Abstract
Multilevel organization of morphometric data (cells are “nested” within patients) requires special methods for studying correlations between karyometric features. The most distinct feature of these methods is that separate correlation (covariance) matrices are produced for every level in the hierarchy. In karyometric research, the cell‐level (i.e., within‐tumor) correlations seem to be of major interest. Beside their biological importance, these correlation coefficients (CC) are compulsory when dimensionality reduction is required. Using MLwiN, a dedicated program for multilevel modeling, we show how to use multivariate multilevel models (MMM) to obtain and interpret CC in each of the levels. A comparison with two usual, “single‐level” statistics shows that MMM represent the only way to obtain correct cell‐level correlation coefficients. The summary statistics method (take average values across each patient) produces patient‐level CC only, and the “pooling” method (merge all cells together and ignore patients as units of analysis) yields incorrect CC at all. We conclude that multilevel modeling is an indispensable tool for studying correlations between morphometric variables.
Highlights
In Part 1, we considered methods of testing hypothesis with the help of multilevel models, using an example of exploring nature and significance of differences between benign and malignant follicular tumors of the thyroid
While the first pair of variables is the same as in Fig. 1A, the other two are completely different and difficult to explain. In any case, such an explanation should refer to tumors, not to cells, since the underlying CC are from the level-2
What CC should be used to explore interrelations between karyometric features? First, it should be noted that there are no “right” or “wrong” CC generated in multivariate multilevel models (MMM)
Summary
In Part 1, we considered methods of testing hypothesis with the help of multilevel models, using an example of exploring nature and significance of differences between benign and malignant follicular tumors of the thyroid. Another type of research questions that can arise in the morphometry is concerned with correlation structure of karyometric features. Correlation (or covariance) matrix can be used to perform dimensionality reduction, i.e., to eliminate redundant variables, which do not convey any more information in addition to others. Dimensionality reduction usually is accomplished by factor or cluster analysis and is sometimes very important before fitting other types of models
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