The analysis of covariance (ANCOVA) can be thought of as an extension of the one-way ANOVA to incorporate a covariate variable. This covariate is linearly related to the dependent variable and its inclusion into the analysis can increase the ability to detect differences between groups of an independent variable. An ANCOVA is used to determine whether there are any statistically significant differences between the adjusted population means of two or more independent (unrelated) groups.

For example, you could use a one-way ANCOVA to determine whether exam performance differed based on test anxiety levels amongst students whilst controlling for revision time (i.e., your dependent variable would be “exam performance”, measured from 0-100, your independent variable would be “test anxiety level”, which has three groups – “low-stressed students”, “moderately-stressed students” and “highly-stressed students” – and your covariate would be “revision time”, measured in hours). You want to control for revision time because you believe that the effect of test anxiety levels on exam performance will depend, to some degree, on the amount of time students spent revising.

## Assumptions

In order to run a one-way ANCOVA, there are ten assumptions that need to be considered. The first four assumptions relate to your choice of study design and the measurements you chose to make, whilst the second six assumptions relate to how your data fits the one-way ANCOVA model. These assumptions are:

- Assumption #1: You have
**one dependent variable**that is measured at the**continuous**level. Examples of**continuous variables**include include height (measured in centimetres), temperature (measured in °C), salary (measured in US dollars), revision time (measured in hours), intelligence (measured using IQ score), age (measured in years), reaction time (measured in milliseconds), grip strength (measured in kg), power output (measured in watts), test performance (measured from 0 to 100), sales (measured in number of transactions per month), academic achievement (measured in terms of GMAT score), and so forth.

- Assumption #2: You have
**one independent variable**that consists of**two or more categorical**,**independent groups**. Example independent variables that meet this criterion include ethnicity (e.g., three groups: Caucasian, African American and Hispanic), physical activity level (e.g., four groups: sedentary, low, moderate and high), profession (e.g., five groups: surgeon, doctor, nurse, dentist, therapist), and so forth.

**Note 1:** The “groups” of the independent variable are also referred to as “categories” or “levels”, but the term “levels” is usually reserved for groups that have an order (e.g., fitness level, with three levels: “low”, “moderate” and “high”).

**Note 2:** If you have **two independent variables** rather than just one, and this second independent variable is not another **covariate** (see Assumption #3 below), you should consider a two-way ANCOVA instead of a one-way ANCOVA.

- Assumption #3: You have
**one covariate variable**that is measured at the**continuous**level (see Assumption #1 for examples of continuous variables). A covariate is simply a continuous independent variable that is added to an ANOVA model to produce an ANCOVA model. This covariate is used to adjust the means of the groups of the categorical independent variable. It acts no differently than in a normal multiple regression, but is usually of less direct importance (i.e., the coefficient and other attributes are often of secondary importance or not at all). In an ANCOVA the covariate is generally only there to provide a better assessment of the differences between the groups of the categorical independent variable on the dependent variable.

**Important:** You can have **many** continuous covariates in a one-way ANCOVA, but we only show you how to analyze a design with **one** continuous covariate in this guide. We will be adding a separate guide to the site to help with **multiple** continuous covariates, so if this is of interest, please contact us and we will email you when the guide becomes available.

**Note:** If your covariate is **not** a **continuous variable**, but is a **categorical variable** with **two or more categorical, independent groups**, like the independent variable in Assumption #2 above, please contact us. Your covariate **does not** have to be continuous, but the analysis is not always then called ANCOVA. Therefore, we will be adding a separate guide to the site to help with this situation.

- Assumption #4: You should have
**independence of observations**, which means that there is no relationship between the observations in each group of the independent variable or between the groups themselves. Indeed, an important distinction is made in statistics when comparing values from either different individuals or from the same individuals. Independent groups (in a one-way ANCOVA) are groups where there is no relationship between the participants in any of the groups. Most often, this occurs simply by having different participants in each group. For example, if you split a group of individuals into four groups based on their physical activity level (e.g., a “sedentary” group, “low” group, “moderate” group and “high” group), no one in the sedentary group can also be in the high group, no one in the moderate group can also be in the high group, and so forth. As another example, you might randomly assign participants to either a control trial or one of two interventions. Again, no participant can be in more than one group (e.g., a participant in the control group cannot be in either of the intervention groups). This will be true of any independent groups you form (i.e., a participant cannot be a member of more than one group). In actual fact, the ‘no relationship’ part extends a little further and requires that participants in different groups are considered unrelated, not just different people (e.g., participants might be considered related if they are husband and wife, or twins). Furthermore, participants in one group cannot influence any of the participants in any other group.It is also fairly common to hear this type of study design, with two or more independent groups, being referred to as “between-subjects” because you are concerned with the differences in the dependent variable between different subjects. An example of where related observations might be a problem is if all the participants in your study (or the participants within each group) were assessed together, such that a participant’s performance affects another participant’s performance (e.g., participants encourage each other to lose more weight in a ‘weight loss intervention’ when assessed as a group compared to being assessed individually; or athletic participants being asked to complete ‘100m sprint tests’ together rather than individually, with the added competition amongst participants resulting in faster times, etc.). Independence of observations is largely a study design issue rather than something you can test for, but it is an important assumption of the one-way ANCOVA. If your study fails this assumption, you will need to use another statistical test instead of the one-way ANCOVA.

- Assumption #5: The covariate should be linearly related to the dependent variable at each level of the independent variable. The first assumption you need to test for is whether there is a linear relationship between the covariate, pre, and the dependent variable, post, for each level of the independent variable, group. In the one-way ANCOVA model, it is assumed that the covariate, pre, is linearly related to the dependent variable, post, for all groups of the independent variable, group. To test this assumption you can plot a grouped scatterplot of the dependent variable, post, against the covariate, pre, grouped on the independent variable, group. You can also add lines of best fit for each group for extra clarity. We show you how to plot a grouped scatterplot using the
**Chart Builder…**procedure in SPSS Statistics, as well as explain how to interpret the output.

- Assumption #6: You should have homogeneity of regression slopes.This assumption checks that there is no interaction between the covariate, pre, and the independent variable, group. Put another way, the regression lines you plotted for
**Assumption #5**above must be parallel (i.e., they must have the same slope). However, whilst this grouped scatterplot will give you an indication of whether the slopes are parallel, you should test this assumption statistically by determining whether there is a statistically significant interaction term, group*pre. One of the reasons for this is that you might not expect the lines to always be parallel as they are plots of the*sample*data and the assumption applies to the*population*regression lines – you will always expect some deviation.By default, SPSS Statistics does not include an interaction term between a covariate and an independent variable in its GLM Univariate procedure. If you contact us, we will show you how to specifically request this term in the model using the**Univariate…**procedure to determine if there is a statistically significant interaction term, before showing you how to determine if you have homogeneity of regression slopes.

## Interpreting Results

After running the one-way ANCOVA procedures and testing that your data meet the assumptions of a one-way ANCOVA in the previous sections, SPSS Statistics will have generated a number of tables that contain all the information you need to report the results of your one-way ANCOVA. We show you how to interpret these results.

The one-way ANCOVA has two main objectives: (1) to determine whether the independent variable is statistically significant in terms of the dependent variable; and (2) if so, determine where any differences in the groups of the independent variable lie. Both of these objectives will be answered in the following sections:

- Descriptive statistics and estimates: You can start your analysis by getting an overall impression of what your data is showing through the descriptive statistics and estimates (the “
**Descriptive Statistics**” and “**Estimates**” tables). The**Descriptive Statistics**table presents the mean, standard deviation and sample size for the dependent variable, post, for the different groups of the independent variable, group. You can use this table to understand certain aspects of your data, such as: (a) whether there are an equal number of participants in each of your groups; (b) which groups had the higher/lower mean score (and what this means for your results); and (c) if the variation in each group is similar. However, these values**do not**include any adjustments made by the use of a covariate in the analysis, which is important. Therefore, you need to consult the**Estimates**table where the mean values of the groups of the independent variable have been adjusted by the covariate, pre. These values are called**adjusted means**because they have been adjusted by the covariate. - One-way ANCOVA results: In evaluating the main one-way ANCOVA results, you can start by determining the overall statistical significance of the model; that is, whether the (adjusted) group means are statistically significantly different (i.e., is the independent variable statistically significant?). In our example, we want to determine whether there was an overall statistically significant difference in post-intervention cholesterol concentration (post) between the different interventions (group) once their means had been adjusted for pre-intervention cholesterol concentrations (pre). This is achieved by interpreting the
**Tests of Between-Subjects Effects**table, which contains the main results of the one-way ANCOVA. - Post hoc tests: If there is a statistically significant difference between the adjusted means (i.e., your independent variable is statistically significant), you can use a Bonferroni post hoc test to determine where exactly the differences lie. By inspecting the
**Pairwise Comparisons**table, you can determine whether cholesterol concentration was, for example, statistically significantly greater or smaller in the control group compared to the low-intensity exercise group, as well as determining what the mean difference was (including 95% confidence intervals). If you contact us, we will explain how to interpret the Bonferroni post hoc test results.

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