New Statistical Algorithms for the Analysis of Mass - FU Berlin, FB MI ...

78 CHAPTER 4. (BIO-)MEDICAL APPLICATIONS
and φ being the CDF of a standard normal random variable. To check the
requirements of normality we use established standard tests, such as Lilliefors
test ((Lilliefors, 1967), needs a large sample), Anderson-Darling test ((Anderson
and Darling, 1952), for small to medium sample size, e.g. 10-200),
Shapiro-Wilk test (Shapiro and Wilk, 1965), or the Jarque-Bera ((Bera and
Carlos, 1980), highly attentive to outliers) test.
(2) If Tn is not a normal distribution the Pivotal Intervals method can
be used. A 1 − α bootstrap pivotal confidence interval can be calculated by:
�
Cn =
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2 · ˆ θn − θ ∗ ((1−α/2),B) , 2 · ˆ θn − θ ∗ ((α/2),B)
where θ ∗ β,B is the β sample quantile of (ˆ θ ∗ n,1 , . . . , ˆ θ ∗ n,B ) and ˆ θn = T (X1, . . . , Xn)
Model Validation by Cross-Validation
Cross-validation is one of several approaches to estimating how well the model
just learned from some training data is going to perform on future (yet unseen)
data. It is better than the widely used residuals approach. The problem with
residual evaluation is that it just gives a indication on how well a model fits
the given data opposed to the a predictions for the performance on data it has
not already seen. Other (more complex) method include Akaike Information
Criterion (AIC, asymptotically equal to CV with k = n − 1) or Bayesian
Information Criterion (BIC, asymptotically equal to CV with k ≈ 10).
Cross-validation (Stone, 1974) partitions the original sample into (two or
many) subsets. The analysis (e.g. model parameter estimation) is initially
performed on one of these subsets (often denoted training set), while the other
subsets (test sets) are used to confirm and validate the initial analysis.
A widely used method is the so called k-fold cross-validation. Here, the
original sample is partitioned into k sub-samples. The cross-validation process
is then repeated k times: in each step one of the k sub-samples (each sample is
used exactly once) is used as the test set and the remaining k − 1 sub-samples
as the training set. The final results is usually computed by taking the average
from the k single results.
4.3 Study Results
This section will explain the results of the algorithmic pipeline described in
the previous chapter when analyzing data such as introduced in section 4.1.
The outcome of the first stage of the analysis pipeline are lists of peaks that
occur in a significant portion of a group at the same m/z value. These peaks
are the basis for further analysis stages that yields three distinct classes of
results:
Correlations: If patient meta-data are available (such as age, weight, blood
parameters etc., see for example Figure 4.3.2) (cor-)relations can be
sought for between peak properties (such as height) and meta-data properties
(see section 4.3.1).
Fingerprints: Peaks of two groups (e.g. cancer vs. healthy) are compared to
find peaks having the same m/z value in both groups but differ in properties.
More formally, peaks in group A at position X that are similar