Direction Method Example Analysis

 Description: The direction method is used to determine the average direction of advance of a spread of cases. A chain of infection is constructed by first sequencing the cases by time of occurrence. The earliest case would be first, followed by the second case and so on. An arrow is then drawn from the location of the first case to the location of the second case, and this is repeated until all cases are connected. The chain of infection has at least two ends (the first and last cases), and branches when cases occur at exactly the same time. The test statistic is a vector whose direction is the average direction of the arrows comprising the chain of infection, and whose magnitude is the angular variance of these arrows. When the arrows all point in the same direction the angular variance is small, and when they point in many directions the angular variance is large. The method is sensitive to a systematic, directional spread of cases. This pattern occurs when an epidemic sweeps through an area. It also arises from geographically localized exposures, with individuals near the source receiving higher doses and showing symptoms before those further from the source. In this section the term `chain of infection' is used loosely to describe the sequence of occurrence of cases. It does not imply that the sequence of cases was the result of an infectious disease.

 Notation:

The vector v points in the average direction of advance of the spread of cases, and its magnitude (termed the angular concentration) represents the variance in the angles between connected cases. When the magnitude is small the variance in the angles is large, and when the magnitude is large the variance in the angles is small. Angles are taken as counter clockwise degrees from horizontal, with East corresponding to 0 and North to 90. Concentration is in the range of 0 to 1, with 1 indicating an angular variance of 0. A consistent direction of spread of cases will result in an angular concentration near 1. A random spread of cases will result in an angular concentration near 0.

 Time connection matrix: Suppose N cases occur at times (t₁,...t_N). The elements of the time connection matrix are determined by the researcher to reflect the suspected temporal scale of the pattern. The alternative directed time measures are:

Use relative when you wish to include the vectors connecting each case to all of the cases that follow it. This is appropriate when you hypothesize a directional process operating on a longer time span. You are connecting cases that may be several links removed in the chain of infection.

Use adjacent when you wish to connect each case only to it's temporal nearest neighbors. These are the cases that occur just before and just after the case. This is appropriate when you hypothesize directional effects of short duration.

Use following when you wish to connect each case only to the case (or cases) that immediately follow it. This is appropriate when you wish to trace the average direction of the chain of infection.

 Null hypothesis: The directions of the vectors connecting pairs of cases are independent of the times at which the cases occur. In other words, the null spatial model states that the direction of the axis drawn between two cases is independent of when the two cases occurred.

 Significance: The significance of the average direction is evaluated through a randomization procedure which holds the sine and cosine matrices constant and randomly assigns connections between pairs of cases. This is equivalent to holding the locations of the cases fixed while randomizing their times of occurrence. This randomization procedure is repeated to generate a distribution of the angular concentration under the null hypothesis. A P-value is determined by comparing the angular concentration from the original (not randomized) data to this null distribution. Let NGE by the number of times the angular concentration under simulation was greater than or equal to the angular concentration obtained from the original data. Nruns is the number of simulations. The P-value is then

 Input screen: Select `Space-time' from the horizontal methods menu, then `Direction' and `Data' to display the direction data entry screen. You need to enter the name of the point file containing the locations and times of occurrence of the cases (`Wave.pnt') and the number of runs to use when generating the null distribution (249). You also must choose the directed time measure that will be used to generate the time connection matrix. The alternatives are relative, adjacent and following. The data in `Wave.pnt' describe an infectious disease that originated in the south-west corner of a study area. You believe the cases spread northward from this initial focus, and you would like to determine the direction of the spread of cases. Choose `adjacent' for the directed time measure. This will include vectors connecting each case to the cases that immediately preceded or followed it.

 Run screen: Press `F10' to exit the data entry screen, and select `Run'. The 249 simulations will begin, and the null distribution of the angular concentration will be constructed in the window on the left of the run screen. Examine the top window while the simulations run. It shows the input file (`wave.pnt'), the directed time measure (`Adjacent'), the number of cases (30) and the number of elements in the sine and cosine matrices (435). It also displays the number of runs used to evaluate significance (249), the average angle (73.08863), the angular concentration (0.26658) and the significance (0.0040).

The chain of infection is shown in the lower right hand window. The first case occurs in the south of the map and the last case occurs in the north. There appears to be a systematic progression of cases in a Northerly direction. The average angle is shown on the map as a vector with the first case at its tail, and pointing in the average direction of advance. But is this pattern statistically significant?

To answer this question we examine the frequency distribution shown in the left window. This is the distribution of concentration expected under the null hypothesis of no association between the angles from case to case and the times at which the cases occurred. An average angle will always be calculated whether or not there is a systematic direction to the spread of cases, and is meaningful only when the concentration is statistically significant. The observed concentration (0.26658 for these data) is shown as a vertical line to the right of the distribution, and is significant (P=0.0040). We conclude the cases spread along an average angle of 73.00863.

 Notes: A significant angular concentration can arise whenever case location depends on the locations of preceding cases, and does not necessarily mean there was an underlying directional process. For example, a simple random walk may result in statistical significance whenever the walk starts in one place and ends up somewhere else. Although the walk is undirected, the spread of cases has a significant, average direction. You cannot infer a directional process from the direction test, you can, however, determine whether the observed spread of cases tends to be in one direction.