One-Way Anova

10

Instructions for Conducting One-Way ANOVA in SPSS

One way ANOVA is used to examine mean differences between two or more groups. It is a bivariate test with one IV and one DV. The IV must be categorical and the DV must be continuous. In this demonstration we describe how to conduct one way ANOVA, and planned and post hoc comparisons. The instructions also include a check for whether the homogeneity of variance has been met. The DV is a measure of base year standardized reading scores (BY2XRSTD); the IV is an indicator of the geographic region of each school (G8REGON). The IV has four values—northeast (coded as 1), north central (coded as 2), south (coded as 3), and west (coded as 4). This analysis is based on 300 randomly selected cases from the NELS database with no missing observations. Although not used in the actual analysis, the data set also includes one weight variable (F2PNWLWT).

Copies of the data set and output are available on the companion website. The data set file is entitled, “ONE WAY ANOVA.sav”. The output file is entitled, “One Way ANOVA results.spv ”.

The following instructions are divided into three sets of steps:

1.  Conduct an exploratory analysis to a) examine descriptive statistics, b) check for outliers, c) check that the normality assumption is met, and d) verify that there are mean differences between groups to justify ANOVA.

2.  Conduct the actual one-way ANOVA to determine whether group means are different form one another (warranting planned or post hoc comparison tests, as described in step 3). Also, check that the homogeneity of variance assumption is met.

3.  Conduct planned or post hoc comparisons if warranted. For illustration purposes, we provide instructions for conducting both planned and post hoc comparisons.

NOTE: There is no way use SPSS to check that the independence assumption has been met. The independence assumption means that the errors associated with each observation are independent from one another. For this particular example, it is the assumption that if two students are in the same region/group, the extent to which the score of one student deviates from the group mean is not correlated with the extent to which the scores of other students deviate from the group mean. The independence assumption is met when case selection is random. Thus it is based on the sampling procedures used to create the sample. Because the NELS study incorporated random sampling, we assume that the independence assumption has been met. If the independence assumption is violated, then one-way ANOVA should not be used.

To get started, open the SPSS data file entitled, ONE WAY ANOVA.sav.

STEP 1: Exploratory Analyses

First we calculate descriptive statistics. At the Analyze menu, select Compare Means. Click on Means. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Dependent List box. Highlight G8REGON (Composite Geographic Region of School) and move it to the Independent List box. Click on Options. In the Statistics box, highlight Variance, Skewness, and Std. Error of Skewness. Click to move them to the Cell Statistics box. By default Mean, Number of Cases, and Standard Deviation are already in the Cell Statistics box. If they are not in the Cell Statistics box, move them over now. Click Continue. Click OK.

In addition to the case processing summary, the output includes a report (see following table) that provides descriptive statistics for each of the factors for the IV. As the report indicates, the means of standardized reading scores for the West, South, and North Central regions are similar to one another while the mean for the North East region is somewhat higher. This suggests that these groups may differ with regard to their average standardized reading scores. These differences, especially between the northeast region and the other regions, warrant an ANOVA.

Skewness statistics provide preliminary information about the existence of outliers. Skewness values outside of -1 and 1 suggest that outliers may be present. The report shows that skewness values for all regions fall within -1 to 1 range, suggesting no outliers. We can examine box plots to confirm this. Select Graphs from the menu, then Legacy Dialogs, then click Box plot. Click Simple. Within the “Data in Chart Are”, select Summaries for groups of cases. Click Define. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Variable box. Highlight G8REGON (Composite Geographic Region of School) and move it to the Category Axis box. Click OK to create a box plot graph.

The output includes the case processing summary and a graph of the box plots of reading scores by region. The box plots indicate that none of the regions include outliers.

If outliers are present, their cases would be displayed at either end of the respective box plot, as illustrated in the following example. As you can see, cases 163 and 237 are outliers.

Next, we check whether the normality assumption has been met. The normality assumption means that the residual errors are assumed to be normally distributed, roughly in the shape of the normal curve. We check this assumption by examining histograms of each group. At the Graphs menu, select Legacy Dialogs. Click Histograms. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Variable box. Highlight G8REGON (Composite Geographic Region of School) and move it to the Rows box. Check the Display normal curve. Click OK.

The following shows histograms of the distributions by region. The distribution of the northeast region is negatively skewed, and the distributions of the other regions (i.e., north central, south, and west) are positively skewed. However, we should not be too concerned about this deviation from normality because (a) our sample size is large enough such that, according to the central limit theorem, the distribution of the sample means should be approximately normal (b) we found no outliers in any group and (c) the skewness statistic for each group fell within the -1 to 1 range.

Finally, to justify the use of ANOVA (i.e., that there are likely to be mean differences between groups), we can examine mean differences between groups by using the error bar graph procedure. At the Graphs menu, select Legacy Dialogs, click Error Bar, and then click Simple. Within the Data in Chart Are dialog box, select Summaries for groups of cases. Click Define. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Variable box. Highlight G8REGON (Composite Geographic Region of School) and move it to Category Axis. Click OK.

The output provides an Error Bar graph that displays the 95% confidence interval of the mean for each group. We can use this graph to examine mean differences between groups (and to justify the use of ANOVA to analyze these data).

If a horizontal line can be drawn such that it goes through all the bars, this indicates that the mean differences between groups are likely to be too close to show differences and to justify ANOVA. Bars that overlap with one another suggest that means for these groups are not significantly different, and the overall F test for ANOVA will not be significant. If at least one bar does not overlap with the others, this suggests a significant mean difference, relative to the other groups, justifying ANOVA.

It appears from Figure 2 that the Northeast region bar does not overlap with the other bars, suggesting a mean difference between this group and the other groups on standardized reading scores. Thus, ANOVA appears to be warranted. The error bars also provide a preliminary impression of whether the homogeneity of variance assumption is satisfied. If the bars appear to be fairly equal in length, it is likely that the equal variance assumption is satisfied. However, we’ll still need to conduct a statistical test for homogeneity of variance to be sure.

STEP 2: One-Way ANOVA

To run the one-way ANOVA, at the Analyze menu, select Compare Means. Click One-Way ANOVA. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Dependent List box. Highlight G8REGON (Composite Geographic Region of School) and move it to the Factor box. Click Options in the lower right corner to open the One-Way ANOVA: Options dialogue box. Check Homogeneity of variance test. Click Continue to return to the One-Way ANOVA dialogue box. Click OK at the bottom of the One-Way ANOVA dialogue box to run the one way ANOVA.

The results of the overall F test in the ANOVA summary table can be examined to determine whether group means are different. As indicated, the overall F test is significant (i.e., p value < 0.05), indicating that means between groups are not equal. However, in order to determine which of the group means are different, it is necessary to conduct planned or post hoc comparison tests.

The Test of Homogeneity of Variances table in the output can be examined to check whether the homogeneity of variance assumption has been met.

Results show that the test for homogeneity of variances is not significant (i.e., p =.733, which is greater than 0.05). This indicates that the homogeneity of variances assumption is met.

STEP 3: Planned or Post Hoc Comparison Tests (if warranted)

First we describe how to conduct planned comparison tests. This test requires equal sample sizes between groups. The sample sizes of the groups in this data set are unequal. The smallest group, with 57 cases, is from the northeast region. In order to conduct a planned comparison (simply for illustration purposes), we randomly selected 57 students from each of the other regions to obtain equal sample sizes. In this example, we compare mean reading scores of students in the northeast region to mean reading scores in the other regions, based on what we learned from earlier preliminary findings about mean differences between groups. The null hypothesis is:

Planned comparisons require that that coefficients (i.e., weights) must sum to 0 (i.e., balance). Because there are three groups on the left side of the equation, we put a ‘3’ besides the northeast for the equation to sum to 0. We can rewrite the equation as:

To conduct the planned comparison test, at the Analyze menu, select Compare Means. Click One-Way ANOVA. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Dependent List box. Highlight G8REGON (Composite Geographic Region of School) and move it to the Factor box. Click Contrasts in the upper right corner to open the One-Way ANOVA: Contrasts dialogue box. Move your cursor to the Coefficients box. Type “-3” in the box; click Add; type “1” in the box; click Add; type “1” in the box again and click Add. Type “1” in the box one more time and click Add. Click Continue to return to the One-Way ANOVA dialogue box. Click OK at the bottom of the One-Way ANOVA dialogue box to run the planned comparison tests.

As the following table shows, the results of the planned comparison test are not significant. Non-significance indicates that the means of the standardized reading scores across groups are equal.

The following describes how to conduct a post hoc comparison using the Tukey HSD test. Click on Analyze in the menu bar. Select Compare Means. Click on One-Way ANOVA to open the One-way ANOVA dialogue box. Highlight BY2XRSTD (Reading Standardized Score) and move it to the Dependent List box. Highlight G8REGON (Composite Geographic Region of School) and move it to the Factor box. Click Post Hoc in the upper right corner to open the One-Way ANOVA: Post Hoc Multiple Comparisons box. Check Tukey. Click Continue to return to the One-Way ANOVA dialogue box. Click OK at the bottom of the One-Way ANOVA dialogue box to run the post hoc comparison test.

As the following table indicates, the mean of the northeast is significantly (p <.05) different from the means of the other regions, confirming what was observed in the error bar plots.

The results also indicate that groups can be put into two subsets, based on mean differences. As the following table indicates, the west, south, and north central regions are grouped together because their means are similar. The northeast region represents the other group.