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TOPIC 4 CORRELATION AND REGRESSION Flipbook PDF

TOPIC 4 CORRELATION AND REGRESSION


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Learning Outcome Solve accurately the statistical problems in life science through calculation Select appropriately the statistical test in data presentation, data description and data analysis by using computer software

TOPIC 4 CORRELATION & REGRESSION BY MRS. HAZWANI BINTI HUSAIN

SUBTOPICS 1. Applying correlation coefficient in research 2. Applying the concept of regression in research 3. Generating scatter diagram and correlation coefficient using computer software 4. Determining linear regression equation using computer software

Applying correlation co-efficient in research • •

Definition of independent & dependent variables

Independent variables – factor that exists before change occurs in the dependent variable



Dependent

variables



the

result/

effect

in

the

relationship between variables

This figure shows independent and dependent persons. Do you have other examples?



Why we need to identify which data is independent & which one is the dependent?

Try this!



Definition of independent & dependent variables

• Thinking skill • Learning style Determine which one is the dependent variable and which one is the independent variable. What is their relationship? e.g. The creative thinking skills affect learning style

Applying correlation co-efficient in research (cont.)



Purpose of scatter diagram



A scatter diagram is a tool for analyzing relationships between two variables.



One variable is plotted on the horizontal axis and the other is plotted on the vertical axis.



The pattern of their intersecting points can graphically show relationship patterns.

Applying correlation co-efficient in research (cont.)



Purpose of scatter diagram

When to use it: •

Use a scatter diagram to examine theories about cause-and-effect relationships and to search for root causes of an identified problem.

Applying correlation co-efficient in research (cont.) • •

The correlation coefficient is used to indicate the relationship of two random variables.

Use of correlation coefficient



It provides a measure of the strength and direction of the correlation varying from -1 to +1.

Applying correlation co-efficient in research (cont.) • •

Positive values indicate that the two variables are positively correlated, meaning the two variables vary in

Use of correlation coefficient

the same direction.



Negative values indicate that the two variables are negatively correlated , meaning the two variables vary in

the contrary direction.



Generally, a correlation greater than 0.8 is considered as strong, whereas less than 0.5 is described as weak.

Applying correlation co-efficient in research (cont.)



Use of correlation coefficient Negative correlation, r =-1

Perfect positive correlation, r =+1

No linear association, r =0

The correlation for this plot is 0.8. It is heavily influenced by the extreme cluster of four points away from the main body.

Applying correlation co-efficient in research (cont.)

The higher the value of X, the lower the value of Y



Use of correlation coefficient Negative correlation, r =-1

The higher the value of X, the higher the value of Y

Perfect positive correlation, r =+1

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Pearson Moment correlation coefficient & Spearman correlation coefficient

There

are

many

types

of

correlation

tests

&

measurement scales Correlation test

Type of measurement

Pearson product-moment coefficient

Relationship between variables using interval & ratio scales

Spearman’s rho or eta coefficient

Relationship between variables when the distribution of data is not normal & both variables are in ordinal scale (arranged into rank)

Point-biserial coefficient

Relationship between variables using interval or ratio scales & a nominal scale

Biserial coefficient

Relationship between variables using interval or ratio scales & a nominal scale

Cramer, Phi & Lambda coefficients

Used when variables are in nominal scale &each variable has more than 2 categories

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Pearson Moment correlation coefficient & Spearman correlation coefficient

Pearson Moment correlation coefficient

-Pearson correlation test is a parametric test -Used to analyse data which are connected linearly

Variable 1 Interval / ratio scale

Pearson correlation test

Variable 2 Interval/ ratio scale

What is the meaning of interval & ratio scale?

Applying correlation co-efficient in research (cont.) • •

Use formula to determine relationship equation through calculation

Determine the correlation

Subject

X

Y

1

42

80

2

55

67

3

48

66

4

66

82

5

92

88

6

88

100

7

90

99

8

60

97

9

72

93

10

54

77

Applying correlation co-efficient in research (cont.) • •

Use formula to determine relationship equation through calculation

Step 1: Calculate the difference between the X score and the mean score (X-Ẋ) Subject

X

1

42

2

55

3

48

4

66

5

92

6

88

7

90

8

60

9

72

10

54

X-Ẋ

Applying correlation co-efficient in research (cont.) •

Step 2: Calculate the difference between the Y score and the mean score (Y-Ẏ)



Use formula to determine relationship equation through calculation

Subject

Y

1

80

2

67

3

66

4

82

5

88

6

100

7

99

8

97

9

93

10

77

Y-Ẏ

Applying correlation co-efficient in research (cont.)





Step 3: Calculate (X-Ẋ)² for each X score



Step 4:Obtain the total. ∑ (X-Ẋ)²

• Use formula to determine • relationship Subject equation 1 through calculation

Step 5: Calculate (X-Ẋ) (Y-Ẏ) for each subject Step 6:Calculate the total ∑(X-Ẋ) (Y-Ẏ) X

Y

42

80

2

55

67

3

48

66

4

66

82

5

92

88

6

88

100

7

90

99

8

60

97

9

72

93

10

54

77

X-Ẋ

Y-Ẏ

(X-Ẋ)²

(X-Ẋ) (Y-Ẏ)

Applying correlation co-efficient in research (cont.) •

Step 7: Calculate the values of b and a using the following formula:



Use formula to determine relationship equation through calculation

b = ∑(X-Ẋ) (Y-Ẏ) ___________ ∑ (X-Ẋ)²

a= Ẏ-bẊ

Applying correlation co-efficient in research (cont.) •



Use formula to determine relationship equation through calculation

Answer:

Y= 51.55 +O.50X

Applying correlation co-efficient in research (cont.)



Use formula to compute Pearson Moment correlation coefficient

YP=51.55+0.50X

Subject

X

Y

1

42

80

2

55

67

3

48

66

4

66

82

5

92

88

6

88

100

7

90

99

8

60

97

9

72

93

10

54

77



Y-Ẏ

(Y-Ẏ) ²

YP

Y-YP

(Y-YP) ²

Applying correlation co-efficient in research (cont.) •

The correlation coefficient, r is calculated by using the following formula:



Use formula to compute Pearson Moment correlation coefficient



r² = 1 – variation around regression variation around mean Y



r² = 1 – ∑(Y-YP ) ² )÷N ∑(Y-Ẏ) ² ) ÷N



*Find r.

Applying correlation co-efficient in research (cont.) •



Use formula to compute Pearson Moment correlation coefficient

Answer:

r=0.72

Based on the calculation the correlation coefficient is 0.72. This means that the correlation between IQ and

performance in mathematics test is strong. Variance r² 0.517 shows that 51.7% of the students’ performance is due to their IQ.

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Spearman correlation coefficient

Spearman correlation coefficient

- Spearman correlation test is a non-parametric test -Used to analyse data which are not correlate linearly

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Spearman correlation coefficient

Spearman correlation coefficient

- Spearman correlation test is a non-parametric test -Used to analyse data which are not normally distributed/ not correlate linearly

Variable 2 Ordinal scale

Variable 1 Ordinal scale Spearman correlation test

What is the meaning of ordinal scale?

Take 5!

Applying correlation co-efficient in research (cont.)



Use formula to compute Spearman correlation coefficient

• Question: A researcher wants to identify the relationship between job satisfaction level and motivation level of a group of accountants. A total of ten respondents, selected randomly from the HMM Accountancy Agency, answer 12-item job satisfaction test measured on an ordinal scale of: 1 = very low 2 = low 3 =average 4 = high 5 = very high And 9-item motivation test measured on an ordinal scale of: 1 = very low 2 = low 3 =average 4 = high 5 = very high

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Spearman correlation coefficient

Question: Respondent

Job satisfaction level (Total score of 12 ordinal scale items)

Motivation level (Total score of 9 ordinal scale items)

1

38

20

2

35

26

3

25

22

4

49

15

5

48

25

6

20

14

7

40

19

8

23

16

9

33

17

10

50

23

Applying correlation co-efficient in research (cont.)

Ranking starts from the lowest total score to highest total score

35

26

3

25

22

4

49

15

5

48

25

6

20

14

7

40

19

8

23

16

9

33

17

10

50

23

Square of ranking difference, P²

2

Ranking difference, P =p1-p2

20

Ranking, p2

38

Motivation level (Total score of 9 ordinal scale items)

1

Ranking, p1

Use formula to compute Spearman correlation coefficient

Job satisfaction level (Total score of 12 ordinal scale items)



Answer: Respondent



∑P²=

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Spearman correlation coefficient

Answer:

The Spearman’s rho test coefficient value is calculated by using the following formula: r = 1 –(6∑P² ) N(N²-1) where, P² = square of ranking difference N = sample size

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Spearman correlation coefficient

Answer:

Refer the critical value of r

Applying correlation co-efficient in research (cont.) • •

Use formula to compute Spearman correlation coefficient

Answer:

The result shows the calculated r value *0.394) is smaller than the critical r value (0.648). Thus there is no significant relationship between job satisfaction level and motivation level of the accountants in the population.

Applying correlation co-efficient in research (cont.) • •

Importance of correlation coefficient

Determination of association, i.e. correlation between phenomena (variables) is an important tool in scientific study.



The associations observed between two phenomena only allow us to pose a hypothesis in a scientific experiment

Take 5!

Apply the concept of regression in research • The structure of linear regression equation and its importance

Simple linear regression is used to describe the relation between one continuous outcome variable—for example, height and age

Apply the concept of regression in research (cont.) Y=a+bX The regression equation & its application in research

where, • • • •

Y= dependent variable X = independent variable a = constant value of the crossing point of the linear regression line at the Y axis b = the regression line gradient which state the relationship between variables

b =p /q

Where, • p = magnitude of the vertical line • q = magnitude of the horizontal line that connects the regression line

Apply the concept of regression in research (cont.) b =p /q Where, • p = magnitude of the vertical line • q = magnitude of the horizontal line that connects the regression line

R

The regression equation & its application in research

Regression line

p q

Apply the concept of regression in research (cont.) Solution: The regression equation & its application in research

Graph sketching method a. Plot the X (horizontal axis) and Y (vertical axis) values on the graph paper b. Sketch the regression line (straight line) that represents these coordinates points, by evenly dividing the points at the top and the bottom of the line c. Find the intersection point of the regression point d. Sketch the vertical line, p and horizontal line, q which connects to the regression line e. Put the derived values into the equation

Apply the concept of regression in research (cont.)

The regression equation & its application in research

Question: A teacher wants to identify whether the intelligence of students influences their capability in carrying out living skill activities. Determine the relationship. Subject

Independent variable Intelligence score (X)

Dependent variable Living skill score (Y)

1

92

98

2

60

88

3

72

95

4

54

79

5

45

82

6

54

70

7

50

67

8

76

83

9

48

88

10

88

99

Apply the concept of regression in research (cont.)

The regression equation & its application in research

Answer:

Y = 63 + 0.91 X

Apply the concept of regression in research (cont.) Application:

The regression equation & its application in research

Can be used to predict further result. For example: Predicting level of stress from the amount of time until you have to give a talk

Apply the concept of regression in research (cont.) Regression analysis is a way of predicting an outcome variable from

The regression equation & its application in research

one predictor variable (simple regression) or several predictor variables (multiple regression).

Outcomei= (model) + errori Whereby, Model : things that define the line that we fit to the data. It can be the slope/ gradient (b1) or the intercept of the line (b0).

i : variable

We want to fit a model that best describes the data.

Apply the concept of regression in research (cont.) Other equation that can be used:

The regression equation & its application in research

Yi = (b0 +b1Xi) + εi Whereby, Yi = outcome b0 = intercept b1 = slope Xi = is the ith participant score on the predictor variable εi = error

Q&A