Trend Analysis

Trend analysis is a statistical procedure performed to evaluate hypothesized linear and nonlinear relationships between two quantitative variables. Typically, it is implemented either as an analysis of variance (ANOVA) for quantitative variables or as a regression analysis. It is commonly used in situations when data have been collected over time or at different levels of a variable; especially when a single independent variable, or factor, has been manipulated to observe its effects on a dependent variable, or response variable (such as in experimental studies). In particular, the means of a dependent variable are observed across conditions, levels, or points of the manipulated independent variable to statistically determine the form, shape, or trend of such relationship.

Examples of a quantitative variable that a survey researcher may be interested in manipulating to measure its effects on another quantitative variable are amount of incentives provided in a survey (e.g., 0, 2, 4, and 6 dollars), interviewer training time (e.g., 0, 1, 2, 3 hours), number of interviews assigned to interviewers, time allowed to perform a memory task, distance of exit polling interviewers from the voting booth, number of callbacks in a telephone survey, time elapsed between the first occasion on which participants were sent questionnaires and follow-up surveys, among others.

Using trend analysis, the researcher evaluates statistically whether the relationship between the dependent and independent variable is linear, quadratic, cubic, or other high-order function. The number of bends to be tested in a polynomial function is determined by the number of conditions, or levels, of the independent variable. For instance, if there are two conditions in an experiment, only a linear trend can be tested; if there are three conditions, only a quadratic trend is testable; if there are four conditions, a cubic trend is possible, and so on.

To implement this analysis, it is common practice to have an ordered and equally spaced metric for levels of the independent variable and an equal number of subjects allocated exclusively to each condition of such variable. In cases where the number of subjects allocated into each condition varies, weighting procedures may be used; however, there may be limitations associated with weighted trend analysis that the researcher should further identify. Additionally, interpretation of trend analysis results requires attention to surrounding aspects such as statistical power and effect size.

If implemented as regression analysis, the independent variable (X) is entered into the equation and followed sequentially by increasing powers of the same variable. Consequently, regression equations for trend analysis may be specified as Y = a1X + b (linear), Y = a1X + a2X2 + b (quadratic), Y = a1X + a2X2 + a3X3 + b (cubic), and so forth, where Y represents the dependent variable, b the intercept, and a1 …ak regression coefficients. The highest-order regression coefficient in the equation being statistically significant settles the shape of the relationship, p-values help determine whether the observed trend is due to a systematic influence of the manipulated variable or by chance alone.

Under ANOVA, trend analysis is conducted using contrast coefficients, also known as polynomial[Page 907]contrasts. These coefficients are numbers usually taken from either available tables or user-designed tables. They help represent a hypothesized trend. For example, contrast coefficients for testing a linear trend would be −1.5, −0.5, 0.5, and 1.5, if the independent variable had four levels. For a quadratic trend they would be 1, −1, −1, 1; for a cubic, −1, 3, −3, 1. Contrast coefficients can be plotted over the v-axis across levels of the independent variable to visualize the hypothesized trend, for example, a straight line, a u-shape line, an s-shape line, or higher-order trends. Departures of the actual trend (plot of actual means) from the hypothesized trend (plot of contrast coefficients) are analyzed in terms of p-values to determine the type of relationship. Currently, commercial statistical packages can perform trend analysis, but they require user-entered contrast coefficients.

...