## One-Way ANCOVA

ANCOVA is a statistical method that extends the one-way ANOVA to include a covariate variable. This covariate is linearly related to the dependent variable, and its inclusion into the analysis can increase the accuracy of detecting differences between groups of an independent variable. ANCOVA can be used in various scenarios. For instance, suppose you want to determine whether the effectiveness of a drug differs based on the dosage levels while controlling for the age of the patients. Here, the dependent variable would be “effectiveness of the drug,” and the independent variable would be “dosage level,” which has three groups – low, moderate, and high dosage levels. The covariate would be the “age of patients,” measured in years. You would want to control for the age of patients because you believe that the effect of dosage levels on the drug’s effectiveness will depend, to some degree, on the age of the patients.

**Assumptions of One-Way ANCOVA**

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

- Assumption #1: You have one continuous dependent variable. Continuous variables can take on infinite values within a given range. For instance, temperature, time, height, weight, distance, age, blood pressure, speed, electricity consumption, and sound level are typical continuous variables. For instance, the temperature in a room can be any value within the limits of the thermometer, such as 22.5°C, 22.51°C, and so on. Similarly, time can be measured to any level of precision, like seconds, milliseconds, or even smaller units. Height and weight can vary infinitely within their possible range, measured in units like meters or feet, and can include fractions (like 1.75 meters). Distance between two points, age measured in years, months, days, and even smaller units, blood pressure measured in millimeters of mercury (mmHg), speed measured in units like kilometers per hour or miles per hour, electricity consumption measured in kilowatt-hours or other units, and sound level measured in decibels are other examples of continuous variables that can take on a range of continuous values.
- Assumption #2: You need an independent variable containing two or more categorical, non-overlapping groups. For instance, if you are researching the effect of diet on weight loss, your independent variable can be diet type, and the groups can be low-carb, high-carb, and balanced diet. Another example can be the study of the impact of social media on mental health, where the independent variable can be social media usage, and the groups can be daily, weekly, monthly, and rarely.
- Assumption #3 states that an ANCOVA model adds a continuous covariate variable to adjust the means of groups of the categorical independent variable. The role of the covariate is similar to that of a continuous independent variable in a normal multiple regression, but it is of lesser importance in an ANCOVA model. The coefficient and other attributes of the covariate are often of secondary importance or not necessary at all. In an ANCOVA model, the covariate is solely used to assess the differences between the groups of the categorical independent variable on the dependent variable. For example, let’s say you are conducting an ANCOVA study to examine the effects of different types of exercise on participants’ weight loss. You might include a continuous covariate variable, such as age or BMI, to adjust the means of the groups of the categorical independent variable (type of exercise). The role of the covariate would be to provide a better assessment of the differences in weight loss between the groups of exercise types.
- Assumption #4 is the independence of observations, which means that there should be no relationship between the observations in different independent variable groups or between the groups themselves. To ensure this independence, researchers often form independent groups where participants in one group cannot be in another. For instance, if you want to compare the effect of different exercise regimes on weight loss, you might form four groups of participants with different activity levels, such as ‘sedentary’, ‘low,’ ‘moderate,’ and ‘high’ activity level groups. In this case, no participant can be in more than one group. Similarly, suppose you want to compare the effect of different treatments on a medical condition. In that case, you might randomly assign participants to either a controlled trial or one of two interventions, and no participant can be in more than one group. It is important to note that the ‘no relationship’ part of the assumption extends beyond having different participants in each group. Participants in different groups should be considered unrelated, not just different people. For instance, participants might be considered related if they are husband and wife or twins. Moreover, participants in one group should not influence any of the participants in any other group. This type of study design with two or more independent groups is often called “between-subjects” because researchers are concerned with the differences in the dependent variable between different subjects. If related observations are a problem, such as when participants encourage each other to perform better or compete with each other, then the independence of observations assumption is violated, and researchers need to use another statistical test instead of the one-way ANCOVA.
- Assumption#5 is that the covariate, pre, should have a linear relationship with the dependent variable, post, for each independent variable group level. In the one-way ANCOVA model, it is assumed that the covariate, pre, has a linear relationship with the dependent variable, post, for all independent variable groups. To verify this assumption, you can create 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 to enhance clarity. We demonstrate how to create a grouped scatterplot using the Chart Builder feature in SPSS Statistics and explain how to interpret the results.
- Assumption #6 is that you should have homogeneity of regression slopes. This assumption checks for the absence of interaction between the covariate, pre, and the independent variable, group. In other words, the regression lines you plotted for Assumption #5 above should be parallel (i.e., they should have the same slope). However, although the grouped scatterplot can indicate whether the slopes are parallel, it is important to test this assumption statistically by determining if there is a statistically significant interaction term, group*pre. One of the reasons for this is that you cannot always expect the lines to be parallel, as the plots of the sample data, and the assumption applies to the population regression lines, which will always show some deviation. By default, SPSS Statistics does not include an interaction term between a covariate and an independent variable in its GLM Univariate procedure. We can show you how to request this term in the model using the Univariate procedure to determine if there is a statistically significant interaction term before guiding you on determining if you have homogeneity of regression slopes.

**Interpret Your One-Way ANCOVA Results **

Once you have completed the one-way ANCOVA procedures and verified that your data meets the assumptions of a one-way ANCOVA, SPSS Statistics will automatically generate several tables containing all the necessary information to report the results of your one-way ANCOVA. We will guide you through the interpretation of these results.

The two primary objectives of the one-way ANCOVA are: (1) to determine the statistical significance of the independent variable about the dependent variable, and (2) if the independent variable is significant, to determine where the differences in the groups of the independent variable lie. We will answer both of these objectives in the following sections.

**Descriptive statistics and estimates:**You can begin analyzing your data by examining the descriptive statistics and estimates (the “Descriptive Statistics” and “Estimates” tables) to get an overall understanding of what your data is showing. The “Descriptive Statistics” table displays 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 determine various aspects of your data, such as whether there are an equal number of participants in each group, which groups had higher/lower mean scores (and what this means for your results), and if the variation in each group is similar. However, these values do not include any adjustments made by using a covariate in the analysis, which is crucial. Therefore, you should refer to the “Estimates” table, which displays the mean values of the independent variable groups adjusted by the covariate, “pre.” These adjusted means are called “adjusted means” since the covariate has adjusted them.**One-way ANCOVA results:**To evaluate the primary one-way ANCOVA results, you should first determine the overall statistical significance of the model, which means choosing if the (adjusted) group means are statistically significantly different (i.e., is the independent variable statistically significant?). In our example, we want to determine if there was a statistically significant difference in post-intervention cholesterol concentration (post) among the different interventions (group) once their means had been adjusted for pre-intervention cholesterol concentrations (pre). This can be done 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., the independent variable is statistically significant), you can use a Bonferroni post hoc test to determine where the differences lie. The “Pairwise Comparisons” table lets you determine whether cholesterol concentration was statistically significantly greater or smaller in the control group compared to the low-intensity exercise group and 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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