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Biostatistics - Correlation explanation.pptx
- 1. Biostatistics Instructor: H M Nawaz Mehar
- 2. Correlation • Two variables are said to be correlated if the change in one variable results in a corresponding change in the other variable. • The correlation is a statistical tool which studies the relationship between two variables. • Correlation is concerned with the measurement of “strength of association between variables”. • The degree of association between two or more variables is termed as correlation. • Correlation analysis helps us to decide the strength of the linear relationship between two variables.
- 3. Correlation Definition: “Correlation is a statistical tool, with the help of which, we can determine whether or not two or more variables are correlate and if they are correlated, what is the degree and direction of correlation.” “The correlation is the measure of the extent and the direction of the relationship between two variables in a bivariate distribution.”
- 4. Correlation Example: • Height and weight of children. • An increase in the price of the commodity by a decrease in the quantity demanded. Types of Correlation: 1. Positive and Negative Correlation. 2. Simple, Partial and Multiple Correlation. 3. Linear and Non-linear Correlation.
- 5. Correlation 1. Positive and Negative correlation: If both the variables are varying in the same direction i.e. if one variable is increasing and the other on an average is also increasing or if as one variable is decreasing, the other on an average, is also decreasing, correlation is said to be positive. If on the other hand, the variable is increasing, the other is decreasing or vice versa, correlation is said to be negative. • Example 1: heights and weights, amount of rainfall and yields of crops, price and supply of a commodity, blood pressure and age. • Example 2: price and demand of commodity, sales of woolen garments and the summer days.
- 6. Correlation 2. Simple, Partial and Multiple Correlation: When only two variables are studied, it is a case of simple correlation. In partial and multiple correlation, three or more variables are studied. In multiple correlation three or more variables are studied simultaneously. In partial correlation, we have more than two variables, but consider only two variables to be influencing each other, the effect of the other variables being kept constant.
- 7. Correlation 3. Linear and Non-linear Correlation: If the change in one variable tends to bear a constant ratio to the change in the other variable, the correlation is said to be linear. Correlation is said to be nonlinear if the amount of change in one variable does nor bear a constant ratio to the amount of change in the other variable.
- 8. Methods of Studying Correlation 1. Scatter diagram • This is a graphic method of finding out relationship between the variables. • Given data are plotted on a graph paper in the form of dots i.e. for each pair of x and y values we put a dot and thus obtain as many points as the number of observations. • The greater the scatter of points over the graph, the lesser the relationship between the variables.
- 9. Methods of Studying Correlation
- 10. Methods of Studying Correlation[Interpretation • If all the points lie in a straight line, there is either perfect positive or perfect negative correlation. • If all the points lie on a straight falling from the lower left hand corner to the upper right hand corner then the correlation is perfect positive. Perfect positive if r = + 1. • If all the points lie on a straight falling from the upper left hand corner to the lower right hand corner then the correlation is perfect negative. Perfect negative if r = -1. • The nearer the points are to be straight line, the higher degree of correlation. • The farthest the points from the straight line, the lower degree of correlation. • If the points are widely scattered and no trend is revealed, the variables may be un-correlated i.e. r = 0.
- 11. Methods of Studying Correlation 1. Karl Pearson’s Coefficient of Correlation • A scatter diagram gives an idea about the type of relationship or association between the variables under study. It does not tell us about the quantification of the association between the two. In order to quantify the relation ship between the variables a measure called correlation coefficient developed by Karl Pearson. • It is defined as the measure of the degree to which there is linear association between two internally scaled variables. • Thus, the coefficient of correlation is a number which indicates to what extent two variables are related , to what extent variations in one go with the variations in the other.
- 12. Methods of Studying Correlation • The symbol ‘r’ or ‘rₓᵧ’ or ‘rᵧₓ’is denoted in this method and is calculated by: r = {Cov(X,Y) ÷ Sₓ Sᵧ} • Where Cov(X, Y) is the Sample Covariance between X and Y. Mathematically it is defined by Cov(X, Y)={Σ(X–X̅)(Y–Y̅)} ÷(n – 1) • Sₓ = Sample standard deviation of X, is given by Sₓ = {Σ(X – X̅)² ÷ (n – 1)}½ • Sᵧ = Sample standard deviation of Y, is given by Sᵧ = {Σ(Y – Y̅)² ÷ (n – 1)}½ 𝑟 = 𝐶𝑜𝑣 𝑋, 𝑌 𝑉𝑎𝑟 𝑋 . 𝑉𝑎𝑟 𝑌 = 𝑋 − 𝑋 . 𝑌 − 𝑌 𝑋 − 𝑋 2 . 𝑌 − 𝑌 2 𝑟𝑥𝑦 = 𝑆𝑥𝑦 𝑆𝑥. 𝑆𝑦 = 𝑛 𝑋𝑌 − 𝑋 𝑌 𝑛 𝑋2 − 𝑋 2 𝑛 𝑌2 − 𝑌 2
- 13. Methods of Studying Correlation[Interpretation • If the covariance is positive, the relationship is positive. • If the covariance is negative, the relationship is negative. • If the covariance is zero, the variables are said to be not correlated. • The coefficient of correlation always lie between +1 to -1. • When r = +1, there is perfect positive correlation between the variables. • When r = -1, there is perfect negative correlation between the variables. • The correlation coefficient is independent of the choice of both origin and scale of observation. • The correlation coefficient is a pure number. It is independent of the units of measurement.
- 14. Methods of Studying Correlation[Properties • The coefficient of correlation always lie between +1 to -1 i.e. -1⩽ r ⩽ +1. • When r = 0, there is no correlation. • When r = 0.7 to 0.999, there is high degree of correlation. • When r = 0.5 to 0.699, there is a moderate degree of correlation. • When r is less than 0.5, there is a low degree of correlation. • If ‘r’ is near to +1 or -1, there is a strong linear association between the variables. • If ‘r’ is small(close to zero), there is low degree of correlation between the variables. • The coefficient of correlation is the geometric mean of the two regression coefficients. Symbolically: r= √(bₓᵧ . bᵧₓ)
- 15. Coefficient of Determination • The coefficient of determination(r²) is the square of the coefficient of correlation. • It is the measure of strength of the relationship between two variables. • It is subject to more precise interpretation because it can be presented as a proportion or as a percentage. • The coefficient of determination gives the ratio of the explained variance to the total variance. • Thus, coefficient of determination: 𝑟2 = 𝑆𝑆𝑅 𝑆𝑆𝑇 = 1 − 𝑆𝑆𝐸 𝑆𝑆𝑇 𝑆𝑆𝑇 = 𝑌𝑖 − 𝑌 2 , 𝑆𝑆𝑅 = 𝑌𝑖 − 𝑌 2 , 𝑆𝑆𝐸 = 𝑌𝑖 − 𝑌𝑖 2 Regression Sum of Squares
- 17. Coefficient of Determination • Definition: “Coefficient of determination shows what amount of variability or change in independent variable is accounted for by the variability of the dependent variable.” “The coefficient of determination, 𝑟2, is used to analyze how differences in one variable can be explained by a difference in a second variable”
- 18. Coefficient of Determination[Properties • The coefficient of determination is the square of the correlation(r), thus it ranges from 0 to 1. • With linear regression, the coefficient of determination is equal to the square of the correlation between the x and y variables. • If r² is equal to 0, then the dependent variable cannot be predicted from the independent variable. • If r² is equal to 1, then the dependent variable can be predicted from the independent variable without any error. • If r² is between 0 and 1, then it indicates the extent that the dependent variable can be predictable. If R2 of 0.10 means, it is 10 percent of the variance in the y variable is predicted from the x variable. • Coefficient of determination (r²) is always non-negative and as such that it does not tell us about the direction of the relationship (whether it is positive or negative) between two series.