Correlation and Simple Linear Regression

Statistical Concepts Series

Kelly H. Zou, PhD Correlation and Simple Linear Kemal Tuncali, MD 1 Stuart G. Silverman, MD Regression

Index terms: In this tutorial article, the concepts of correlation and regression are reviewed and adiology Data analysis demonstrated. The authors review and compare two correlation coefficients, the Statistical analysis R Pearson correlation coefficient and the Spearman ␳, for measuring linear and non- Published online linear relationships between two continuous variables. In the case of measuring the 10.1148/radiol.2273011499 linear relationship between a predictor and an outcome variable, simple linear Radiology 2003; 227:617–628 regression analysis is conducted. These statistical concepts are illustrated by using a data set from published literature to assess a computed tomography–guided inter- 1 From the Department of Radiology, ventional technique. These statistical methods are important for exploring the Brigham and Women’s Hospital (K.H.Z., relationships between variables and can be applied to many radiologic studies. K.T., S.G.S.) and Department of Health © RSNA, 2003 Care Policy (K.H.Z.), Harvard Medical School, 180 Longwood Ave, Boston, MA 02115. Received September 10, 2001; revision requested October 31; revision received December 26; ac- cepted January 21, 2002. Address correspondence to K.H.Z. (e-mail: Results of clinical studies frequently yield data that are dependent of each other (eg, total [email protected]). procedure time versus the dose in computed tomographic [CT] fluoroscopy, signal-to- © RSNA, 2003 noise ratio versus number of signals acquired during magnetic resonance imaging, and cigarette smoking versus lung cancer). The statistical concepts correlation and regression, which are used to evaluate the relationship between two continuous variables, are reviewed and demonstrated in this article. Analyses between two variables may focus on (a) any association between the variables, (b) the value of one variable in predicting the other, and (c) the amount of agreement. Agreement will be discussed in a future article. Regression analysis focuses on the form of the relationship between variables, while the objective of correlation analysis is to gain insight into the strength of the relationship (1,2). Note that these two techniques are used to investigate relationships between continuous variables, whereas the ␹2 test is an example of a test for association between categorical variables. Continuous variables, such as procedure time, patient age, and number of lesions, have no gaps on the measurement scale. In contrast, categorical variables, such as patient sex and tissue classification based on segmentation, have gaps in their possible values. These two types of variables and the assumptions about their measurement scales were reviewed and distinguished in an article by Applegate and Crewson (3) published earlier in this Statistical Concepts Series in Radiology. Specifically, the topics covered herein include two commonly used correlation coefficients, the Pearson correlation coefficient (4,5) and the Spearman ␳ (6–10) for measuring linear and nonlinear relationship, respectively, between two continuous variables. Corre- lation analysis is often conducted in a retrospective or observational study. In a clinical trial, on the other hand, the investigator may also wish to manipulate the values of one variable and assess the changes in values of another variable. To evaluate the relative impact of the predictor variable on the particular outcome, simple regression analysis is preferred. We illustrate these statistical concepts with existing data to assess patient skin dose based on total procedure time by using a quick-check method in CT fluoroscopy– guided abdominal interventions (11). These statistical methods are useful tools for assessing the relationships between continuous variables collected from a clinical study. However, it is also important to distin- guish these statistical methods. While they are similar mathematically, their purposes are different. Correlation analysis is generally overused. It is often interpreted incorrectly (to establish “causation”) and should be reserved for generating hypotheses rather than for testing them. On the other hand, regression modeling is a more useful statistical technique that allows us to assess the strength of the relationships in the data and the uncertainty in the model by using confidence intervals (12,13).

617 Figure 1. Scatterplots of four sets of data generated by means of the following Pearson correlation coefﬁcients (from left to right): r ϭ 0 adiology (uncorrelated data), r ϭ 0.8 (strongly positively correlated), r ϭ 1.0 (perfectly positively correlated), and r ϭϪ1 (perfectly negatively correlated). R

CORRELATION Rank Correlation TABLE 1 The Spearman ␳ is the sample correla- Interpretation of Correlation The purpose of correlation analysis is tion coefficient (r ) of the ranks (the rel- Coefficient to measure and interpret the strength of s ative order) based on continuous data a linear or nonlinear (eg, exponential, Correlation Direction and Strength (19,20). It was first introduced by Spear- polynomial, and logistic) relationship be- Coefficient Value of Correlation man in 1904 (6). The Spearman ␳ is used tween two continuous variables. When Ϫ to measure the monotonic relationship 1.0 Perfectly negative conducting correlation analysis, we use Ϫ0.8 Strongly negative between two variables (ie, whether one Ϫ the term association to mean “linear asso- 0.5 Moderately negative variable tends to take either a larger or Ϫ0.2 Weakly negative ciation” (1,2). Herein, we focus on the smaller value, though not necessarily lin- 0.0 No association Pearson and Spearman ␳ correlation co- ϩ early) by increasing the value of the other 0.2 Weakly positive efficients. Both correlation coefficients ϩ0.5 Moderately positive variable. ϩ take on values between Ϫ1 and ϩ1, rang- 0.8 Strongly positive ϩ1.0 Perfectly positive ing from being negatively correlated (Ϫ1) Linear versus Rank Correlation to uncorrelated (0) to positively corre- Note.—The sign of the correlation coefficient Coefficients lated (ϩ1). The sign of the correlation (ie, positive or negative) defines the direction of the relationship. The absolute value indi- coefficient (ie, positive or negative) de- The Pearson correlation coefficient ne- cates the strength of the correlation. fines the direction of the relationship. cessitates use of interval or continuous The absolute value indicates the strength measurement scales of the measured out- of the correlation (Table 1, Fig 1). We come in the study population. In con- elaborate on two correlation coefficients, trast, rank correlations also work well ␳ linear (eg, Pearson) and rank (eg, Spear- with ordinal rating data, and continuous Spearman can be found in the literature man), that are commonly used for mea- data are reduced to their ranks (Appendix (20,21). suring linear and general relationships C) (20,21). The rank procedure will also between two variables. be illustrated briefly with our example Limitations and Precautions data. The smallest value in the sample It is worth noting that even if two vari- Linear Correlation has rank 1, and the largest has the high- ables (eg, cigarette smoking and lung est rank. In general, rank correlations are The Pearson correlation coefficient is cancer) are highly correlated, it is not not easily influenced by the presence of also known as the sample correlation co- sufficient proof of causation. One vari- skewed data or data that are highly vari- efficient (r), product-moment correlation able may cause the other or vice versa, or able. coefficient, or coefficient of correlation a third factor is involved, or a rare event (14). It was introduced by Galton in 1877 may have occurred. To conclude causa- Statistical Hypothesis Tests for a (15,16) and developed later by Pearson tion, the causal variables must precede Correlation Coefficient (17). It measures the linear relationship the variable it causes, and several con- between two random variables. For ex- The null hypothesis states that the un- ditions must be met (eg, reversibility, ample, when the value of the predictor is derlying linear correlation has a hypoth- strength, and exposure response on the ␳ basis of the Bradford-Hill criteria or the manipulated (increased or decreased) by esized value, 0. The one-sided alterna- a fixed amount, the outcome variable tive hypothesis is that the underlying Rubin causal model) (23–26). ␳ changes proportionally (linearly). A lin- value exceeds (or is less than) 0. When ear correlation coefficient can be com- the sample size (n) of the paired data is SIMPLE LINEAR REGRESSION puted by means of the data and their large (n Ն 30 for each variable), the stan- sample means (Appendix A). When a sci- dard error (s) of the linear correlation (r) The purpose of simple regression analysis entific study is planned, the required is approximately s(r) ϭ (1 Ϫ r2)/͌n. The is to evaluate the relative impact of a Ϫ␳ sample size may be computed on the ba- test statistic value (r 0)/s(r) may be predictor variable on a particular out- sis of a certain hypothesized value with computed by means of the z test (22). If come. This is different from a correlation the desired statistical power at a specified the P value is below .05, the null hypoth- analysis, where the purpose is to examine level of significance (Appendix B) (18). esis is rejected. The P value based on the the strength and direction of the rela-

618 ⅐ Radiology ⅐ June 2003 Zou et al the relationship for prediction and estima- performed. (a) To understand whether tion, and (d) assess whether the data ﬁt the assumptions have been met, deter- these criteria before the equation is applied mine the magnitude of the gap between for prediction and estimation. the data and the assumptions of the model. (b) No matter how strong a rela- Least Squares Method tionship is demonstrated with regression analysis, it should not be interpreted as The main goal of linear regression is to causation (as in the correlation analysis). ﬁt a straight line through the data that (c) The regression should not be used to predicts Y based on X. To estimate the in- predict or estimate outside the range of tercept and slope regression parameters values of the independent variable of the that determine this line, the least squares sample (eg, extrapolation of radiation method is commonly used. It is not neces- cancer risk from the Hiroshima data to sary for the errors to have a normal distri- adiology that of diagnostic radiologic tests). bution, although the regression analysis is

R more efficient with this assumption (27). With this regression method, a set of re- AN EXAMPLE: DOSE VERSUS Figure 2. Simple linear regression model gression parameters are found such that TOTAL PROCEDURE TIME shows that the expectation of the dependent the sum of squared residuals (ie, the differ- IN CT FLUOROSCOPY variable Y is linear in the independent variable ences between the observed values of the ϭ ϭ X, with an intercept a 1.0 and a slope b outcome variable and the fitted values) are 2.0. We applied these statistical methods to minimized (14). The fitted y value is then help assess the benefit of the use of CT computed as a function of the given x fluoroscopy to guide interventions in the value and the estimated intercept and abdomen (11). During CT fluoroscopy– slope regression parameter (Appendix D). tionship between two random variables. guided interventions, one might postu- For example, in Equation (1), once the es- In this article, we deal with only linear late that the radiation dose received by a timates of a and b are obtained from the regression of one continuous variable on patient is related to (or correlated with) regression analysis, the predicted y value at another continuous variable with no the total procedure time, because the any given x value is calculated as a ϩ bx. gaps on each measurement scale (3). more difficult the procedure is, the more There are other types of regression (eg, CT fluoroscopic scanning is required, Coefficient of Determination, R2 multiple linear, logistic, and ordinal) which means a longer procedure time. analyses, which will be provided in a fu- It is meaningful to interpret the value of The rationale was to assess whether radi- ture article in this Statistical Concepts the Pearson correlation coefficient r by ation dose could be estimated by simply Series in Radiology. squaring it; hence, the term R-square (R2) measuring the total CT fluoroscopic pro- A simple regression model contains or coefficient of determination. This mea- cedure time, with the null hypothesis only one independent (explanatory) vari- sure (with a range of 0–1) is the fraction of that the slope of the regression line is 0. ϭ able, Xi, for i 1,...,n subjects, and is the variability in Y that can be explained Earlier, we discussed two methods to linear with respect to both the regression by the variability in X through their linear target lesions with CT fluoroscopy. In parameters and the dependent variable. relationship, or vice versa. That is, R2 ϭ one method, continuous CT scanning is

The corresponding dependent (outcome) SSregression/SStotal, where SS stands for the used during needle placement. In the variable is labeled. The model is ex- sum of squares. Note that R2 is calculated other method, short CT scanning is used pressed as only on the basis of the Pearson correlation to image the needle after it is placed. The coefficient in the linear regression analysis. latter method, the so-called quick-check Y ϭ a ϩ bX ϩ e , (1) i i i Thus, it is not appropriate to compute R2 method, has been adopted almost exclu- where the regression parameter a is the in- on the basis of rank correlation coefficients sively at our institution. Now, we demon- tercept (on the y axis), and the regression such as the Spearman ␳. strate correlation and regression analyses parameter b is the slope of the regression based on a subset of the interventional ϭ line (Fig 2). The random error term ei is Statistical Hypothesis Tests procedures (n 19). With the quick- assumed to be uncorrelated, with a mean check method, we examine the relation- There are several hypotheses in the of 0 and constant variance. For conve- ship between total procedure time (in context of regression analysis, for exam- nience in inference and improved effi- minutes) and dose (in rads) on a natural ple, to test if the slope of the regression ciency in estimation (27), analyses often log scale. We also examine the marginal line is b ϭ 0 (hypothesis, there is no lin- incur an additional assumption that the ranks of the x (log of total time) and y ear association between Y and X). One errors are distributed normally. Transfor- (log of dose) components (Table 2). For may also test whether intercept a takes mation of the data to achieve normality convenience, the x data are given in as- on a certain value. The significance of the may be applied (28,29). Thus, the word line cending order. effects of the intercept and slope may (linear, independent, normal, equal vari- In Table 2, each set of rank data is also be computed by means of a Student ance) summarizes these requirements. derived by first placing the 19 observa- t statistic introduced earlier in this Statis- Typical steps for regression model anal- tions in each sample in ascending order tical Concepts Series in Radiology (30). ysis are the following: (a) determine if the and then assigning ranks 1–19. Ties are assumptions underlying a normal relation- broken by means of averaging the respec- Limitations and Precautions ship are met in the data, (b) obtain the tive adjacent ranks. Finally, the ranks are equation that best fits the data, (c) evaluate The following understandings should identified for the observations of each of the equation to determine the strength of be considered when regression analysis is the paired x and y samples.

Volume 227 ⅐ Number 3 Correlation and Simple Linear Regression ⅐ 619 The natural log (ln) transformation of the total time is used to make the data TABLE 2 Total Procedure Time and Dose of CT Fluoroscopy–guided Procedures, by appear normal, for more efﬁcient analy- Means of the Quick-Check Method sis (Appendix D), with normality veriﬁed statistically (31). However, normality is Subject x Data: Log Time Ranks of y Data: Log Dose Ranks of not necessary in the subsequent regres- No. (ln[min]) x Data (ln[rad]) y Data sion analysis. We created a scatterplot of 1 3.61 1 1.48 2 the data, with the log of dose (ln[rad]) on 2 3.87 2 1.24 1 the x axis and the log of total time (ln- 3 3.95 3 2.08 5.5 4 4.04 4 1.70 3 [minutes]) on the y axis (Fig 3). 5 4.06 5 2.08 5.5 For illustration purposes, we will con- 6 4.11 6 2.94 10 duct both correlation and regression 7 4.19 7 2.24 7 analyses; however, the choice of analysis 8 4.20 8 1.85 4

adiology 9 4.32 9.5 2.84 9 depends on the aim of research. For ex- 10 4.32 9.5 3.93 16

R ample, if the investigators wish to assess 11 4.42 11.5 3.03 11 whether there is a relationship between 12 4.42 11.5 3.23 13 time and dose, then correlation analysis 13 4.45 13 3.87 15 is appropriate. In comparison, if the in- 14 4.50 14 3.55 14 15 4.52 15 2.81 8 vestigators wish to evaluate the impact of 16 4.57 16 4.07 17 the total time on the resulting dose, then 17 4.58 17 4.44 19 regression analysis is preferred. 18 4.61 18 3.16 12 19 4.74 19 4.19 18 Source.—Reference 11. Correlations Note.—Paired x and y data are sorted according to the x component; therefore, the log of the To compute the Spearman ␳ with a Pear- total procedure time and the log of the corresponding rank have an increasing order. When ties are ϭ present in the data, the average of their adjacent ranks is used. Pearson correlation coefficient son correlation coefficient of r 0.85, the between log time and log dose, r ϭ 0.85; Spearman ␳ϭ0.84. marginal ranks of time and dose were de- ϭ rived separately; consequently, rs 0.84. Both correlation coefficients confirm that the log of total time and the log of dose are correlated strongly and positively.

Regression We first conducted a simple linear regression analysis of the data on a log scale (n ϭ 19); results are shown in Table 3. The value calculated for R2 was 0.73, which suggests that 73% of the variability of the data could be explained by the linear regression. Figure 3. Scatterplot of the log of dose (y Figure 4. Scatterplot of the log of dose (y The regression line, expressed in the axis) versus the log of total time (x axis). Each axis) versus the log of total time (x axis). The form given in Equation (1), is Y ϭ point in the scatterplot represents the values of regression line has the intercept a ϭϪ9.28 and Ϫ9.28 ϩ 2.83X, where the predictor vari- two variables for a given observation. slope b ϭ 2.83. We conclude that there is a able X represents the log of total time, possible association between the radiation and the outcome variable Y represents dose and the total time of the procedure. the log of dose. The estimated regression ϭ parameters are a ϭϪ9.28 (intercept) and total time is specified at x 4 (translated to e4 ϭ 55 minutes, approximately), then the b ϭ 2.83 (slope) (Fig 4). This regression TABLE 3 line can be interpreted as follows: At X ϭ log dose that is to be applied is approxi- Results based on Correlation and ϭϪ ϩ ϫ ϭ 0, the value of Y is Ϫ9.28. For every one- mately y 9.28 2.83 4 2.04 (trans- Regression Analysis for Example 2.04 ϭ unit increase in X, the value of Y will lated to e 7.69 rad). On the other Data ϭ increase on average by 2.83. Effects of hand, if the log total time is specified at x 4.5 ϭ Regression Statistic Numerical Result both the intercept and slope are statisti- 4.5 (translated to e 90 minutes, ap- cally significant (P Ͻ .005) (Excel; Mi- proximately), then the log dose that is to Correlation coefficient r 0.85 ϭϪ ϩ R-square (R2) 0.73 crosoft, Redmond, Wash); therefore, the be applied is approximately y 9.28 2.83 ϫ 4.5 ϭ 3.46 (translated to e3.46 ϭ Regression parameter null hypothesis (H0, the dose remains Intercept Ϫ9.28 constant as the total procedure time in- 31.82 rad). Such prediction can be useful Slope 2.83 for future clinical practice. creases) is rejected. Thus, we confirm the Source.—Reference 11. alternative hypothesis (H1, the dose in- creases in the total procedure time). SUMMARY AND REMARKS The regression line may be used to give predicted values of Y. For example, if in a Two important statistical concepts, cor- commonly in radiology research, are re- future CT fluoroscopy procedure, the log relation and regression, which are used viewed and demonstrated herein. Addi-

620 ⅐ Radiology ⅐ June 2003 Zou et al tional sources of information and elec- ware program (R Software. Available at: lib ranks of the two marginal data by using the tronic textbooks on statistical analysis .stat.cmu.edu/R). Correlation option in Data Analysis Tools methods found on the World Wide Web (Excel; Microsoft) or by using the Cor func- are listed in Appendix E. A glossary of the APPENDIX B tion (Insightful; MathSoft) or the free soft- statistical terms used in this article is pre- ware. sented in Appendix F. Total sample size based on the Pearson When correlation analysis is con- correlation coefficient: Specify r ϭ expected APPENDIX D ducted to measure the association be- correlation coefficient, C ϭ 0.5 ϫ ln[(1 ϩ tween two random variables, either the r)/(1 Ϫ r)], N ϭ total number of subjects Simple regression analysis: Regression Pearson linear correlation coefficient or required, ␣ϭtype I error (ie, significance analysis may be performed by using the the Spearman rank correlation coeffi- level, typically fixed at 0.05), ␤ϭtype II “Regression” option with Data Analysis cient ␳ may be adopted. The former coef- error (ie, 1 minus statistical power, typically Tools (Excel; Microsoft). This regression 2 ficient is used to measure the linear rela- fixed at 0.10). Then N ϭ [(Z␣ ϩ Z␤)/C] ϩ 3, analysis tool yields the sample correlation 2 adiology tionship but is not recommended for use where Z␣ is the inverse of the cumulative R ; estimates of the regression parameters, probability of a standard normal distribu- along with their statistical significance on

R with skewed data or data with extremely large or small values (often called the tion with the tail probability of ␣. Similarly, the basis of the Student t test; residuals; and standardized residuals. Scatter, line fit, and outliers). In contrast, the latter coeffi- Z␤ is the inverse of the cumulative proba- residual plots may also be created. Alterna- cient is used to measures a general asso- bility of a standard normal distribution ␤ tively, the analyses can be performed by ciation, and it is recommended for use with the tail probability of . Consequently, using the function “lsfit” (Insightful; Math- with data that are skewed or that have compute the smallest integer, n, such that n Ն N, as the required sample size. Soft) or the free software. outliers. For example, an investigator wishes to With either program, one may choose to When simple regression analysis is conduct a clinical trial of a paired design transform the data or exclude outliers be- conducted to assess the linear relation- based on a one-tailed hypothesis test of the fore conducting a simple regression analy- ship of a dependent variable as a func- correlation coefficient. The null hypothesis sis. A commonly used variance-stabilizing tion of the independent variable, caution is that the correlation between two vari- transformation is the natural log function must be used when determining which ables is r ϭ 0.60 (ie, C ϭ 0.693) in the (ln) applied to one or both variables. Other of the two variables is viewed as the in- population of interest. The alternative hy- transformation (eg, Box-Cox transforma- dependent variable that makes sense pothesis is that the correlation is r Ͼ 0.60. tion) and weighting methods in regression clinically. A useful graphical aid is a scat- Type I error is fixed to be 0.05 (ie, Z␣ ϭ analysis may also be used (28,29). terplot. Once the regression line is ob- 1.645), while type II error is fixed to be 0.10 ϭ tained, caution should also be used to (ie, Z␤ 1.282). Thus, the required sample APPENDIX E avoid prediction of a y value for any size is N ϭ 21 subjects. A sample size table value of x that is outside the range of the may also be found in reference 18. Uniform resource locator, or URL, links to data. Finally, correlation and regression electronic statistics textbooks: www.davidm analyses do not infer causality, and more APPENDIX C lane.com/hyperstat/index.html, www.statsoft rigorous analyses are required if causal .com/textbook/stathome.html, www.ruf.rice.edu ϳ inference is to be made (23–26). Formula for computing Spearman ␳ and / lane/rvls.html, www.bmj.com/collections /statsbk/index.shtml, espse.ed.psu.edu/statistics Pearson rs: Replace bivariate data, Xi and Yi ϭ ϭ /investigating.htm. (i 1,...,n), by their respective ranks Ri APPENDIX A ϭ rank(Xi) and Si rank(Yi). Rank correlation coefficient, r ,isdefined as the Pearson cor- APPENDIX F Formula for computing the Pearson cor- s relation coefficient between the R and S relation coefficient, r: The formula for com- i i values, which can be computed by means of Glossary of statistical terms: puting r between bivariate data, X and Y i i the formula given in Appendix A. An alter- Bivariate data.—Measurements obtained values (i ϭ 1,...,n)is native direct formula was given by Hett- on more than one variable for the same unit mansperger (19). or subject. n ␳ ͸ ͑ Ϫ ៮͒͑ Ϫ ៮͒ The Spearman may also be computed by Correlation coefficient.—A statistic be- Xi X Yi Y first reducing the continuous data to their tween Ϫ1 and 1 that measures the associa- iϭ1 r ϭ , marginal ranks by using the “rank and per- tion between two variables. n n centile” option with Data Analysis Tools Intercept.—The constant a in the regres- ͸ ͑ Ϫ ៮͒2 ͸͑ Ϫ ៮͒2 (Excel; Microsoft) or the “rank” function sion equation, which is the value for y when ͱ Xi X Yi Y ϭ iϭ1 iϭ1 (Insightful; MathSoft) or the free software. x 0. Both software programs correctly rank the Least squares method.—The regression line where X and Y are the sample means of the data in ascending order. However, the rank that is the best fit to the data for which the

Xi and Yi values, respectively. and percentile option in Excel ranks the sum of the squared residuals is minimized. The Pearson correlation coefficient may data in descending order (the largest is 1). Outlier.—An extreme observation far be computed by means of a computer-based Therefore, to compute the correct ranks, away from the bulk of the data, often statistics program (Excel; Microsoft) by us- one may first multiply all of the data by Ϫ1 caused by faulty measuring equipment or ing the option “Correlation” under the op- and then apply the rank function. Excel recording error. tion “Data Analysis Tools”. Alternatively, it also gives integer ranks in the presence of Pearson correlation coefficient.—Sample may also be computed by means of a ties compared with the methods that yield correlation coefficient for measuring the built-in software function “Cor” (Insightful; possible noninteger ranks, as described in linear relationship between two variables. MathSoft, Seattle, Wash [MathSoft S-Plus 4 the standard statistics literature (19). R2.—The square of the Pearson correla- guide to statistics, 1997; 89–96]. Available Subsequently, the sample correlation co- tion coefficient r, which is the fraction of at: www.insightful.com) or with a free soft- efficient is computed on the basis of the the variability in Y that can be explained by

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