One-Way Analysis of Variance (One-Way ANOVA) The objectives of this lesson are to learn: •
the definition/purpose of One-Way Analysis of Variance
•
...
One-Way Analysis of Variance (One-Way ANOVA) The objectives of this lesson are to learn: •
the definition/purpose of One-Way Analysis of Variance
•
the use of One-Way ANOVA
•
the use of SPSS to complete a One-Way Analysis of Variance
•
the interpretations of results
Definition One-Way ANOVA involves examination of the significant differences between means of three or more groups on one factor or dimension. For example, you might want to know whether five groups of people (below 20, 21-30, 31-40, 41-50, above 50) differ in their living expenses (per month). When to Use One-Way ANOVA Any analysis where: •
There is only one dimension or factor (dependent variable)
•
There are three or more groups of the factor (independent variable)
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One is interested in looking at mean differences across these groups
Assumptions: • Steps for a One-Way ANOVA Using SPSS We will use a step-by-step approach to go through the steps for a one-way ANOVA using SPSS statistical analysis package. Here is the background information of the sample data we are using here. Number of subjects: 60 Independent variable (Factor): Primary disability types including Physical (code = 1), mental (code = 2), and intellectual (code = 3) disabilities.
Dependent variable: Vocational rehabilitation service cost (VRS cost) Data: Table 1 Table 1 Physical D isability
Steps for One-Way ANOVA: Step 1 : A statement of statistical hypothesis H 0 : µ1 = µ 2 = µ3 or means for all groups are equal OR all µ j s are equal There is no significant difference in vocational rehabilitation service cost between disability types H a : at least one mean differs from the rest.
Step 2 : Setting the α level of risk associated with the null hypothesis (or Type I error) The level of Type I error is .05. Step 3: Assumptions testing Assumption 1: Normally distributed data: It is assumed that the data are from a normally distributed population. The rationale behind hypothesis testing relies on having normally distributed populations and so if this assumption is not met then the logic behind hypothesis testing is flawed. Most researchers eyeball their sample data by using a histogram on SPSS. Analyze ⇒ Descriptive Statistics ⇒ Frequencies (Figure 1) Figure 1
** Ignore the frequency table**
Interpretation: The curve demonstrates a bell-shape curve. The data appear to be normally distributed. Assumption 2: Homogeneity of variance: This assumption means that the variances should not change systematically throughout the data. In designs in which you examine several groups of subjects this
assumption means that each of these groups should have the same variance. In other words, the variances of scores in different populations are equal. This means that the unsystematic variation in a population is the same for each treatment condition. Analyze ⇒ Compare Means ⇒ One-Way ANOVA (Figure 2) Figure 2
Levene Test is the test used to examine the homogeneity of variances
Test of Homogeneity of Variances VR Cost Levene Statistic 2.636
df1
df2 2
57
Sig. .080
Interpretation: •
One-way ANOVA assumes that the variances of the groups are all equal.
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This table displays the result of the Levene test for homogeneity of variances.
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The significance value .080 exceeds .05, suggesting that the variances for the three groups of subjects are equal; therefore, the assumption is justified.
Assumption 3: Independence: This assumption is the data from different subjects are independent, which means that the behavior of one subject does not influence the behavior of another. Yes, the data that we use here are from different/independent subjects. Step 4 & 5: Test statistic using SPSS/ interpreting results After the assumptions are tested and justified, we now can begin to test the statistic. Analyze ⇒ Compare Means ⇒ One-Way ANOVA (Figure 3) Figure 3
Tukey and Scheffe are the most commonly used Post Hoc tests which compare means
SPSS Output and Interpretation Oneway Descriptives VR Cost
N Physical Disability Mental Disability Intellectual Disability Total
95% Confidence Interval for Mean Lower Bound Upper Bound 495.7755 618.5245 871.5995 965.4005 179.0677 260.6323 485.9731 644.3603
Minimum 340.00 625.00 99.00 99.00
Maximum 755.00 1050.00 365.00 1050.00
Interpretation: This table displays descriptive statistics for each group and for the entire data set. N indicates the size of each group. The effects of unequal variances will be reduced if the group sizes are approximately equal. Mean shows the average values. One-Way ANOVA compares these sample estimates to determine if the population means differ. The standard deviation indicates the amount of variability of the scores in each group. These values should be similar to each other for ANOVA to be appropriate. Equality can be inspected via the Levene test [refer to Step 2 testing assumption (2)]. The 95% confidence interval for the mean indicates the upper and lower bounds which contain the true value of the population mean 95% of the time. None of the disability group overlap with either of the other two groups. Maximun and minimum values indicate the highest and lowest VRS costs for each type of disabilities.
ANOVA VR Cost
Between Groups Within Groups Total
Sum of Squares 4883046 661822.1 5544868
df 2 57 59
Mean Square 2441523.117 11610.914
F 210.278
Sig. .000
Interpretation:
The results of the analysis are presented in an ANOVA table. In one-way ANOVA, the total variation is partitioned into two components: Between Groups and Within Groups. Between Groups represents variation of the group means around the overall mean. Within Groups represents variation of the individual scores around their respective group means. Sig indicates the significance level of the F-test (F-test is the test used to determine whether the ANOVA is significant). The significance value .000
