For the Pearson r correlation, both variables should be normally distributed normally distributed variables have a bell-shaped curve. Other assumptions include linearity and homoscedasticity.
Linearity assumes a straight line relationship between each of the two variables and homoscedasticity assumes that data is equally distributed about the regression line. Correlation coefficients between. Continuous data: Data that is interval or ratio level. This type of data possesses the properties of magnitude and equal intervals between adjacent units.
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Equal intervals between adjacent units means that there are equal amounts of the variable being measured between adjacent units on the scale. An example would be age. An increase in age from 21 to 22 would be the same as an increase in age from 60 to Kendall rank correlation : Kendall rank correlation is a non-parametric test that measures the strength of dependence between two variables. The following formula is used to calculate the value of Kendall rank correlation:. Key Terms. Spearman rank correlation : Spearman rank correlation is a non-parametric test that is used to measure the degree of association between two variables.
The Spearman rank correlation test does not carry any assumptions about the distribution of the data and is the appropriate correlation analysis when the variables are measured on a scale that is at least ordinal. The assumptions of the Spearman correlation are that data must be at least ordinal and the scores on one variable must be monotonically related to the other variable. However, the magnitude of the difference between levels is not necessarily known. An example would be rank ordering levels of education.
Algina, J. Comparing squared multiple correlation coefficients: Examination of a confidence interval and a test significance. Psychological Methods, 4 1 , Bobko, P. Correlation and regression: Applications for industrial organizational psychology and management 2nd ed.
Bonett, D. Meta-analytic interval estimation for bivariate correlations. This requirement will be fully explained in the example of the calculation of the statistic in the case study example. The owner of a laboratory wants to keep sick leave as low as possible by keeping employees healthy through disease prevention programs. Many employees have contracted pneumonia leading to productivity problems due to sick leave from the disease. There is a vaccine for pneumococcal pneumonia, and the owner believes that it is important to get as many employees vaccinated as possible.
Due to a production problem at the company that produces the vaccine, there is only enough vaccine for half the employees. In effect, there are two groups; employees who received the vaccine and employees who did not receive the vaccine.
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The company sent a nurse to every employee who contracted pneumonia to provide home health care and to take a sputum sample for culture to determine the causative agent. They kept track of the number of employees who contracted pneumonia and which type of pneumonia each had. The data were organized as follows:. In this case, the independent variable is vaccination status vaccinated versus unvaccinated.
The dependent variable is health outcome with three levels:. The company wanted to know if providing the vaccine made a difference. To answer this question, they must choose a statistic that can test for differences when all the variables are nominal. The formula for calculating a Chi-Square is:. The marginal values for the case study data are presented in Table 2. The second step is to calculate the expected values for each cell.
Expected values must reflect both the incidence of cases in each category and the unbiased distribution of cases if there is no vaccine effect. This means the statistic cannot just count the total N and divide by 6 for the expected number in each cell. That would not take account of the fact that more subjects stayed healthy regardless of whether they were vaccinated or not. Chi-Square expecteds are calculated as follows:.
Specifically, for each cell, its row marginal is multiplied by its column marginal, and that product is divided by the sample size. Table 3 provides the results of this calculation for each cell. A Chi-square table of significances is available in many elementary statistics texts and on many Internet sites. This is a result of the observed value being 23 while only Therefore, this cell has a much larger number of observed cases than would be expected by chance. Cell 1 reflects the number of unvaccinated employees who contracted pneumococcal pneumonia. This means that the number of unvaccinated people who contracted pneumococcal pneumonia was significantly greater than expected.
This means that a significantly lower number of vaccinated subjects contracted pneumococcal pneumonia than would be expected if the vaccine had no effect. Therefore the company can conclude that there was no difference between the two groups for incidence of non-pneumococcal pneumonia. It can be seen that for both groups, the majority of employees stayed healthy. The meaningful result was that there were significantly fewer cases of pneumococcal pneumonia among the vaccinated employees and significantly more cases among the unvaccinated employees. As a result, the company should conclude that the vaccination program did reduce the incidence of pneumoccal pneumonia.
Most researchers inspect the table to estimate which cells are overrepresented with a large number of cases versus those which have a small number of cases. One might ask if, in this case, the Chi-square was the best or only test the researcher could have used. Nominal variables require the use of non-parametric tests, and there are three commonly used significance tests that can be used for this type of nominal data.
The first and most commonly used is the Chi-square. The third test is the maximum likelihood ratio Chi-square test which is most often used when the data set is too small to meet the sample size assumption of the Chi-square test.
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As exhibited by the table of expected values for the case study, the cell expected requirements of the Chi-square were met by the data in the example. Specifically, there are 6 cells in the table. This table meets the requirement that at least 5 of the 6 cells must have cell expected of 5 or more, and so there is no need to use the maximum likelihood ratio chi-square. Suppose the sample size were much smaller. Suppose the sample size was smaller and the table had the data in Table 4.
Sample raw data presented first, sample expected values in parentheses, and cell follow the slash. This table should be tested with a maximum likelihood ratio Chi-square test. When researchers use the Chi-square test in violation of one or more assumptions, the result may or may not be reliable.
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Second, the appropriate test may produce a significant result while the inappropriate test provides a result that is not statistically significant, which is a Type II error. Third, the appropriate test may provide a non-significant result while the inappropriate test may provide a significant result, which is a Type I error. Blackwell Science. Barnhart HX, Barborial DP Applications of the repeatability of quantitative imaging biomarkers: a review of statistical analysis of repeat data sets.
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The Chi-square test of independence
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