Background. research questions 25/09/2012. Application of Correlation and Regression Analyses in Pharmacy. Which Approach Is Appropriate When?

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1 Background Let s say we would like to know: Application of Correlation and Regression Analyses in Pharmacy 25 th September, 2012 Mohamed Izham MI, PhD the association between life satisfaction and stress level does income inequality cause health and social problems? is the grade student receives relate with the number of hours he or she uses to study research questions Let s say we would like to know: Which Approach Is Appropriate When? Choosing the right statistical method for the data is the key statistical expertise that you need to have. the association between life satisfaction and stress level does income inequality cause health and social problems? is the grade student receives relate with the number of hours he or she uses to study 1

2 Do I Need to Know the Formulas? You do need to understand the concept behind them and the general statistical concepts imbedded in the use of the formulas. Questions: What does it mean to say that two variables are associated with one another? How can we statistically formalize the concept of association? HAPPINESS Type of tests for measuring association: Pearson s correlation Spearman s rho correlation (or rank correlation) Point-biserial correlation Risk ratio Kappa ( ) Chi-square test Fisher s exact test COFFEE CONSUMPTION 2

3 What is the Purpose of Correlation? Correlation determines whether values of one variable are related to another. A measure of how strong of a LINEAR relationship the explanatory (x) and response (y) variable have: the correlation coefficient (r). Pearson s Correlation Properties The sign of a correlation coefficient gives the direction of the association. Correlation is always between -1 and +1 Strong correlation: close to -1 or +1 Weak correlation: near zero So how do we decide whether correlation test is appropriate? Research question? Hypothesis? Study objective? Are you looking for an association/a relationship? Variables involved; identify your. Independent variable (x): is a variable that can be controlled or manipulated. Dependent/Outcome variable (y): is a variable that cannot be controlled or manipulated. Its values are predicted from the independent variable. Scale of measurement? The two variables must be quantitative (interval/ratio) variables and normally distributed Pearson s Correlation Before you use correlation, you must check several conditions: Quantitative Variables Condition Both are continuous and normally distributed Use Shapiro-Wilk or Kolmogorov Smirnov test Straight Enough Condition Outlier Condition HOW??? 3

4 Performing normality test with S-W or K-S test SPSS Choose Descriptive Statistics Explore: choose both statistics and plots Click Normality plots with tests If p 0.05, conclude that the data comes from a normal distribution If one of the variables is not normally distributed, Pearson s correlation is not appropriate Use Spearman s rho correlation Use Spearman s rho correlation if you have ordinal data Looking at Scatterplots The Use of Scatter Plots 4

5 Looking at Scatterplots (cont.) Looking at Scatterplots (cont.) Linear Not Linear It is a curve Looking at Scatterplots (cont.) Looking at Scatterplots (cont.) Unusual features: Look for the unexpected. Are thereoutliers? Are there clusters or subgroups? Variables: The x-axis: explanatory variable (independent) The y-axis: the response variable (dependent) 5

6 Looking at Scatterplots (cont.) Looking at Scatterplots (cont.) Outliers outlier how to build scatterplot? SPSS Go to Graphs, the Chart Builder Under Gallery tab, select Scatter/Dot Drag into Chart Preview Area Transfer the independent and dependent variables into the x-axis and y-axis, respectively»ok 6

7 Correlation examples Correlation is sensitive to outliers. A single outlying value can make a small correlation large or make a large one small. What should you do? Ensure the data is not an error Run the test with the outlier The run a test without the outlier Correlation coefficient, r The value of r can range between -1 and + 1. If r = 0, then there is no correlation between the two variables. If r = 1 (or -1), then there is a perfect positive (or negative) relationship between the two variables. Crude rule of thumb (Colton, 1974) 0 to 0.25 (or 0.25): little/no correlation 0.26 to 0.50 (or -025 to -0.50): moderate 0.51 to 0.75 (or -051 to -075): moderate to good 0.76 and greater (-0.75 or greater): very good to excellent Test of Correlation Null hypothesis: correlation is zero Test statistic is t = r [(n-2)/(1-r 2 )] 0.5 The statistic is distributed as Student t distribution with n-2 degrees of freedom Rule: reject null hypothesis if p 0.05; conclusion: correlation is not zero; there is a significant association between x and y Then interpret r value 28 7

8 Hands-on session with SPSS First check for normality, then draw scatterplot. Finally, run correlation analysis. Your statistics teacher tells you that the correlation between Exam 1 and Exam 2 is 0.75 In general, if one did poorly in exam 1, would he/she do poorly in exam 2? A. Certainly. The strong positive correlation assures the person will perform poorly in the 2 nd exam as well B. Likely. The strong positive correlation means the person is likely to perform poorly in the 2 nd exam as well C. No. Performance in 1 exam has no influence on the other. Correlation Causation Correlation Causation Whenever we have a strong correlation, it is tempting to explain it by imagining that the predictor variable has caused the response to help. Scatterplots and correlation coefficients never prove causation. A hidden variable that stands behind a relationship and determines it by simultaneously affecting the other two variables is called a lurking variable (a confounder). 8

9 Complexity of interrelationships among many variables; Relationship between pharmacy student s high school grade and college grades. But others variables are involved too such as: IQ, hours of study, influence of parents, motivation, age, and instructors. Take Home Lesson Correlation measures association and not causation. Correlation assumes linear relationship. Values range between 1 and +1 and measure the strength and direction of the relationship. Need to plot the data to check for linearity and outlier. 34 Background Let s say we would like to know: Linear Regression Analysis does job diversity of QU-CPH faculty members affect their health and quality of life? does income inequality cause health and social problems? does number of hours student will study a day affect his/her final grade? 9

10 What is Regression Analysis? When can we use Regression Analysis? Concerns prediction or estimation of outcome variable, based on value of another variable (or variables) Regression analysis involves identifying the relationship between a dependent variable and one or more independent variables. To find cause and effect; causal relationship When correlation analysis shows significant relationship; there is an association between x (independent variable) and y (dependent variable) then proceed with causal relationship For simple linear regression (or least squares regression): describe the relationship between a single dependent variable y and a single independent variable x example We use model to describe the relationship between x and y ; use x to predict y y = mx + c c = y-intercept (value y when x=0) m = slope of the line y = outcome variable x = explanatory variable Suppose QU clinic is investigating the relationship between stress and blood pressure. Assume that both a stress test score and a blood pressure reading have been recorded for a sample of 20 patients. n=20 x = stress level (score = continuous) y = blood pressure (mmhg = continuous) Model, y= x m = 0.49 c =

11 What are the required conditions? Both variables are continuous data. Both variables are normally distributed (bivariate normal). Each variable will be considered normal if its skewness and kurtosis statistics fall between 1.0 and +1.0 or if the sample size is sufficiently large to apply the Central Limit theorem. Or run S-W or K-S test If data is not normal, option for data transformation. The relationship between the two variables is linear. test of linearity desired outcome: fail to reject null hypothesis The variance of the values of the dependent variable is uniform for all values of the independent variable (equality of variance). Assumption of homoscedasticity 11

12 Female Life Expectancy Assumption of Homoscedasticity Homoscedasticity (equality of variances) means that the points are evenly dispersed on either side of the regression line for the linear relationship. In this scatterplot, the points extend about the same distance above and below the regression line for most of the length of the regression line. This scatterplot meets the assumption of homoscedasticity Infant Mortality Rate Infant Mortality Rate In this scatterplot, the spread of the points around the regression line is narrower at the left end of the regression line than at the right end of the regression line. This funnel shape is typical of a scatterplot showing violations of the assumption of homoscedasticity Birth Rate Test of Homoscedasticity When we compared groups, we used the Levene test of population variances to test for the assumption that the group variances were equal. In order to use this test for the assumption of homoscedasity, we will convert the interval level independent variable into a dichotomous variable with low scores in one group and high scores in the other group. We can then compare the variances of the two groups derived from the independent variable. Desired outcome: fail to reject the null hypothesis Next issue Residuals are potential errors (difference between the observed and predicted by the model) Elements of variation unexplained by the fitted model Check if residuals are too large Need to minimize the residual as small as possible Plot of the residuals must show normal distribution Example: data obtained from a chemical process Yield of the process is thought to be related to the reaction temperature 12

13 Hands-on session with SPSS First check for normality, then draw scatterplot. Next, run correlation analysis, finally run regression. Findings Model: y = 1.995x

14 "Do the independent variables (reaction temperature, x) reliably predict the dependent variable (product of the chemical process, y)? YES; y = 1.995x We can now predict (y) given any value of (x) within the range (check min and max values). Significantly correlated, p 0.05; r = 0.992, a very strong positive correlation R-Square is the proportion of variance in the dependent variable (product of chemical process) which can be predicted from the independent variable (reaction temp.) it indicates that 98.4% of the variance in y can be predicted from the variable x Take Home Lesson Regression measures causal relationship. Regression assumes linear relationship. Both variables, x and y must show normal distribution. No point of running regression analysis if correlation analysis does not show any significant relationship. Need to plot the data to check for linearity and outlier. Need to ensure for equality of variance Able to interpret the output from the 3 tables: correlation, model summary, R-square 54 14