Missing covariate data commonly occur in epidemiological and clinical research, and are often dealt with using multiple imputation (MI). Imputation of partially observed covariates is complicated if the substantive model is non-linear (e.g. Cox propo
rtional hazards model), or contains non-linear (e.g. squared) or interaction terms, and standard software implementations of MI may impute covariates from models that are incompatible with such substantive models. We show how imputation by fully conditional specification, a popular approach for performing MI, can be modified so that covariates are imputed from models which are compatible with the substantive model. We investigate through simulation the performance of this proposal, and compare it to existing approaches. Simulation results suggest our proposal gives consistent estimates for a range of common substantive models, including models which contain non-linear covariate effects or interactions, provided data are missing at random and the assumed imputation models are correctly specified and mutually compatible.
We describe a compression method for floating-point astronomical images that gives compression ratios of 6 -- 10 while still preserving the scientifically important information in the image. The pixel values are first preprocessed by quantizing them
into scaled integer intensity levels, which removes some of the uncompressible noise in the image. The integers are then losslessly compressed using the fast and efficient Rice algorithm and stored in a portable FITS format file. Quantizing an image more coarsely gives greater image compression, but it also increases the noise and degrades the precision of the photometric and astrometric measurements in the quantized image. Dithering the pixel values during the quantization process can greatly improve the precision of measurements in the images. This is especially important if the analysis algorithm relies on the mode or the median which would be similarly quantized if the pixel values are not dithered. We perform a series of experiments on both synthetic and real astronomical CCD images to quantitatively demonstrate that the magnitudes and positions of stars in the quantized images can be measured with the predicted amount of precision. In order to encourage wider use of these image compression methods, we have made available a pair of general-purpose image compression programs, called fpack and funpack, which can be used to compress any FITS format image.
We compare a variety of lossless image compression methods on a large sample of astronomical images and show how the compression ratios and speeds of the algorithms are affected by the amount of noise in the images. In the ideal case where the image
pixel values have a random Gaussian distribution, the equivalent number of uncompressible noise bits per pixel is given by Nbits =log2(sigma * sqrt(12)) and the lossless compression ratio is given by R = BITPIX / Nbits + K where BITPIX is the bit length of the pixel values and K is a measure of the efficiency of the compression algorithm. We perform image compression tests on a large sample of integer astronomical CCD images using the GZIP compression program and using a newer FITS tiled-image compression method that currently supports 4 compression algorithms: Rice, Hcompress, PLIO, and GZIP. Overall, the Rice compression algorithm strikes the best balance of compression and computational efficiency; it is 2--3 times faster and produces about 1.4 times greater compression than GZIP. The Rice algorithm produces 75%--90% (depending on the amount of noise in the image) as much compression as an ideal algorithm with K = 0. The image compression and uncompression utility programs used in this study (called fpack and funpack) are publicly available from the HEASARC web site. A simple command-line interface may be used to compress or uncompress any FITS image file.
