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I have lots of repeated measures data for a dataset of 95 subjects. The data concerns medical-imaging measures of morphology.

medical-imaging is a tricky kind of data - the same body part imaged on two different days could look quite different, due random differences in the positioning of the subject.

So, I have taken extra measurements from each image which show (and quantify) differences in body position between images. This means that in addition to my morphology measurements, I now have 2 explicit 'measurement error' measurements (I'll call them MEMs) .

I can assume that all my morphology measurements are influenced, by these known MEMs. But there will also be other kinds of measurement error (probably spatial) that I have not (perhaps cannot?) directly quantify.

I suspect some morphology measurements will be more influenced by MEMs (and unseen MEMs) than others. In an ideal world - the morphology measurements would not change over time - they would remain constant. So we can assume fluctuations in morphology are largely driven by measurement error.

If I look at variations in the measurements, I can see the MEMs change a lot. So do my morphology measurements:

d (7 observations over time for a single subject The MEMs are the dashed lines. The morphology measurements are the solid lines. All values have been normalised)

Currently, I am trying to figure out what type of analysis could help me figure out the contribution my MEMs are making to the overall measurement error. Particularly, what morphology measurements are most influenced by the MEMs. Since these errors are largely spatial, there may be interactions between different measurements as the position and orientation changes in 3D space.

All I can think of is to look at peaks in variation of certain measurements, and see if spikes in variability correlate with spikes in MEMs.

Can anyone suggest a method to approach this? Could LMER be specified to explore this? Many thanks,

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Consider each morphology characteristic as a random variable $X$. This random variable fluctuates with each measurement by the subject's positioning and other reasons.

The random variable $x$ is measured in $n=6$ different days as it looks in the plot in the question. We obtain in this way the sample $(x_1,\cdots,x_6)$. It makes sense to estimate the mean of $x$ (let us call it $\mu$). You can only know $\mu$ from averaging the measured $X$ in a huge number of images. However, we only have a few images available. So, what we do in statistics is to estimate $\mu$ with the sample mean $\bar{x} = \sum_{i=1}^n x_i$. The error provides an idea of the difference between $\mu$ and its estimator $\bar{x}$. The error is $s/\sqrt{n}$ with $s^2 = \sum_{i=1}^n (x_i - \bar{x})^2 / (n-1)$, the sample variance.

As a technicality, you need to multiply this error by a factor of 1.16 to account for the fact that there are only six measurements. This is known as the Student's error. To reduce the error you just need to measure more times (increase n).

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