A study on covariance matrix and its applications in microarray data analysis

Abstract

Any physical quantity that is measured in an experiment is subject to some error and newlineuncertainty. In a cDNA microarray experiment the measurement result is the expression level of each gene which is measured in terms of intensity ratio. The true intensity ratio at a spot is unknown and hence we estimate it with the associated uncertainty. The uncertainty associated with intensity ratio at a spot depends on uncertainty in measuring the average intensities of red and green foreground as well as background pixels in that spot. The objectives of this thesis study are (a) To study uncertainty associated with background corrected intensity ratio at each spot in microarray experiment (b) To study the variance covariance matrix of background corrected intensity ratio at each spot in microarray experiment and use it for estimating propagated uncertainty (c) To see how correlation between pixels within a spot affects uncertainty, and (d) To find how the estimated uncertainty at each spot affects statistical inference. We use theory of error propagation for estimating the uncertainty associated with background corrected intensity ratio at a spot as well as to estimate the variance covariance matrix of intensity ratios at all spots in a newlinemicroarray chip. To explore how correlation affects the statistical inferences we estimate uncertainty associated with mean intensity ratio of each spot under three different situations(a) assuming positive correlation (b) negative correlation and, (c ) zero correlation between intensities of red and green pixels in a spot. The inverse of these estimated uncertainties were used as weights in downstream analysis and obtained the weighted mean and weighted standard deviation We used a dataset from the GEO database for verifying our results. We obtained 95% confidence interval for the mean background corrected intensity ratio of each spot and also the p values using the weighted mean and weighted standard deviations obtained under the three different situations.