The one-way ANCOVA (analysis of covariance) can be thought of as an extension of the one-way ANOVA to incorporate a covariate. Like the one-way ANOVA, the one-way ANCOVA is used to determine whether there are any significant differences between two or more independent (unrelated) groups on a dependent variable. However, whereas the ANOVA looks for differences in the group means, the ANCOVA looks for differences in adjusted means (i.e., adjusted for the covariate). As such, compared to the one-way ANOVA, the one-way ANCOVA has the additional benefit of allowing you to "statistically control" for a third variable (sometimes known as a "confounding variable"), which you believe will affect your results. This third variable that could be confounding your results is called the covariate and you include it in your one-way ANCOVA analysis.
Note: You can have more than one covariate and although covariates are traditionally measured on a continuous scale, they can also be categorical. However, when the covariates are categorical, the analysis is not often called ANCOVA. In addition, the "one-way" part of one-way ANCOVA refers to the number of independent variables. If you have two independent variables rather than one, you could run a two-way ANCOVA.
If you are familiar with the one-way ANCOVA, you can skip to the Assumptions section. On the other hand, if you are not familiar with the one-way ANCOVA, the example below should help provide some clarity.
Researchers wanted to investigate the effect of three different types of exercise intervention on systolic blood pressure. To do this, they recruited 60 participants to their study. They randomly allocated 20 participants to each of three interventions: a "low-intensity exercise intervention", a "moderate-intensity exercise intervention" and a "high-intensity exercise intervention". The exercise in all interventions burned the same number of calories. Each participant had their "systolic blood pressure" measured before the intervention and immediately after the intervention. The researcher wanted to know if the different exercise interventions had different effects on systolic blood pressure. To answer this question, the researchers wanted to determine whether there were any differences in mean systolic blood pressure after the exercise interventions (i.e., whether post-intervention mean systolic blood pressure different between the different interventions). However, the researchers expected that the impact of the three different exercise interventions on mean systolic blood pressure would be affected by the participants' starting systolic blood pressure (i.e., their systolic blood pressure before the interventions). To control the post-intervention systolic blood pressure for the differences in pre-intervention systolic blood pressure, you can run a one-way ANCOVA with pre-intervention systolic blood pressure as the covariate, intervention as the independent variable and post-intervention systolic blood pressure as the dependent variable. If you find a statistically significant difference between interventions, you can follow up a one-way ANCOVA with a post hoc test to determine which specific exercise interventions differed in terms of their effect on systolic blood pressure (e.g., whether the high-intensity exercise intervention had a greater effect on systolic blood pressure than the low-intensity exercise intervention).
This "quick start" guide shows you how to carry out a one-way ANCOVA (with one covariate) using SPSS Statistics, as well as interpret and report the results from this test. Since the one-way ANCOVA is often followed up with a post hoc test, we also show you how to carry out a post hoc test using SPSS Statistics. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a one-way ANCOVA to give you a valid result. We discuss these assumptions next.
When you choose to analyse your data using a one-way ANCOVA, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a one-way ANCOVA. You need to do this because it is only appropriate to use a one-way ANCOVA if your data "passes" nine assumptions that are required for a one-way ANCOVA to give you a valid result. In practice, checking for these nine assumptions just adds a little bit more time to your analysis, requiring you to click a few more buttons in SPSS Statistics when performing your analysis, as well as think a little bit more about your data, but it is not a difficult task.
Before we introduce you to these nine assumptions, do not be surprised if, when analysing your own data using SPSS Statistics, one or more of these assumptions is violated (i.e., is not met). This is not uncommon when working with real-world data rather than textbook examples, which often only show you how to carry out a one-way ANCOVA when everything goes well! However, don’t worry. Even when your data fails certain assumptions, there is often a solution to overcome this. First, let’s take a look at these nine assumptions:
You can check assumptions #4, #5, #6, #7, #8 and #9 using SPSS Statistics. Before doing this, you should make sure that your data meets assumptions #1, #2 and #3, although you don't need SPSS Statistics to do this. Remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a one-way ANCOVA might not be valid. This is why we dedicate a number of sections of our enhanced one-way ANCOVA guide to help you get this right. You can find out about our enhanced content as a whole here, or more specifically, learn how we help with testing assumptions here.
In the section, Test Procedure in SPSS Statistics, we illustrate the SPSS Statistics procedure to perform a one-way ANCOVA, assuming that no assumptions have been violated. First, we set out the example we use to explain the one-way ANCOVA procedure in SPSS Statistics.
A researcher was interested in determining whether a six-week low- or high-intensity exercise-training programme was best at reducing blood cholesterol concentrations in middle-aged men. Both exercise programmes were designed so that the same number of calories was expended in the low- and high-intensity groups. As such, the duration of exercise differed between groups. The researcher expected that any reduction in cholesterol concentration elicited by the interventions would also depend on the participant's initial cholesterol concentration. As such, the researcher wanted to use pre-intervention cholesterol concentration as a covariate when comparing the post-intervention cholesterol concentrations between the interventions and a control group. Therefore, the researcher ran a one-way ANCOVA with: (a) post-intervention cholesterol concentration (post) as the dependent variable; (b) the control and two intervention groups as levels of the independent variable, group; and (c) the pre-intervention cholesterol concentrations as the covariate, pre.
In SPSS Statistics, we entered three variables: (1) the dependent variable, post, which is the post-intervention cholesterol concentration; (2) the independent variable, group, which has three categories: "control", "Int_1" (representing the low-intensity exercise intervention), and "Int_2" (representing the high-intensity exercise intervention); and (3) pre, which represents the pre-intervention cholesterol concentrations. In our enhanced one-way ANCOVA guide, we show you how to correctly enter data in SPSS Statistics to run a one-way ANCOVA. You can learn about our enhanced data setup content in general here. Alternately, we have a generic, "quick start" guide to show you how to enter data into SPSS Statistics, available here.
The nine steps below show you how to analyse your data using a one-way ANCOVA in SPSS Statistics when the nine assumptions in the Assumptions section have not been violated. At the end of these nine steps, we show you how to interpret the results from this test. If you are looking for help to make sure your data meets assumptions #4, #5, #6, #7, #8 and #9, which are required when using a one-way ANCOVA and can be tested using SPSS Statistics, you can learn more about our enhanced content here.
Click Analyze > General Linear Model > Univariate... on the main menu, as shown below:
Published with written permission from SPSS Statistics, IBM Corporation.
You will be presented with the Univariate dialogue box, as shown below:
Published with written permission from SPSS Statistics, IBM Corporation.
Transfer the dependent variable, post, into the Dependent Variable: box, the independent variable, group, into the Fixed Factor(s): box, and the covariate, pre, into the Covariate(s): box, by selecting each variable (by clicking on it) and clicking the relevant button. You will be end up with the screen below:
Published with written permission from SPSS Statistics, IBM Corporation.
Click the button. You will be presented with the Univariate: Options dialogue box, as shown below:
Published with written permission from SPSS Statistics, IBM Corporation.
Transfer the variable, group, from the Factor(s) and Factor Interactions: box to the Display Means for: box using the button. Then, check Compare main effects, which will activate the Confidence interval adjustment: option. From this drop-down menu, select the option. Also, select Descriptive statistics and Estimates of effect size in the –Display– area, as shown below:
Published with written permission from SPSS Statistics, IBM Corporation.
Click the button and you will be returned to the Univariate dialogue box.
Click the button. This will generate your output.
Go to the next page for the SPSS Statistics output and an explanation of the output.