Analysis of Variance

The statistical technique used to test the null hypothesis that several means are equal is called analysis of variance. It is called that because it examines the variability in the sample and, based on the variability, it determines whether there is reason to believe the population means are not equal. In analysis of variance, the observed variability in the sample is divided, or partitioned, into two parts: the variability of observations within a group (around the group mean), and variability between the group means. If the two estimates are substantially different, you can reject the null hypothesis that the population means are equal. The statistical test for null hypothesis that all of the groups have the same mean in the population is based on computing the ratio of the two estimates, called an F statistic. The observed significance level is obtained by comparing the calculated F value to the F distribution (the distribution of the F statistic when the null hypothesis is true).

A significant F value only tells you that the means are probably not all equal. It does not tell you which pairs of groups appear to have different means. To pinpoint exactly where the differences are, multiple comparisons may be performed.

In the following exercise you will perform an ANOVA with canx as the dependent variable and 'exp' as the factor variable. To perform a pairwise mean comparisons to identify which means differ from others a Tukey HSD test has been employed.

• From the Statistics menu select Analysis of Variance (ANOVA)/Estimate Model(Selecting General Linear Model/Estimate Model from Statistics menu will result in the same procedure.)

• Highlight CANX as Dependent(s): and click [Add-->]
• Highlight EXP as Factor(s): and click [Add-->]
• Select Post hoc Tests and choose Tukey from drop-down list. (Selecting General Linear Model/Pairwise Comparisons from Statistics menu will result in the same procedure plus Dunnett's test. But this option becomes active only after you run your ANOVA.)
• Click OK

The output as shown below will be displayed on the Main window. In the Graph window, Quick Graph includes a plot of residuals from each estimated cell mean versus the estimated cell mean.

Effects coding used for categorical variables in model.
 
Categorical values encountered during processing are:
EXP (3 levels)
          1,        2,        3
 
Dep Var: CANX   N: 40   Multiple R: 0.406   Squared multiple R: 0.165
 
 
                             Analysis of Variance
 
Source             Sum-of-Squares   df  Mean-Square     F-ratio       P
 
EXP                       71.466     2       35.733       3.659       0.035
 
Error                    361.309    37        9.765
 

------------------------------------------------------------------------------

------------------------------------------------------------------------------
 
*** WARNING ***
Case           39 is an outlier        (Studentized Residual =        3.385)
 
Durbin-Watson D Statistic     1.769
First Order Autocorrelation   0.043
COL/
ROW EXP
  1  1
  2  2
  3  3
Using least squares means.
Post Hoc test of CANX
------------------------------------------------------------------------------
 
Using model MSE of 9.765 with 37 df.
Matrix of pairwise mean differences:
 
                         1           2           3
              1          0.000
              2          2.800       0.000
              3          2.709      -0.091       0.000
 
Tukey HSD Multiple Comparisons.
Matrix of pairwise comparison probabilities:
 
                         1           2           3
              1          1.000
              2          0.054       1.000
              3          0.087       0.997       1.000
------------------------------------------------------------------------------

The output shows that there is a significant difference among groups with different levels of computer experience at least at .05 probability level.

The output for pairwise comparisons include a table of mean differences and another table of probabilities. To determine significant differences, examine the pairs and their probability level. From the output it is evident that there is a marginally significant difference between group 1 (exp=1) and group 2 (exp=2). None of the other combinations produced a significant difference at least at the 0.05 level.

Using SYSTAT's Graph Menu

SYSTAT provides a wide selection of graphics for every stage of your project: exploration, research, and presentation. The graphics capabilities of SYSTAT include:

• histograms with curve fitting
• bar graphs, box plots, stem-and-leaf diagrams, pie charts
• 3-D rotation, maps with geographic projections
• mathematical function plots, log and power scales
• confidence intervals, ellipses, and centroids
• contour plots, control charts
• case coding of labels and symbols
• linear, quadratic, step, spline, polynomial, LOWESS, exponential, and DWLS smoothing in two and three dimensions
• rectangular, spherical, polar, cylindrical, and triangular coordinates, perspective depth and projections.

Plotting Two Variables with SYSTAT

Looking at a plot is one of the best ways to examine relationships and patterns. For example, a scatterplot allows the visual representation of two separate distributions on a single diagram.

In the following task you will plot the variables CANX (dependent variable) by COMPSCOR (independent variable). We will also fit the data points on the scatterplot based on the least-squares principle.

• From Graph menu (Main window) select Plots and then choose Scatterplot

• Highlight COMPSCOR as X-variable: and click Add -->
• Highlight CANX as Y-variable(s): and click Add -->
• Choose Linear as smoother method from Options/Smoother and click Continue
• Click OK

The plot, as shown below, will be displayed on the screen

To print the graphics output, select File/Print..., and respond to the queries appear in subsequent dialog boxes. You may save your graphics streams to a file using File/Save as... To remove the Graphics window select File/Close Window.

Detailed discussion of all the graphics capabilities of SYSTAT is not possible through this document. You may refer to SYSTAT's Graphics document for learning more about the graphics capability of SYSTAT.