# Two-Way ANOVA

The two-way ANOVA is an extension of the one-way ANOVA that assesses the interaction effect between two independent variables on a continuous dependent variable. It is also called a “factorial ANOVA” or, more specifically, a “two-way between-subjects ANOVA.” In the context of an experiment, the two-way ANOVA can be extremely useful in understanding how different variables interact. For instance, in a study where two drugs are being given to elderly patients to treat heart disease, the two independent variables are “drug” with two levels (“Current” and “Experimental”) and “risk” with two levels (“Low” and “High”). In contrast, the dependent variable is “cholesterol” (concentration in the blood). The researchers aim to determine whether the experimental drug is better or worse than the current drug at lowering cholesterol and whether the effect of the drugs differs depending on whether elderly patients are classified as low or high risk. To answer these questions, the researchers use the two-way ANOVA, which prioritizes determining whether there is a statistically significant interaction effect. This result will decide if the researchers’ first aim is misleading or incomplete. If a statistically significant interaction effect is found, it indicates that the two drugs have different effects in low and high-risk elderly patients. The researchers can then interpret the effects in the two-way ANOVA and run any post hoc tests as required. In describing a two-way ANOVA, the number of groups in each independent variable can be used. For example, a two-way ANOVA with gender and transport type as independent variables could be described as a 2 x 3 ANOVA.

Assumptions

Six assumptions need to be considered to run a two-way ANOVA. The first three assumptions relate to your choice of study design and the measurements you chose to make, while the second three assumptions relate to how your data fits the two-way ANOVA 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 is that you have two independent variables. 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.

Note: In the context of a two-way ANOVA, these independent variables are referred to as “fixed factors” or “fixed effects” if the groups represent all categories of interest. For example, if you’re studying the impact of teaching style (with three groups: lecture, discussion, interactive) and class size (with three groups: small, medium, large) on student engagement, and these specific styles and sizes are your sole focus, they are fixed factors. However, if these groups are randomly selected to represent a larger population, they become “random factors,” necessitating a different statistical approach. This distinction is crucial as it influences the choice of statistical test and the interpretation of the results.

• Assumption #3: 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 different 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 have no influence over 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. When observations are not independent, such as when the same participants are used in each group or when participants are inherently related (like family members), a different statistical approach, like a two-way repeated measures ANOVA or a two-way mixed ANOVA, would be more appropriate. The independence of observations is mainly a matter of study design, and it is crucial to ensure this independence to apply a valid two-way ANOVA. If this assumption is not met, the results of the ANOVA could be misleading, necessitating an alternative statistical method.
• Assumption #4 emphasizes eliminating significant outliers within any treatment group in a rigorous experimental setup. It is crucial to do so to ensure that the results are not skewed and accurately represent the treatment’s effectiveness. For instance, consider a study involving 150 individuals examining the impact of different diets on cholesterol levels. This study will be an atypical result if one individual shows a drastic deviation by exhibiting a 70 mg/dL reduction. In contrast, the rest show minimal variation with an average 20 mg/dL reduction. Such outliers might suggest unique physiological responses or external factors affecting that individual, which are not representative of the diet’s effectiveness. These outliers can have a negative impact on the three-way repeated measures ANOVA by distorting the differences between cells of the design and causing problems when generalizing the results to the population. Therefore, it is essential to identify and understand outliers in data analysis and choose whether to keep them, remove them, or alter their value in some way, given their potential impact on the results.
• Assumption #5 states that for each cell of the design, your dependent variable (residuals) should be approximately normally distributed. This normality assumption is necessary for significant statistical testing using a two-way ANOVA. However, the ANOVA is considered “robust” to deviations from normality, meaning some violations of this assumption can still be tolerated while the test provides valid results. As the sample size increases, the distribution can be non-normal, and the two-way ANOVA can still provide valid outcomes thanks to the Central Limit Theorem. Unfortunately, the exact sample size required to meet this assumption is not well known. Moreover, if the distributions are all skewed similarly, it is less troublesome than when groups have differently shaped distributions. It is important to note that, technically, the normality assumption concerns the residuals and not the raw data. Therefore, in this example, it is necessary to investigate whether the residuals, RES_1, are normally distributed in each design cell.
• Assumption #6, the homogeneity of variances, requires that the variance of your dependent variable (residuals) is the same in each design cell. This assumption is crucial for statistical significance testing in a two-way ANOVA. Although this assumption can be violated slightly in studies with equal sample sizes in each design cell, it is still considered an important assumption. To determine whether this assumption is met, you can use Levene’s test for equality of variances.

Interpreting Results

When interpreting the results of a two-way ANOVA, a systematic approach is necessary to understand the relationships between your independent and dependent variables. This process can be broken down into two main steps, which I will illustrate using different examples.

**Step #1: Determine if a Statistically Significant Interaction Effect Exists**

The first step in interpreting your two-way ANOVA results is checking for a significant interaction effect between your independent and dependent variables. This interaction shows how the effect of one independent variable may change across the levels of the other independent variable. For instance, you might be investigating the interaction between types of dietary supplements (e.g., vitamin C, vitamin D, placebo) and age groups (e.g., young adults, middle-aged adults, elderly) on immune system strength. Here, the focal variable could be the type of supplement, and the moderator variable might be the age group. If a significant interaction effect is found, you proceed to Step #2A. If there is no significant interaction, you move to Step #2B.

**Step #2A: Explore Statistically Significant Simple Main Effects or Interaction Contrasts**

If your analysis reveals a significant interaction effect, the next step is to examine the simple main effects or interaction contrasts. This step involves looking at how one independent variable influences the dependent variable at each level of the other independent variable. For example, using the dietary supplement and age group scenario, you would compare the effect of each type of supplement (e.g., vitamin C, vitamin D, placebo) on immune system strength within each age group. The aim is to see if, for instance, vitamin C has a different impact on young adults compared to elderly individuals. This detailed examination helps to understand the nuances of the interaction effect.

**Step #2B: Assess Statistically Significant Main Effects**

In cases where there isn’t a significant interaction effect, the focus shifts to the main effects of each independent variable. This step examines if the effect of one independent variable is consistent across all levels of the other independent variable. For instance, if there is no significant interaction between dietary supplements and age groups, you would look at the overall effect of dietary supplements on immune system strength, irrespective of age group. If a main effect is found to be significant, you can further analyze it with post hoc tests to identify specific differences between groups. This step is crucial for understanding the overall impact of each independent variable on the dependent variable when their interaction isn’t significant.

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