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**Correlation and regression Dr. Ghada Abo-Zaid**

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Outline Once you have finished studying this chapter, you will be able to: Draw a scatter diagram, and explain the relationship between two variables from the plot. Understand the definition of covariance. Calculate the covariance, and interpret the results. Calculate the coefficient of correlation and interpret the results. Clarify the difference between the covariance and correlation.

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Outline Identify the assumptions and limitations of correlation coefficient. Test the hypothesis of coefficient of correlation. Understand the definition of Spearman's rank correlation coefficient. Calculate Spearman's rank correlation coefficient and interpret the results. Identify the difference between the correlation coefficient and Spearman's rank correlation coefficient. Test the hypothesis of Spearman's rank correlation coefficient.

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Scatter Plot Exploring the dataset before starting any statistical analysis is considered currently as one of the most important steps in the statistical analysis, especially in social science research. A scatter plot or scatter diagram might be used for examining initially whether there is an association between two variables, and shows the direction of this association.

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**Possible scatter plots association between X and Y variables.**

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Negative association

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**No association and non-linear association**

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Covariance Basically, covariance is used for detecting the direction of an association between two random variables. If the two variables are moved at the same direction, it is named as a positive covariance. If the two variables are moved at the reverse directions, it is named as a negative covariance.

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**Covariance is denoted as**

In other words, A covariance is a positive or a negative single number that help in detecting the association between two variables by its sign. For example if the single number is minus, this refers to an indirect association between two variables and vice versa. Covariance is denoted as Cov (X,Y).

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Sampling Covariance The sample covariance between X and Y is defined by two formulas: Second: Short calculation formula where and are the sampling means for X and Y respectively.

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Example In the stock market, the interest of the analyst is to select stocks that: a) reduce the risk taken for the same amount of return. b) select the stocks that are working well together. Table 4.1 shows the daily returns for two stocks using the closing prices, say NSGB bank, , and Sidi krier petroleum, , in 2014, for a sequence of 10 days.

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**Table 4.1: Gives the daily returns for two stocks using the closing prices.**

Calculate the covariance between X and Y by using a) Long calculation formula b) Short calculation formula

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**Short calculation formula**

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**Short calculation formula**

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**Short calculation formula**

Interpretation: The result indicates to a positive relationship between the two variables (return of the two stocks) X and Y.

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Correlation Though a covariance measure gives the direct association between two variables, it is still not capable of measuring the size or strength of an association. A correlation is a statistical measure that determines the strength of an association between two variables and detect their direction. It is also named as Pearson's correlation coefficient in honour of Karl Person ( )

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**Short calculation formula**

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**Note that the coefficient of correlation lies between +1 and -1.**

if r = +1 this indicates a perfect positive correlation between X and Y. if r = -1 , this indicates a perfect negative association between X and Y. If r = 0 , this is an indicator of no correlation between X and Y.

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**Assumptions of Person's Correlation**

The variables X and Y must be continuous random variables. The data for X and Y variables must tend to a normal distribution ( bell shape).

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Example A sample of 8 students was selected randomly to examine the association between the number of hours a student spent studying for an exam (X) and the score that a student obtained on that exam (Y). The data are given below

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Example Find the linear correlation coefficient between the number of hours a student spent in studying and the score a student obtained in the exam, and interpret the result.

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**Solution by using short calculation formula**

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**Solution by using short calculation formula**

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**Solution by using short calculation formula**

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**Solution by using short calculation formula**

Interpretation: This indicates there is a very strong positive association between the number of hours a student spent in studying and the number of the score on the exam. Interpretation: This means that the more hours a student spent in studying, the better score he or she will obtain.

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**Hypothesis Test for a Linear Correlation Coefficient**

Hypothesis test for a linear correlation coefficient is basically used to detect whether the sample correlation coefficient r is the estimator of population correlation coefficient r (rho) or not by using the Student t distribution. The student t statistic formula is given below: which is distributed as with degree of freedom

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**listed the steps of the hypothesis t- test for a linear correlation coefficient**

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**the steps of the hypothesis t- test for a linear correlation coefficient**

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Example A sample of 7 observations was taken randomly to examine the association between the income per thousand pounds, X, and the number of breads con consumed for person per day, Y

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Example Find the linear correlation coefficient between X and Y and interpret the result Test the significant of the linear correlation coefficient at significant level, , equals 5%

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Solution Find the linear correlation coefficient between X and Y and interpret the result

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Solution

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Interpretation: This indicates there is a very strong negative association between the income and the number of bread consumed for person. This means that the more income a person earns the less money spent on bread.

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**Test the significant of the linear correlation coefficient at significant level, , equals 5%**

Step 1: Let Step 2 :

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**Test the significant of the linear correlation coefficient at significant level, , equals 5%**

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**Test the significant of the linear correlation coefficient at significant level, , equals 5%**

We conclude that there is a sufficient evidence to support that there is a linear correlation coefficient between the two variables.

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Rank Correlation Coefficient of correlation is used to measure the association between two variables, but this is under certain conditions. One of these conditions is that X and Y random variables should be continuous. In addition, the data of X and Y variables are underlying the normal distribution. What happen if one of those conditions is not achieved?

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Rank Correlation this basically lead to think of another measure of correlation called rank correlation coefficient It is also named as Spearman's rank correlation coefficient. Spearman's rank correlation coefficient, is a non- parametric statistics measure that is equivalent to Pearson's correlation coefficient, r.

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It is also undertaken to measure the association between two variables, even if these variables do not underlying normal distribution or they are not continuous variables. Spearman's Rank correlation coefficient is undertaken if the data are in orders or can be ranked in orders.

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Rank Correlation The formula of Spearman's Rank correlation coefficient, is given as:

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**Steps for calculating Spearman's Rank Correlation Coefficient**

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**Steps for calculating Spearman's Rank Correlation Coefficient**

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Example The following table gives the grades of 8 students in linear algebra course , X, and probability course, Y where : E , V.G, G, and P are excellent, very good, good, and pass respectively. Find the correlation between X and Y.

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Solution

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Solution Interpretation: This indicates that there is a moderate positive association between the evaluation grades of linear algebra and probability course.

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**Hypothesis Significant test of Spearman's Rank Correlation Coefficient**

Hypothesis test is also undertaken for a Spearman's Rank Correlation Coefficient to detect whether the sample rank correlation coefficient is an estimator of population correlation coefficient r (rho) or not by using the Student t distribution.

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**the steps of the hypothesis t- test for Spearman's Rank Correlation Coefficient**

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**the steps of the hypothesis t- test for Spearman's Rank Correlation Coefficient**

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Example In the previous Example test the significance of Spearman's Rank Correlation Coefficient

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Solution

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Solution

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Solution

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**SIMPLE REGRESSION ANALYSIS**

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**Outlines Goals of chapter five simple linear regression analysis**

Once you have finished studying this chapter, you will be able to: Identify the difference between the dependent and independent variable. Fit simple linear egression model and interpret the results. Understand the assumption of simple linear regression model. know the BLUE method and the advantage of this method for estimating the unknown parameters of simple linear regression model.

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**Outlines Assessing the best fitted simple linear regression model.**

Construct ANOVA table and calculate F-value. Understand hypothesis tests of individual regression coefficients. Calculate the coefficient of determination and interpret the result.

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**Simple Regression Analysis**

The Correlation coefficient is a statistics measure that examines an association between two variables and determine the direction and the strength of there relation. However, it does not indicate causation. For instance, if two variables, say X and Y show high positive correlation, this does not mean that if Y increase by a certain value, X will increase by the same amount or more. In addition, it does not give any information about one of the variables predict one another.

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**Simple Regression Analysis**

Simple linear regression model is a simple statistical model that indicates causality by determining the dependent variable, Y and an independent variable, X . Least square estimation method is used for fitting a model and estimate the effect size of the independent variable, X that influence on the dependent variable, Y.

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**Simple Regression Analysis**

In social sciences, it is difficult to estimate the exact relationship between two variables. It is often assumed that there is an acceptable measurement of error. In statistical analysis, the error term is included in the model as a random factor called the error term or disturbance term.

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**Simple Regression Analysis**

The simple linear regression model is written mathematically as:

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Estimation of and

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The sum of square error

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Example A chain of stores sells treadmills; the manager of the stores wants to know whether advertising of the product increase the sales. The manger decided to examine the association between an advertising expenditures and sales for 6 months. Note that advertising expenditures is the independent variable, X and is measured by thousand Egyptian pounds per month, and Y is the number of treadmills sold per month. The data are given in the following table.

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Example Draw the scatter plot and depict the association between X and Y. Fit the simple linear regression model. Calculate the error term by using: a). Long calculation formula. b). Short calculation formula.

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Solution Draw the scatter plot

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Depict of Scatter plot The scatter plot shows a direct association between the advertising expenditures, X and the number of sold treadmill, Y. This indicates that the more amount of money spending on advertising, the more expected number of treadmills are sold.

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**2. Fitting the simple linear regression model.**

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Estimate b0 and b1

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Estimate b0 and b1

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**Calculate the error term**

First: Long calculation formula

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**Calculate the error term**

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**Assessing the Best Fitted Simple Linear Regression Model**

Testing the significance of the linear regression model can be undertaken through three methods. The first method is Analysis of variance (ANOVA) table, that aimed to test the full model by calculating F-value. The second method is used hypotheses test to test whether a regression parameter are significant or not. The last one is the coefficient of determination, In the following sections, the three assessing methods are explained in details.

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**First method: Analysis of Variance (ANOVA table)**

To construct ANOVA table, three types of variation should be defined , which are given in the following equation: SST is named as total sum of squares variation, and it calculated by using the following formula

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**SSE is named as sum of square error which is the error that not explained by regression.**

It can be calculated as follows

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**SSR is named as sum of square regression**

SSR is named as sum of square regression. It is variation that explained by regression. it can be written mathematically as follows:

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**Construct an ANOVA table**

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Decision

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**Hypothesis Tests of Individual Regression Coefficients**

Test the significance of slope

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**Test the significance of slope b1**

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**test the significance of b0**

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**test the significance of b0**

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**Third Method: The Coefficient of Determination,**

The coefficient of determination is one of the statistical tool that assess linear regression model. This indicates how well data fit the line. The coefficient of determination lies between zero and one. If the coefficient of determination, equals one, this indicates the perfect linear relationship between X and Y. While, if =0 this indicates that there is no linear relationship between X and Y. In general, if the value of is close to one, this indicates that the model is a good fit and vice versa

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**The formula of coefficient of determination is given below**

In other words, the coefficient of determination is the square of the Person's correlation coefficient between X and Y, and it is written mathematically as follows:

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Example In previous example , the estimate simple linear regression model were as follows:

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**Example From the dataset at previous example find the following:**

Test the full estimate simple regression model by constructing ANOVA table , find F-value and interpret the results. Test the significance of and individually, at 5% significant level. Calculate the coefficient of determination, and interpret the results.

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Solution Test the full estimate simple regression model by constructing ANOVA table , find F-value and interpret the results.

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Solution

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Solution

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**Solution Step 5: Decision**

Since F-calculated> F-tabulated, then, this indicates the significance of the full simple regression model.

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Solution

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Solution

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Solution

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**Second : test the significance of b0**

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**Second : test the significance of b0**

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**3) Calculate the coefficient of determination, and interpret the results.**

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