trend analysis in analysis of variance

may be used in analysis of variance to determine the shape of the relationship between the dependent variable and an independent variable which is quantitative in that it represents increasing or decreasing levels of that variable. Examples of such a quantitative independent variable include increasing quantities of a drug such as nicotine or alcohol, increasing levels of a state such as sleep deprivation or increasing intensity of a variable such as noise. If the F ratio of the analysis of variance is statistically significant, we may use trend analysis to find out if the relationship between the dependent and the independent variable is a linear or non-linear one and, if it is non-linear, what kind of nonlinear relationship it is.

The shape of the relationship between two variables can be described in terms of a polynomial equation. A first-degree or linear polynomial represents a linear or straight line relationship between two variables where the values of one variable either increase or decrease as the values of the other variable increase. For example, performance may decrease as sleep deprivation increases. A linear trend can be defined by the following equation, where y represents the dependent variable, b the slope of the line, x the independent variable and a a constant.

A minimum of two groups or levels is necessary to define a linear trend.

A second-order or quadratic relationship represents a curvilinear relationship in which the values of one variable increase (or decrease) and then decrease (or increase) as the values of the other variable increase. For instance, performance may increase as background noise increases and then decrease as it becomes too loud. A quadratic trend can be defined by the following equation:

A minimum of three groups or levels is necessary to define a quadratic trend.

A third-order or cubic relationship represents a curvilinear relationship in which the values of one variable first increase (or decrease), then decrease (or increase) and then increase again (or decrease). A cubic trend can be defined by the following equation:

A minimum of four groups or levels is necessary to define a cubic trend.

[Page 170]A fourth-order or quartic relationship represents a curvilinear relationship in which the values of one variable first increase (or decrease), then decrease (or increase), then increase again (or decrease) and then finally decrease again (or increase). A quartic trend can be defined by the following equation:

A minimum of five groups or levels are necessary to define a quartic trend.

These equations can be represented by orthogonal polynomial coefficients which are a special case of contrasts. Where the number of cases in each group is the same and where the values of the independent variable are equally spaced (such as 4, 8, 12 and 16 hours of sleep deprivation), the value of these coefficients can be obtained from a table which is available in some statistics textbooks such as the one listed below. The coefficients for three to five groups are presented in Table T.3. The procedure for calculating orthogonal coefficients for unequal intervals and unequal group sizes is described in the reference listed below. The number of orthogonal polynomial contrasts is always one less the number of groups. So, if there are three groups, there are two orthogonal polynomial contrasts which represent a linear and a quadratic trend.

...