Multiple Linear Regression Calculator

How to Use the Multiple Linear Regression Calculator

This calculator performs multiple linear regression using one numerical response variable and two or more predictors. The predictors can be numerical, categorical, or a mixture of both.

To use the calculator:

1. Enter the response variable in the Y column.
2. Enter the predictor variables in the X1, X2, X3, … columns. You may rename the variables where necessary. If a predictor is categorical, select the category you want to use as the base category.
3. Click + Add Predictor if you need more predictor columns.
4. Select the number of decimal places you want to display.
5. Click Calculate.

The calculator instantly returns the fitted multiple linear regression equation and other related statistics such as R², adjusted R², F-statistic, p-value, and standard error. The calculator also produces regression outputs similar to SPSS regression outputs, with a Model Summary Table, ANOVA table, Regression Coefficients table, Collinearity Statistics Table, and Categorical Variable Coding Table. You can also copy-paste these tables to a Word document for interpretation.

Want to do prediction using the fitted regression model? Just click the “predict Y+” button to enter the given values and make the predictions.

Tip. You do not have to enter the data manually. You can paste several columns directly from Excel or Google Sheets. Alternatively, upload a CSV, Excel, or TXT file. When pasting or uploading data, place the response variable first, followed by the predictors.

The calculator also applies listwise deletion to missing data. This means that if one value is missing from a row, that entire observation is excluded from the regression analysis.

What Is Multiple Linear Regression?

Multiple linear regression is a statistical model used when you want to predict one numerical response variable from two or more predictor variables.

For example, suppose a researcher wants to predict students’ examination scores using Study Hours, Attendance, and Program Type. In this case:

• Examination Score is the response variable, Y.
• Study Hours, Attendance, and Program Type are the predictors.

Unlike simple linear regression, multiple regression allows us to examine each predictor while taking the other predictors in the model into account.

For instance, suppose Study Hours has a positive coefficient. That coefficient tells us how much Exam Score is expected to change for a one-unit increase in Study Hours after controlling for Attendance and Program Type.

As such, the coefficient does not simply describe the relationship between Study Hours and Exam Score on its own. It describes the relationship after the other predictors in the model have been accounted for.

Multiple Linear Regression Equation

The general multiple linear regression equation is:

ŷ = b₀ + b₁x₁ + b₂x₂ + … + bₖxₖ

Where:

• ŷ is the predicted value of Y.
• b₀ is the intercept.
• b₁, b₂, …, bₖ are the regression coefficients.
• x₁, x₂, …, xₖ are the predictor values.
• k is the number of regression terms in the model.

For example, suppose the fitted equation is: ŷ = 24.6 + 2.8(Study Hours) + 0.35(Attendance)

The Study Hours coefficient is 2.8. Therefore, for every additional study hour, Y is predicted to increase by 2.8 units, assuming Attendance remains constant.

Similarly, the Attendance coefficient is 0.35. Therefore, a one-unit increase in Attendance is associated with an estimated increase of 0.35 units in Y, assuming Study Hours remains constant.

This is why you will often see the phrase holding the other predictors constant when interpreting multiple regression coefficients.

Using Categorical Predictors

The calculator also allows you to include categorical predictors such as Treatment Group, Program Type, Region, or Employment Status.

Suppose Program Type contains two categories:

• Campus
• Online

If you select Campus as the base category, the calculator creates a term for Program [Online]. As such, Campus becomes the comparison group. Therefore, if the coefficient for Program [Online] is B = 3.20, then online participants are predicted to have Y values that are 3.20 units higher than Campus participants, after controlling for the other predictors in the model. On the other hand, if B = −3.20, Online participants are predicted to have Y values 3.20 units lower than Campus participants.

What Is the Base Category?

The base category is simply the category used as the reference when the other categories are compared.

For example, suppose Treatment Group has three categories:

• Control
• Treatment A
• Treatment B

If Control is selected as the base category, the calculator estimates coefficients for Treatment A versus Control and Treatment B versus Control.

Changing the base category changes the way the coefficients are expressed. However, it does not change the overall fit of the regression model.

Note. If categories are entered as numbers such as 1, 2, and 3, the calculator will treat them as numerical values. Therefore, use text labels if the variable is categorical.

Understanding the Multiple Linear Regression Outputs

After you click Calculate, the calculator first displays the fitted regression equation together with a short summary of the model. It then provides the detailed results in the Model Summary, ANOVA, Coefficients, and Collinearity Statistics tables. If the model contains a categorical predictor, you will also see a Categorical Variable Coding table.

These tables answer different questions. The Model Summary tells you how well the model fits the data; the ANOVA table tests the model as a whole, while the Coefficients table shows how each predictor relates to Y after the other predictors have been taken into account.

1) Model Summary Table

Start with the Model Summary because it gives you a quick picture of how well the predictors explain the response variable.

The table reports R, R², Adjusted R², SE, and N.

R gives the correlation between the observed Y values and the values predicted by the regression model. Since the calculator reports multiple R, the value is non-negative.

More importantly, R² tells you how much of the variation in Y is accounted for by the predictors in the model. For example, if R² = 0.72, the predictors account for approximately 72% of the variation in Y.

However, R² tends to increase when more predictors are added, even when some of those predictors add very little useful information. For this reason, the calculator also reports Adjusted R². Adjusted R² takes both sample size and the number of regression terms into account. As such, Adjusted R² is often more useful when comparing models with different numbers of predictors.

The table also reports SE, which is the standard error of the estimate. It gives an idea of the typical size of the prediction errors, or residuals. A residual is the difference between the observed Y value and the value predicted by the model.

Finally, N shows the number of complete observations included in the regression analysis.

2) ANOVA Table

Once you have looked at the overall fit, the next question is whether the regression model is statistically significant. The ANOVA table answers this question.

It separates the total variation in Y into two parts:

• variation accounted for by the regression model; and
• variation that remains unexplained.

The table reports Regression Sum of Squares, Residual Sum of Squares, Total Sum of Squares, degrees of freedom, mean squares, the F statistic, and its p-value.

Therefore, using the results from the ANOVA table, you can test the overall fit of the multiple regression. The hypotheses are:

• H₀: All slope coefficients are equal to zero
• H₁: At least one slope coefficient is not equal to zero

Therefore, if the ANOVA table gives p < .05, you would normally reject the null hypothesis and conclude that the predictors, taken together, significantly predict Y.

For example, F(3, 46) = 9.84, p < .001 shows that the overall regression model is statistically significant. However, this does not mean that all three predictors are significant. The F-test looks at the model as a whole. To determine which predictors make a statistically significant contribution after controlling for the others, you need to look at the Coefficients table.

3) Coefficients Table

The Coefficients table helps you determine how each predictor contributes to the model after controlling for the other predictors. For each regression term, the calculator reports B, SE, β, t, p, and the 95% confidence interval for B.

• B (unstandardized coefficient): This is usually the main coefficient to interpret. For a numerical predictor, B shows the expected change in Y for a one-unit increase in that predictor, assuming the other predictors remain constant. For example, if Study Hours has B = 2.40, each additional study hour is associated with an estimated 2.40-unit increase in Y after controlling for the other predictors. For a categorical predictor, B instead shows the estimated difference between that category and the selected base category.
• SE (standard error): The standard error shows the amount of uncertainty around the estimated B coefficient. In general, a smaller SE indicates a more precise estimate.
• β (standardized coefficient): Unlike B, β expresses the relationship in standard deviation units. This makes it useful when comparing numerical predictors measured on different scales. However, you should not use β on its own to decide whether a predictor is statistically significant.
• t and p: These statistics test whether the regression coefficient differs significantly from zero after holding the other predictors constant. For example, B = 2.84, SE = 0.61, t = 4.66, p < .001 provides strong evidence that the predictor makes a statistically significant contribution to the model.
• 95% CI for B: The confidence interval gives a range of plausible values for the population regression coefficient. If the interval does not include zero, the coefficient will normally be statistically significant at the 5% level for a two-sided test.

Tip. Use the unstandardized B coefficients when writing the regression equation. The standardized β coefficients are mainly useful when comparing the relative size of numerical predictors.

4) Collinearity Statistics

After interpreting the coefficients, it is also important to check whether the predictors overlap too strongly with one another. This is because multiple regression tries to estimate the unique contribution of each predictor. If two or more predictors contain nearly the same information, it becomes difficult to separate their individual effects. This problem is known as multicollinearity.

To assess multicollinearity, the calculator reports Tolerance and VIF for each regression term. A VIF close to 1 suggests very little multicollinearity. As VIF increases, the coefficient may become less stable, and its standard error may become larger. You will often see VIF values above 5 or 10 used as warning levels. However, you should treat them as guidelines rather than automatic rules.

Therefore, if a predictor has a high VIF, look at the variables involved before deciding what to do. In some studies, strongly related predictors may still need to remain in the model for theoretical reasons.

5) Categorical Variable Coding

If you include a categorical predictor, the calculator adds a Categorical Variable Coding table below the main regression output. This table is useful because it shows exactly how the categories were converted into regression terms.

Suppose Program contains Campus and Online, with Campus selected as the base category. The table may show:

Therefore, the coefficient for Program [Online] compares Online participants with Campus participants. If Program contained three categories, the calculator would create two dummy terms because one category serves as the base group.

Example: Multiple Linear Regression

Suppose a college researcher wants to determine whether Study Hours, Attendance, and Program Type can be used to predict students’ Exam Scores. The following data were collected from 12 students.

Perform a multiple linear regression to determine whether Study Hours, Attendance, and Program Type significantly predict Exam Score.

Solution

To perform a multiple linear regression analysis using the calculator:

1. Enter Exam Score in the Y column.
2. Enter Study Hours in X1.
3. Enter Attendance in X2.
4. Enter Program in X3.
5. Rename the variables accordingly.
6. Select Campus as the base category for Program.
7. Click Calculate.

The calculator gives the following regression equation: ŷ = 16.0157 + 2.9673(Study Hours) + 0.4663(Attendance) + 2.9294(Program [Online]). Since Campus is the base category, Program [Online] equals 1 for Online students and 0 for Campus students.

The calculator will also provide the following regression tables.

Model Summary

From the table, R² = 0.9887. Therefore, Study Hours, Attendance, and Program Type account for approximately 98.87% of the variation in Exam Score. Also, after adjusting for the number of regression terms and the sample size, Adjusted R² = 0.9845. The standard error of the estimate is 1.0639.

ANOVA Table

The ANOVA table shows that the overall regression model is statistically significant, F(3, 8) = 233.5400, p < .001. Therefore, there is sufficient evidence that Study Hours, Attendance, and Program Type collectively predict Exam Score. However, we still need to look at the Coefficients table to determine the contribution of each predictor.

Coefficients Table

From the Coefficients table, all three predictors make a statistically significant contribution to the model. Using the unstandardized coefficients, we can interpret the results as follows:

• Study Hours: B = 2.9673, p < .001. Holding Attendance and Program Type constant, every additional study hour is associated with an estimated 2.97-point increase in Exam Score.
• Attendance: B = 0.4663, p < .001. Holding Study Hours and Program Type constant, a one-unit increase in Attendance is associated with an estimated 0.47-point increase in Exam Score.
• Program Type: Program [Online] has B = 2.9294, p = .002. Since Campus is the base category, Online students are predicted to score approximately 2.93 points higher than Campus students, after controlling for Study Hours and Attendance.

The intercept is 16.0157. This represents the predicted Exam Score when Study Hours and Attendance are both zero and the student belongs to the Campus group. However, the intercept may not have a useful practical interpretation if those predictor values are outside the range of the observed data.

Collinearity Statistics

The VIF values are all close to 1. Therefore, there is no evidence of problematic multicollinearity among the predictors in this example.

Making a Prediction From the Regression Model

Once the regression model has been fitted, you can use the equation to predict Y for a new combination of predictor values.

From the previous example, ŷ = 16.0157 + 2.9673(Study Hours) + 0.4663(Attendance) + 2.9294(Program [Online])

Suppose an Online student studies for 8 hours and has an Attendance value of 85; we can predict the exam score as follows:

ŷ = 16.0157 + 2.9673(8) + 0.4663(85) + 2.9294(1)

= 82.32

Therefore, the predicted Exam Score is approximately 82.32.

You can make this calculation directly using the Predict Y option in our multiple linear regression calculator.

How to Report Multiple Linear Regression Results in APA Style

When reporting multiple linear regression results, start with the overall model. Then report the coefficients that answer your research question.

Using the example above, you could write:

A multiple linear regression was conducted to determine whether Study Hours, Attendance, and Program Type predicted Exam Score. The overall regression model was statistically significant, F(3, 8) = 233.54, p < .001, R² = .989, adjusted R² = .984.

You could then report the predictor results as follows:

Study Hours was a significant positive predictor of Exam Score, B = 2.97, SE = 0.12, β = .984, t(8) = 24.95, p < .001, 95% CI [2.69, 3.24]. Attendance was also a significant positive predictor, B = 0.47, SE = 0.03, β = .547, t(8) = 14.02, p < .001, 95% CI [0.39, 0.54]. After controlling for Study Hours and Attendance, Online students scored significantly higher than Campus students, B = 2.93, SE = 0.63, t(8) = 4.63, p = .002, 95% CI [1.47, 4.39].

As you can see, when reporting multiple linear regression results in APA, you do not need to report every number from the calculator. Instead, you only need to report the overall model and the coefficients that are relevant to your research question.

Assumptions of Multiple Linear Regression

Before interpreting the p-values and confidence intervals, the following assumptions should be satisfied.

• Linearity. The relationship between each numerical predictor and Y should be reasonably linear after accounting for the other predictors.
• Independence. The observations should be independent of one another.
• Homoscedasticity. The spread of the residuals should remain reasonably constant across the fitted values.
• Normality of residuals. The residuals should be approximately normally distributed when using the usual t-tests and F-test, particularly in small samples.
• No perfect multicollinearity. One predictor should not be an exact linear combination of the other predictors.
• No highly influential observations. A small number of unusual observations should not dominate the fitted regression model.

Tip. The predictor variables themselves do not need to follow a normal distribution. Also, categorical predictors do not need to have a linear relationship with Y. Their coefficients represent differences between categories.

When Should You Use Multiple Linear Regression?

Multiple linear regression is appropriate when your response variable is numerical, and you have two or more predictors. For example, you could use multiple regression to predict:

• House price from floor area, age, bedrooms, and location.
• Examination score from study hours, attendance, and program type.
• Sales from advertising expenditure, product price, and store size.
• Salary from years of experience, education level, and job category.

However, multiple linear regression is not suitable for every response variable. For example, if Y has only two possible outcomes such as Yes/No, Pass/Fail, or Disease/No Disease, logistic regression is usually more appropriate.

Frequently Asked Questions