A random variable $\chi$ is a functional variable if it takes values in a functional space $F$ (full normed or semi-normed). A particular case occurs when the functional variable $\chi = {\chi(t) : t ∈ T}$, where $T$ is an interval $T ⊂ R$ that belongs to a Hilbert space, as is the case of continuous functions in an interval.
A set of functional data $\chi_1, ....., \chi_n$ is the observation of $n$ functional variables $\chi_1, ....., \chi_n$ with the same distribution as $\chi$. Where $\chi$ is usually assumed to be an element of:
$$L_2(T) = \{f : T → R, \int_T f(t)^2dt < ∞\}$$ With the inner product $(f, g) = \int_T f (t) g (t) dt$. The norm of $\chi(t)$ is defined by: $$||\chi(t)||=\left(\int_a^b X(t)^2 dt\right)^{\frac{1}{2}}$$In this context, the `ILS` package is used to apply consistency tests (outlying detection) in an Interlaboratory Study. For this purpose, the `TG` dataset composed of the Thermogravimetric (TG) curves described in Examples of Interlaboratory Studies is used.
In this vignettes, we use the `ILS` package to perform the estimations and graphical representation of the statistics $H(t)$, $K(t)$, $d_H$ and $d_K$, with the aim to perform a r&R study for the datasets composed of functional data `TG` and `DSC` that are also included in the `ILS` package.
## `r colorize("3.1. Hypothesis of reproducibility and repeatability","#1CA666")`In the `ILS` studies, each laboratory performs n samples experimentally, obtaining n different curves of observations $\{X_1^l(t),\ldots,X_n^l(t)\}$, which are obtained for each, $l=1,\ldots,L$. Functional statistics $H_l(t)$ and $K_l(t)$ are calculated for each laboratory assuming the corresponding null hypothesis that there are no statistically different measurements between the laboratories.
The null hypothesis of reproducibility states that: $$H_0: \mu_1(t)=\mu_2(t)=\cdots=\mu_p(t)$$Where $\mu_l(t), l = 1,\ldots,L$ is the functional mean of the population for each laboratory $l$. To evaluate the reproducibility of the laboratory results, the $H(t)$ statistic is calculated as follows:
$$H_l(t)=\frac{X_i^l(t)-\bar{X}(t)}{S_l(t)}; l=1,\ldots,L$$Where $\bar{X}(t)$ and $S_l(t)$ are the mean and the functional point-to-point variance calculated for the $l$ laboratory.
The null hypothesis of repeatability can be defined by: $$H_0=\sigma_1^2(t)=\sigma_2^2(t)=\ldots=\sigma_L^2(t)$$Where $\sigma_l(t), l = 1,\ldots,L$ are the theoretical functional variances corresponding to each laboratory $l$. The repeatability test is based on the statistic ($K(t)$), expressed as:
$$K_l(t)=\frac{S_l(t)}{\sqrt{\bar{S}^2(t)}}; l=1,\ldots,L $$ Where, $\bar{S}^2(t)=\frac{1}{L}\displaystyle\sum_{l=1}^LS_l^2(t)$On the other hand, to test the reproducibility hypothesis, the test statistic $d_H$ is defined as:
$$d_l^H=||H_l(t)||=\left(\int_a^b H_l(t)^2 dt\right)^{\frac{1}{2}}$$Considering that the larger values of $d_K$ correspond to non-consistent laboratories, for the repeatability hypothesis, we define $d_l^K=||K(t)||$ and likewise, the large values of $d_K$ correspond to non-consistent laboratories.
## `r colorize("3.2. ILS: Thermogravimetric Study","#1CA666")`The techniques developed to check if inconsistent laboratories are detected either by outliers in the within-laboratory or in between-laboratory variability, have been implemented in the `ILS` package. As above mentioned, laboratories 1, 5 and 6 have provided different results from the remaining laboratories and should be detected as outliers. We use the datasets described in 2.2, the `TG` dataset that contains Thermogravimetric test results from 7 laboratories, while the `DSC` dataset contains results from 6 laboratories (excluding laboratory 1). First you estimate the functional statistics $H(t)$ and $K(t)$ by the function `mandel.fqcs()`, then you make the corresponding graphs in the defined functional space.
```{r warning=FALSE, message=FALSE} library(ILS) data(TG, package = "ILS") delta <- seq(from = 40 ,to = 850 ,length.out = 1000 ) fqcdata <- ils.fqcdata(TG, p = 7, argvals = delta) mandel.tg <- mandel.fqcs(fqcdata,nb = 10) plot(mandel.tg,legend = T,col=c(rep(3,5),1,1)) ```Figure 7, shows both the $K(t)$ and $H(t)$ statistics for each laboratory, as well as the $d_K$ and $d_H$ contrast statistics. The control limit between short lines is constructed at a significance level $\alpha = 0.01$ corresponding to the critical values $c_K$ and $c_H$. The following code refers to the use of the `ILS` package into the `TG` dataset.
```{r} data(DSC, package = "ILS") fqcdata.dsc <- ils.fqcdata(DSC, p = 6, index.laboratory = paste("Lab",2:7), argvals = delta) mandel.dsc <- mandel.fqcs(fqcdata.dsc,nb = 10) plot(mandel.dsc,legend = F,col=c(rep(3,4),1,3)) ```Interlaboratory Study defined by the `DSC` dataset. Thus, Figure 8 shows that repeatability hypothesis was not reject. Otherwise, the reproducibility’s hypothesis was rejected in the case of laboratory 6 (see Figure 8), that is properly detected as an outlier.