# Two-Way ANCOVA

The two-way ANCOVA is a statistical test to assess whether there is an interaction effect between two distinct, independent variables on a continuous dependent variable. In simpler terms, it helps us understand if these two variables have a combined influence on the outcome. This analysis considers one or more continuous covariates and additional factors that might impact the dependent variable.

To illustrate this concept further, imagine a study where researchers wanted to evaluate the impact of two teaching methods (the traditional approach and the other a new experimental method) on student test scores. However, they also wanted to consider the effect of students’ prior knowledge levels, which varied across the participants. In this scenario, the two independent variables are the teaching method (with two groups: “Traditional” and “Experimental”) and prior knowledge (with two levels: “Low” and “High”). The dependent variable is the test score, and the continuous covariate is the students’ age. The researchers aim to determine if the experimental teaching method performs differently compared to the traditional method, taking into account students’ prior knowledge levels. Additionally, they want to explore whether the impact of teaching methods varies based on prior knowledge levels. This situation aligns with the typical application of a two-way ANCOVA.

It’s worth noting that the researchers’ primary focus is on detecting a statistically significant interaction effect because this will provide crucial insights into how teaching methods affect test scores, considering students’ prior knowledge and age. Depending on the presence and nature of this interaction effect, the researchers will decide how to interpret the results and may conduct further post hoc tests. This process is explained in detail as you navigate through the analysis, ensuring a comprehensive understanding of the statistical outcomes and how to report them accurately.

In terms of describing a two-way ANCOVA, it can be summarized by specifying the number of groups in each independent variable. For instance, if you were conducting a two-way ANCOVA to examine the impact of gender (with two groups: “male” and “female”) and mode of transportation (with three groups: “bus,” “train,” and “car”) on salary, with age as a covariate, you could label it as a “2 x 3 ANCOVA.” Similarly, if your analysis involved three groups in one independent variable and five groups in the other, it would be termed a “3 x 5 ANCOVA.” This nomenclature simplifies the description of ANCOVA designs with multiple independent variables.

Assumptions

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

• Assumption #1: You have one dependent variable measured at the continuous level (i.e., the interval or ratio level). 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.

Note: You should note that SPSS Statistics refers to continuous variables as Scale variables.

• Assumption #2: You have two independent variables where each independent variable consists of two or more categoricalindependent groups. These variables are typically categorized into groups, which can be dichotomous (two groups) or polytomous (three or more groups). The groups within an independent variable are often called “categories” or “levels.” The term “levels” is generally used for groups that have a natural ordering, such as education level, with categories like “high school,” “undergraduate,” and “graduate.” For instance, you might examine the effects of education level (with three levels: high school, undergraduate, and graduate) and employment status (with two groups: employed and unemployed) on financial literacy. While these terms can be used interchangeably, we often use “groups” for clarity and simplicity but might switch to “levels” when it fits better contextually. Independent variables in an ANOVA are also known as factors. In a two-way ANOVA, we deal with two factors or independent variables. When these factors comprise independent groups, they are termed “between-subjects factors,” as the focus is on differences between distinct subjects. For example, in a study comparing dietary habits (with three groups: omnivore, vegetarian, vegan) and exercise frequency (with three levels: none, moderate, regular) on health outcomes, each variable is a factor with independent groups.
• Assumption #3 states that you should have one continuous covariate in your ANCOVA model. A covariate is an independent variable added to the ANOVA model to produce an ANCOVA model. It is used to adjust the means of the groups of categorical independent variables. The covariate acts no differently than in a normal multiple regression but is usually of less direct importance. In a two-way ANCOVA, the covariate is generally included to better assess the differences between the groups of categorical independent variables on the dependent variable.
• 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 different individuals or the same individuals. Independent groups (in a two-way ANOVA) are groups with no relationship between the participants in any of the groups. Most often, this occurs simply by having different participants in each group. For example, consider a study where participants are divided into groups based on their preferred learning style (e.g., visual, auditory, kinesthetic, and reading/writing). In this scenario, each participant belongs exclusively to one group. A person who prefers visual learning will not be in the auditory or any other group. The individuals in one group must not influence those in another. This separation guarantees that one participant’s learning style does not affect another’s preferences or responses. Another example could involve dividing participants into groups based on their favorite type of music (e.g., classical, rock, pop, and jazz). Each participant’s preference is unique, and their choice does not impact the preferences of others in different groups. In this way, the observations remain independent. It is important to note that the independence of observations extends beyond mere physical separation. For instance, if participants in a weight loss study are grouped based on the type of diet they follow (e.g., vegan, keto, Mediterranean, and paleo), it’s not enough for participants to be in different groups. They also should not influence each other’s dietary adherence or weight loss outcomes, as this would violate the assumption of independent observations.
• Assumption #5 states that the covariate, weight, should have a linear relationship with the dependent variable, cholesterol, for each combination of groups of the two independent variables (i.e., for each cell of the design). To test this assumption, you can create a grouped scatterplot of cholesterol against weight, grouped by diet and exercise, and add a loess line to each scatterplot to determine whether the relationships are linear.
• Assumption #6 is about the homogeneity of regression slopes, meaning that the relationship between weight and cholesterol, as assessed by the regression slope, should be the same in each design cell. This assumption ensures that the linear relationships in Assumption #5 are the same. To test this assumption, you must add interaction terms between the covariate and the independent variables and compare the model with interaction terms to the two-way ANCOVA model without interaction terms. However, this can be tricky in SPSS Statistics because there are multiple interaction terms, and SPSS Statistics only allows you to evaluate them all at different times. If you need help, please sign up, and our free guide will show you how to do it. Our free guide will also explain how to interpret the results and what to do if your data fails to meet this assumption.

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**Interpreting Results:**
After performing the two-way ANCOVA analysis and confirming that your dataset meets the required assumptions for this analysis, SPSS Statistics generates various tables and graphs containing all the information necessary to report your findings. Here, we guide you through the steps to interpret these results.

**Step 1: Assessing the Significance of the Two-Way Interaction Effect**
The first step in interpreting your two-way ANCOVA results is determining whether a statistically significant two-way interaction effect exists. The primary aim of a two-way ANCOVA is to ascertain if there is an interaction effect between the two independent variables concerning the dependent variable while considering a continuous covariate. Let’s consider an example:
Imagine you’re investigating the impact of two factors, “temperature” and “humidity,” on plant growth, with the covariate “soil quality.” A significant two-way interaction effect would suggest that the effect of temperature on plant growth (after accounting for soil quality) depends on the humidity level and vice versa. In essence, temperature affects plant growth differently at various humidity levels.

– **If yes** – indicating a statistically significant two-way interaction effect – proceed to **Step 2A**.
– **If no** – suggesting the absence of a statistically significant two-way interaction effect – continue to **Step 2B**.

**Step 2A: Exploring Simple Main Effects**
When you find a statistically significant two-way interaction effect, the next step is to investigate whether there are statistically significant simple main effects (or interaction contrasts). This interaction effect signifies that the effect of one independent variable on the dependent variable depends on the level of the other independent variable while considering the continuous covariate.
For instance, in our plant growth example, if you have a significant two-way interaction effect, you’d examine the simple main effects of temperature on plant growth within each level of humidity (e.g., the effect of temperature on plant growth for low humidity, moderate humidity, and high humidity). Similarly, you’d explore the simple main effects of humidity on plant growth within each temperature level.

**Step 2B: Examining Main Effects**
Suppose there is no statistically significant two-way interaction effect. In that case, it indicates that the effect of an independent variable is consistent across all levels of the other independent variable after accounting for the continuous covariate. In such cases, it’s customary to report the main effects. Let’s consider another scenario:
Suppose you’re investigating the impact of “study time” and “caffeine consumption” on exam scores while controlling for “sleep duration” as a covariate. If there is no significant two-way interaction effect, it suggests that the effect of study time on exam scores is the same regardless of caffeine consumption levels. In other words, the impact of study time on exam scores is consistent, taking into account sleep duration.
– A significant **main effect of study time** implies that study time influences exam scores, irrespective of caffeine consumption levels.
– A significant **main effect of caffeine consumption** suggests that caffeine consumption impacts exam scores, irrespective of study time.
It’s worth noting that when dealing with more than two levels in an independent variable (e.g., caffeine consumption with three levels: “no caffeine,” “low caffeine,” and “high caffeine”), pairwise comparisons can be employed to understand specific differences between these levels.
In summary, interpreting the results of a two-way ANCOVA involves determining the presence or absence of a significant two-way interaction effect and then exploring simple main effects or main effects accordingly, all while considering the continuous covariate.

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