Statnote 32 : the partial correlation coefficient

In previous statnotes, the application of correlation and regression methods to the analysis of two variables (X,Y) was described. The most important statistic used to measure the degree of correlation between two variables is Pearson’s ‘product moment correlation coefficient’ (‘r’). The correlation between two variables may be due to their common relation to other variables. Hence, investigators using correlation studies need to be alert to the possibilities of spurious correlation and the methods of ‘partial correlation’ are one method of taking this into account. This statnote applies the methods of partial correlation to three scenarios. First, to a fairly obvious example of a spurious correlation resulting from the ‘size effect’ involving the relationship between the number of general practitioners (GP) and the number of deaths of patients in a town. Second, to the relationship between the abundance of the nitrogen-fixing bacterium Azotobacter in soil and three soil variables, and finally, to a more complex scenario, first introduced in Statnote 24involving the relationship between the growth of lichens in the field and climate.

Statnote 33:the intra-class correlation coefficient

The intra-class correlation coefficient (ICC or ri) is a method of measuring correlation when the data are paired and therefore, should be used when experimental units are organised into groups. A useful analogy is with the unpaired or paired ‘t’ test to compare the differences between the means of two groups. In studies of reproducibility, there may actually be little difference between the ICC and Pearson’s ‘r’ for ‘true’ repeated measurements. If, however, there is a systematic change in the measurements made on the first compared with the second occasion, then the ICC will be significantly less than ‘r’, and less confidence would be placed in the reproducibility of the results.

The use of correlation and regression methods in optometry

Correlation and regression are two of the statistical procedures most widely used by optometrists. However, these tests are often misused or interpreted incorrectly, leading to erroneous conclusions from clinical experiments. This review examines the major statistical tests concerned with correlation and regression that are most likely to arise in clinical investigations in optometry. First, the use, interpretation and limitations of Pearson's product moment correlation coefficient are described. Second, the least squares method of fitting a linear regression to data and for testing how well a regression line fits the data are described. Third, the problems of using linear regression methods in observational studies, if there are errors associated in measuring the independent variable and for predicting a new value of Y for a given X, are discussed. Finally, methods for testing whether a non-linear relationship provides a better fit to the data and for comparing two or more regression lines are considered.

Statnote 14: the correlation of two variables (Pearson's 'r')

Pearson's correlation coefficient (‘r’) is one of the most widely used of all statistics. Nevertheless, care needs to be used in interpreting the results because with large numbers of observations, quite small values of ‘r’ become significant and the X variable may only account for a small proportion of the variance in Y. Hence, ‘r squared’ should always be calculated and included in a discussion of the significance of ‘r’. The use of ‘r’ also assumes that the data follow a bivariate normal distribution (see Statnote 17) and this assumption should be examined prior to the study. If the data do not conform to such a distribution, the use of a non-parametric correlation coefficient should be considered. A significant correlation should not be interpreted as indicating ‘causation’ especially in observational studies, in which the two variables may be correlated because of their mutual correlations with other confounding variables.

Statnote 15: the non-parametric correlation coefficients

If in a correlation test, one or both variables are small whole numbers, scores based on a limited scale, or percentages, a non-parametric correlation coefficient should be considered as an alternative to Pearson’s ‘r’. Kendall’s t and Spearman’s rs are similar tests but the former should be considered if the analysis is to be extended to include partial correlations. If the data contain many tied values, then gamma should be considered as a suitable test.

Data analysis methods in optometry. Part 5: correlation

1. Pearson's correlation coefficient only tests whether the data fit a linear model. With large numbers of observations, quite small values of r become significant and the X variable may only account for a minute proportion of the variance in Y. Hence, the value of r squared should always be calculated and included in a discussion of the significance of r. 2. The use of r assumes that a bivariate normal distribution is present and this assumption should be examined prior to the study. If Pearson's r is not appropriate, then a non-parametric correlation coefficient such as Spearman's rs may be used. 3. A significant correlation should not be interpreted as indicating causation especially in observational studies in which there is a high probability that the two variables are correlated because of their mutual correlations with other variables. 4. In studies of measurement error, there are problems in using r as a test of reliability and the ‘intra-class correlation coefficient’ should be used as an alternative. A correlation test provides only limited information as to the relationship between two variables. Fitting a regression line to the data using the method known as ‘least square’ provides much more information and the methods of regression and their application in optometry will be discussed in the next article.

Statnote 22: Measuring 'agreement'

If a quantity has been measured by two different methods, the degree of agreement between them should not be tested using Pearson’s correlation coefficient ‘r’. Instead the differences between the two methods should be compared with their mean difference using a Bland and Altman plot. Such a plot illustrates the level of agreement between two methods and enables the degree of bias of one method over the other to be calculated and applied if necessary as a correction factor.

Development of a questionnaire to assess the relative subjective benefits of presbyopia correction

Purpose To develop a standardized questionnaire of near visual function and satisfaction to complement visual function evaluations of presbyopic corrections. Setting Eye Clinic, School of Life and Health Sciences, Aston University, Midland Eye Institute and Solihull Hospital, Birmingham, United Kingdom. Design Questionnaire development. Methods A preliminary 26-item questionnaire of previously used near visual function items was completed by patients with monofocal intraocular lenses (IOLs), multifocal IOLs, accommodating IOLs, multifocal contact lenses, or varifocal spectacles. Rasch analysis was used for item reduction, after which internal and test–retest reliabilities were determined. Construct validity was determined by correlating the resulting Near Activity Visual Questionnaire (NAVQ) scores with near visual acuity and critical print size (CPS), which was measured using the Minnesota Low Vision Reading Test chart. Discrimination ability was assessed through receiver-operating characteristic (ROC) curve analysis. Results One hundred fifty patients completed the questionnaire. Item reduction resulted in a 10-item NAVQ with excellent separation (2.92), internal consistency (Cronbach a = 0.95), and test–retest reliability (intraclass correlation coefficient = 0.72). Correlations of questionnaire scores with near visual acuity (r = 0.32) and CPS (r = 0.27) provided evidence of validity, and discrimination ability was excellent (area under ROC curve = 0.91). Conclusion Results show the NAVQ is a reliable, valid instrument that can be incorporated into the evaluation of presbyopic corrections.

Increased expression of phosphorylated forms of RNA-dependent protein kinase and eukaryotic initiation factor 2 alpha may signal skeletal muscle atrophy in weight-losing cancer patients

Previous studies suggest that the activation (autophosphorylation) of dsRNA-dependent protein kinase (PKR) can stimulate protein degradation, and depress protein synthesis in skeletal muscle through phosphorylation of the translation initiation factor 2 (eIF2) on the alpha-subunit. To understand whether these mediators are important in muscle wasting in cancer patients, levels of the phospho forms of PKR and eIF2alpha have been determined in rectus abdominus muscle of weight losing patients with oesophago-gastric cancer, in comparison with healthy controls. Levels of both phospho PKR and phospho eIF2alpha were significantly enhanced in muscle of cancer patients with weight loss irrespective of the amount and there was a linear relationship between phosphorylation of PKR and phosphorylation of eIF2alpha (correlation coefficient 0.76, P=0.005). This suggests that phosphorylation of PKR led to phosphorylation of eIF2alpha. Myosin levels decreased as the weight loss increased, and there was a linear relationship between myosin expression and the extent of phosphorylation of eIF2alpha (correlation coefficient 0.77, P=0.004). These results suggest that phosphorylation of PKR may be an important initiator of muscle wasting in cancer patients.

Estimation of functional connectivity from electromagnetic signals and the amount of empirical data required

An increasing number of neuroimaging studies are concerned with the identification of interactions or statistical dependencies between brain areas. Dependencies between the activities of different brain regions can be quantified with functional connectivity measures such as the cross-correlation coefficient. An important factor limiting the accuracy of such measures is the amount of empirical data available. For event-related protocols, the amount of data also affects the temporal resolution of the analysis. We use analytical expressions to calculate the amount of empirical data needed to establish whether a certain level of dependency is significant when the time series are autocorrelated, as is the case for biological signals. These analytical results are then contrasted with estimates from simulations based on real data recorded with magnetoencephalography during a resting-state paradigm and during the presentation of visual stimuli. Results indicate that, for broadband signals, 50-100 s of data is required to detect a true underlying cross-correlations coefficient of 0.05. This corresponds to a resolution of a few hundred milliseconds for typical event-related recordings. The required time window increases for narrow band signals as frequency decreases. For instance, approximately 3 times as much data is necessary for signals in the alpha band. Important implications can be derived for the design and interpretation of experiments to characterize weak interactions, which are potentially important for brain processing.

Reproducibility-repeatability of choroidal thickness calculation using optical coherence tomography

Purpose. To evaluate the repeatability and reproducibility of subfoveal choroidal thickness (CT) calculations performed manually using optical coherence tomography (OCT). Methods. The CT was imaged in vivo at each of two visits on 11 healthy volunteers (mean age, 35.72 ± 13.19 years) using the spectral domain OCT. CT was manually measured after applying ImageJ processing filters on 15 radial subfoveal scans. Each radial scan was spaced 12° from each other and contained 2500 A-scans. The coefficient of variability, coefficient of repeatability (CoR), coefficient of reproducibility, and intraclass correlation coefficient determined the reproducibility and repeatability of the calculation. Axial length (AL) and mean spherical equivalent refractive error were measured with the IOLMaster and an open view autorefractor to study their potential relationship with CT. Results. The within-visit and between-visit coefficient of variability, CoR, coefficient of reproducibility, and intraclass correlation coefficient were 0.80, 2.97% 2.44%, and 99%, respectively. The subfoveal CT correlated significantly with AL (R = -0.60, p = 0.05). Conclusions. The subfoveal CT could be measured manually in vivo using OCT and the readings obtained from the healthy subjects evaluated were repeatable and reproducible. It is proposed that OCT could be a useful instrument to perform in vivo assessment and monitoring of CT changes in retinal disease. The preliminary results suggest a negative correlation between subfoveal CT and AL in such a way that it decreases with increasing AL but not with refractive error.

Development and validation of the Attitudes Towards the Homeless questionnaire

CONTEXT: The homeless are a significant group within society, which is increasing in size. They have demonstrably greater physical and mental health needs than the housed, and yet often have difficulty accessing primary health care. Medical 'reluctance' to look after homeless people is increasingly suggested as part of the problem. Medical education may have a role in ameliorating this. OBJECTIVES: This paper reports on the development and validation of a questionnaire specifically developed to measure medical students' attitudes towards the homeless. METHOD AND RESULTS: The Attitudes Towards the Homeless Questionnaire, developed using the views of over 370 medical students, was shown to have a Pearson test-retest reliability correlation coefficient of 0.8 and a Cronbach's alpha coefficient of 0.74. CONCLUSIONS: The Attitudes Towards the Homeless Questionnaire appears to be a valid and reliable instrument, which can measure students' attitudes towards the homeless. It could be a useful tool in assessing the effectiveness of educational interventions.

Development of a near activity visual questionnaire to assess accommodating intraocular lenses

Purpose: To develop a questionnaire that subjectively assesses near visual function in patients with 'accommodating' intraocular lenses (IOLs). Methods: A literature search of existing vision-related quality-of-life instruments identified all questions relating to near visual tasks. Questions were combined if repeated in multiple instruments. Further relevant questions were added and item interpretation confirmed through multidisciplinary consultation and focus groups. A preliminary 19-item questionnaire was presented to 22 subjects at their 4-week visit post first eye phacoemulsification with 'accommodative' IOL implantation, and again 6 and 12 weeks post-operatively. Rasch Analysis, Frequency of Endorsement, and tests of normality (skew and kurtosis) were used to reduce the instrument. Cronbach's alpha and test-retest reliability (intraclass correlation coefficient, ICC) were determined for the final questionnaire. Construct validity was obtained by Pearson's product moment correlation (PPMC) of questionnaire scores to reading acuity (RA) and to Critical Print Size (CPS) reading speed. Criterion validity was obtained by receiver operating characteristic (ROC) curve analysis and dimensionality of the questionnaire was assessed by factor analysis. Results: Rasch Analysis eliminated nine items due to poor fit statistics. The final items have good separation (2.55), internal consistency (Cronbach's α = 0.97) and test-retest reliability (ICC = 0.66). PPMC of questionnaire scores with RA was 0.33, and with CPS reading speed was 0.08. Area under the ROC curve was 0.88 and Factor Analysis revealed one principal factor. Conclusion: The pilot data indicates the questionnaire to be internally consistent, reliable and a valid instrument that could be useful for assessing near visual function in patients with 'accommodating' IOLS. The questionnaire will now be expanded to include other types of presbyopic correction. © 2007 British Contact Lens Association.

Is beta-amyloid (Abeta) deposition in the frontal cortex of patients with Alzheimer’s disease related to blood vessels?

The density of the diffuse, primitive and classic beta-amyloid (Abeta) deposits and the incidence of large and small diameter blood vessels was studied in the upper laminae of the frontal cortex of 10 patients with sporadic Alzheimer’s disease (AD). The data were analysed using the partial correlation coefficient to determine whether variations in the density of Abeta deposit subtypes along the cortex were related to blood vessels. Significant correlations between the density of the diffuse or primitive Abeta deposits and blood vessels were found in only a small number of patients. However, the classic Abeta deposits were positively correlated with the large blood vessels in all 10 patients, the correlations remaining when the effects of gyral location and mutual correlations between Abeta deposits were removed. These results suggest that the larger blood vessels are involved specifically in the formation of the classic Abeta deposits and are less important in the formation of the diffuse and primitive deposits.

Perceptions of technological change : the attitude of managers towards group technology in the United States food equipment industry

In the center of today's continued and rapid technological change and ever competitive environment of the next millennium, manufacturers must realize that unless they are ready to consider and evaluate new technologies brought onto them, they may fail to adequately respond to the challenges that lie ahead of them. This research was designed to determine the consistency of the perceptions of technical and non-technical administrators, in manufacturing environment, towards technological change and group technology as an advanced manufacturing system. This research has included a review of literature with references to technological change, justification and implementation processes, and various manufacturing systems including group technology and its benefits. This research has used the research method of empirical analysis (quantitative) and case studies (qualitative) to research perceptions of technical and non-technical administrators towards technological change and group technology. Sixty-four (64) technical and fifty-one (51) nontechnical administrators from fifty (50) manufacturing organizations in the United States of America responded to the mail survey questionnaire used in this research. Responses were analyzed using the Repeated Measures ANOVA procedure to compare mean responses of each group. Two correlation analyses, Cronback Coefficient Alpha and Pearson Correlation Coefficient, were also performed to determine the reliability of the questionnaire as well as the degree of correlation of perceptions between these two groups. This research, through the empirical analysis, has found that perceptions of the technical and non-technical administrators towards group technology were not consistent. In other words, they did not perceive the benefits of group technology in the same manner to the overall organizational performance. This finding was significant since it provided the first clear and comprehensive view of the technical and non-technical administrators' perception towards group technology and technological change, in Food Equipment Manufacturer Industry, in United States of America. In addition, a number of cases were analyzed and the results have supported those of the quantitative analysis. Therefore, this research not only has provided basic data, which was unavailable prior to this investigation, but it also provided a basis for future studies.