In this problem, you will examine a statistical technique for comparing two or *more* independent groups on the dependent variable. The appropriate statistic, called **One-Way ANOVA**, compares the *means* of the samples or groups in order to make inferences about the population means. Oneway ANOVA also is called single factor analysis of variance because there is only one independent variable or factor. The independent variable has nominal levels or a few ordered levels. The overall ANOVA test does not take into account the order of the levels, but additional tests (contrasts) can be done that do consider the order of the levels. More information regarding contrasts can be found in Leech et al. (in press).

Remember that, in Chapter 9, we used the independent samples *t* test to compare two groups (males and females). The one-way ANOVA *may* be used to compare two groups, but ANOVA is necessary if you want to compare three or more groups (e.g., three levels of *father’s education*) in a single analysis. Review Fig. 6.1 and Table 6.1 to see how these statistics fit into the overall selection of an appropriate statistic.

**Assumptions of ANOVA**

- Observations are independent. The value of one observation is not related to any other observation. In other words, one person’s score should not provide any clue as to how any of the other people would score. Each person is in only one group and has only one score on each measure; there are no repeated or within-subjects measures.
- Variances on the dependent variable are equal across groups.
- The dependent variable is normally distributed for each group.

Because ANOVA is robust, it can be used when variances are only approximately equal if the number of subjects in each group is approximately equal. ANOVA also is robust if the dependent variable data are approximately normally distributed. Thus, if assumption #2, or, even more so, #3 is not fully met, you may still be able to use ANOVA. There are also several choices of post hoc tests to use depending on whether the assumption of equal variances has been violated.

**Dunnett’s C **and **Games-Howell **are appropriate post hoc tests if the assumption of equal variances is violated.

10.1 Are there differences among the three *father’s education revised* groups on *grades in h.s.*, *visualization test* scores, and *math achievement*?

We will use the **One-Way ANOVA **procedure because we have one independent variable with three levels. We can do several one-way ANOVAs at a time so we will do three ANOVAs in this problem, one for each of the three dependent variables. Note that you could do MANOVA (see Fig. 6.1) instead of three ANOVAs, especially if the dependent variables are correlated and conceptually related, but that is beyond the scope of this book. See our companion book (Leech et al., in press).

To do the three one-way ANOVAs, use the following commands:

**Analyze →****Compare Means →****One-Way ANOVA…**- Move
*grades in h.s., visualization test,*and*math achievement*into the**Dependent List:**box in Fig. 10.1. - Click on
*father’s educ revised*and move it to the**Factor**(independent variable) box. - Click on
**Options**to get Fig. 10.2. - Under
**Statistics,**choose**Descriptive**and**Homogeneity of variance test**.

- Under
**Missing Values**, choose**Exclude cases analysis by analysis**. - Click on
**Continue**then**OK**. Compare your output to Output 10.1.

**Output 10.1: One-Way ANOVA**

ONEWAY grades visual mathach BY faedRevis

/STATISTICS DESCRIPTIVES HOMOGENEITY

/MISSING ANALYSIS.

The between-groups differences for *grades in high school* and *math achievement* are significant *(p* < .05) whereas those for *visualization* are not.

**Interpretation of Output 10.1**

The first table, **Descriptives, **provides familiar descriptive statistics for the three father’s education groups on each of the three dependent variables (*grades in h.s.*, *visualization test,* and *math achievement*) that we requested for these analyses. Remember that, although these three dependent variables appear together in each of the tables, __we have really computed three separate one-way ANOVAs__.

The second table (**Test of Homogeneity of Variances**) provides the __Levene’s test to check the assumption__ that the variances of the three *father’s education* groups are equal for each of the dependent variables. Notice that for *grades in h.s.* (*p* = .220) and *visualization test* (*p* = .153) the Levene’s tests are *not* significant. Thus, the assumption is *not* violated. However, for *math achievement*, *p =* .049; therefore, the Levene’s test is significant and thus the assumption of equal variances is violated. In this latter case, we could use the similar nonparametric test (Kruskal- Wallis). Or, if the overall *F* is significant (as you can see it was in the ANOVA table), you could use a post hoc test designed for situations in which the variances are unequal. We will do the latter in Problem 2 and the former in Problem 3 for *math achievement*.

The **ANOVA **table in Output 10.1 is the key table because it shows whether the overall *F*s for these three ANOVAs were significant. Note that the three *father’s education* groups differ significantly on *grades in h.s.* and *math achievement* but not *visualization test*. When reporting these findings one should write, for example, *F* (2, 70) = 4.09, *p* = .021, for *grades in h.s*. The 2, 70 (circled for *grades in h.s.* in the ANOVA table) are the degrees of freedom (*df*) for the between-groups “effect” and within-groups “error,” respectively. *F tables* also usually include the mean squares, which indicate the amount of variance (sums of squares) for that “effect” divided by the degrees of freedom for that “effect.” You also should report the means (and SDs) so that one can see which groups were high and low. Remember, however, that if you have three or more groups you will not know which specific pairs of means are significantly different unless you do a priori (beforehand) contrasts (see Fig. 10.1) or post hoc tests, as shown in Problem 10.2. We provide an example of appropriate APA-format tables and how to write about these ANOVAs after Problem 10.2.

Source: Morgan George A, Leech Nancy L., Gloeckner Gene W., Barrett Karen C.

(2012), *IBM SPSS for Introductory Statistics: Use and Interpretation*, Routledge; 5th edition; download Datasets and Materials.

29 Mar 2023

28 Mar 2023

15 Sep 2022

27 Mar 2023

19 Sep 2022

14 Sep 2022