Measuring relationships

MEASURING RELATIONSHIPSPearson r Product Moment Coefficient of Correlation

Mike Arieh P. Medina, PhD

Outline

• What is coefficient of correlation• How to calculate a correlation• Assumptions about correlation calculations• Interpretation of the correlation• Evaluating the significance of the relationship

The correlation coefficient

• Indicated by r• Ranges from -1.00 to +1.00– The value indicates the strength of the

relationship– The closer to 0, the weaker the relationship– The closer to +/- 1.00 the stronger the relationship

• The sign indicates whether the relationship is positive or negative

Question

• Which of the following coefficient of correlation indicates the strongest relationship?

• A. + 0.45• B. – 0.33• C. + 0.58• D. – 0.67

The Equation

Problem: Is no. of hospitals related to MMR?

City No. of Hospitals Maternal Mortality Rate

A 5 5B 10 2C 4 8D 8 3E 2 8F 7 5G 9 5H 6 7I 1 10J 12 3

Hypothesis

• H0: There is no significant relationship between no. of hospitals and MMR.

• H1: There is a significant relationship between no. of hospitals and MMR.

0 2 4 6 8 10 12 140

2

4

6

8

10

12

Scatterplot of MMR vs. Hospitals

Hospitals

MM

R

Types of Correlation

Positive Correlation Negative Correlation

No CorrelationPerfect Correlation

Types of Correlation

Strong Correlation Weak Correlation

Sample Problem

x y xy x2 y2

5 5 25 25 2510 2 20 100 44 8 32 16 648 3 24 64 92 8 16 4 647 5 35 49 259 5 45 81 256 7 42 36 491 10 10 1 100

12 3 36 144 9Σx = 64 Σy = 56 Σxy = 285 Σx2 = 520 Σy2 = 374

Computation

y

Computation

Computation

Correlation Assumptions

• Before you can compute for Pearson r make sure that the following assumptions are satisfied:– The relationship is linear– The variables are measured on an interval or ratio

scale– The distribution of the variable is normal– The distribution is continuous

Interpretation

Magnitude of r

Range of r Descriptive label for r+/- 0.80 – 0.90 Very Strong+/- 0.60 – 0.79 Strong+/- 0.40 – 0.59 Moderate+/- 0.20 – 0.39 Weak+/- 0.00 – 0.19 Very Weak

Interpretation

• r2 = Coefficient of Determination• Percent variance that is shared between variable x and

variable y

• 1- r2 = Coefficient of Alienation• Percent variance that is not shared between variable x

and variable y

• Now, compute for the r2 of the previous example and interpret.

InterpretationCoefficient Value Interpretation

r 0.899Very Strong negative relationship between no. of hospitals and MMR (the more hospitals in a city the lesser its MMR)

r2 0.80880.8% of the variation in MMR is influenced by the variation in the no. of hospitals

1-r2 0.19219.2% of the variation in MMR is influenced by factors other than no. of hospitals

How do we know if it’s a significant relationship? (Ex. at 0.05)

df = N-2

DOWNLOAD table of critical values of the Pearson Product-Moment correlation

Exercise

• Now, compute for the coefficient of correlation using the population and weight of waste hypothetical data.

• Population (thousands): 8, 6, 7, 4.5, 5• Solid Waste (thousand tons): 2, 1.8, 1.5, 1.1, 1.25• Create a scatterplot• Interpret the results• ANS: r = 0.858