One-Way Repeated Measures ANOVA in SPSS
A repeated measures ANOVA is a statistical method that is used to test if the mean measurements across two or more than two times or conditions are significantly different from each other. In this way, a repeated measures ANOVA can be viewed as an extension to a dependent samples t-test.
Introduction to One-Way Repeated Measures ANOVA
Sometimes measurements are repeated at several time points to better understand the effect of an independent variable. In such studies, there is dependency across observations because an observation at times T1, T2, and T3 belong to the same person (or experimental unit), and therefore we need statistical methods that take into account the dependency between observations. Repeated measures ANOVA is such a method that accounts for dependency between data and can produce a measure of change across the time points or conditions.
Similar to the one-way ANOVA, a one-way repeated measures ANOVA includes one independent categorical or nominal variable and one continuous dependent variable which is measured at two, three, or more times or occasions. The independent variable in one-way repeated ANOVA can be Time (e.g., time 1, time 2, time 3) or condition / exposure, such as Dosage (dosage 1, dosage 2, dosage 3 with different amounts on the same group of patients). Because a one-way repeated measures ANOVA analyzes the mean measurements within the same participants across different times or occasions, it is also called within-subject ANOVA (since we are interested in knowing the change within the same group across times or conditions).
In the following sections, we present an example research scenario where a one-way repeated measures ANOVA will be used to analyze the data. We will demonstrate how to perform a one-way repeated measures ANOVA in SPSS step-by-step and how to interpret the SPSS results output..
One-way Repeated Measures ANOVA Example
Does providing a school-based yoga program to school children reduce their math anxiety?
A school psychologist devised a study to examine whether incorporating yoga practice within the school setting could alleviate students' math anxiety. To quantify math anxiety levels, a standardized and psychometrically sound math anxiety questionnaire was utilized. Following the administration of the test, the staff identified and randomly selected 25 students whose scores indicated a notable presence of math anxiety. These students subsequently participated in a structured yoga program implemented within the school environment over a six-month period. During the course of the program, math anxiety levels were assessed at three distinct intervals: at the program's initiation, midway at three months, and upon completion at six months.
This research employed a repeated-measures design wherein a single cohort of students underwent three assessments of math anxiety. As the measurements were obtained from the same participants, the data is inherently dependent. To determine whether there were statistically significant changes in the mean math anxiety scores across the three time points, a one-way repeated measures analysis of variance (ANOVA) was deemed an appropriate statistical method. Table 1 below provides the math anxiety scores of five participants measured at the three specified intervals.
| Student | Time 1 | Time 2 | Time 3 |
|---|---|---|---|
| S1 | 71 | 49 | 63 |
| S2 | 83 | 59 | 43 |
| S3 | 58 | 52 | 37 |
| S4 | 68 | 48 | 50 |
| S5 | 81 | 60 | 36 | ... | ... | ... |
The researcher aims to determine whether the implementation of a yoga program influences math anxiety levels among students and, if so, to identify the extent of its effectiveness over time. Data collected from the math anxiety assessments at three intervals are entered into the SPSS software in the school's computer lab for analysis. The data for this example can be downloaded in the SPSS format or in CSV format.
Entering Data into SPSS
To enter the data in the SPSS program for the math anxiety example, begin by clicking on the Variable View tab (located at the bottom left of SPSS window) and create the following variables under the "Name" column: Student, Anxiety_Time1, Anxiety_Time2, and Anxiety_Time3. Assign the following attributes to each variable:
- Student: Data type is String. Measurement level is Nominal
- Anxiety_Time1: Data type is Numeric. Measurement level is Scale
- Anxiety_Time2: Data type is Numeric. Measurement level is Scale
- Anxiety_Time3: Data type is Numeric. Measurement level is Scale
This setup will allow for efficient data entry and analysis in SPSS, particularly for examining the changes in math anxiety scores across the three time points using the repeated measures design.
When creating variables in SPSS, it is essential to define both the data type and the measurement level. The data type determines how the software processes the input data, while the measurement level guides the statistical algorithms used for analysis.
In this example, the Student variable consists of students’ names or ID’s and is not involved in the computation, so we select "string" as the data type and "nominal" as the measurement level. The data type for the variables Anxiety_Time1, Anxiety_Time2, and Anxiety_Time3 is also "Numeric," and for the measurement level, we select "Scale." SPSS uses the term scale for interval or ratio measurement levels. After creating all variables, the Variable View panel of SPSS for our dataset should look like Figure 1.
Once the variables are created, we switch to the Data View tab of the SPSS program and enter the data into the columns Student, Anxiety_Time1, Anxiety_Time2, and Anxiety_Time3. For the column Student, we can enter their names or ID's. For the Anxiety_Time1, Anxiety_Time2, and Anxiety_Time3, we enter the student’s score on the math anxiety test measured at times 1, 2, and 3, respectively. Figure 2 shows how the data for all four variables should look like in the Data View tab.
Now we are ready to conduct the one-way repeated measures ANOVA in SPSS.
Analysis: One-way Repeated Measures ANOVA in SPSS
A one-way repeated measures ANOVA is a statistical technique used to compare means across multiple time points for the same group of participants. In the context of our study, the school psychologist aims to determine whether practicing yoga has a significant effect on students' math anxiety over time. Math anxiety levels are measured at three intervals: Time 1 (program initiation), Time 2 (three months into the program), and Time 3 (six months into the program). Since all measurements are derived from the same group of students, a repeated measures design is employed. This approach enables the researcher to evaluate whether there is a significant change in mean math anxiety scores across the three time points.
In SPSS, the one-way repeated measures ANOVA test can be accessed from the menu Analyze / General Linear Model / Repeated Measures. So, as Figure 3 demonstrates, we click on Analyze and then choose General Linear Model and then click on Repeated Measures.
Once we click on Repeated Measures item, a window will pop out that asks for the name of Within-Subject Factor and the Number of Levels the factor has, and the name of the Measure (dependent variable). Figure 4 shows the Repeated Measures Define Factor(s) window.
For Factor name, we choose Time, as the repeated measurements occurred over times, and press Add. For the number of Levels, we enter 3 because we have three time points and measurements. We press Add button for Levels. For Measure Name, we choose the name math_Anxiety and press the Add button. Measure is synonymous with dependent variable. We then click on the Define button for defining the variables we just created. The previous window disappears and a new window (Figure 5) appears.
In the new window (Figure 5), we can see our three Anxiety Time measures and in the opposite window we can see three slots with question marks and the numbers 1, 2, and 3. We pass each measurement level from the left panel into the corresponding slot in the opposite panel. We can click on the level name and use the blue right-arrow button to pass them to the right panel. After completing moving the levels to their respective time slots, the completed window should look like Figure 6 below.
While we are in the window, we can request for some additional results. If we want to know the mean math anxiety scores across three time points, we can click on Plots and ask for a plot for Time on the horizontal axis (Figure 7).
After selecting Time, we click on the blue right-arrow button to place it in the Horizontal Axis box and then click on Add button. If you do not click on Add button and just click on Continue no plot will be produced. We click on Continue (after Adding the plot) and close this window.
In one-way repeated measures ANOVA, we are also interested to know if the mean values are significantly different from each other. So, we click on the Options button and by selecting Time from the left panel we move it to the Display Means for panel. Just below that check Compare Main Effect and from the menu below choose Bonferroni. In the Display panel below, we also check Descriptive statistics option. Your window should look like Figure 8 below.
Next, we press Continue and then press the OK button to run the one-way repeated measures ANOVA.
Interpretation: One-way Repeated Measures ANOVA in SPSS
In this study, the researcher aims to evaluate the effectiveness of yoga practice in reducing math anxiety among students over time. Math anxiety levels are measured at three intervals: Time 1 (at the start of the yoga program), Time 2 (three months into the program), and Time 3 (six months into the program). Since the same group of students participated throughout the study, the researcher employs a one-way repeated measures ANOVA to compare the mean math anxiety scores across the three time points. The mean math anxiety scores for these intervals are illustrated in Figure 9 below.
As Figure 9 shows, the mean math anxiety shows a decrease from Time 1 to Time 2 and from Time 2 to Time 3. To understand if the change overall is significant, a one-way repeated measures ANOVA can show that. And to determine if the difference between the means at three time points are significant, pairwise Bonferroni tests can tell us. We now go through the requested analysis results in the SPSS output one by one.
The first table (Figure 10) shows the names of the levels of the dependent variable.
One point to note is that in this table, the three levels of the time correspond to three dependent variables under column Dependent Variable. Although we only have one dependent variable, SPSS regards the different measurements of math anxiety at three time points as separate but related dependent variables. When a study has several related dependent variables, we call it multivariate analysis. Therefore, SPSS outputs several multivariate tests, as well.
The next table in the SPSS output (Figure 11) shows some descriptive statistics for each time points, such as the mean, standard deviation, and sample size (N).
The Descriptive Statistics table shows that the mean values are decreasing over the three measurements (75.96, 55.44, 45.00, respectively).
The next table shows the Multivariate Tests. As we discussed above, SPSS considers the three measurements as three dependent variables and therefore produces several multivariate tests (Figure 12).
Although we can ignore this table, it hints at the statistical significance of our model (effect of time on three math anxiety scores). All four multivariate tests show that the overall model is statistically significant (p < 0.01).
The next table shows if the assumption of sphericity is met (Figure 13).
An assumption underlying the repeated measures ANOVA is the sphericity which means that the variances of difference scores (i.e., subtracting Time 2 from Time 1 scores, Time 3 from Time 1 scores, and Time 3 from Time 2 scores) are equal. The Mauchly’s test provides information if the variances are equal. As the results in Figure 13 show, the Mauchly’s W (0.819) is not statistically significant (p = 0.101), which means that the variances are not different. So, we assume the sphericity is met; otherwise, we should use other test results (shown in the next table).
In the next table (Figure 14), we can see the results of one-way repeated measures ANOVA for both when sphericity is met and when sphericity is not met.
Because the sphericity assumption is met in this analysis, we can look at the first row (Sphericity Assumed) in the table to see the results of the one-way repeated measures ANOVA. According to this table, the overall model is statistically significant with an F = 101.318 and p < 0.01. Therefore, we conclude that the overall change in mean math anxiety scores is statistically significant.
We can ignore some of the tables produced and go straight to the Pairwise Comparisons table (Figure 15). As we discussed above, when the overall model is statistically significant, our next question would be which time points are statistically different from each other?
The Pairwise Comparison table compares the means of math anxiety scores in pairs:
- A comparison between Time 1 and Time 2 scores reveals a mean difference of 20.520, which is statistically significant (p < 0.01).
- A comparison between Time 1 and Time 3 scores reveals a mean difference of 30.960, which is statistically significant (p < 0.01).
- A comparison between Time 2 and Time 3 scores reveals a mean difference of 10.440, which is statistically significant (p < 0.01).
In summary, we conclude that after three months, students' math anxiety significantly decreased (by 20.520 points) and after six months it further decreased by 10.440 points (for a total of 30.960 points). This shows the yoga program was effective even in three months into the program.
Reporting the Results of One-way Repeated Measures ANOVA
In this research, the focus was on evaluating the effectiveness of a yoga intervention in reducing math anxiety among students. A random sample of 25 participants was selected, and math anxiety levels were assessed at three distinct time points: Time 1 (program initiation), Time 2 (three months into the program), and Time 3 (six months into the program). Math anxiety was measured using a standardized questionnaire.
At Time 1, the group had a mean math anxiety score (M) of 75.96, with a standard deviation (SD) of 11.054. At Time 2, the mean score decreased to 55.44, with a standard deviation (SD) of 3.979. By Time 3, the mean score further declined to 45.00, with a standard deviation (SD) of 11.265.
A one-way repeated measures ANOVA was conducted to compare the mean math anxiety scores across the three time points. The Mauchly test showed that the sphericity assumption is met. The one-way-repeated measures ANOVA analysis revealed a statistically significant effect of the yoga intervention on mean math anxiety levels at the p < 0.05 level for the three time intervals [F(2)=101.318, p < .05]. Post hoc comparisons using a pairwise analysis with Bonferroni correction showed that the mean score at Time 3 was significantly lower than the scores at both Time 2 and Time 1. Additionally, the mean score at Time 2 was significantly lower than that at Time 1.
These results suggest that the yoga program is highly effective in reducing math anxiety over time. The findings provide strong support for incorporating structured yoga interventions within educational settings as a strategy to alleviate math anxiety among students.
In light of these findings, future research could explore long-term effects of yoga interventions on math anxiety and investigate additional supportive strategies that can complement yoga in educational contexts.