The clinical nurse specialist is frequently the person within the nursing department who leads nursing research studies. Because there are so many options available and these options are predicated on the research questions (Table 1 ) and the level of measurement, it is important that the clinical nurse specialist is aware of the options and criteria used to make the best option selections. Research designs are typically designated as qualitative or quantitative. The 4 major quantitative designs are classified as descriptive, correlation, quasi-experimental, and experimental.^{1} The purpose of this article is to provide important information needed to design, conduct, and analyze a correlation study.

Table 1: Research Questions and Specific Aims for Correlation Research Designs

WHAT IS IN A NAME?
Confusion sometimes arises because the term correlation is used to describe a type of analysis, types of research methods, and a research design. Many different studies (descriptive and experimental) might use correlation analysis, but this does not make them correlation studies.^{1} Correlation studies should have a sound theoretical base that explains why the researcher would expect certain variables to be related to each other.^{1,2} This is a critical but often overlooked step to consider when designing a correlational study. This is the stage where the researcher identifies and clearly defines (conceptually and operationally) all potentially relevant and confounding variables.

WHY DO A CORRELATION STUDY?
There are 2 reasons why a researcher might want to conduct a correlation study. The first is when causation is not suspected.^{1,2} One example would be, if the researcher wanted to compare the scores of students on 2 versions of the same test, it would not be logical to think that the score on 1 version caused the score on the other version. Thus, cause and effect are not a consideration. The second reason a researcher might conduct a correlation study is when causation is suspected but the suspected cause (independent variable [IV]) cannot be manipulated because it is not possible, it is not practical or feasible, or it would be unethical.^{1–3} Correlation studies can be simple or predictive and can look at similarities or differences. Data analysis choices are dependent upon the research questions (Table 1 ) or specific aim of the study.

STRENGTHS AND LIMITATIONS OF CORRELATION DESIGNS
Strengths
Perhaps, the biggest advantage of a correlation design is the ability to assess relationships between many variables in a single study.^{1} Correlation designs are excellent to use when little is known about a problem or phenomenon. It provides information that would be useful for designing more rigorous studies.^{2}

Limitations
The major limitation of correlation research is that the results do not really indicate cause and effect.^{1,3} Correlation can only identify relationships; it cannot explain why relationships exist. Two variables could be genuinely related to each other, or a third variable could act on both variables (see Figure 1 ) in the same way, making it appear they are related (spurious relationship).^{2}

FIGURE 1: Correlation relationships.

CORRELATION DESIGNS
Contrary to what is commonly thought, correlation studies do not use only correlation analysis. There are basically 2 types of correlation studies. One seeks to find relationships between variables (simple correlation), and the other seeks to predict 1 variable based on knowledge of another variable (predictive correlation).^{1,2} The most obvious difference between these 2 designs is seen in the way the research questions are worded (see Table 1 ).

Simple Correlation
A simple correlation study, sometimes called descriptive correlation , answers the question about relationships between variables but cannot be used to determine cause-and-effect.^{1} In simple correlation designs, there are no IVs and dependent variables (DVs), just variables. The researcher does not manipulate the variables, and data are collected after the phenomenon has occurred. Correlation analysis is used to determine whether a relationship exists and how strong that relationship is.^{1,3}

Predictive Correlation
Predictive correlation is used to determine whether 1 variable can predict (IV) another variable (DV). Unlike simple correlation, simple linear regression is used to examine the relationship between an IV and a DV.^{1,4} This design and analysis expand upon simple correlation by allowing the researcher to predict a DV if the IV is known.^{1,4} An extension of this design looks at multiple IVs and various models to determine which combination of variables best predicts the DV (multiple or logistic regression).^{1,4}

SAMPLE SIZE
Too often, sample size is not considered when designing correlation studies, but it is critical. As with any study design, a sample that is too small can result in a type II error (finding no significant difference when there is one).^{1} Correlations found in a small sample tend to be unreliable and have a large confidence interval (CI). As a general rule, for 2 variables considering a medium effect size (0.30), an α of .05, and a power (1 − β) of 0.80, a simple correlation study would require at least 85 subjects.^{5,6} For assistance in calculating sample size, there are several easy-to-use calculators online (Table 2 ).

Table 2: Sample Size Calculators

Box 1 Sample Size Calculators for Correlation Studies
https://www.sample-size.net/correlation-sample-size/

https://www.masc.org.au/stats/PowerCalculator/PowerCorrelation

The rules for sample size for linear regression are less clear. In general, most resources suggest a minimum of 30 observations for 1 IV and 10 to 20 additional observations for each additional IV. One resource suggested that a sample of 100 is an adequate sample, 200 is a good sample, and 400 is a great sample.^{6}

CORRELATION ANALYSIS
Because neither nurses nor physicians receive a significant amount of training related to statistical analysis, it is not surprising that correlation analysis and its interpretation are often misapplied.^{7} Common errors include inferring causation (called causal overreach) when there is no way to assess for causation, misunderstanding the importance or lack of importance of the level of significance, forgetting to consider alpha inflation, or not considering sample size bias issues.^{7} Skipping data cleaning or assumption testing could impact the fidelity of the data and could alter the outcome of the analyses, especially if the researcher selected the wrong statistical test to assess for correlation.^{7,8}

Looking at the Data
It is important to clean the data; even 1 error can make a big difference in the analysis. Look especially for outliers.^{8} These are often data entry errors that can be identified and corrected.^{8} However, if outliers are not errors due to data entry, consider removing the outliers from the database before analysis, especially if considering using a parametric correlation test, such as Pearson product moment correlation.^{2,8} Once the researcher is confident of the data fidelity, examination of each variable is indicated to test the assumptions and select the correct correlation test.^{3,8}

Testing Assumptions
Data Level
The first assumption relates to the level of data (ordinal, interval, or ratio) at which the variable is measured. There are 3 correlation tests (Pearson's, Spearman's, and Kendall's) used to assess relationships or associations between variables^{1,8} (see Table 3 ). Pearson's can only be used if both variables are measured at the interval or ratio level (also called scale or continuous data), and data must be free from outliers.^{4} Spearman's and Kendall's can be used if the data are measured at the ordinal, interval, or ratio level and are not impacted by outliers. Spearman's usually produces a larger rho (r) than Kendall's, but Kendall's is the best choice if the sample is small.^{8}

Table 3: Assumptions for Correlation Tests

Distribution
The next assumption is about the distribution of the data.^{7} Initially, the researcher should look at the distribution of each variable independently.^{1,8} There are 4 types of distributions (Figure 2 ). Figure 2 A depicts a normal distribution. The data are evenly distributed, with most clustered around the mean. In a normal distribution, 95% of the data are within 1 SD of the mean, and 99% are within 2 SDs of the mean. Figures 2 B and C show data that are skewed to the right or left. The direction is based upon the tail of the curve.^{1,8} For example, in Figure 2 B, the tail is on the right, so this is skewed to the right. If the data represented test scores for a class, the student data in Figure 2 C show students scored higher in general than the students whose data are depicted in Figure 2 B. Distribution around the mean is described as skewness.^{4,8} If using a statistical program such SPSS, skewness is reported as a positive or negative number. In general, any number between −1 and +1 is considered normal enough to meet the normality assumption.^{3,5,8}

FIGURE 2: Assessing data distributions.

Another type of distribution is called kurtosis .^{1} One example is the platykurtic or flat distribution (Figure 2 D).^{8} If the data represented first postoperative pain score for a sample of patients, it would appear that they nearly all experienced a similar amount of pain. In contrast to the flat distribution, the data could demonstrate a peaked distribution. This curve is referred to as a leptokurtic curve (Figure 2 E).^{8} It is similar to the normal curve, but more of the scores are centered on the mean than in the normal distribution. In the example pictured, if the data represented test scores, it would appear that most students scored in the middle or midrange. The last example is a bimodal distribution (Figure 2 F). When using a statistical program such as SPSS, kurtosis such as skewness is reported as a positive or negative number.^{7} In general, any number between −2 and +2 is considered normal enough to meet the normality requirements.^{3,5,8}

The final distribution is a bimodal distribution (Figure 2 F). In this distribution, there seems to be 2 groups.^{8} If the data represented test scores, it would appear that there is a group of students who scored high and a group that scored low on a test. It would not be appropriate to run any correlation tests on the total sample in this distribution, but to run each as a separate group.^{7} Often, the bimodal distribution is due to another variable that is not being accounted for, such as gender or age. Knowing the distribution of each variable will allow the researcher to select the best correlation test when the analysis is run.

Relationships
The next assumption to test is related to linearity or monotonic relationships between variables (bivariate).^{8} Although all 3 tests (Pearson's, Spearman's, and Kendall's) require a monotonic relationship between variables, only Pearson's requires a linear relationship.^{1,4,8} In a monotonic relationship, the scores on both variables tend to move in the same general direction, but they could move at different rates, resulting in a curved rather than a straight line.^{8} In contrast, in a linear relationship, the scores move in the same direction and the same rate, resulting in a straight line.^{4,8} A curved distribution of pairs of scores could be monotonic but would not be linear.^{4,8} A curved distribution could be analyzed with the Spearman or Kendall test, because it only requires a monotonic relationship.^{1,8} To use Pearson's, linearity must be established. Figure 3 depicts various bivariate relationships. The first 4 (Figures 3 A-D) are examples of linear relationships. The fifth example would be a linear relationship if the outlier in the upper right corner were removed. Figures 3 F and G are curved and thus are monotonic but not linear. The final figure panel depicts 2 variables that are not related to each other in any fashion.^{8}

FIGURE 3: Assessing linearity: scatterplots.

Homoscedasticity
The last assumption is that the variance of the errors is similar.^{8} That means, the scatter pattern around the linear line should be constant and not appear in patterns nor fan out as the scores go from right to left. Figure 4 provides an example of homoscedasticity and heteroscedasticity. If data violate this assumption, either Spearman or Kendall test could be used to assess correlation.^{8}

FIGURE 4: Assessing linearity: residual plots.

REGRESSION ANALYSIS
Unlike simple correlation, simple linear regression allows the researcher to designate 1 of the 2 variables as the IV and the other as the DV.^{1,8} The goal is to predict the DV based on knowledge of the IV. Regression is a parametric test, so all of the assumptions associated with Pearson's are applicable to linear regression.^{3,8} However, regression has a few additional assumptions related to residuals. Residuals (random error) are the difference between the expected score and the actual score for each pair.^{8} To simplify this, if the data are linear, then there should be no pattern associated with the residuals.^{3,8} One easy way to look at residuals is seen in Figure 4 . The first picture is an example of linearity of residuals (it looks like a random pattern, because this is random error). The other 3 pictures show some type of pattern, which indicates the residuals are not linear. If this assumption is violated, then a different statistical test will need to be used.^{8}

Interpreting Correlation
It is assumed that, at this point, the data are sent off to the statistician, but it is possible to do these simple calculations yourself using a program such as SPSS. Regardless of who runs the analysis, the researcher will need to understand how to interpret the findings. In correlation, the researcher looks at both the P value (statistical significance, usually set at <.05) and rho (r). Rho indicates the strength of a relationship and can range from −1.0 to +1.0.^{8} Although different resources provide different interpretations of rho (r), the most common guidelines considers that a small association or a weak association would be a r of 0.1 to 0.3, a moderate association would be a r between 0.30 and 0.50, and a strong association would be a r greater than 0.50.^{8} To determine how much of the variance 2 variables share, square r and multiply by 100. Thus, for a r of 0.75 between 2 variables, it would be correct to say that the proportion of the variance they share is 56% (0.75 × 0.75).

One issue that is often not addressed is the CI associated with various relationships. If using SPSS, this is not an option that can be selected when running the analysis. However, there are various other resources to make this calculation. To use the CI calculators (Table 4 ), the researcher will enter the rho (r) and the sample size.^{9} In the example (Table 5 ) provided, it is possible to see that a larger sample size and a larger rho (r) result in a narrower CI. Thus, the researcher can be 95% or 99% confident that the true population mean for this correlation lies between these 2 numbers.^{10}

Table 4: Calculators to Determine Confidence Intervals for Correlation Studies

Table 5: Confidence Interval (CI) Table

Interpreting regression is a bit trickier. The statistical data-run will report a rho (r), which is interpreted as in correlation.^{1,8} Rho squared is the percentage of variance shared by the variables.^{8} To determine significance (P value), an analysis of variance is reported (F test).^{1,8} Finally, to predict the DV from the IV, a coefficient model is reported.^{8} The number for the IV is reported as an unstandardized coefficient and indicates how much the DV will increase based on 1 unit of the IV.^{1,2,8} For example, if the DV was a quality of life score and the IV was hours of sleep and the unstandardized coefficient for hours of sleep was 4, then the researcher would say that, for every hour of sleep, the quality of life score increased by 4 points.

A SPECIAL CASE: BLAND-ALTMAN
It is worth mentioning that there are times when simple correlations or linear regression are not appropriate ways to assess agreement between 2 similar measures of the same construct or 2 similar devices (such as scales or thermometers).^{9} For example, correlation would be appropriate to determine if there is a relationship between height and weight, but it would not be appropriate if the researcher wanted to know whether 2 scales weighed people the same. Because both devices (scales) provide weight data, they should be highly correlated, but that does not mean they are equal or the same. To determine the degree of agreement between 2 scales, the researcher would evaluate the differences between the 2 scales. This requires a special test called the Bland-Altman.^{9} Although it is not important to understand how to run a Bland-Altman, it is important to remember that, when comparing 2 things that are supposed to be the same, correlation is not the best choice. Thus, to compare the accuracy and precision of 1 device compared with those of a similar device, the researcher would need to test for differences using Bland-Altman.^{9} It is important to note that this test will require statistical assistance.

CONCLUSION
Correlation design is very useful either when there is no reason to assume cause and effect or when assessing for cause and effect would not be feasible or ethical. Correlation (simple and predictive) research study designs provide a relatively easy and inexpensive design for the clinical nurse specialist to implement. The type of analysis used for a correlation study will depend upon the research questions (correlation, regression, Bland-Altman), the level of the data (nominal, ordinal, interval, or ratio), and the assumption tests (normality, linearity, and homoscedasticity). It is important for the clinical nurse specialist to not only know how to design a correlation study but also understand how to correctly interpret the findings. Identifying relationships or comparing differences is an essential step in discovering new knowledge that will impact patient outcomes and nursing care.

References
1. Adams KA, Lawrence EK.

Research Methods, Statistics, and Applications . 2nd ed. Los Angeles, CA: Sage; 2019.

2. Polit DF, Beck CT.

Nursing Research : Generating and Assessing Evidence for Nursing Practice. 10th ed. Philadelphia, PA: Wolters Kluwer; 2017.

3. Marrin WE, Bridgmon KD.

Quantitative and Statistical Research Methods: From Hypothesis to Results . San Francisco, CA: Jossey-Bass; 2012.

4. Grove SK, Cipher DJ.

Statistics for Nursing Research : A Workbook for Evidence-Based Practice . 2nd ed. St Louis, MO: Elsevier; 2017.

5. Schonbrodt FD, Perugini M. At what sample size do correlations stabilize?

J Res Personality . 2013;47(5):609–612.

6. Cohen J.

Statistical Power Analysis for Behavioral Sciences . New York, NY: Academic Press; 2013.

7. Hung M, Bounsanga J, Voss MW. Interpretation of correlations in clinical research.

Postgrad Med . 2017;129(8):902–906. doi:.

8. Mertler CA, Reinhart RV.

Advanced and Multivariate Statistical Methods: Practical Application and Interpretation . New York, NY: Routledge; 2017.

9. Kalra Aakshi. Decoding the Bland–Altman plot: basic review.

Journal of the Practice of Cardiovascular Sciences . 2017;3(1):36.

10. Moinester M, Gottfried R. Sample size estimation for correlations with pre-specified confidence intervals.

The Quantitative Methods for Psychology . 2014;10(2):124–130.