Correlation & Linear Regression
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This document discusses correlation and linear regression. It defines correlation as the analysis of the relationship between two quantitative variables. Pearson's r is used to calculate correlation and can range from -1 to 1, with values closer to those extremes indicating a stronger linear relationship. A positive r represents a positive correlation while a negative r represents an inverse relationship. The coefficient of determination (r-squared) provides the proportion of variance shared between variables. However, correlation does not imply causation. Linear regression finds the best fitting straight line through data points to predict the value of one variable based on the other.
Correlation & Linear Regression
- 2. Correlation • 2 Continuous Variables • linear relationship • analysis of the relationship between two quantitative outcomes, e.g., height and weight.
- 4. Bivariate Correlation Positive and Linear Negative and Linear
- 5. How to calculate r? a b c
- 6. Example •∑ x = 4631 ∑ x2 = 688837 •∑ y = 2863 ∑ y2 = 264527 •∑ xy = 424780 n = 32 •a=424780-(4631*2863/32)=10,450.22 •b=688837-46312/32=18,644.47 •c=264527-28632/32=8,377.969 •r=a/(b*c)0.5 =10,450.22/(18,644.47*83,77.969)0.5 =0.836144 •t= 0.836144*((32-2)/(1-0.8361442))0.5 t = 8.349436 & d.f. = n - 2 = 30, p < 0.001
- 7. Please refer to Table A3. We use df = 30. t = 8.3494 > 3.65 (p=0.001) Therefore if t=8.3494, p<0.001.
- 8. How to interpret the value of r? • r lies between -1 and 1. Values near 0 means no (linear) correlation and values near ± 1 means very strong correlation. • The negative sign means that the two variables are inversely related, that is, as one variable increases the other variable decreases.
- 9. How to interpret the value of r?
- 10. Pearson’s r • A 0.9 is a strong positive association (as one variable rises, so does the other) • A -0.9 is a strong negative association (as one variable rises, the other falls) r=0.9 has nothing to do with 90% r=correlation coefficient
- 11. Coefficient of Determination Defined • Pearson’s r can be squared , r 2, to derive a coefficient of determination. • Coefficient of determination – the portion of variability in one of the variables that can be accounted for by variability in the second variable
- 12. Coefficient of Determination • Pearson’s r can be squared , r 2, to derive a coefficient of determination. • Example of depression and CGPA – Pearson’s r shows negative correlation, r=-0.5 – r2=0.25 – In this example we can say that 1/4 or 0.25 of the variability in CGPA scores can be accounted for by depression (remaining 75% of variability is other factors, habits, ability, motivation, courses studied, etc)
- 13. Coefficient of Determination and Pearson’s r • Pearson’s r can be squared , r 2 • If r=0.5, then r2=0.25 • If r=0.7 then r2=0.49 • Thus while r=0.5 versus 0.7 might not look so different in terms of strength, r2 tells us that r=0.7 accounts for about twice the variability relative to r=0.5
- 14. Causal Silence: Correlation Does Not Imply Causality Causality – must demonstrate that variance in one variable can only be due to influence of the other variable • Directionality of Effect Problem • Third Variable Problem
- 15. Correlation In SPSS • For this exercise, we will be using the data from the CD, under Chapter 8, korelasi.sav • This data is a subset of a case-control study on factors affecting SGA in Kelantan. • Open the data & select - >Analyse >Correlate >Bivariate…
- 16. Correlation in SPSS • We want to see whether there is any association between the mothers’ weight and the babies’weight. So select the variables (weight2 & birthwgt) into ‘Variables’. • Select ‘Pearson’ Correlation Coefficients. • Click the ‘OK’ button.
- 17. Correlations Correlation Results WEIGHT2 Pearson Correlation WEIGHT2 BIRTHWGT 1 .431* Sig. (2-tailed) . .017 N 30 30 BIRTHWGT Pearson Correlation .431* 1 Sig. (2-tailed) .017 . N 30 30 *. Correlation is significant at the 0.05 level (2-tailed). • The r = 0.431 and the p value is significant at 0.017. • The r value indicates a fair and positive linear relationship.
- 18. Scatter Diagram 3.6 3.4 3.2 • If the correlation is 3.0 significant, it is best 2.8 2.6 to include the 2.4 scatter diagram. 2.2 2.0 • The r square 1.8 1.6 indicated mothers’ 1.4 weight contribute 1.2 19% of the BIRTHWEIGHT 1.0 .8 .6 variability of the .4 babies’ weight. .2 0.0 Rsq = 0.1861 0 10 20 30 40 50 60 70 80 90 100 MOTHERS' WEIGHT
- 20. Linear Regression • Come up with a Linear Regression Model to predict a continuous outcome with a continuous risk factor, i.e. predict BP with age. Usually the next step after correlation is found to be strongly significant. • y = a + bx – e.g. BP = constant (a) + regression coefficient (b) * age • b=
- 21. Regression Line • In a scatterplot showing the association between 2 variables, the regression line is the “best-fit” line and has the formula y=a + bx a=place where line crosses Y axis b=slope of line (rise/run) Thus, given a value of X, we can predict a value of Y
- 22. Regression Line (Defined) Regression line is the line where absolute values of vertical distances between points on scatterplot and a line form a minimum sum (relative to other possible lines) Positive and Linear Negative and Linear
- 23. Example b= ∑x = 6426 ∑ x2 = 1338088 ∑y = 4631 ∑ xy = 929701 n = 32 b = (929701-(6426*4631/32))/ (1338088-(64262/32)) = -0.00549 Mean x = 6426/32=200.8125 mean y = 4631/32=144.71875 y = a + bx a = y – bx (replace the x, y & b value) a = 144.71875+(0.00549*200.8125) = 145.8212106 Systolic BP = 144.71875 - 0.00549.chol