are the independent variables, β0 is the constant, β1 to βk are the regression coefficients, and ε is the error term. When you write the equation from SPSS, you normally take the values from the Unstandardized B column in the Coefficients table. The Standardized Beta column is used to compare relative effects. It does not directly replace the coefficient in an equation written in the original measurement units.
For example, if the dependent variable is Purchase Intention and the three independent variables are Quality, Perceived Value, and Trust, an illustrative equation may look like this:
Y = 0,842 + 0,315X1 + 0,228X2 + 0,401X3
The numbers above are illustrative output, not the results of a specific study. When you work on your thesis, replace them with the values in your own .sav file.
If you are looking for related material, you can visit the Statistics topic. Regression is usually run after you have developed your what is a scale, coded the variables, and calculated a representative score for each scale.
The meaning of a regression equation in quantitative research
A regression equation connects your research model to hypothesis testing. If the coefficient of X1 is positive and its Sig. value is below the significance level you selected, the data support a positive relationship between X1 and Y, while the other variables remain in the model.
You can use the equation to read four main pieces of information. The first is the direction of the effect, shown by the positive or negative sign of the coefficient. The second is the effect size in the original measurement scale, shown by B. The third is the relative comparison between variables, usually based on Standardized Beta. The fourth is the statistical significance of each coefficient, shown by t and Sig..
For example, B = 0,315 for X1 means that when X1 increases by one unit, the predicted value of Y increases by an average of 0,315 units, assuming that the other independent variables remain unchanged. Keep the condition “the other variables remain unchanged” in your explanation. Without it, the sentence can be read as an absolute causal claim.
The equation also helps you present the results of hypotheses H1, H2, and H3. The equation itself does not prove a causal relationship. Your conclusion must also consider the research design, scale quality, sample size, assumption checks, and theoretical reasoning. If the model contains a mediator, distinguish between the direct and indirect effects. The concept of a mediator needs an appropriate analytical procedure, rather than a simple reading of one regression equation.
What makes a regression equation acceptable
There is no single number that determines whether a regression equation is acceptable. You need to assess model fit, the significance of each coefficient, multicollinearity, and the regression assumptions together. The table below gives commonly used reference points in quantitative research, with the relevant source so you can explain them in your thesis.
| Check | Reference point | How to use it when reading the output | Source |
|---|---|---|---|
| Regression sample size | n ≥ 50 + 8m | m is the number of independent variables in the model | (Tabachnick and Fidell, 2013) |
| Durbin-Watson | From 1 to 3 | Check autocorrelation in the residuals | (Field, 2013) |
| VIF | Below 5 | Assess multicollinearity among the independent variables | (Hair et al., 2019) |
| R² in PLS-SEM | 0,25 weak; 0,50 moderate; 0,75 substantial | Use these reference points only when reporting PLS-SEM | (Hair et al., 2011) |
| f² | 0,02 small; 0,15 medium; 0,35 large | Assess effect size | (Cohen, 1988) |
| Sig. of the coefficient | Below 0,05 when α = 5% is selected | The coefficient is statistically significant at the selected significance level | (Field, 2013) |
Interpret the Sig. value according to the significance level you specified in advance. If Sig. = 0,032, the coefficient is significant at the 5% level. If Sig. = 0,218, you do not have enough statistical evidence to conclude that the coefficient differs from 0 at the 5% level. Do not write that the hypothesis is completely wrong simply because one coefficient is not significant. The result may reflect the data, sample size, measurement approach, or sample characteristics.
R² shows the proportion of variation in Y explained by the independent variables in the model. A higher R² does not automatically mean that the model is better, because adding variables can increase R². You should also examine Adjusted R Square, especially when comparing models with different numbers of variables.
How to read the regression equation in the output
After you run the regression in SPSS, the most important table for writing the equation is Coefficients. You should also examine Model Summary, ANOVA, and the diagnostic tables. The tables below are labelled illustrative output and follow the structure of the software output so you can see where to read each value.
Model Summary, illustrative output
| Model | R | R Square | Adjusted R Square | Std. Error of the Estimate | Durbin-Watson |
|---|---|---|---|---|---|
| 1 | 0,748 | 0,559 | 0,548 | 0,512 | 1,842 |
ANOVA, illustrative output
| Model | Sum of Squares | df | Mean Square | F | Sig. |
|---|---|---|---|---|---|
| Regression | 48,215 | 3 | 16,072 | 61,284 | 0,000 |
| Residual | 38,764 | 148 | 0,262 | ||
| Total | 86,979 | 151 |
Coefficients, illustrative output
| Model | Unstandardized B | Std. Error | Standardized Beta | t | Sig. | VIF |
|---|---|---|---|---|---|---|
| (Constant) | 0,842 | 0,214 | 3,935 | 0,000 | ||
| Quality | 0,315 | 0,071 | 0,302 | 4,437 | 0,000 | 1,684 |
| Perceived Value | 0,228 | 0,068 | 0,221 | 3,353 | 0,001 | 1,529 |
| Trust | 0,401 | 0,074 | 0,376 | 5,419 | 0,000 | 1,772 |
From the Unstandardized B column, the illustrative equation is:
Y = 0,842 + 0,315X1 + 0,228X2 + 0,401X3
Here, X1 is Quality, X2 is Perceived Value, X3 is Trust, and Y is Purchase Intention. The coefficients of all three variables are positive. In the illustrative table, all Sig. values are below 0,05, so the three coefficients are statistically significant at the 5% level. All VIF values are below 5, which is consistent with the reference point for multicollinearity in (Hair et al., 2019).
The Standardized Beta column is used to compare the relative effects of the variables after they have been placed on the same standardized scale. In this example, Trust has Beta = 0,376, which is higher than Quality and Perceived Value. You can write that Trust has the strongest relative effect in the illustrative model. Avoid calling this an absolute effect or claiming causation if the research design does not support that wording.
What to do when the regression equation does not meet the criteria
First, identify exactly which part does not meet the criterion. If ANOVA has a Sig. value above the selected significance level, the overall model is not statistically significant according to that criterion. If only one variable has a large Sig. value, the issue concerns that variable's individual coefficient, not necessarily the entire model. If VIF is high, you have a multicollinearity problem. If the residuals violate an assumption, check the diagnostics before interpreting the coefficients.
Start the troubleshooting process with the data. Check whether a variable was entered with the wrong code, whether missing values remain, whether reverse-coded items were handled correctly, or whether the scale score was calculated incorrectly. You should also examine unusual observations, because outliers can strongly affect the coefficients. Removing an observation only because it makes the result look better is difficult to defend. If you remove one, you need a clear criterion and a record of the number removed, the reason, and the results before and after the procedure.
Next, check the correlations among the independent variables and examine VIF, Tolerance, and residual plots. Do not delete a variable immediately just because its Sig. value is large. The variable may be theoretically important, or its result may be affected by a high correlation with another variable. Discuss the possible approach with your supervisor, then rerun the model for a reason that you can explain in the research methods section.
If a coefficient has the opposite sign from the hypothesis, check the coding and the wording of any reverse-coded variable first. When the data contain no error, the opposite sign is a result to report, not a number to correct. You can discuss that the result does not support the hypothesis, while avoiding claims that go beyond the data.
Distinguishing the regression equation from easily confused statistics
A regression equation and a regression model are related, but they are not exactly the same. The model is the structure of the relationships among the variables and hypotheses. The equation is the quantitative representation of those relationships after the coefficients have been estimated from the data.
R Square is also not a coefficient in the equation. R² describes the explanatory power of the overall model, while B describes the predicted change in Y when one independent variable increases by one unit. Similarly, Standardized Beta does not replace Unstandardized B when you are writing the equation in the original measurement scale.
Sig. is not an effect size. A large sample can produce a small p-value for a very small effect, while a practically meaningful effect may not reach statistical significance because the sample size is limited. When reporting the result, place the coefficient, p-value, confidence interval if available, and model R² next to one another.
Linear regression is also different from moderator analysis. With a moderator, the model usually includes an interaction variable to test whether one variable changes the strength or direction of the relationship between X and Y. If your topic includes a moderation hypothesis, do not read only the basic regression equation and then conclude that the moderator has an effect.
Common mistakes
The first mistake is taking the wrong column. Many students use Beta to write an equation that includes a constant and original mean scores. Use Unstandardized B for an equation in the original units, then use Standardized Beta when comparing the variables relatively.
The second mistake is writing the equation without naming the variables. The symbols X1 and X2 are useful only when the preceding table explains them. In Chapter 4, include both the equation and an explanation of what X1, X2, and Y represent.
The third mistake is concluding from the sign while ignoring Sig. A positive coefficient that is not statistically significant is not enough to say that the hypothesis is supported. State the decision criterion and the p-value clearly.
The fourth mistake is treating R² as the percentage effect of each variable. R² belongs to the entire model. To compare relative effects, examine Beta, while still interpreting it in the context of the measurement scale and research model.
The final mistake is running the regression repeatedly and keeping only the most attractive table. Save each version of the data, the variable list, the criteria for removing variables, and the corresponding output. The committee may ask why a variable was removed or why the final model differs from the model proposed in Chapter 2 or Chapter 3.
Frequently asked questions
What is a regression equation in SPSS?
It is a formula that uses the estimated coefficients from SPSS output to describe the relationship between a dependent variable and the independent variables. In multiple regression, the equation usually includes a constant and a coefficient for each independent variable.
Which column do I use to write a regression equation?
Use the Unstandardized B column in the Coefficients table. The coefficient in the (Constant) row is the constant, and each independent-variable row provides its corresponding coefficient. The Standardized Beta column is used to compare relative effects. It is not the default choice for writing an equation in the original units.
What makes a regression equation acceptable?
There is no single threshold for the whole equation. Examine the model ANOVA, the Sig. value for each coefficient, R² or Adjusted R², VIF, Durbin-Watson, and the residual diagnostics. Explain the reference points according to your research design and the appropriate methodological sources.
What should I do when the Sig. value of the regression equation is above 0,05?
If Sig. in the ANOVA table is above 0,05, the overall model is not statistically significant at the 5% level. Check the data, variable calculations, sample size, and assumptions, then report the result honestly if it still does not meet the criterion. Do not remove variables or change the significance level simply to obtain a smaller p-value.
Do I need to check VIF when writing a regression equation?
Yes. VIF helps identify multicollinearity among the independent variables. A VIF below 5 is commonly used in PLS-SEM reporting and related methodological discussions, as cited in (Hair et al., 2019). For SPSS regression, you should still explain how you checked multicollinearity and why you selected the criterion.
How do I write a regression equation in my thesis?
You can adapt the following paragraph: “The regression analysis shows that the model contains [number of independent variables] variables and is [statistically significant/not statistically significant], with [R² or Adjusted R²] equal to [value]. The regression equation based on the unstandardized coefficients is written as follows: [Y = β0 + β1X1 + ...]. In this equation, [X] has a [positive/negative] effect on [Y], with B = [value] and Sig. = [value].” Replace every part in square brackets with the values and variable names from your output.
Open your SPSS file, check Model Summary, ANOVA, and Coefficients in that order, and record the equation using Unstandardized B before interpreting the hypotheses. If you need to run the regression on your own .sav or .csv file and turn the output into a results section, DoThesis M4 analysis is the module for this step.