What Is Ordinary Least Squares? How to Read and Evaluate OLS Regression

What Is Ordinary Least Squares

You open your SPSS output and see the Model Summary, ANOVA, and Coefficients tables, but you are not sure where the B coefficient comes from. Ordinary Least Squares, usually abbreviated as OLS, estimates the coefficients in a linear regression model by minimizing the sum of squared residuals between the observed and predicted values.

With one dependent variable Y and one or more independent variables X, the model is:

Y = β0 + β1X1 + β2X2 + ... + βkXk + ε

Here, β0 is the constant, β1 through βk are the regression coefficients, and ε is the error term. OLS selects coefficient values that make the total ε² across all observations as small as possible. Because the residuals are squared, positive and negative errors do not cancel each other out during estimation.

In quantitative research, OLS usually appears when you run linear regression in SPSS. For example, you may want to test whether service quality, convenience, and price affect purchase intention. OLS estimates how much purchase intention is expected to change when one factor changes, while the other factors in the model are held constant.

Why Ordinary Least Squares Matters in Quantitative Research

OLS answers two groups of questions. The first is whether the model explains variation in the dependent variable. The second is whether each independent variable has a statistically significant relationship with the dependent variable.

If the regression coefficient for X1 is positive, the predicted value of Y tends to increase as X1 increases, after the other independent variables in the model have been held constant. If the coefficient is negative, the relationship runs in the opposite direction. The sign of the coefficient alone is not enough to conclude that a hypothesis is supported. You also need to examine Sig. or the p-value, the confidence interval, overall model fit, and issues such as multicollinearity.

OLS is particularly suitable for linear regression when the dependent variable is quantitative and the research question focuses on a linear relationship. When the dependent variable is binary, such as whether someone buys or does not buy, logistic regression is usually more appropriate. When your aim is to test a mediator or moderator, you need a separate model and testing procedure. You can review the differences between mediator, moderator, and moderating before adding these variables to your model.

OLS does not automatically prove a causal relationship. A small p-value provides evidence of an association in the data and the specified model. To interpret the result as a causal effect, you also need to consider the research design, time order, sampling method, and theoretical foundation.

In a thesis, OLS is often used after you complete the measurement stage. If the independent and dependent variables are measured with multiple items, you can combine them into factor scores or mean scores after checking the scales. See what is a scale if you are distinguishing between the research concept, the scale, and the variable used in your SPSS file.

What Counts as an Acceptable Ordinary Least Squares Result

There is no single number that determines whether OLS is acceptable. You need to read several indicators in the output. The thresholds below are commonly used reference points, but you must interpret them alongside your research question, sample size, and measurement method.

For example, an R² of 0.50 means that the model explains approximately 50% of the variation in Y in the sample. It does not mean that the model is 50% correct or that every prediction is 50% accurate. A low R² is also not enough to reject a model in behavioral fields, where the dependent variable is often affected by many factors outside the model.

For a regression with several independent variables, you should also examine Adjusted R Square. This indicator adjusts R² for the number of variables and the sample size, so it is often more useful when comparing models with different numbers of variables. When reporting the result, state clearly whether you are using R Square or Adjusted R Square.

Read VIF together with Tolerance and the correlation matrix. A high VIF can make regression coefficients unstable, increase standard errors, and make a variable appear nonsignificant even when its simple correlation with Y is fairly clear. The reference point of VIF less than 5 is proposed by Hair et al. in reporting guidance for PLS-SEM, but in thesis work you still need to explain the context rather than treating this value as the only decision rule.

How to Read Ordinary Least Squares in the Output

In SPSS, go to Analyze > Regression > Linear. Move the dependent variable into Dependent and the independent variables into Independent(s). Then, under Statistics, select Estimates, Model fit, Collinearity diagnostics, and Durbin-Watson. Under Plots, you can select residual plots to examine the residual pattern and distribution more closely.

The table below is illustrative output, not the result of a real study. Assume that Y is Purchase Intention, X1 is Service Quality, X2 is Convenience, and X3 is Perceived Value.

Read the tables in order to avoid confusion. In Model Summary, R Square of 0.468 means that the three independent variables explain 46.8% of the variation in Purchase Intention in the illustrative sample. Adjusted R Square of 0.451 is the figure you should report when the model contains several independent variables. Durbin-Watson of 1.842 falls within the reference range from 1 to 3 according to (Field, 2013).

Next, the ANOVA table indicates whether the overall model fits the data better than a model containing only the constant. If the Sig. value for the F test is less than 0.05, you can conclude that the overall regression model is statistically significant at the 5% level, provided this interpretation matches your testing design.

In the Coefficients table, B is the unstandardized coefficient. With X1 equal to 0.318, a one-unit increase in Service Quality is associated with an expected 0.318-unit increase in Purchase Intention, while X2 and X3 are held constant. Beta is the standardized coefficient, which is commonly used to compare the relative size of independent variables within the same model.

The Sig. value for X1 is less than 0.001, for X2 it is 0.003, and for X3 it is 0.038. In this example, all three variables have statistical evidence at the 5% level. You still need to check whether the sign of B matches the hypothesis. A small p-value with a coefficient in the opposite direction from the prediction requires an explanation. You should not automatically record the hypothesis as supported.

All VIF values in the example are less than 5. This indicates no serious multicollinearity according to the reference point in (Hair et al., 2019). When your output shows a high VIF, review the troubleshooting section below instead of deleting the variable immediately.

What to Do When Ordinary Least Squares Results Are Not Acceptable

First, identify exactly which indicator is causing the problem. A low R², a large p-value, a high VIF, unsuitable residuals, and an unreliable scale are different situations. One correction cannot be applied to all of them.

If Cronbach's Alpha or Corrected Item-Total Correlation does not meet your chosen criterion, return to the Reliability Statistics and Item-Total Statistics tables. Check whether reverse-coded items were coded correctly, then examine Alpha if Item Deleted. Remove an item only when you have both statistical and measurement-content reasons. Repeatedly deleting items to increase Alpha can leave the construct underrepresented, which is difficult to defend to your supervisor.

If VIF is high, check whether the independent variables are measuring almost the same content. Examine their correlations, check how you created the composite variables, and review the data entry. You may consider removing a duplicate variable if the theory permits it, combining constructs only when the theoretical model justifies that decision, or collecting new data if the problem lies in the questionnaire design. Do not delete a variable simply because it makes the hypothesis nonsignificant.

If a variable has a p-value greater than your selected level, report that the result is not statistically significant and compare it with the hypothesis. You can check the sample size, scale quality, outliers, and model specification. Outliers can strongly affect OLS coefficients, but removing an observation requires a stated criterion and a clear record. Do not base the decision only on whether removal makes the result look better.

If the residuals show unequal variance, severe skewness, or a nonlinear relationship between X and Y, stop interpreting the coefficients as if all assumptions had been met. Check the relevant plots and diagnostics in SPSS, then discuss transforming variables, respecifying the model, or using another estimation method. You cannot fix a data problem simply by renaming a table in Chapter 4.

If the model is unacceptable because the sample is too small, rerunning the same file does not create new information. Review the number of valid observations, the number of independent variables, and your data collection plan. Never create additional data rows to make the p-value smaller.

Distinguishing Ordinary Least Squares from Linear Regression and Easily Confused Statistics

Linear regression is the name of the model describing a linear relationship between a dependent variable and independent variables. OLS is a commonly used method for estimating the coefficients of that model. The two terms often appear together in a thesis, but they are not identical.

R Square is also not OLS. R Square describes the amount of variation in Y explained by the model. OLS is the method used to find coefficients that minimize the sum of squared residuals. An OLS model can have a low or high R² depending on the data.

Pearson correlation measures the strength and direction of a linear relationship between two variables, usually without controlling for other independent variables. The B coefficient in OLS regression represents the partial relationship of one variable while the other variables in the model are held constant. As a result, the correlation coefficient and regression coefficient can differ in size and can even have opposite signs when the independent variables are correlated.

Standardized Beta and B are also easy to mix up. B retains the measurement units of the variables, so it is suitable for writing a prediction equation. Beta is standardized and is more convenient for comparing the relative importance of independent variables. Both appear in the Coefficients table, but they answer different questions.

OLS in linear regression differs from CB-SEM and PLS-SEM. When you run SmartPLS, you will see outer loading, CR, AVE, HTMT, path coefficient, and bootstrapping. Do not bring CFI, TLI, or RMSEA thresholds from CB-SEM into an OLS or SmartPLS analysis. If you are reviewing other statistical topics, see the Statistics topic to match the indicator to the correct model before choosing what to report.

Common Errors

The first error is writing that “OLS is acceptable” as though OLS had one universal threshold. State clearly whether you are evaluating the model through R², the F test, individual coefficients, VIF, residuals, and sample size.

The second error is looking only at Sig. and ignoring the coefficient sign. A variable with a small Sig. value but a negative coefficient does not support a positive hypothesis. Report B or Beta, t, Sig., and the direction of the relationship.

The third error is using R² to claim that the model explains the cause of Y. R² describes the model's explanatory level in terms of variation within the sample. Your conclusion needs to be more cautious, particularly with cross-sectional survey data.

The fourth error is removing an outlier or independent variable after repeated trials until the result becomes significant. Save each file version, record the removal criterion, and retain the theoretical basis. If you change the model after seeing the results, state clearly whether the analysis is supplementary or whether it was the original model.

You can adapt the following wording for your thesis: “The OLS regression results show that the model has an Adjusted R Square of [value], meaning that the independent variables explain [percentage]% of the variation in [dependent variable]. [Variable name] has a [positive/negative] effect on [dependent variable], with a Beta coefficient of [value] and a Sig. value of [value]. Therefore, hypothesis [H] is [supported/not supported] at the [selected level] significance level.” Replace the placeholders only with the actual values in your output.

Frequently asked questions

What is Ordinary Least Squares and what is OLS used for?

OLS is a method for estimating regression coefficients by minimizing the sum of squared residuals. In a quantitative thesis, OLS is commonly used to evaluate a linear relationship between one dependent variable and one or more independent variables.

Is OLS the same as linear regression?

OLS is an estimation method commonly used in linear regression. Linear regression describes the model form, while OLS describes how the coefficients of that model are estimated.

What counts as an acceptable Ordinary Least Squares result?

There is no single threshold for OLS. You need to examine Adjusted R Square, the F test, B or Beta, Sig., VIF, Durbin-Watson, and residual diagnostics together. Reference points such as VIF less than 5 according to (Hair et al., 2019) and Durbin-Watson from 1 to 3 according to (Field, 2013) apply to particular aspects of the analysis.

Why can R² be low while some variables still have small Sig. values?

R² reflects the explanatory level of the full model, while the Sig. value for an individual variable reflects statistical evidence for that variable's partial coefficient. A model can explain only a small part of the variation in Y while one independent variable still has a clear statistical relationship with Y in the sample.

Should I delete a variable when OLS results are not acceptable?

Delete a variable only when you have a basis in theory, measurement quality, multicollinearity, or data diagnostics. Do not remove it simply because the p-value is large or because you want to increase R². Save and explain every model change in the thesis.

Open your .sav or .csv file again and check the Model Summary, ANOVA, and Coefficients tables in that order. Then record B, Beta, Sig., Adjusted R Square, and VIF in your results table. If you need to run the analysis on that data, DoThesis M4 analysis supports this step.