You cannot use standard point pattern statistics because the assumed spatial process is being constrained to a linear process. The assumption of a Poisson distributed Complete Spatial Randomness (CSR) does not hold. The entire problem becomes one-dimensional and expectations based on area need to be constrained to the linear feature thus, the single dimension of the process.
Through the years there has been some work published on point pattern analysis and kernel density estimation on networks and there is even a quasi-commercial ArcGIS toolbox, SANET available. Otherwise you will have to code solutions based on the literature.
The R library spatstat has some limited functions that support network based point pattern statistics, namely "linnet" (defines a network), "lpp" (projects point pattern to a network), "linearK (K-hat statistic constrained to one dimension)" and "envelope.lpp" (for simulation envelopes).
Here are some references to start with.
Borruso, G., (2008) Network Density Estimation: A GIS Approach for Analysing Point Patterns in a Network Space. Transactions in GIS, 2(3):377–402
Okabe, A., Satoh, T. and Sugihara, K. (2009) A kernel density estimation method for networks, its computational method and a GIS-based tool. International Journal of Geographical Information Science, 23(1):7-32.
Okabe, A., Yomono, H. and Kitamura, M. (1995) Statistical analysis of the distribution of points on a network, Geographical Analysis, 27(2):152-175
Spooner P.G., Lunt I.D., Okabe A. and Shiode S. (2004) Spatial analysis of roadside Acacia populations on a road network using the network K-function, Landscape Ecology, 19(5):491-499.