7.3
Literature review on correlation methods
With the data for touch-feel perception, surface physical characteristics and friction data available, the next step is to perform correlation analysis between them. There are many correlation methods from the simplest regression models to the complex statistical models. Here, the advantage and disadvantage of several typical correlation methods are illustrated as follows.
7.3.1 Linear regression and partial correlation
Partial correlation is the marginal contribution of a single predictor to reduce the unexplained variation in the outcome of linear regression [169]. Partial correlation indicates the explanatory value attributable to a single predictor after taking into account all of other predictors. In linear regression, it is explained in terms of the reduction of the sum of the squared errors attributable to an individual predictor. Due to the nature of the ranking data, the partial correlation method is only feasible if the averaged ranks are used, in which case it cannot provide information on the significance of correlations between the various physical characteristics. A partial correlation statistic for logistic regression has been proposed [170], based on Wald chi-square statistic for individual coefficients and the likelihood of an intercept- only model. While this statistic has the same range as partial correlation in linear regression and there are some similarities in interpretation. However, the Wald chi-square statistic may be a poor estimator in small-to-medium size samples [171]. In order to illustrate the weakness mentioned, the analysis results from previous work are shown in Fig.7.1.
As shown in Fig. 7.1, the correlation between different average perceived ranking and relative surface physical parameters are expressed. The results can prove that perceived roughness was highly correlated with Sm, the perceived softness was
correlated withH/E, and the human in vivo friction coefficients are highly correlated with the ‘cool/warm’ rankings and the ‘slippery/sticky’ rankings. However, there are several problems with this method. Firstly, much of the inter-subject information may be lost by using the averaged perceived ranking. Secondly, the model ignores any covariance between different physical parameters by looking at the correlation one variable at a time. For example, the perceived rough/smooth ranking may be a combination of many parameters such as the hardness and the friction of the surfaces. Lastly, the method can not solve the small-to-medium size data. Most importantly, this method cannot assess the correlation significance, unless the Spearman’s rank correlation analysis is also performed.
7.3. Literature review on correlation methods
Figure 7.1: Comparison of the tactile sensorial rankings and sensitive physical parameters Yue [32]
7.3.2 Kendall’s W test, Wilcoxon signed-rank test and Spearman’s rank correlation analysis
Before establishing the relationship between touch-feel perception and surface physical parameters, an effective evaluation to differentiate the difference among the samples was carried out in [32]. The ranking data were analysed using non-parametric statistics. Kendall’s W test [172] is conventionally employed to test the samples and study the concordance or effectiveness of the evaluation.
The Kendall’s W value (also known as Kendall’s coefficient of concordance) is defined as W = 12S k2(n3−N) (7.1) whereS is defined as S= N X i=1 (Ri−R)2 = N X i=1 Ri2− (PN i=1Ri)2 N (7.2)
where Ri is the rank sum of the samplei evaluated by all the subjects andN is the
total number of the samples. The Kendall’s W value ranges from 0 to 1, where zero represents the evaluation is not effective, and one means there is a great concordance among the subjects. An asymptotic chi-square value can be calculated to assess the correlation significance based on Kendall’s W. If the asymptotic Chi-square value calculated from Kendall’s W is greater than the critical chi-square value at a targeted
7.3. Literature review on correlation methods confidence level [173], e.g. 0.05, such that
χ2
r =k(N −1)W > χ0.05/22×(N−1), (7.3)
then significant differences exist among the samples in the candidate evaluation mode. It also means at least one sample is effectively perceived to be different. The advantage of this method is the ease of computation and is able to quantify the correlation significance with a p-value. However, the results only indicate the evidence that there is some correlation between the rankings, not the magnitude of the influence. Althoughp-value decreases as the magnitude of an influence increases, the magnitude is not quantifiable nor interpretable, especially with categorical ranking data.
In addition, Wilcoxon signed-rank test was applied to further explore the pairwise difference among the samples. In the Wilcoxon test, the pairwise difference in ranking is associated with a positive or negative sign. The output contains two parameters: one is the normalisedz-value (standard score) calculated from the rank sum of the less frequent sign, the other is thep-value (2-tailed) for examining the confidence level at the significant difference among the samples. In our case, by comparing the samples in pairs, their tactile evaluation difference could be statistically confirmed. However, the results still can not give a clear, interpretable relationship between the perceived touch-feel perception and surface physical parameters.
For the tactile evaluation data, the evaluation process can be treated as a case of k people evaluating N samples in m terms of tactile senses (described by sensorial/affective adjectives). So the evaluation rankings are assembled to a N ×m×k data array. In previous work [32], each type of touch-feel perception ranking was reduced to averaged ranking data for each sample and perceptual items. Spearman’s rank correlation analysis was then performed on the combined matrix. Correlation between each pair of the measured surface properties and the evaluated items was then calculated. The correlation between the physical parameters and the tactile perceptions was marked significant when the correlation coefficient |rs|>0.5
and p-value < 0.05. Comparisons of the correlation coefficient rs could assist in
selecting the most effective physical factors. However, similar to the other methods, the magnitude of the correlation is not directly interpretable and cannot be used to write an analytical relationship that links the ranking to a specific variable.
7.3.3 Factor Analysis and PCA
Factor analysis is a statistical process in which the values of observed data are expressed as functions of a number of possible causes in order to find which are the
7.3. Literature review on correlation methods most important. It is to determine the variability among the observed, correlated variables in terms of a potentially lower number of unobserved variables called latent factors. Factor analysis originated in psychometrics and is used in behavioural science, social science, marketing product management, operations research. The aim of this method is to reduce the dimensionality of data while maximising the information preserved, collapsing from large numbers of observed variables to a smaller number of underlying latent variables.
The most common form of factor analysis, PCA (principal component ana- lysis), seeks a linear combination of variables such that the maximum variance is extracted from the observed variables. PCA is available from the SPSS software [172]. It involves the calculation of the eigenvalue decomposition of a data covariance matrix, usually after mean centring the data for each factor. PCA results in a reduction of interdependent variables, typically to two or three independent variables called principal components, from which a majority of the data variance can be explained. PCA has been applied in the study of semantic components of affective words and seeking explanation of relations between affective and sensorial words [8]. Although the results have shown that the tactile perception is correlated with the surface physical characteristics, the relationship cannot be interpreted or quantified physically.
Figure 7.2: Biplot of the loadings of characteristic variables and the scores of the samples (Yue [32])
Two factors (PC1 and PC2) were extracted using SPSS software in Yue [32]. PC1 accounted for 63% of variance and PC2 for 19%, totalling 82%. PC1 is physically dominated by roughness and in-vivo friction, while PC2 largely depends on the compliance and dry sliding friction. The factor scores were shown in Fig.7.2. In this case, ‘smooth/rough’, ‘cool/warm’ and ‘soft-feel’ are highly independent and they contributed to the major psychophysical loadings on PC1 at the psychophysical