# Hypothesis Testing: ANOVA Tests

Hypothesis Testing uses statistics to choose between hypotheses regarding whether data is statistically significant or occurred by chance alone. One type of hypothesis tests are ANOVA tests, which are tests that examine whether two or more means are statistically significantly different from each other or whether the difference between them simply occurred by chance. ANOVA stands for Analysis of Variance. A One-Way ANOVA compares the means of two or more groups. A Factorial ANOVA compares the means of two or more groups while examining the interaction of and between two independent variables. (However, the ANOVA tests do not specify which groups differ significantly, and since there are more than two groups, in order to determine which groups differ, further statistical analyses and Post Hoc tests must be done and can be added to the ANOVA procedure in SPSS.)

### One-Way ANOVA

A One-Way ANOVA compares the means of two or more groups. A One-Way ANOVA thus requires one categorical variable consisting of two or more groups, serving as the independent variable, and one continuous variable, serving as the dependent variable.
In this example, the variable 'Subjective Class Identification, class' will be serving as the categorical variable with 4 groups, and the variable 'Number of College-Level Sci Courses R Have Taken, colscinm' will be serving as the continuous variable. The One-Way ANOVA is specifically looking at whether respondents of different subjective class identifications differ significantly in the mean number of college-level science classes taken.

To generate a One-Way ANOVA, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'One-Way ANOVA...' in the side menu.

In the One-Way ANOVA dialog box that pops up, select the dependent variable of interest (Number of College-Level Sci Courses R Have Taken, colscinm) from the list of variables and bring it over to the 'Dependent List:' field. Then, select the nominal grouping variable of interest (Subjective Class Identification, class) from the list of variables and bring it over to the 'Factor' field. To include Post Hoc tests in the ANOVA output, click 'Post Hoc...'

In the One-Way ANOVA: Post Hoc Multiple Comparisons dialog box that pops up, select the desired Post Hoc test. In this example, we will be using Least Significant Difference (LSD) tests. Then, click 'Continue.'

Back in the One-Way ANOVA dialog box, click 'Options' if you would like to add any other statistics to the ANOVA output. Here, we selected to include a Descriptives table and a Means plot in the output. Click 'Continue.' Then, back in the One-Way ANOVA dialog box, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of four parts. The first table, Descriptives, contains statistical information about the dependent variable, Number of College-Level Sci Courses R Have Taken, split by the independent variable groupings, the respondents' Subjective Class Identification. For each group of respondents, the N, the Mean, the Standard Deviation, the Standard Error of the Mean, a 95% confidence Interval for the Mean, the Minimum value, and the Maximum value are displayed. The second table, ANOVA, contains information about the ANOVA test comparing means both between and within groups and includes the Sum of Squares (a measure of variance), df (degrees of freedom), Mean Square, the F value, and the Sig. value (when the Sig. value is .05 or less, the probability that the difference between the groups was due to chance is 5% or less). In this case, some of the means are significantly different from each other. However, the results of the ANOVA alone do not indicate which groups differ significantly. Thus, the Multiple Comparisons Output which displays the results of the Post Hoc LSD test is necessary.

The next table, Multiple Comparisons Output, displays the results of the LSD test. The LSD test compares each group (class category) to all other groups (class categories). Thus, please note that this table displays some comparisons more than once, since, in every row, each group is compared to all other groups. Each comparison is denoted by a differnet color, and lines of the same color represent repeated comparisons. For each comparison, the table displays the Mean Difference (the difference between the groups' mean number of college-level science classes taken), the Standard Error, the Sig. value (when the Sig. value is .05 or less, the probability that the difference between the groups was due to chance is 5% or less), and a 95% Confidence Interval for the Mean Difference. In this example, the Upper Class differs significantly from the Lower, Working, and Middle Classes, and the Middle Class differs significantly from the Working Class in the number of college-level science courses taken.

The last section of the output, the Means Plot, is a graphical display of how the mean number of college-level science courses the respondents have taken depends on subjective class identification.

### Factorial ANOVA

A Factorial ANOVA compares the means of two or more groups while examining the interaction of and between two independent variables. A Factorial ANOVA thus requires one continuous variable to serve as the dependent variable and more than one categorical variable (each consisting of two or more groups) to serve as the independent variables.
In this example, the variable 'Subjective Class Identification, class' will be serving as the first categorical variable with 4 groups, and the variable 'Not Married, absingle' will be serving as the second categorical variable with 2 groups. The variable 'Hours Per Day Watching TV, tvhours' will serve as the continuous variable. The Factorial ANOVA is specifically looking at whether respondents of different subjective class identifications and of different marital statuses differ significantly in the mean number hours spent watching tv per day.

To generate a Factorial ANOVA, click 'Analyze' in the top toolbar of the Data Editor window. Click 'General Linear Model' in the dropdown menu, and click 'Univariate...' in the side menu.

In the Univariate dialog box that pops up, select the dependent variable of interest (Hours Per Day Watching TV, tvhours) from the list of variables and bring it over to the 'Dependent Variable:' field. Then, select the nominal independent grouping variables of interest (Subjective Class Identification, class, and Not Married, absingle) from the list of variables and bring them over to the 'Fixed Factor(s):' field. To include Post Hoc tests in the ANOVA output, click 'Post Hoc...'

In the Univariate: Post Hoc Multiple Comparisons for Observed Means dialog box that pops up, select the desired variable for which you wish to run a Post Hoc Test (any variable with more than two groups, which, in this case, is the 'class' variable). In this example, we will be using Least Significant Difference (LSD) tests. Then, click 'Continue.'

Back in the Univariate dialog box, click 'Options' if you would like to add any other statistics to the ANOVA output. Here, we selected to include a Descriptives table in the output. Click 'Continue.' Then, back in the One-Way ANOVA dialog box, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of four parts. The first table, Between-Subjects Factors, lists the independent variables Value Labels and N, the group size, for each group. The second table, Descriptive Statistics, contains statistical information about the dependent variable, Hours Per Day Watching TV, displayed corresponding with each of the indepedent variable groups: the respondents' Subjective Class Identification and marital status. For each group of respondents, the Mean, the Standard Deviation, and the N are displayed.

The third table, Tests of Between-Subjects Effects, contains information about the Factorial ANOVA test and includes the Sum of Squares (a measure of variance), df (degrees of freedom), Mean Square, the F value, and the Sig. (when the Sig. value is .05 or less, the probability that the difference between the means was due to chance is 5% or less). The first two rows of the table, Corrected Model and Intercept are advanced statistics and will not be addressed in this tutorial. The next rows correspond with the independent variables, and their significance levels are measured separately, demonstrating the influence of each independent variable individually on the number of hours per day spent watching TV. There is also a row showing the interaction between the two independent variables, demonstrating whether the two independent variables interacted to significantly impact the results. In this case, class influenced the number of hours respondents spent watching TV but marital status did not. Additionally, there was a significant interaction between class and marital status.

The next table, Multiple Comparisons, displays the results of the LSD test. The results of the ANOVA alone did not indicate which class groups differ significantly. Thus, the Multiple Comparisons table, which displays the results of the Post Hoc LSD test, is necessary. The LSD test compares each group (class category) to all other groups (class categories). Thus, please note that this table displays some comparisons more than once, since, in every row, each group is compared to all other groups. Each comparison is denoted by a differnet color, and lines of the same color represent repeated comparisons. For each comparison, the table displays the Mean Difference (the difference between the groups' mean number of hours spent watching TV), the Standard Error, the Sig. value (when the Sig. value is .05 or less, the probability that the difference between the means was due to chance is 5% or less), and a 95% Confidence Interval for the Mean Difference. In this example, the Lower Class differs significantly from the Working, Middle, and Upper Classes in the amount of time spent watching TV per day.