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 Yaxis (usually response

variable) and Xaxis (explanatory variable) and click on Titles to write titles in Line 1. After writing title click on ‘Continue’.

Step 4: Click on ‘Ok’. R:

plot(x, y, main= “Title name”, xlab=“x axis label name”, ylab=“y axis label name” )

where x and y are the data of two variables and for our example x is age and y is hgt.

Scatterplots

60 80 100 120 140

4045

5055

Scatterplot of Age vs Height

Age

Hei

ght

60 80 100 120 140

4045

5055

Scatterplot of varibles X and Y

X

Y

6 8 10 12 14 16 18

3050

7090

Scaterplot of variables X1 and Y1

X1

Y1

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, sx and sy 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.

y

in

i x

i

s

yy

s

xx

nr

11

1

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 0.9829632, which is very close to 1.

Correlation of two variables

60 80 100 120 140

4045

5055

Scatterplot of Age vs Height

Age

Hei

ght

60 80 100 120 140

4045

5055

Scatterplot of varibles X and Y

X

Y

6 8 10 12 14 16 18

3050

7090

Scaterplot of variables X1 and Y1

X1

Y1

r=0.02

r=-0.96r=0.98

Correlation of two variables MS-Excel:

Step 1: Click on function Wizard fx 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 R: cor( variable 1, variable 2) e.g. age and hgt are the variable 1

and variable 2 in our example

Example 1: Calculating Correlation Coefficient of the variables Height and Age in our data set. MS-Excel: Click on function Wizard fx > 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.

age hgt

age 1

hgt 0.982 1

Example 1: Calculating Correlation Coefficient of the variables Height and Age in our data set.

SPSS: Analyze > Correlate > Bivariate > select age and hgt > ok. It gives the following result

age hgt

age Pearson Correlation sig (- tailed) N

1

60

.983 .000 60

Hgt Pearson Correlation sig (- tailed) N

.983 000 60

1

60

Example 1: Calculating Correlation Coefficient of the variables Height and Age in our data set.

R: >cor(age, hgt). It gives the correlation coefficient 0.98296

Strong Association but no Correlation

20 30 40 50 60

20

25

30

35

Scatter Plot

Speed

MP

GGas mileage of an auto mobile first increases than decreases as the speed increases like the following data:

Speed 20 30 40 50 60 MPG 24 28 30 28 24

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 = -0.237 After dropping the last pair,

r = 0.996

X 20 30 40 50 60 80 Y 24 28 30 34 37 15

X 20 30 40 50 60 Y 24 28 30 34 37

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= b0 + b1X + e

where, ‘Y’ is the response variable, ‘X’ is the explanatory variable, ‘e’ is the residual (error), and b0 and b1 are two parameters. Basically, bo is the intercept and b1 is the slope of a straight line y= b0 + b1X.

Simple Linear RegressionA simple regression line is fitted for height on age. The intercept is 31.019 and the slope (regression coefficient) is .1877.

Regresion line of Height on Age

Height = 0.1877Age + 31.019

0

10

20

30

40

50

60

70

0 50 100 150 200

Age (month)

Hei

gh

t (i

nch

)

Simple Linear RegressionAssumptions:

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 b0 and b1

Least Square method estimates b0 and b1 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:

n

ii

n

iii

XX

YYXXb

1

2

11

)(

))((ˆ XbYb 10

ˆˆ

Simple Linear Regression Fitted Least Square Regression line

Fitted Line:

Where is the fitted / predicted value of ith observation (Yi) of the response variable.

Estimated Residual:

Least square method estimates b0 and b1 to minimize the summed error:

iYii XbbY 10

ˆˆˆ

iii YYe ˆˆ

n

iie

1

2ˆ

Simple Linear Regression

Average Income versus % with a College Degree (by State)

22,000

22,500

23,000

23,500

24,000

24,500

25,000

25,500

26,000

15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20

Percentage of Population with College Degree or Higher

Ave

rag

e In

co

me

Le

vel (

$ p

er

yea

r)

e1e3

e2

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 (b1) reflects the

change in the response variable Y for a unit change in the explanatory variable X. That is, the slope of the regression line.

Intercept (b0) estimates the 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 R: lm(y~x, data)

summary(lm(y~x, data)) where, y is the dependent variable, x is the independent

variable and data is a data frame containing the variables in the model.

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

SUMMARY OUTPUT

Regression Statistics Multiple R 0.982963 R Square 0.966217 Adjusted R Square 0.965634 Standard Error 5.604001 Observations 60 ANOVA

df SS MS F Significance

F Regression 1 52095.1 52095.1 1658.825 2.28E-44 Residual 58 1821.48 31.40483 Total 59 53916.58

Coefficients Standard

Error t Stat P-value Lower 95% Upper 95%

Lower 95.0%

Upper 95.0%

Intercept -156.589 6.107669 -25.6381 2.48E-33 -168.815 -144.363 -168.815 -144.363 hgt 5.146693 0.126365 40.72867 2.28E-44 4.893746 5.399641 4.893746 5.399641

Example 2: SPSS Linear Regression output: Height on age

Model Summary

Model R R

Square

Adjusted R

Square Std. Error of the Estimate Change Statistics

R Square Change

F Change df1 df2

Sig. F Change

1 .983(a) .966 .966 1.0703 .966 1658.825 1 58 .000

a Predictors: (Constant), age Coefficients(a)

Unstandardized Coefficients

Standardized Coefficients

Model B Std. Error Beta t Sig.

(Constant) 31.019 .439 70.645 .000 1

age .188 .005 .983 40.729 .000

a Dependent Variable: hgt

Example 2: R Linear Regression output: Height on age>lm (hgt ~ age) gives the following output Call:lm(formula = hgt ~ age)Coefficients:(Intercept) age 31.0187 0.1877 R code summary (lm (hgt ~ age) ) gives the following outputCall:lm(formula = hgt ~ age)Residuals: Min 1Q Median 3Q Max -2.53975 -0.55722 0.08105 0.68147 2.24326 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 31.018720 0.439077 70.64 <2e-16 ***age 0.187735 0.004609 40.73 <2e-16 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.07 on 58 degrees of freedomMultiple R-Squared: 0.9662, Adjusted R-squared: 0.9656 F-statistic: 1659 on 1 and 58 DF, p-value: < 2.2e-16

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 = b0 + b1X1 + b2X2 +… +bpXp +ewhere bi (i=1,2, …, p) is the regression coefficient of Xi

Multiple Regression Fitted Y is given by,

The estimated residual error is the same as that in the simple linear regression,

ii

pp

bb

XbXbXbbY

of estimate theis ˆ Where,

ˆ...ˆˆˆˆ22110

iii YYe ˆˆ

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.

R: Every thing is the same as for the simple regression except adding more independent variables in the model. i.e. lm(y~x1 + x2 , data)

summary(lm(y~x1 + x2 , data)) where, y is the dependent variable, x1 and x2 are independent variables

and data is a data frame containing the variables in the model.

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

SUMMARY OUTPUT

Regression Statistics Multiple R 0.983103 R Square 0.966491 Adjusted R Square 0.965315 Standard Error 5.629974 Observations 60 ANOVA

df SS MS F Significance

F Regression 2 52109.88 26054.94 822.0105 9.27E-43 Residual 57 1806.706 31.6966 Total 59 53916.58

Coefficients Standard

Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

Intercept -158.474 6.728226 -23.5536 4.65E-31 -171.947 -145.001 -171.947 -145.001 hgt 5.136708 0.127791 40.19623 1.6E-43 4.880811 5.392605 4.880811 5.392605 PLUC.post 0.194634 0.28509 0.68271 0.497556 -0.37625 0.765517 -0.37625 0.765517

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

Model R R

Square Adjusted R

Square Std. Error of the Estimate Change Statistics

R Square Change

F Change df1 df2

Sig. F Change

1 .983(a) .966 .965

1.077191687455185

.966 818.970 2 57 .000

a Predictors: (Constant), PLUC.post, age Coefficients(a)

Unstandardized Coefficients

Standardized Coefficients

Model B Std. Error Beta t Sig.

(Constant) 31.330 .752 41.635 .000 age .188 .005 .985 40.196 .000

1

PLUC.post -.028 .055 -.013 -.511 .612

a Dependent Variable: hgt

Example 3: R Multiple Regression output: Height on Age and PLUC.postR code lm(hgt ~ age + PLUC.post) gives the following outputCoefficients:(Intercept) age PLUC.post 31.3297 0.1880 -0.0279

R code summary(lm(hgt~age+PLUC.post)) gives the following outputResiduals: Min 1Q Median 3Q Max -2.52512 -0.57195 0.02261 0.74912 2.21156 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 31.329680 0.752489 41.635 <2e-16 ***age 0.188043 0.004678 40.196 <2e-16 ***PLUC.post -0.027899 0.054644 -0.511 0.612 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.077 on 57 degrees of freedomMultiple R-Squared: 0.9664, Adjusted R-squared: 0.9652 F-statistic: 819 on 2 and 57 DF, p-value: < 2.2e-16

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,

(Y) Total

Residual)(1

(Y) Total

)Regression(2

SS

SS

SS

SSR

Coefficient of Determination (Multiple R-squared)

R2 lies between 0 and 1. R2 = 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.

R: It’s in the R output of Example 3.