1. Amsterdam Rehabilitation Research Center | Reade Correlation and linear regression analysis Martin van der Esch, Phd

2. Amsterdam Rehabilitation Research Center | Reade Content Correlation and linear regression analysis Association research However, also used in experimental studies

3. Amsterdam Rehabilitation Research Center | Reade Correlation and regression -Interested in relationship/association/correlation -Direction and magnitude of relationship -Dependent or independent variables -Association does not imply a ‘cause and effect’ relationship

4. Amsterdam Rehabilitation Research Center | Reade Correlation

5. Amsterdam Rehabilitation Research Center | Reade Correlation Expressed as productmomentcorrelation Pearson coefficent (r) when data are not skewed or rank order correlation Spearman (rs) when data are ordinal, skewed or in case of presence of outliers. Dimensionless Rage between +1 and –1 (0 = no correlation) Magnitude indicates how close the points are to a straight line (the strength of an association)  +1 or –1: perfect correlation: all points lying on the line 4

6. Amsterdam Rehabilitation Research Center | Reade Between -1 to 1.

7. Amsterdam Rehabilitation Research Center | Reade

9. Correlation coefficient Range: -1 ≤ r ≤ 1. In SPSS

10. Amsterdam Rehabilitation Research Center | Reade Formula correlation

11. Amsterdam Rehabilitation Research Center | Reade Regression analysis

12. Amsterdam Rehabilitation Research Center | Reade Statistical analysis Data were analyzed with SPSS for Windows 16.0 (SPSS Inc). According to their distribution, the various parameters are expressed as mean (± standard deviation) or median (interquartile range). Data with a non-Gaussian distribution was log transformed for analysis if possible. To compare the groups, student’s T-test or Mann-Whitney U test was used when appropriate. Furthermore, correlations between variables were analyzed by using Pearson correlation or Spearman’s rho tests. Univariate linear regression analyses were performed on log-transformed data to investigate the influence of possible confounders (i.e. sex, smoking status, systolic blood pressure and body mass index (BMI) on the results. Wilcoxon signed-rank test was used to investigate the differences in values at baseline and at 8 weeks in the prospectively followed subgroup of patients (n=9). P-values less than 0.05 were considered statistically significant. I C van Eijk, M E Tushuizen, A Sturk, B A C Dijkmans, M Boers, A E Voskuyl, M Diamant, G.J. Wolbink, R Nieuwland and M T Nurmohamed Circulating microparticles remain associated with complement activation despite intensive anti-inflammatory therapy in early rheumatoid arthritis Ann Rheum Dis published online 16 Nov 2009;

13. Amsterdam Rehabilitation Research Center | Reade Typical association question Research question: is there an association between age and pain in patients with …? Hypothesis: pain increases in older patients Y = a + bX + e

14. Amsterdam Rehabilitation Research Center | Reade age pain 50 

15. Amsterdam Rehabilitation Research Center | Reade 14 Simple (uni) linear regression analysis  Difference with correlation analysis: prediction  line that gives the best description of the scatter plot, best fitting line  difficult to draw line by hand  solve problem with mathematical equation

16. Amsterdam Rehabilitation Research Center | Reade Simple (uni) linear regression analysis We use the ‘Method of Least Squares’ to fit the best line Minimal distance between the data and the fitting line

17. Amsterdam Rehabilitation Research Center | Reade age pain 50 

18. Amsterdam Rehabilitation Research Center | Reade Simple regression analysis  1 = difference between age 0 and age 1 difference between age 1 and age 2 ----------------------------------- difference between age 30 and age 31 Pain =  0 +  1 * age What is  0 ? What is  1 ?  1 = Beta=b  0 = pain at age is 0

19. Amsterdam Rehabilitation Research Center | Reade 18 Mathematical equation to describe the relationship y = a + b*x  y is called the dependent (outcome) variable  x is called the independent (predictor, explanatory) variable  a is the intercept: value of y when x=0  b (unstandardized beta) is the ´slope´: it represents the amount by which Y increases on average if we increase x by one unit  a and b are called regression coefficients

20. Amsterdam Rehabilitation Research Center | Reade Simple regression analysis Regression coefficient is equal to the difference in the outcome variable when the determinant one unit changes

21. Amsterdam Rehabilitation Research Center | Reade age pain 50  11 1

22. Amsterdam Rehabilitation Research Center | Reade Simple regression analysis pain = - 20 + 0,5 * age What is –20? What is 0,5?

23. Amsterdam Rehabilitation Research Center | Reade You can also analyse difference between two groups with simple regression analysis.

24. Amsterdam Rehabilitation Research Center | Reade Back to 2 groups and analysis of pain grouppreafter medication75.8 (6.8)65.8 (10.1) placebo75.4 (7.1)68.2 (9.0)

25. Amsterdam Rehabilitation Research Center | Reade

26. Now analysed by simple regression analysis placeboMedication 1 Continuous outcome Pain

27. Amsterdam Rehabilitation Research Center | Reade placeboMedication 1 Continuous outcome Pain

28. Amsterdam Rehabilitation Research Center | Reade Simple regression analysis Regression coefficient is equal to the difference in mean between two comparable groups

29. Amsterdam Rehabilitation Research Center | Reade Simpe (uni) linear regression analysis  1 = mean difference between placebo and medication Pain =  0 +  1 * group placebo = 0; medication = 1  0 = mean in controlegroup

30. Amsterdam Rehabilitation Research Center | Reade 00 11

31. Hypothesis test for β N-2 degrees of freedom

32. Amsterdam Rehabilitation Research Center | Reade P value? t -3,598

33. Amsterdam Rehabilitation Research Center | Reade Back to the example Experimental design Including another medicine Three comparable groups

34. Amsterdam Rehabilitation Research Center | Reade Pain groupToT1 medication175.8 (6.8)65.8 (10.1) medication276.8 (7.5)61.9 (11.7) placebo75.4 (7.1)68.2 (9.0)

35. Amsterdam Rehabilitation Research Center | Reade placebomedication1medication2 Continuous outcome

36. Amsterdam Rehabilitation Research Center | Reade Group analysed as continuous variabele

37. Amsterdam Rehabilitation Research Center | Reade But…, group isn’t a continous variable: a categorical variable Therefore it needs to be analysed by dummy-variables

38. Amsterdam Rehabilitation Research Center | Reade

39. Dummy variables Dummy 1: new medication - placebo Dummy 2: alt. medication - placebo Placebo: controle / control groep

40. Amsterdam Rehabilitation Research Center | Reade Simple regression analysis Pain =  0 +  1 * medicationgroup1 +  2 * medicatiogroup2 What is  0?  0 = mean of placebogroup What is  1?  1 = difference between placebo and medication1 What is  2?  2 = difference between placebo and medication2

41. Amsterdam Rehabilitation Research Center | Reade placebomedication1medication2 Continuous outcome

42. Amsterdam Rehabilitation Research Center | Reade Pain =  0 +  1 * medicationgroup1 +  2 * medicationgroup2

43. Amsterdam Rehabilitation Research Center | Reade Intermezzo Little excercise.. Is there a relationship between your height (cm) and shoesize (european size)… Estimate relationcoefficient… What does that mean? Estimate formula Height = ? + ? * shoesize Group: men/woman. Groups: occupational therapy, physiotherapy, other.

44. Amsterdam Rehabilitation Research Center | Reade Assumption linear regression analysis Linear relationship between x en y Scatter diagram (otherwise Logaritmic transformation (next week) For each value of x, there is a distribution of values of y in the population; this distribution is Normal Analyses of the residuals Variability of the distribution of y values in the population is the same for all values of x, i.e. the variance is constant (s 2 / sd) Analyses of the residuals

45. Amsterdam Rehabilitation Research Center | Reade Checking for linearity Scatterplot Adding a quadratic term Splitting exposure variable into groups (4-5)

46. Amsterdam Rehabilitation Research Center | Reade Adding a quadratic term pain age

47. Amsterdam Rehabilitation Research Center | Reade Checking for linearity Splitting exposure variable into groups

48. Amsterdam Rehabilitation Research Center | Reade Splitting exposure variable into groups pain age 1 2 3 4

49. Amsterdam Rehabilitation Research Center | Reade Example in SPSS Examine the association between age and pain score at baseline. Scatterplot Linear regression analysis Checking for linearity Adding a quadratic term Splitting exposure variable into groups

50. Amsterdam Rehabilitation Research Center | Reade Scatter plot

51. Amsterdam Rehabilitation Research Center | Reade Lineair regression analysis Pain (at baseline) = 56.2 + 0.23 * age

52. Amsterdam Rehabilitation Research Center | Reade Adding a quadratic term

53. Amsterdam Rehabilitation Research Center | Reade Splitting exposure variable into groups Produce categorical age variable Recode to dummy variables Perform linear regression analysis with dummies Are the B’s increasing in a linear order with comparable distance between the dummies?

54. Amsterdam Rehabilitation Research Center | Reade Splitting exposure variable into groups

55. Amsterdam Rehabilitation Research Center | Reade Questions? 54