Part V: Analysis of Variance (ANOVA)
Trend Tests
| Overview and Procedure | A Trend Test Example |
 
 
The trend analysis procedures are designed to help assess whether there is a functional relationship between the IV and the DV. The functional relationship describes the general trend or nature of relationship between the DV and IV. Using this procedure, we can precisely describe the trend in the data in terms of its component parts. We can decompose the relationship between the IV and DV into as many trend components as you have degrees of freedom for the IV. To perform a Trend Analysis, the IV must be quantitative and there should be equal spacing between the levels. Computationally, it is really no different than computing a SScontrast value. The only difference is that your weight coefficients are predetermined by the table of orthogonal polynomials.

The procedure for these tests is as follows:

  • Look up coefficients in a table for the given trend (based on the # of levels for the IV).
  • Compute SScontrast for a given trend using the SScontrast formula for either means or totals. The tabled coefficient values, together with their signs, are applied to the appropriate totals or means of the IV.
  • Compute Ftrend by dividing the obtained MScontrast by the MSerror from the omnibus F. Determine whether a significant trend exists.
  • In order to control for compounding error rate, the following procedure could be adopted in sequentially testing trends:

a) Assume we have an IV with 5 levels (4 df's)

b) Test linear trend (1 df) and test residual (3df). Proceed if and only if residual is significant.

c) Test linear (1df), quadratic (1 df) and residual (2 df). Proceed if and only if the residual is significant.

d) Continue in the above manner with the next highest trend.

Trend components include, in increasing complexity, the linear, quadratic, cubic, quartic and quintic components. More trends exist beyond this, but they are much more complex to interpret.
 
 
Given the following data, we will analyze whether a trend exists.

Picture (580x207, 4.1Kb)

After looking at the plot of the means it appears that a quadratic trend exists in the data. To test the quadratic trend, we use the following weights:

Picture (297x97, 2Kb)

And the planned comparisons formula:

Picture (518x131, 4.4Kb)

The following table contains the calculated sums of squares for all of the possible trend tests for the above data set.

Source SS df MS F
(Between) (590.34) (3)    
Linear 223.50 1 223.50 11.45**
Quadratic 365.51 1 365.51 18.72**
Cubic 1.32 1 1.32 <1.00
         
Error (Within) 1483.55 76 19.52