In a meta-analysis, graphical displays can be used to check statistical assumptions for numerical procedures and they can be used to discover important patterns in the data. The authors propose the normal quantile plot as a preferred alternative to the funnel plot for such purposes. The normal quantile plot, like the funnel plot, can be used to investigate whether all studies come from a single population and to search for publication bias. However, the normal quantile plot is easier to interpret than the funnel plot, especially when it includes 95% confidence bands. In addition, the normal quantile plot can be used to check the normality assumption for numerical procedures. The funnel plot cannot be used for this latter purpose. Most people have heard the phrase, "A picture is worth a thousand words." This phrase seems especially applicable to producers and consumers of metaanalytic reviews. In a meta-analysis, graphical displays (figures, plots) can be used to enhance numerical analyses in at least two ways. First, graphical displays can be used to discover patterns and relations among variables in a meta-analysis. Second, graphical displays can be used to check statistical assumptions on which numerical analyses are based. One of the most popular graphical displays for exploring meta-analytic data sets is the funnel plot (Light & Pillemer, 1984). In this article, we begin by describing the funnel plot and its uses. We then propose the normal quantile plot as a preferred alternative to the funnel plot. The Funnel Plot A funnel plot is a two-dimensional graph with sample size on one axis and effect-size estimate on the other axis. Most statisticians would recognize a funnel plot as a special type of scatter plot. The funnel plot capitalizes on the well-known statistical principle that sampling error decreases as sample size increases. In
Using archival data from Minneapolis recorded in 3-hr time intervals, E. G. Cohn and J. Rotton concluded that there is an inverted U-shaped relationship between temperature and assault, with the maximum assault rate occurring at 74.9 degrees F. They depicted this relationship by plotting temperature against assault. This plot, however, fails to take into account time of day. Time of day was strongly related to both temperature and assault, but in opposite directions. Between 9:00 p.m. and 2:59 a.m. of the next day, when most assaults occurred, there was a positive linear relationship between temperature and assault. The Minneapolis data actually provide stronger support of a positive linear (or monotonic) relationship between temperature and assault than of an inverted U-shaped relationship.
The application of a nonlinear time series model to the prediction of traffic parameters on a freeway network is investigated. The nonlinear time series approach is a statistical technique that has strong potential for on-line implementation. A new approach for predicting corridor travel times is developed and tested with travel-time data. The travel-time data are derived from observed speed data, which are collected from an 18-km (11.2-mi) freeway section in Orlando, Florida. The westbound Interstate-4 morning peak period (6:00 to 10:00 a.m.) for 20 incident-free days is tested with the goal of predicting recurrent congestion. The problem is addressed from the perspectives of single-variable and multiple-variable prediction of corridor travel times. In single-variable prediction, speed time-series data are used to forecast travel times along the freeway corridor. A calibrated single-variable prediction model is developed through the application of decay factors to smooth out the input data and the establishment of a threshold on the minimum speed prediction permitted. Multivariable prediction schemes are developed using speed, occupancy, and volume data provided by inductive loop detectors on the study section. The prediction performance of the calibrated single-variable model is shown to be superior to the multivariable prediction schemes. This new approach produces reasonable errors for short-term (5-min) travel-time predictions. The developed model can be implemented on-line with minimal effort.
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