## One-way ANOVA with Trend Analysis

(WINKS SDA Professional Edition Example)

**Trend Analysis Example:** A wholesale nursery is experimenting with a flower supplement that is designed to increase the number of flowers produced on a plant. To determine whether the supplement works and what strength to use, the nursery randomly selects 12 plants for 5 different commercially available strengths (1 to 5) of the supplement. After two weeks, the number of opened flowers per plant is counted. The following figure shows a portion of the data set for this analysis (TREND.SDA).

The data set contains one subject (plant) per row and contains a “grouping” variable called Strength to indicate the concentration of the supplement that was used for that plant For this data set, there are 60 rows, one for each plant.

What makes this analysis different from a typical one-way ANOVA is that the groups have an order (i.e., the strength 1 to 5). For this case you may want to perform a trend analysis. A trend analysis tests the hypotheses that the means of the ordered groups change in a linear or higher order (e.g., quadratic or cubic) fashion. To perform the example on the TREND.SAV dataset follow these instructions:

**Step 1. **Open the data set **TREND.SDA **and select Analyze, Advanced ANOVA, One –Way ANOVA with Trend.

**Step 2.** Select *Strength* for Group and Flowers for Data field.

**Step 3.** **Click Continue and OK**, and the following (abbreviated) output appears. (Output also contains One-Way ANOVA results.)

Trend Analysis Results

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

GROUP 1 2 3 4 5

TOTAL 431. 456. 536. 623. 591.

N 12 12 12 12 12

MSE= 36.34091

ANOVA For Trend

TERM DF MS F Appx p

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

LINEAR 1 1976.41 54.39 <.001

QUAD 1 68.15 1.88 0.174

CUBIC 1 252.3 6.94 0.0109

**Step 4.** Select **Graph, Display Graph** to display a preliminary graph. Select **Edit, Options** and check only the box titled "Display Means". Click OK to display the following plot:

It appears that the number of flowers increases as the strength of the supplement increases (up to a point). A test for a linear, quadratic and cubic trend is shown in the ANOVA table. A quadratic trend is one that has a consistent curving pattern either upward or downward while a cubic trend is characterized by a shift in curvature from upward to downward or vice versa.

In this analysis, there is a significant linear trend (p <0.001.) The quadratic term is not a significant quadratic (single curve) trend (p =0.174). The cubic term (p=0.011) is significant. The results of this analysis suggest that there is a somewhat linear increase in mean flower production as the level of the supplement increases. The cubic term suggests (from examination of the plot) that there is a faster increase as the strength is increased from 2 to 3 and a downturn in flower production when the strength reaches 5.

The program will also handle cases when the Ns are not the same. For example, in the TREND.SDA data set,

**Step 1.** Highlight the first 2 rows and choose Edit, Filter Records within selected range. Notice that the first record numbers have a asterisk (*) next to the numbers indicating that they will not be used in an analysis.

**Step 2. **Select Analyze, Advanced ANOVA, One –Way ANOVA with Trend.

**Step 3.** Select *Strength* for Group and Flowers for Data field.

**Step 4.** **Click Continue and OK**, and the following (abbreviated) output appears. (Output also contains One-Way ANOVA results.)

Notice that the results are slightly different than the first example, reflecting only 10 observations in Group 1.

Trend Analysis Results

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

GROUP 1 2 3 4 5

TOTAL 370. 456. 536. 623. 591.

N 10 12 12 12 12

MSE= 35.24214

ANOVA For Trend

TERM DF MS F Appx p

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

LINEAR 1 1871.19 53.1 <.001

QUAD 1 465.21 13.2 <.001

CUBIC 1 98.62 2.8 0.1002

- End of Tutorial-