Association, variables correlational analysis

Figure 2.15(a) shows the relationship between and Cp for the component characteristics analysed. Note, there are six points at q = 9, Cp = 0. The correlation coefficient, r, between two sets of variables is a measure of the degree of (linear) association. A correlation coefficient of 1 indicates that the association is deterministic. A negative value indicates an inverse relationship. The data points have a correlation coefficient, r = —0.984. It is evident that the component manufacturing variability risks analysis is satisfactorily modelling the occurrence of manufacturing variability for the components tested. 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

How can multivariate methods be used to avoid the problems associated with the OVAT approach In general, multivariate methods use the information contained in the relation between the variables (correlations or covariances) and therefore data like those in Figure 6.3 present no problem. The risk of type I errors is kept under control in multivariate analysis by considering all variables simultaneously. To consider all variables simultaneously involves a... 

The statistical techniques that have been developed to measure the amount of association between variables are called correlation methods. A statistical analysis performed to determine the degree of correlation is called a correlation analysis. For... 

Canonical correlation analysis identifies and quantifies the associations between two sets of variables [126]. Canonical correlation analysis is conducted by using canonical variates. Consider n observations of two random vectors X and y of dimensions p and q forming data sets Xpxn and Y xn with... 

Correlation analysis permits an objective evaluation of the degree of association between two variables. If a value of variable. is produced by a process that also produces a corresponding value of variable y, a correlation analysis will indicate if the association between the two variables is significant at some selected level of confidence. A significant correlation does not imply that y may be predicted from. or vice versa it simply indicates that there is some significant association. Association does not imply a cause and effect relationship, i.e., that. v physically generates y. The associated values of. y and y may be the result of some unknown underlying cause and effect process that involves both variables. 

A number of different approaches have been taken by researchers examining the relationship between accident statistics and employee job tenure (how long the employee has worked in the job). In some studies, researchers have formed groups of employees based on their job tenure and compared accident rates across the groups. Unfortunately, not all studies that have used this group comparison approach to study the relationship between job tenure and accidents have attempted to control for employee age across the groups. Other studies have used job tenure as a predictor variable in regression analysis or simply correlation analysis in an attempt to find associations between an employee s job tenure and accidents. 

Carroll, J. D. (1968) Generalization of canonical correlation analysis to three or more sets of variables, Proceedings of the 76th Convention of the American Psychological Association, 3, 227-228. 

Monomer concentration dynamics are presented in Figure 5. Additional observations for Run 5 are accurately correlated during the reactor startup and at final steady state. The observation at one residence time, Run 4, may be in error. The total cummu-lative, molar concentrations of macromolecules as a function of time are presented in Figure 6. The errors associated with this dependent variable are also evident during the steady state analysis of initiation... 

In adults, a study of 75 autopsies of persons who had resided in a soft-water, leached soil region of North Carolina found a positive correlation between lead level in the aorta and death from heart-related disease (Voors et al. 1982). The association persisted after adjustment for the effect of age. A similar correlation was found between cadmium levels in the liver and death from heart-related disease. (Aortic lead and liver cadmium levels were considered to be suitable indices of exposure.) The effects of the two metals appeared to be additive. Potential confounding variables other than age were not included in the analysis. The investigators stated that fatty liver (indicative of alcohol consumption) and cigarette smoking did not account for the correlations between lead, cadmium and heart-disease death. 

A later analysis (Emhart et al. 1987) related PbB levels obtained at delivery (maternal and cord blood) and at 6 months, 2 years, and 3 years of age to developmental tests (MDI, PDI, Kent Infant Development Scale [KID], and Stanford-Binet IQ) administered at 6 months, 1 year, 2 years, and 3 years of age, as appropriate. After controlling for covariates and confounding risk factors, the only significant associations of blood lead with concurrent or later development were an inverse association between maternal (but not cord) blood lead and MDI, PDI, and KID at 6 months, and a positive association between 6-month PbB and 6-month KID. The investigators concluded that, taken as a whole, the results of the 21 analyses of correlation between blood lead and developmental test scores were "reasonably consistent with what might be expected on the basis of sampling variability," that any association of blood lead level with measures of development was likely to be due to the dependence of both PbB and... 

The paper by Cochrane, St. Leger, and Moore (1978) typifies the issues associated with many early studies. Specifically, they relied on cross sections with multiple countries and often limited the analysis to simple correlations. Because determinants of life expectancy are multifactorial, national studies are more likely to detect differences than international studies. It is also critically important to include adequate control variables. In fact, a later study (Cremieux, Ouellette, and Meilleur 1999) based on extensive national data suggests that a 10% increase in health care spending reduces infant mortality by 0.5% for males and 0.4% for females while increasing life expectancy by half a year for males and three months for females. The current study uses similar modeling and data hence, results on the effect of pharmaceuticals reported below can be put in perspective relative to the overall effect of health care spending from that earlier research. 

---