One-Way ANOVA

If you aim to investigate whether there are any statistically significant distinctions in the means of two or more distinct groups, you can employ a one-way analysis of variance (ANOVA). For instance, consider a situation where you wish to determine if there are variations in the performance of athletes in a track event based on their preferred running surface (i.e., your dependent variable would be “race performance,” measured in seconds, and your independent variable would be “running surface,” comprising three groups: “grass track,” “cinder track,” and “synthetic track”). Alternatively, a one-way ANOVA could be used to explore whether there are differences in customer satisfaction scores across different service channels (e.g., in-person, phone, online), where your dependent variable would be “satisfaction score,” and your independent variable would be “service channel,” encompassing multiple groups.

It’s important to note that the one-way ANOVA is also known as a between-subjects ANOVA or one-factor ANOVA. While it can technically be applied to an independent variable with only two groups, the independent-samples t-test is preferred in such cases. Hence, the one-way ANOVA is commonly described as a test used when you have three or more groups rather than two or more.

It’s crucial to understand that the one-way ANOVA provides an omnibus test statistic, which can establish differences among at least two groups but cannot pinpoint the specific groups that differ. Discerning which groups exhibit significant differences from one another is essential, given the potential for multiple groups in your study design. This task is accomplished through follow-up tests, which may include post hoc tests or custom contrasts. If you are new to one-way ANOVA, it’s advisable to familiarize yourself with the fundamental prerequisites for conducting it, the null and alternative hypotheses under examination, the choice between post hoc tests and custom contrasts, as well as considerations related to effect sizes and sample sizes, especially in the context of balanced or unbalanced designs. However, suppose you are already well-versed in these aspects. In that case, you may want to proceed with the example presented in this guide, which includes an SPSS Statistics data file for hands-on practice before applying one-way ANOVA to your own dataset.

Assumptions of One-Way ANOVA

Six assumptions need to be considered to run a one-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 one-way ANOVA 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 have one independent variable, including two or more categorical, independent 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: 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. 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
Assumption #4 states that when analyzing the relationship between the independent and dependent variables, it is crucial to ensure that there are no significant outliers present in any group of the independent variable. Outliers are data points that significantly differ from the rest of the group regarding size or value. For example, consider a survey that measures the satisfaction level of customers regarding a product. In one particular group, most customers rate the product between 4 and 5 out of 5, but one customer rates it 1 out of 5. Such an outlier can significantly impact your analysis, leading to incorrect conclusions. This conclusion is because it can affect that group’s mean and standard deviation, which can impact the results of the statistical tests. It is necessary to identify and address outliers in each independent variable group to avoid this, especially when dealing with smaller sample sizes. For instance, in this case, you should check whether each group in the independent variable, “age,” is free from outliers regarding the dependent variable, “satisfaction_level,” to ensure accurate results.
Assumption #5: The dependent variable must be approximately normally distributed for each independent variable group. However, the test is considered “robust” to violations of normality. This robustness means that some violations of the normality assumption can still be tolerated, and the test can still provide valid results. As the sample size increases, the distribution can be very non-normal, and the one-way ANOVA can still provide valid results, thanks to the Central Limit Theorem. Moreover, if the distributions are skewed similarly, it is less troublesome than when you have groups with differently-shaped distributions.
Assumption #6 is that you have homogeneity of variances, which means that the population variance for each independent variable group should be the same. Violating this assumption is not often too serious if the sample size in each group is similar. However, if the sample sizes differ, the one-way ANOVA is sensitive to violating this assumption. SPSS Statistics uses Levene’s test of equality of variances and two differently-calculated one-way ANOVAs, giving you a valid result irrespective of whether you met or violated this assumption. In addition, if your data fails this assumption, we show you how to run a Games-Howell post hoc test for your multiple comparisons instead of the Tukey post hoc test. If you need help carrying out Levene’s test for equality of variances, please do not hesitate to contact us.

Interpret Your One-Way ANOVA Results

In our free guide, we provide a detailed understanding of interpreting and reporting the results of one-way ANOVA, post hoc tests, and custom contrasts. For one-way ANOVA with post hoc testing, you can interpret the SPSS Statistics output for the standard one-way ANOVA, including means, standard deviations, F-value, degrees of freedom, and p-value. You can then determine which group means are significantly different, decide whether to reject or fail to reject the null hypothesis, and summarize your results in a single paragraph. For one-way ANOVA with contrasts, if you’re interested in custom contrasts instead of post hoc tests, we explain the SPSS Statistics procedures required to run custom contrasts. You can then interpret the output to determine which group means are significantly different based on the simple or complex contrasts you ran, decide whether to reject or fail to reject the null hypothesis, and summarize your results in a single paragraph.

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