One-Way Independent Measures ANOVA

SPSS for Windows

A researcher would like to evaluate the effects of four teaching methods. Twenty-eight junior high students were randomly selected. They were then randomly assigned to one of  four teaching methods: A, B, C, and D. Below are their scores on the final examination. Different subjects are used for all conditions of the experiment.

Data Set

Independent variable and Dependent variable

• The independent variable is ______. It has ______ levels.

• The dependent variable is ______.

Appropriate Tests of Significance

Can you use the t test to compare differences among four sample groups?

No. The t test is limited to the comparison of two groups at a time. The F-test is the appropriate statistical method to be called for.

Assumptions

Analysis-of-variance procedures require the following assumptions:

• Each of the groups is an independent random sample from a normal population.

• In the population, the variances of the groups are equal.

Check the assumption of equal variances. 

To test the null hypothesis that the four groups come from populations with the same variance, you can use the Levene test.

Hypotheses

Are there significant differences among the four teaching methods?

          a. State the null hypothesis

      All groups have the same means in the populations.

          b. State the alternative hypothesis.

      The population means are not all equal

Set a significance level

Use a .05 significant level.

Data Input

The Graphs Menu

Produce a boxplot. A boxplot plots the 25th percentile, the median (the 50th percentile), and the 75th percentile, and values that far from the rest.

What to look for

a. The horizontal line inside the box identifies the group median. Examine the medians for the four methods. Note that Method 4 had the lowest scores. Scores on the other three groups were about the same.

b. If the median is not in the center of the box, the distribution may be skewed.

c. The larger the box, the greater the spread of the observations. Examine the vertical length of the box ( a measure of the spread). Method 2 had the largest variability. There were no outliers or extreme values.   

The Analyze Menu

1. To obtain a one-way analysis of variance, from the menus choose:

Analyze
Compare Means
One-Way ANOVA

Select the dependent variable (final) and the factor variable (method).

2. Check the assumption of equal variances

To test the null hypothesis that the four groups come from populations with the same variance, you can use the Levene test.  

To obtain descriptive statistics, the Levene statistic, and means plot, click on Options.

Select Descriptive, Homogeneity-of-variance, and Means plot. Click Continue and OK.

 

SPSS Output

Check the assumption of equal variances

Do four groups come from populations with the same variance?

To test the null hypothesis that groups come from populations with the same variances, you can use the Levene test.

The observed significance level is larger than .05. The null hypothesis is not rejected. The assumption of equal variances was met. 

The F Test

For this example, F(3,24) = 6.663. The observed significance level was .002. There were significant differences among the four teaching methods, p < .05. The differences among groups represented systematic effects. About 45% of the variation in the scores was accounted for by the teaching methods.

The sums of squares are additive. SSb + SSw = SSt 


 

How to Compute? 

Variance has been traditionally partitioned using squared sums of scores and sums of squared scores. This approach was advantageous at a time when the sole computational device for statistical analyses was a mechanical calculator capable only of elementary algebraic operations. By giving priority to computational rather than to conceptual simplicity, this approach tends to obscure the issues involved in the analysis of variance.
 

 

One-way ANOVA

• HyperStat Online Textbook by David Lane

http://www.davidmlane.com/hyperstat/intro_ANOVA.html

• Analysis of Variance by Jason Newsom

 

Visualize four group means

Means Plot

The Graphs Menu: Produce Error Bars

To visualize the 95% of confidence interval for the mean, choose Graphs \ Error Bar. Click the icon next to Simple. In the Data in Chart Are area, select the default: Summaries for groups of cases. Click the Define button. Move "final" to the Variable textbox. Move "method" to the Category Axis. In the Bars Represent area, choose the default: Confidence interval for mean. Click OK

Note that method 4 had the lowest scores and its confidence interval overlapped very little with the other three confidence intervals.