33851 - ‫بسم الله الرحمن‬...

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Unformatted text preview: ‫بسم الله الرحمن‬ ‫الرحيم‬ Correlation & Regression Dr. Moataza Mahmoud Abdel Wahab Lecturer of Biostatistics High Institute of Public Health University of Alexandria Correlation Finding the relationship between two quantitative variables without being able to infer causal relationships Correlation is a statistical technique used to determine the degree to which two variables are related Scatter diagram • Rectangular coordinate • Two quantitative variables • One variable is called independent (X) and the second is called dependent (Y) • Points are not joined • No frequency table Example Scatter diagram of weight and systolic blood pressure Scatter diagram of weight and systolic blood pressure Scatter plots The pattern of data is indicative of the type of relationship between your two variables: positive relationship negative relationship no relationship Positive relationship Negative relationship Reliability Age of Car No relation Correlation Coefficient Statistic showing the degree of relation between two variables Simple Correlation coefficient (r) It is also called Pearson's correlation or product moment correlation coefficient. It measures the nature and strength between two variables of the quantitative type. The sign of r denotes the nature of association while the value of r denotes the strength of association. If the sign is +ve this means the relation is direct (an increase in one variable is associated with an increase in the other variable and a decrease in one variable is associated with a decrease in the other variable). While if the sign is -ve this means an inverse or indirect relationship (which means an increase in one variable is associated with a decrease in the other). The value of r ranges between ( -1) and ( +1) The value of r denotes the strength of the association as illustrated by the following diagram. strong 1- intermediate -0.75 -0.25 weak weak 0 indirect perfect correlation intermediate 0.25 strong 0.75 1 Direct no relation perfect correlation If r = Zero this means no association or correlation between the two variables. If 0 < r < 0.25 = weak correlation. If 0.25 ≤ r < 0.75 = intermediate correlation. If 0.75 ≤ r < 1 = strong correlation. If r = l = perfect correlation. How to compute the simple correlation coefficient (r) r xy ( x) x n 2 2 x y n 2 ( y) 2 . y n :Example A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as shown in the following table . It is required to find the correlation between age and weight. serial No Age (years) Weight (Kg) 1 7 12 2 6 8 3 8 12 4 5 10 5 6 11 6 9 13 These 2 variables are of the quantitative type, one variable (Age) is called the independent and denoted as (X) variable and the other (weight) is called the dependent and denoted as (Y) variables to find the relation between age and weight compute the simple correlation coefficient using the following formula: r xy ( x) 2 2 x n x y n ( y)2 2 . y n Serial .n Age (years) (x) Weight (Kg) (y) xy X2 Y2 1 7 12 84 49 144 2 6 8 48 36 64 3 8 12 96 64 144 4 5 10 50 25 100 5 6 11 66 36 121 6 9 13 117 81 169 Total =x∑ 41 =y∑ 66 xy=∑ 461 =x2∑ 291 =y2∑ 742 41 66 461 6 r (41) 2 (66) 2 291 .742 6 6 r = 0.759 strong direct correlation EXAMPLE: Relationship between Anxiety and Test Scores Anxiety )X( 10 8 2 1 5 6 X = 32∑ Test score (Y) X2 Y2 XY 2 100 4 20 3 64 9 24 9 4 81 18 7 1 49 7 6 25 36 30 5 36 25 30 Y = 32∑ X2 = 230∑ Y2 = 204∑ XY=129∑ Calculating Correlation Coefficient r (6)(129) (32)(32) 6(230) 32 6(204) 32 2 2 774 1024 .94 (356)(200) r = - 0.94 Indirect strong correlation Spearman Rank Correlation Coefficient (rs) It is a non-parametric measure of correlation. This procedure makes use of the two sets of ranks that may be assigned to the sample values of x and Y. Spearman Rank correlation coefficient could be computed in the following cases: Both variables are quantitative. Both variables are qualitative ordinal. One variable is quantitative and the other is qualitative ordinal. :Procedure 1. 2. 3. 4. Rank the values of X from 1 to n where n is the numbers of pairs of values of X and Y in the sample. Rank the values of Y from 1 to n. Compute the value of di for each pair of observation by subtracting the rank of Yi from the rank of Xi Square each di and compute ∑di2 which is the sum of the squared values. 5. Apply the following formula 2 6 (di) rs 1 n(n 2 1) The value of rs denotes the magnitude and nature of association giving the same interpretation as simple r. Example In a study of the relationship between level education and income the following data was obtained. Find the relationship between them and comment. sample numbers A B C D E level education (X) Income (Y) Preparatory. Primary. 25 University. secondary 8 10 10 15 F secondary illiterate G University. 60 50 Answer: (X) (Y) Rank X Rank Y di di2 A Preparatory 25 5 3 2 4 B Primary. 10 6 5.5 0.5 0.25 C University. 8 1.5 7 -5.5 30.25 D secondary 10 3.5 5.5 -2 4 E secondary 15 3.5 4 -0.5 0.25 F illiterate 50 7 2 5 25 G university. 60 1.5 1 0.5 0.25 ∑ di2=64 6 64 rs 1 0.1 7(48) Comment: There is an indirect weak correlation between level of education and income. exercise Regression Analyses Regression: technique concerned with predicting some variables by knowing others The process of predicting variable Y using variable X Regression Uses a variable (x) to predict some outcome variable (y) Tells you how values in y change as a function of changes in values of x Correlation and Regression Correlation describes the strength of a linear relationship between two variables Linear means “straight line” Regression tells us how to draw the straight line described by the correlation Regression Calculates the “best-fit” line for a certain set of data The regression line makes the sum of the squares of the residuals smaller than for any other line Regression minimizes residuals By using the least squares method (a procedure that minimizes the vertical deviations of plotted points surrounding a straight line) we are able to construct a best fitting straight line to the scatter diagram points and then formulate a regression equation in the form of: ˆ y a bX ˆ y y b(x x) b1 b xy x2 x y n ( x) 2 n Regression Equation Regression equation describes the regression line mathematically Intercept Slope Linear Equations Linear Equations Y ˆ y =baX bX Y + a b = S lo p e Change in Y C h a n g e in X a = Y -in te r c e p t X Hours studying and grades Regressing grades on hours Linear Regression Final gra de in co urse = 59 .95 + 3.17 * st udy R-Square = 0 .8 8 90.00 80.00 7 0.00 2.00 4.00 6.00 8.00 10.00 Number of hours spent st udying Predicted final grade in class = 59.95 + 3.17*(number of hours you study per week) Predicted final grade in class = 59.95 + 3.17*(hours of study) …Predict the final grade of Someone who studies for 12 hours Final grade = 59.95 + (3.17*12) Final grade = 97.99 Someone who studies for 1 hour: Final grade = 59.95 + (3.17*1) Final grade = 63.12 Exercise A sample of 6 persons was selected the value of their age ( x variable) and their weight is demonstrated in the following table. Find the regression equation and what is the predicted weight when age is 8.5 years. .Serial no Age (x) Weight (y) 1 2 3 4 5 6 7 6 8 5 6 9 12 8 12 10 11 13 Answer .Serial no Age (x) Weight (y) xy X2 Y2 49 144 36 64 64 144 25 100 36 121 81 169 1 2 3 4 5 6 7 6 8 5 6 9 12 8 12 10 11 13 84 48 96 50 66 117 Total 41 66 461 291 742 41 x 6.83 6 66 y 11 6 41 66 461 6 b 0.92 2 (41) 291 6 Regression equation ˆ y (x) 11 0.9(x 6.83) ˆ y (x) 4.675 0.92x ˆ y (8.5) 4.675 0.92 * 8.5 12.50Kg ˆ y (7.5) 4.675 0.92 * 7.5 11 .58Kg we create a regression line by plotting two estimated values for y against their X component, then extending the line right and left. Exercise 2 The following are the age (in years) and systolic blood pressure of 20 apparently healthy adults. Age B.P Age B.P (x) (y) (x) (y) 20 43 63 26 53 31 58 46 58 70 120 128 141 126 134 128 136 132 140 144 46 53 60 20 63 43 26 19 31 23 128 136 146 124 143 130 124 121 126 123 Find the correlation between age and blood pressure using simple and Spearman's correlation coefficients, and comment. Find the regression equation? What is the predicted blood pressure for a man aging 25 years? Serial 1 x 20 y 120 xy 2400 x2 400 2 43 128 5504 1849 3 63 141 8883 3969 4 26 126 3276 676 5 53 134 7102 2809 6 31 128 3968 961 7 58 136 7888 3364 8 46 132 6072 2116 9 58 140 8120 3364 10 70 144 10080 4900 Serial 11 12 13 14 15 16 17 18 19 20 x 46 53 60 20 63 43 26 19 31 23 y 128 136 146 124 143 130 124 121 126 123 xy 5888 7208 8760 2480 9009 5590 3224 2299 3906 2829 x2 2116 2809 3600 400 3969 1849 676 361 961 529 11448 4167 b1 xy x y n ( x) 2 x2 n ˆ y = 852 2630 114486 20 0.4547 2 852 41678 20 =112.13 + 0.4547 x for age 25 B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg Multiple Regression Multiple regression analysis is a straightforward extension of simple regression analysis which allows more than one independent variable. ...
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