Detection of 'Candidatus Liberibacter asiaticus' represents one of the most difficult, yet critical, steps of controlling Huanglongbing disease. Efficient detection relies on understanding the underlying distribution of bacteria within trees. To that end, we studied the distribution of 'Ca. L. asiaticus' in leaves of 'Rio Red' grapefruit trees and in roots of 'Valencia' sweet orange trees grafted onto sour orange rootstock. We performed two sets of leaf collection on grapefruit trees; the first a selective sampling targeting symptomatic leaves and their neighbors and the second a systematic collection disregarding symptomology. From uprooted orange trees, we exhaustively sampled fibrous roots. In this study, the presence of 'Ca. L. asiaticus' was detected in leaves using real-time polymerase chain reaction (PCR) targeting the 16S ribosomal gene and in roots using the rpIJ/rpIL ribosomal protein genes and was confirmed with conventional PCR and sequencing of the rpIJ/rpIL gene in both tissues. Among randomly collected leaves, 'Ca. L. asiaticus' was distributed in a patchy fashion. Detection of 'Ca. L. asiaticus' varied with leaf symptomology with symptomatic leaves showing the highest frequency (74%) followed by their neighboring asymptomatic leaves (30%), while randomly distributed asymptomatic leaves had the lowest frequency (20%). Among symptomatic leaves, we found statistically significant differences in mean number of bacterial cells with respect to both increasing distance of the leaf from the trunk and cardinal direction. The titer of 'Ca. L. asiaticus' cells was significantly greater on the north side of trees than on the south and west sides. Moreover, these directions showed different spatial distributions of 'Ca. L. asiaticus' with higher titers near the trunk on the south and west sides as opposed to further from the trunk on the north side. Similarly, we found spatial variation in 'Ca. L. asiaticus' distribution among root samples. 'Ca. L. asiaticus' was detected more frequently and bacterial abundances were higher among horizontally growing roots just under the soil surface (96%) than among deeper vertically growing roots (78%). Bacterial abundance declined slightly with distance from the trunk. These results point to paths of research that will likely prove useful to combating this devastating disease.
We consider random sample splitting for estimation and inference in high dimensional generalized linear models, where we first apply the lasso to select a submodel using one subsample and then apply the debiased lasso to fit the selected model using the remaining subsample. We show that, no matter including a prespecified subset of regression coefficients or not, the debiased lasso estimation of the selected submodel after a single splitting follows a normal distribution asymptotically. Furthermore, for a set of prespecified regression coefficients, we show that a multiple splitting procedure based on the debiased lasso can address the loss of efficiency associated with sample splitting and produce asymptotically normal estimates under mild conditions. Our simulation results indicate that using the debiased lasso instead of the standard maximum likelihood estimator in the estimation stage can vastly reduce the bias and variance of the resulting estimates. We illustrate the proposed multiple splitting debiased lasso method with an analysis of the smoking data of the Mid-South Tobacco Case-Control Study.
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