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Trend Analysis
Trend analysis is the assessment of linear and curvilinear associations among variables. Linear trends are straight-line associations commonly evaluated in ordinary least squares (OLS) regression. Curvilinear trends include U- or inverted U-shape associations known as quadratic or parabolic trends, as well as sideways S-shape associations known as cubic trends. Other types of trends are available but rarely pursued because justifications for their use are often questionable. After describing some applied examples and the background of trend analysis, this entry discusses how trend analysis is applied in research.
Examples and Background
Applied examples of trend analysis within the research literature are prevalent. For example, Debra Murphy and William Marelich report linear trend declines in depression during an 18-month period for both resilient and nonresilient children whose mothers are HIV positive. Jessica Kahn and colleagues investigated patterns of adolescents’ physical activity, noting a quadratic trend as physical activity peaks at the age of 13 then declines (an inverted U-shape). Adele Hayes and her associates found that the effects of exposure-based cognitive therapy (EBCT) on depression during the course of therapy could best be summarized as a cubic trend, with depression initially declining after treatment, followed by an increase or depression spike, then another decline in depression (a sideways S-shape).
Trend analysis and assessment of curvilinear associations emerged because of the limitations in assuming variable relationships are linear. In the latter two examples noted earlier, assessing a linear trend alone would not have been sufficient to explain the variable associations. To illustrate, say an investigation was performed examining the association between task performance and arousal level. Using the Yerkes-Dodson law as the theoretical framework, performance and arousal should have an inverted U-shape; moderate levels of arousal should lead to optimal performance, whereas low and high levels of arousal should show arrested performance. A bivariate scatter plot of hypothetical task performance and arousal data reveals the expected curvilinear association. However, a bivariate OLS regression between these two variables would not capture this inverted U-shape association because only the linear trend is being assessed, and the multiple R value would be close to zero. The variables are related to each other but in a curvilinear fashion. To reveal this relationship, a quadratic trend should be assessed to summarize the inverted U-shape association between the two variables.
Trend Analysis in Application
Trend analysis is commonly performed in OLS multiple regression and analysis of variance (ANOVA) applications. It is also used in statistical applications designed to assess associations such as structural equation modeling and multilevel modeling. Trend analysis is performed when there is a hypothesized linear and/or curvilinear association between two or more variables, with one of the variables acting as a criterion or dependent variable and the others acting as predictors. In addition, the predictors must be scale measured to allow the shape of the association to have meaning. For example, a predictor might be a continuous score on a measured psychological scale, such as a score on the depression inventory. Or, the predictor might be a discrete measure such as group membership, with each group representing participants receiving different doses of a drug or possibly each group representing additive intensity levels of an independent variable. As long as the discrete variable is scale measured, it might be used for trend analysis.
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