RESEARCH STATISTICS Jobayer Hossain Larry Holmes, Jr November 6, 2008 Examining Relationship of Variables

Response (dependent) variable - measures the outcome of a study. Explanatory (Independent) variable - explains or influences the changes in a response variable Outlier - an observation that falls outside the overall pattern of the relationship. Positive Association - An increase in an independent variable is associated with an increase in a dependent variable. Negative Association - An increase in an independent variable is associated with a decrease in a dependent variable.

Scatterplot Shows relationship between two variables (age and height in this case). Reveals form, direction, and strength of the relationship.

Scatterplot - Demo MS-Excel: – Step 1: Select Chart wizard – Step 2: From the chart type select XY(Scatter) – Step 3: Select Chart from chart sub type and click next – Step 4: Data range: range for variable in the x axis (age in our case), range for the variable in the y axis (hgt in our case) and click on next – Step 5: Click on titles to write title and labels of the chart, click on axes and mark on Value (X) and Value (Y) axis, click on Gridlines and take all the mark off( if you want no gridlines), click on legend and put options of legend (if you don’t want legend) then click on Finish

Scatterplot - Demo SPSS: – Step 1: Click on Graphs and select Scatter/Dot – Step 2: Select simple scatter – Step 3: Select variables for Y axis (usually response variable) and X axis (explanatory variable) and click on Titles to write titles in Line 1. After writing title click on ‘Continue’. – Step 4: Click on ‘Ok’.

Scatterplots Strong positive association Points are scattered with a poor association Strong negative association

Correlation of two variables Correlation measures the direction and strength of the relationship between two quantitative variables. Suppose that we have data on variables x and y for n individuals. Then the correlation coefficient r between x and y is defined as, Where, s x and s y are the standard deviations of x and y. r is always a number between -1 and 1. Values of r near 0 indicate little or no linear relationship. Values of r near -1 or 1 indicate a very strong linear relationship. The extreme values r=1 or r=-1 occur only in the case of a perfect linear relationship, when the points lie exactly along a straight line.

Correlation of two variables Positive r indicates positive association i.e. association between two variables in the same direction, and negative r indicates negative association. Scatterplot of Height and Age shows that these two variables possess a strong, positive linear relationship. The correlation coefficient of these two variables is , which is very close to 1.

Correlation of two variables r=0.02 r=-0.96r=0.98

Correlation of two variables MS-Excel: – Step 1: Click on function Wizard f x – Step 2: Select function Correl – Step 3: Input range Array 1 and Array 2 (for our example age and hgt) and click on Ok. Another option: Check if the data of two variables are in a side by side position. If not, copy and paste them side by side in a new worksheet and follow the following steps- – Step 1: Select Data Analysis from the menu Tools – Step 2: Select Correlation and type or select Input and Output range and click on ok.

Correlation of two variables  SPSS: – Step 1: Select Correlate from the menu Analyze – Step2: Click on Bivariate and select the variables (age and hgt) and click on Ok

Example 1: Calculating Correlation Coefficient of the variables Height and Age in our data set1. MS-Excel: Click on function Wizard f x > Correl > Input range Array 1: age Input range Array 1: hgt > ok. It gives correlation coefficient, r=.98. or Tools > data analysis > correlation > range for variables age and hgt > ok. It gives you the following results. agehgt age1 hgt0.9821

Example 1: Calculating Correlation Coefficient of the variables Height and Age in our data set1 SPSS: Analyze > Correlate > Bivariate > select age and hgt > ok. It gives the following result age hgt age Pearson Correlation sig (- tailed) N Hgt Pearson Correlation sig (- tailed) N

Strong Association but no Correlation Gas mileage of an auto mobile first increases than decreases as the speed increases like the following data: Scatter plot shows an strong association. But calculated, r = 0, why? r = 0 ? It’s because the relationship is not linear and r measures the linear relationship between two variables.

Influence of an outlier Consider the following data set of two variables X and Y: r = After dropping the last pair, r = 0.996

Simple Linear Regression Regression refers to the value of a response variable as a function of the value of an explanatory variable. A regression model is a function that describes the relationship between response and explanatory variables. A simple linear regression has one explanatory variable and the regression line is straight. The linear relationship of variable Y and X can be written as in the following regression model form Y= b 0 + b 1 X + e where, ‘Y’ is the response variable, ‘X’ is the explanatory variable, ‘e’ is the residual (error), and b 0 and b 1 are two parameters. Basically, b o is the intercept and b 1 is the slope of a straight line y= b 0 + b 1 X.

Simple Linear Regression A simple regression line is fitted for height on age. The intercept is and the slope (regression coefficient) is.1877.

Simple Linear Regression Assumptions: 1. Response variable is normally distributed. 2. Relationship between the two variables is linear. 3. Observations of response variable are independent. 4. Residual error is normally distributed with mean 0 and constant standard deviation.

Simple Linear Regression Estimating Parameters b 0 and b 1 – Least Square method estimates b 0 and b 1 by fitting a straight line through the data points so that it minimizes the sum of square of the deviation from each data point. – Formula:

Simple Linear Regression Fitted Least Square Regression line – Fitted Line: – Where is the fitted / predicted value of i th observation (Y i ) of the response variable. – Estimated Residual: – Least square method estimates b 0 and b 1 to minimize the summed error:

Simple Linear Regression e1e1 e3e3 e2e2 In this example, a regression line (red line) has been fitted to a series of observations (blue diamonds) and residuals are shown for a few observations (arrows).  Fitted Least Square Regression line

Simple Linear Regression Interpretation of the Regression Coefficient and Intercept – Regression coefficient (b 1 ) reflects the average change in the response variable Y for a unit change in the explanatory variable X. That is, the slope of the regression line. E.g. – Intercept (b 0 ) estimates the average value of the response variable Y without the influence of the explanatory variable X. That is, when the explanatory variable = 0.0.

Simple Linear Regression MS-Excel: – Step1: Select Data Analysis from the menu Tools – Step2: Input Y range: range for the response (dependent) variable, Input X range: range for the independent variable – Step 3: Make selections of out puts you want and click on ok SPSS: – Step 1: Select Regression from the menu Analyze – Step 2: Select Linear and then for Dependent: Select dependent variable and for Independent(s): select Independent variable – Step 3: Click on Statistics and select options, clik on Plots and select variables for scatter plots (if you want), click on continue an d then ok

Example 2: MS-Excel Linear Regression output: Height on age

Example 2: SPSS Linear Regression output: Height on age

Multiple Regression Two or more independent variables to predict a single dependent variable. Multiple regression model of Y on p number of explanatory variables can be written as, Y = b 0 + b 1 X 1 + b 2 X 2 +… +b p X p +e where b i (i=1,2, …, p) is the regression coefficient of X i

Multiple Regression Fitted Y is given by, The estimated residual error is the same as that in the simple linear regression,

Multiple Regression MS-Excel: Every thing is the same as for the simple regression except we need to select multiple columns for the range of independent variables. If the columns of independent variables are not side by side, then we need to have them side by side. For that we can insert another excel sheet and copy the data from original sheet and paste the data in the new sheet. We first copy the dependent variable and paste that in one of the columns and then copy independent variables and paste them in side by side columns. SPSS: Everything is the same as for the simple regression except we select more than one independent variables.

Example 3: MS Excel Multiple Regression output: Height on Age and PLUC.post

Example 3: SPSS Multiple Regression output: Height on Age and PLUC.post

Coefficient of Determination (Multiple R-squared) Total variation in the response variable Y is due to (i) regression of all variables in the model (ii) residual (error). Total variation of y, SS (y) = SS(Regression) +SS(Residual) The Coefficient of Determination is,

Coefficient of Determination (Multiple R-squared) R 2 lies between 0 and 1. R 2 = 0.8 implies that 80% of the total variation in the response variable Y is due to the contribution of all explanatory variables in the model. That is, the fitted regression model explains 80% of the variance in the response variable.

Example 4: Calculating coefficient of determination for the variables in example 3. MS-Excel: It’s in the MS-Excel output of Example 3. SPSS: It’s in the SPSS output of Example 3.

Questions