Learning Objectives. IQ Scores of Students. Correlation. Correlation and Simple Linear Regression

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1 NURS56 INTRODUCTION TO STATISTICS Learning Objectives Correlation and Simple Linear Regression Patsy P.H. Chau ( To examine how two continuous variables are related Adapted from Dr Daniel Fong when we do not have data from the population School of Nursing, The University of Hong Kong 2 IQ Scores of Students ~ Willerman et al. (99) Correlation - To summarize the relationship between two continuous variables A sample of 40 right-handed students were randomly selected The following IQ scores were measured: Full Scale IQ (FSIQ) Verbal IQ (VIQ) Performance IQ (PIQ) Other measurements Gender, height, weight, MRI counts of non-gray pixels in the brain Is VIQ and PIQ associated? 3 IQ scores are approximately continuous 4

2 Scatter Plot of VIQ against PIQ VIQ PIQ Do they look associated? VIQ PIQ Pearson Correlation Coefficient It is good to have a single number, in addition to a graph, to measure the strength of a linear relationship Correlation measures the linear association between two continuous variables. It takes values between - and. There are many types of correlation coefficient 6 Example Continued Measuring Linear Association between VIQ and PIQ Let Y be VIQ, and X be PIQ Let n be the sample size Pearson correlation coefficient Is an Association Strong? Size of r Interpretation Very high correlation High correlation Moderate correlation Low correlation Little if any correlation Is the association strong? VIQ PIQ 7 Perfect -ve linear association No linear association Perfect +ve linear association 8

3 Correlation measures linear association Some Bivariate Relationships Another Illustrating Example (a) Positive linear association No relationship (b) Negative linear relationship Nonlinear relationship A sample of 5 patients were gathered Their depression and anxiety levels were measured Is depression and anxiety associated? Depression X i Anxiety Y i (c) (d) 9 r 0.69 Is it a high correlation? 0 Use and Testing of Pearson Correlation Coefficient H 0 : Two variables are not linearly associated H A : Two variables are linearly associated Assumptions. both variables follow a Normal distribution 2. data obtained from different subjects are independent, i.e. data measured from one subject will not increase our knowledge of the data measured from the other subjects. P-value Q & A. Decide true or false for each of the followings - indicates no linear relationship 0.9 indicates a very high association 0 indicates no relationship at all Pearson s correlation coefficient assumes a Normal distribution for each of the two variables True or False? 2

4 Example Revisited How does IQ Change with Brain Size? Simple Linear Regression - To examine the effects of one variable on an outcome variable of interest FSIQ denotes full scale IQ and MRI_C denotes brain size FSIQ MRI _ C FSIQ For every 00 units increase in MRI_C, we expect FSIQ increases by.2 units How is the above equation obtained? 3 MRI_C 4 Least Squares Estimation 30 0 Errors Some Terminologies Simple Linear Regression of Y on X Regression equation for Y regressing on X y = a + bx FSIQ = (Brain size) FSIQ y = a + bx MRI_C a and b are determined such that the sum of all squared errors is minimized 5 Dependent variable Independent variable The independent variable is the hypothesized cause of, or influence on, the dependent variable 6

5 Another Illustrating Example A sample of 5 patients were gathered Their depression levels and age were measured Estimating the Depression Level of a Person of Age 20? Regression equation: Y X Depression Y i Age X i Does depression change with age? Regression equation: Y X 7 DEPRESSI AGE Expected depression level of a person of age 20 is (20) = Taller Students Had Larger Brain Size? student did not have height measured Only 39 students were analyzed How Does Brain Size Change with Height? X = height Y = brain size Regression equation: Y X 0000 MRI_C Pearson correlation coefficient (r) = 0. HEIGHT Strength of the association? 9 20

6 What do you expect their difference in brain size? Regression equation: Y X What do you expect their difference in brain size? Regression equation: Y X Student A Height = inches Student B Height = 75 inches Student C Height = h inches Student D Height = h+2 inches () Expected (75) = brain size = 958 The brain size of student A is expected to be ( ) = 5505 units smaller than that of student 2 B Expected ( h) ( h 2) brain size The brain size of student C is expected to be 02(2) = 352 units smaller than that of student D Simple Linear Regression Q & A Assumptions. the errors follow a Normal distribution 2. data obtained from different subjects are independent, i.e. data measured from one subject will not increase our knowledge of the data measured from the other subjects. 2. In a simple linear regression, estimates of a regression equation are determined by minimizing the total prediction errors The linear relationship between a dependent variable and an independent variable is examined True or False? 23 24