Coyne and Messina Articles Part 3 Spearman Coefficient Review

It is important to recognize the role of the normal distribution as a mathematical model for errors occurring by chance. It also provides a way to describe a sample variable based on measurements of the sample, and to compare it to other samples.

The Bell Curve's Significance

One of the goals of good evidence-based practice is to implement practices that bring better outcomes to patients as a whole (the population). To do this, you need to know how the sample matches the population. In most cases, the larger population will not be available to measure, but we know that the larger a data set is, the more likely it is to be in the form of a normal distribution. This is the basic argument for using the largest feasible numbers of subject in the sample.

Characteristics of a Sample

The characteristics of a sample are described by measuring its central tendency, and its dispersion (variability from that central tendency).

Measures of Central Tendency

The measures of central tendency include the mean, the median, and themode. The mean is the average of all the measures in the data set, and is calculated by taking the sum of all the variables (α) and dividing it by the total sample size (N), so that M(mean) = b. The mean is the most commonly used measure of central tendency, but it has its flaws. If you are trying to describe accurately a set of scores, you can see one of the problems with the mean in the following data set:

      2, 2, 4, 5, 75

The N of this sample is 5 because there are five numbers in the sample. The mean of the sample is 17.6, as follows: (2+2+4+5+75)/5 = 17.6

As you can see, the mean is very distorted from the bulk of the data set by the presence of one large outlying score.

The median is the point on which half of the scores in the sample fall below, and half fall above. In the data set above, the median is 4. You can see the difference from the mean score. The median is used for ordinal data primarily, although it can be used for interval and ratio data also. It is more useful when one wants to depict the center point of a data set without worrying about the distortion on the central point caused by outliers in the data set. Use of the median does require that all values in the data set be rank ordered in numerical fashion.

The mode is the measure that occurs most often in the data set. In the set above, the mode is 2. It is used with nominal data where subjects are grouped by categories which are all equivalent to each other. In health care research, the mode is rarely used, but you may see reference to it.

Measures of Dispersion

The nature of all variables is that they vary (hence the term "variables"), giving different measures for different individuals. In a data set, knowing where the middle, or central measure is, and how much variability exists inside the data set is essential. While there are a number of measures of dispersion, we will concentrate on the two most commonly used: the variance and the standard deviation. The variance is defined as the mean of the squared deviations from the mean of the data set. To explain this, let's consider a data set:

      1, 2, 3, 4, 5

The N, or sample size, is 5, the sum of the numbers is 15, and the mean is 15/5 = 3. If you take each number and subtract it from the mean of the data set, it will look like this:

      1-3 -2

      2-3 -1

      3-3  0

      4-3  1

      5-3  2

             --

             0

When you add the numbers subtracted from the mean, they will always equal zero when added together on a normal distribution. So, in order to measure variance, we square the numbers to get rid of the negative numbers. It looks like this:

      1-3 -2         4

      2-3 -1         1

      3-3 0          0

      4-3 1          1

      5-3 2          4

                        10 (this result is called the Sum of Squares)

Now that the negative numbers have been removed, you can take the mean of the Sum of Squares by SS/N (Sum of Squares divided by N) or 10/5 = 2, which is the variability of the sample around its mean of 3. However, the numbers are not equivalent since the mean is a raw number and the variance of 2 is a squared number. In order to get them both represented the same way, you convert the variance to the standard deviation, which is simply the square root of the variance, that is: 

In this data set, the standard deviation is the square root of 2. So the numerical descriptions of the data set above are a mean (central tendency) of 3 and a standard deviation (variability) of 1.

Non-Parametric Statistics

This group of statistics is used with nominal and ordinal data sets. The most common one is the Chi-Square test. Chi-Squares are useful to determine whether two groups or measures are related or whether there is significant difference between them. As an example:

You can commit one new educator to a single unit. Is there a difference between the types of education taken between clinical staff in respiratory care and in the lab, which would affect this decision?

The analysis model is set up in table form, as follows:

In order to calculate the Chi-Square, you must first calculate the expected values in each cell. This is done by taking the formula E = RC/N, or Expected value = row total*column total/ total sample size. For the first cell, the observed number is 21. The expected number is E = 31*50 "row total of 31 * column total of 50," divided by total sample size of 100. The expected value of the first row, first column cell is 15.5. Use this formula to calculate the remaining expected values for each cell.

Blue is the observed or actual finding.Red is the calculated expected finding.

In a Chi-Square calculation, you are checking the relationship between the observed finding and the expected finding to see if there are differences. The next formula calculates the Chi-Square itself.

      Chi-Square = Sum of (observed-expected)squared

                                                     Expected

Or, for cell one, (21 − 15.5)squared   which equals 1.95

                                    15.5

Calculate the remaining cells in the same way, add the results, and you will get the Chi-Square number.

The Chi-Square calculation is 5.63. Now that you have the number, what does it mean? In order to determine if it is statistically significant, thus answering the research question, you need to know two things: where to find the table and what degrees of freedom means.

Degrees of freedom is a statistical concept that is complex and a full definition is beyond the scope of this module. However, you can determine degrees of freedom by taking the sample size − 1, or df = N-1. In the case listed above, the degrees of freedom will equal 99, or 100-1, since the total education events in the data set equal 100.

Once you know the degrees of freedom, go to your handy textbook on statistics (Broyles, 2006) to the table that depicts the Chi-Square distribution (p. 362). Select the alpha (p<.05) on the top row, then go down the left side and find your degree of freedom (df). Since the table only goes to 30, use that one. The alpha for 0.05, 5th column from the left and df of 30, is 43.773. This means that the Chi-Square value needs to be greater than 43.773 in order to be significant. Since it is not, what does that tell you? Where will you put your educator?

* The Answer is below the References for this Module.

Spearman Rank Order

Another common non-parametric test is the Spearman Rank Order correlation coefficient. This non-parametric test can be used with ordinal data to describe the relationship between two variables. The Spearman may be used when the goal is to order the rankings of the data rather than the actual scores.

For example, two OR nurses ranked five instrument sets for their efficiency of use, as follows:

Rankings                      Diff                  Diff Squared

      1                            0                      0

      3                            1                      1

      5                            2                      4

      4                            0                      0

      2                            3                      9

14 total of differences squared to remove negative numbers

Spearman calculation:

R = 1-6*sum of differences squared, divided by N times itself 3 times-N

            OR

R= 1-6(14)/125-5 = 84/120 = 0.7. The Spearman coefficient for this data set is a correlation of 0.7. This could be interpreted as a relatively high rate of correlation between the 2 OR nurse's ranking of the instrument set efficiency. For another example of the use of the Spearman coefficient, review the article by Messina, Scotti, Ganey, and Zipp (2009) again.

Conclusion

There are times in performing research on health care variables when the researcher will not have the luxury of working with interval or ratio data sets. When that situation occurs, the non-parametric tests will become very useful. In addition, when describing a data set, knowing the measure of central tendency and the measure of dispersion is essential. To recap, the Chi-Square measure answers the question of whether two variables are significantly different from each other, and requires the use of nominal or ordinal data sets and a minimum sample size of 4. The Spearman correlation coefficient answers the question of how two variables relate to each other and requires the use of ordinal data and a minimum sample size of 4.

References

Broyles, RW. (2006). Fundamentals of statistics in health administration. Boston: Jones and Bartlett.

Harris, MB. (2004). Medical statistics made easy. New York: Martin Dunitz.

Messina, D.J., Scotti, D.J., Ganey, R., & Zipp, G.P. (May/June 2009). The relationship between patient satisfaction and inpatient admissions across teaching and nonteaching hospitals. Journal of Healthcare Management, 54,3. pp. 177-190.

* Answer: there is no statistically significant difference between the groups.  Statistically, it won't matter where you put your educator. However, pragmatically, the lab does more on-site education, so that may be where you get the most benefit from an on-site educator.

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