8. Correlation

KNOWLEDGE FOR THE BENEFIT OF HUMANITYKNOWLEDGE FOR THE BENEFIT OF HUMANITY

BIOSTATISTICS (HFS3283)

CORRELATION

Dr.Dr. MohdMohd RazifRazif ShahrilShahril

School of Nutrition & Dietetics School of Nutrition & Dietetics

Faculty of Health SciencesFaculty of Health Sciences

UniversitiUniversiti Sultan Sultan ZainalZainal AbidinAbidin

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S C H O O L O F N U T R I T I O N A N D D I E T E T I C S • U N I V E R S I T I S U L T A N Z A I N A L A B I D I N

Topic Learning Outcomes At the end of this lecture, students should be able to;

• identify types of correlation analysis and when they are

used.

• explain assumptions to be met when using Pearson and

Spearman correlation.

• perform Pearson and Spearman correlation analysis

using SPSS.

• explain how to interpret the SPSS outputs from Pearson

and Spearman correlation analysis.

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Correlation

• A measurement of the magnitude and direction

of two numerical variables

• The estimated correlation value may not be

100% accurate but the stronger is the

relationship, the more accurate the estimate.

• Involves mainly observation of a certain

relationship – No control of manipulation imparted

– E.g. what is the correlation between sleeping hours

and BMI among UniSZA students

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Aims for correlation

• To show three (3) important aspects involved in

an association:

– Type of correlation

– Direction of correlation

– Magnitude of correlation

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Uses of correlation

• Prediction – E.g. The SPM Trial exam results are used as selection

criteria into universities

• Validity – To ensure an instrument measures what it should measure

– E.g. new digital thermometer vs. clinic thermometer

• Reliability – To ensure an instrument always give consistent reading (if

other parameters are stable)

• Theory verification – E.g. is it true that a relationship exists between an

individual’s IQ level and their brain size.

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Bivariate distribution

• A distribution showing the association between

two (2) numerical variables

• Usually illustrated via the Scatter diagram/ plot

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Pearson’s r

• The Pearson’s correlation coefficient, r provides

the magnitude of a correlation.

• According to Pearson’s mathematical procedure,

a correlation is linear.

• r ranges from 0 (no correlation) to +/- 1.0

(strongest correlation)

– The higher the value of r, the stronger is the

magnitude of correlation

– The symbols + or – indicate the direction of the

correlation

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Characteristics of r

• Describes only linear correlation

• Sensitive towards variability (spread) and range

• Can be influenced by sampling variation

• Can be influenced by extreme/ outlier values

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Types of correlation

• Three main types;

– Positive correlation (r = + ; variables change in the

same direction)

– Negative correlation (r = - ; variables change in the

opposite direction)

– Zero correlation (r = 0 ; both variable show no

association and direction)

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Types of correlation (cont.)

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Types of correlation (cont.)

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Interpretation

• A relationship exits between two variables; X & Y

• Does not mean any change in X will also alter Y

• Does not involve causal relationship (e.g. X causes Y to

occur)

• If r = 0.50 does not mean 50% relationship

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Interpretation (cont.)

CorrelationCorrelation InterpretationInterpretation

--1.00 to 1.00 to --0.760.76 Strong negative correlationStrong negative correlation

--0.510.51 to to --0.750.75 Good negative correlationGood negative correlation

--0.50 to 0.50 to --0.260.26 Fair negative correlationFair negative correlation

--0.25 to 0.010.25 to 0.01 Poor negative correlationPoor negative correlation

00 No correlationNo correlation

0.010.01 to 0.25to 0.25 Poor positive correlationPoor positive correlation

0.26 to 0.500.26 to 0.50 Fair positive correlationFair positive correlation

0.51 to 0.750.51 to 0.75 Good positive correlationGood positive correlation

0.76 to 1.000.76 to 1.00 Strong positive correlationStrong positive correlation

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Linear vs. Non-linear correlation

• Not all correlations are linear

• Sometimes correlation can be non-linear e.g.

Curvilinear

– Score are concentrated on a curved line

• The Pearson’s coefficient is not suitable to

quantify this relationship

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Linear vs. Non-linear correlation

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Positive linear r = 0.82

Negative linear r = -0.70

Independent r = 0.00

Curvilinear r = 0.00

Curvilinear r = 0.00


Simple example on correlation

• Research Question: Is sleeping duration (hrs) correlated with exam score (marks)?

• Null Hypothesis: There is no correlation between sleeping duration (hrs) and exam score (marks)

• Results: If r = -0.89; p < 0.05; n = 220

• Interpretation: The correlation is negative i.e. as the sleeping duration increase, the exam score will decrease and vise versa. – The large magnitude of correlation (nearing 1.00)

means the association is strong

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Assumptions

• The data is drawn from a random sample of

population.

• The data is independent observation to each

other.

• The variables of interest are having bivariate

normal distribution.

• The relationship between the two variables

must be linear.

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1 1

2 2

3 3

4 4


Assumption 3 – Linearity

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1 1

2 2

3 3


Assumption 3 – Linearity (cont.)

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4 4

5 5

6 6

7 7


Assumption 3 – Linearity (cont.)

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The relationship between the two variables is linear

based on the elliptical shape.


If assumptions are not met?

• If both variable does not meet all assumptions,

then use Spearmen’s correlation analysis. – Values obtained would be Spearmen’s correlation

coefficient (rho).

• If any one of the variable meet all assumption,

proceed with Pearson’s correlation analysis – Values obtained is Pearson’s correlation coefficient r.

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Correlation in SPSS

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1 1

2 2 3 3

4 4

6 6

5 5

For normal distribution, we can use Pearson’s test. For non-normal distribution, we can proceed to Spearman’s test.


SPSS Output

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SPSS Output Interpretation

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• In correlation, look at the correlation coefficient

which is the strength of relationship instead of

the p value. – Even though the coefficient is weak, the p value is

usually significant depending upon the sample size.

• In this analysis the p value is < 0.05, therefore

we reject the null hypothesis.


SPSS Output Interpretation (cont.)

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• There is a significant (linear) correlation between

sleeping duration and exam score (p< 0.001).

The observed correlation coefficient (r) is 0.337,

which suggests positive and fair correlation. It

suggests that higher the duration of sleeping,

the higher the exam scores.


Results presentation

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Exam score (marks)

r p value*

Sleeping duration (hrs) 0. 337 < 0.001

* Pearson’s correlation

Table: Correlation between sleeping duration (hrs) and exam score (marks)

Thank YouThank You

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Date post:	13-Jan-2017
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8. Correlation

Health & Medicine