In the analysis of spatial point patterns, an important role is played by statistical tests based on simulation envelopes, such as the envelope of simulations of Ripley's K function. Recent ecological literature has correctly pointed out a common error in the interpretation of simulation envelopes. However, this has led to a widespread belief that the tests themselves are invalid. On the contrary, envelope‐based statistical tests are correct statistical procedures, under appropriate conditions. In this paper, we explain the principles of Monte Carlo tests and their correct interpretation, canvas the benefits of graphical procedures, measure the statistical performance of several popular tests, and make practical recommendations. There are several caveats including the under‐recognized problem that Monte Carlo tests of goodness of fit are probably conservative if the model parameters have to be estimated from data. Finally, we discuss whether graphs of simulation envelopes can be used to infer the scale of spatial interaction.

In Geographical Information Systems, spatial point pattern data are often analysed by dividing space into pixels, recording the presence or absence of points in each pixel, and fitting a logistic regression. We study weaknesses of this approach, propose improvements, and demonstrate an application to prospective geology in Western Australia. Models based on different pixel grids are incompatible (a 'change-of-support' problem) unless the pixels are very small. On a fine pixel grid, a spatial logistic 1151 A. Baddeley et al./Spatial logistic regression regression is approximately a Poisson point process with loglinear intensity; we give explicit distributional bounds. For a loglinear Poisson process, the optimal parameter estimator from pixel data is not spatial logistic regression, but complementary log-log regression with an offset depending on pixel area. If the pixel raster is randomly subsampled, logistic regression is conditionally optimal. Bias and efficiency depend strongly on the spatial regularity of the covariates. For discontinuous covariates, we propose a new algorithmic strategy in which pixels are subdivided, and demonstrate its efficiency.

We develop statistical methods for analysing a pattern of points on a region of the sphere, including intensity modelling and estimation, summary functions such as the K function, point process models, and model-fitting techniques. The methods are demonstrated by analysing a dataset giving the sky positions of galaxies.

SUMMARY From certain points of view, the range of probability models currently available for describing the joint behaviour of two point processes is rather limited. In this paper we explore the structure of some further models and apply our results to the statistical analysis of bivariate spatial point patterns.

Summary This paper reviews some interesting but scattered results that are known about correlation in bivariate Poisson distributions and processes and presents some new results. A particular concern in both contexts is with results and examples relating to negative correlation.