Quiz STAT

Correlation Analysis

Correlation Analysis

Correlation measures the relationship between two quantitative variables

Linear correlation measures if the ordered paired data follow a straight-line relationship between quantitative variables.

The correlation coefficient (r) computed from the sample data measures the strength and the direction of a linear relationship between two variables.

The range of correlation coefficient is -1 to +1. When there is no linear relationship between the two variables or only a weak relationship, the value of correlation coefficient will be close to 0.

 

Things to Remember

Correlation coefficient cutoff points

 

 

 

 

 

 

+0.30 to + 0.49 weak positive association.

 

 

+ 0.5 to +0.69 medium positive association.

 

 

+0.7to + 1.0 strong positive association.

 

 

 

 

 

– 0.5 to – 0.69 medium negative association.

 

 

– 0.7 to – 1.0 strong negative association.

 

 

– 0.30 to – 0.49 weak negative association.

 

 

0 to – 0.29 little or no association.

 

 

0 to + 0.29 little or no association.

 

 

 

 

 

 

 

 

Relationships of Linear Correlation

As x increases, no definite shift in y: no correlation.

As x increase, a definite shift in y: correlation.

Positive correlation: x increases, y increases.

Negative correlation: x increases, y decreases.

If the points exhibit some other nonlinear pattern: no linear relationship.

Example: No correlation.

 

As x increases, there is no definite shift in y.

Example: Positive/direct correlation.

As x increases, y also increases.

Example: Negative/indirect/inverse correlation.

As x increases, y decreases.

 

Coefficient of linear correlation: r, measures the strength of the linear relationship between two variables.

 

Pearson Correlation formula:

 

 

 

Note:

 

r = +1: perfect positive correlation

r = -1 : perfect negative correlation

Use the calculated value of the coefficient of linear correlation, r, to make an inference about the population correlation coefficient r.

 

 

 

Example 1: Is there a relationship between age of the children and their score on the Child Medical Fear Scale (CMFS), using the data shown in Table 1?

H0: There is no significant relationship between the age of the children and their score on the CMFS

Or

H0: r = 0

 

 

 

 

 

 

 

ID Age (x) CMFS (y)
1 8 31
2 9 25
3 9 40
4 10 27
5 11 35
6 9 29
7 8 25
8 9 34
9 8 44
10 11 19
11 7 28
12 6 47
13 6 42
14 8 37
15 9 35
16 12 16
17 15 12
18 13 23
19 10 26
20 10 36

 

Table 1

Scattergram (Scatterplot) Age (x) = Independent variable, CMFS (y)= Dependent variable

 

 

 

 

 

Correlation Coefficient

The Results:

a. Decision: Reject H0.

b. Conclusion: There is evidence to suggest that there is a significant linear relationship between the age of the child and the score on the CMFS.

Answers the question of whether there is a significant linear relationship or not

 

Simple Linear Regression Analysis

Linear Regression Analysis

Linear Regression analysis finds the equation of the line that predicts the dependent variable based on the independent variable.

 

210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150Drug A (dose in mg)

 

Symptom Index

 

210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150

Drug A (dose in mg)

 

Symptom Index

 

 

 

 

 

 

y = dependent (predicted )variable

a = y intercept (constant)

b = slope (regression coefficient) of line

x = independent (predictor) variable

y = a + bx

 

 

Y

X

Simple Linear Regression Assumptions:

Normality

Equal variances

Independence

Linear relationship

Regression analysis establishes a regression equation for predictions

For a given value of x, we can predict a value of y

 

How good is the predictor?

Very good predictor

Moderate predictor

210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150Drug A (dose in mg)

 

Symptom Index

 

210 190 165 150 130 115 100 90 70 60 40 50 30 60 80 60 110 90 140 110 140 130

Drug B (dose in mg)

 

Symptom Index

 

 

How good is the predictor? R2

For simple regression, Coefficient of determination (R2) is the square of the correlation coefficient

Reflects variance accounted for in data by the best-fit line

Takes values between 0 (0%) and 1 (100%)

Frequently expressed as percentage, rather than decimal

Coefficient of Determination (R2)

Good fit  R2 high

High variance explained

Moderate fit  R2 lower

Less variance explained

210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150Drug A (dose in mg)

 

Symptom Index

 

210 190 165 150 130 115 100 90 70 60 40 50 30 60 80 60 110 90 140 110 140 130

Drug B (dose in mg)

 

Symptom Index

 

 

Previous example: Scatter gram (Scatterplot) Age (x) = Independent variable, CMFS (y)= Dependent variable

 

 

95% Confidence Interval

Regression Line

 

 

 

Correlation Coefficient

Coefficient of Determination (R2)

(Gives the % of variation)

Example 2: A recent article measured the job satisfaction of subjects. The data below represents the job satisfaction scores, y, and the salaries (in thousands), x, for a sample of similar individuals.

 

 

 

 

 

 

 

 

 

1. Draw a scatter diagram for this data.

2. Find the equation of the line of best fit.

 

 

ID Salaries (x) Scores (y)
1 31 17
2 33 20
3 22 13
4 24 15
5 35 18
6 29 17
7 23 12
8 37 21

 

Scatter gram:

The Regression Equation:

Thus a salary of $30,000 will result in a score of 17.

Example 3: Is high school GPA a useful predictor of college GPA, using the data shown in Table 2?

ID HS GPA College GPA
1 4.00 3.80
2 3.70 2.70
3 2.20 2.30
4 3.80 3.20
5 3.80 3.50
6 2.80 2.40
7 3.00 2.60
8 3.40 3.00
9 3.30 2.70
10 3.00 2.80

Table 2:

Scattergram:

Results:

Correlation Analysis

There is a significant linear relationship between high school GPA and College GPA.

Regression Analysis

72.1% of the variation in college GPA is explained by high school GPA

Regression equation: College GPA = 0.50 + 0.73 HS GPA

Conclusion: High school GPA is a useful predictor of college GPA

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Model Summary

.748

a

.560

.535

6.3442

Model

1

R

R Square

Adjusted

R Square

Std. Error of

the Estimate

Predictors: (Constant), Age

a.

0

20

40

60

80

100

120

140

160

180

200

050100150200250

1.49.517()

1.49.517(30)

Scoresalaries

Score

=+

=+

Coefficients

a

1.490

2.327

.640

.546

.517

.078

.938

6.613

.001

(Constant)

SALARIES

Model

1

B

Std. Error

Unstandardized

Coefficients

Beta

Standardi

zed

Coefficien

ts

t

Sig.

Dependent Variable: Job Satisfaction

a.

High school GPA

4.54.03.53.02.52.0

College GPA

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

Model Summary

.849

a

.721

.687

.2678

Model

1

R

R Square

Adjusted

R Square

Std. Error of

the Estimate

Predictors: (Constant), High school GPA

a.

ANOVA

b

1.486

1

1.486

20.725

.002

a

.574

8

7.171E-02

2.060

9

Regression

Residual

Total

Model

1

Sum of

Squares

df

Mean Square

F

Sig.

Predictors: (Constant), High school GPA

a.

Dependent Variable: College GPA

b.

Coefficients

a

.496

.535

.927

.381

.729

.160

.849

4.552

.002

(Constant)

High school GPA

Model

1

B

Std. Error

Unstandardized

Coefficients

Beta

Standardi

zed

Coefficien

ts

t

Sig.

Dependent Variable: College GPA

a.

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