Measuring relationships
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This document discusses the Pearson r correlation coefficient and how to calculate, interpret, and evaluate the significance of relationships between variables. The Pearson r ranges from -1 to 1, with values closer to these extremes indicating a stronger linear relationship and values near 0 a weaker relationship. A correlation of -0.67 represents the strongest relationship among the options given. The correlation coefficient is calculated using the formula presented. An example calculates the correlation between number of hospitals and maternal mortality rate in various cities. Interpretation of the correlation involves evaluating the magnitude, sign, strength of relationship, and significance.
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Correlation is a statistical technique used to determine the strength and direction of the relationship between two variables. The coefficient of correlation (r) measures the statistical correlation and ranges from +1 to -1, where +1 is a perfect positive correlation, -1 is a perfect negative correlation, and 0 is no correlation. Common guidelines for interpreting the coefficient define strong, moderate, and weak correlations based on the absolute value of r. While correlation can examine relationships between quantifiable data, it does not prove causation between variables due to potential confounding factors.
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1) The document analyzes correlations between various physical and psychological measures based on statistical data. It reports correlation coefficients (r values) and significance values (p values) between different variable pairs.
2) Several correlations are identified as statistically significant based on p values less than 0.01 or 0.05, with moderate to strong correlation coefficients ranging from around 0.3 to over 0.5.
3) The strongest correlations explain over 30% of the variance between variables, which is considered clinically significant, while weaker correlations explain only around 1% of variance or less.Correlation analysis notes
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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
IDAge (x)CMFS (y)183129253940410275113569297825893498441011191172812647136421483715935161216171512181323191026201036
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 6.
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This document discusses correlation and related statistical concepts. Correlation measures the strength and direction of association between two quantitative variables. A correlation of 0 means no association, 1 means perfect positive association, and -1 means perfect negative association. Correlation is independent of measurement units and scaling of variables. Hypothesis testing is used to make inferences about the population correlation based on a sample correlation. The null hypothesis is that the population correlation is 0, and alternative hypotheses specify a non-zero correlation. The test statistic used is Student's t distribution. The null is rejected if the calculated t exceeds the critical value or if the p-value is less than the significance level.
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Measuring relationships
- 1. MEASURING RELATIONSHIPS Pearson r Product Moment Coefficient of Correlation Mike Arieh P. Medina, PhD
- 2. 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
- 3. 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
- 4. Question • Which of the following coefficient of correlation indicates the strongest relationship? • A. + 0.45 • B. – 0.33 • C. + 0.58 • D. – 0.67
- 5. The Equation
- 6. Problem: Is no. of hospitals related to MMR? City No. of Hospitals Maternal Mortality Rate A 5 5 B 10 2 C 4 8 D 8 3 E 2 8 F 7 5 G 9 5 H 6 7 I 1 10 J 12 3
- 7. 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.
- 8. 0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 MMR Hospitals Scatterplot of MMR vs. Hospitals
- 9. Types of Correlation Positive Correlation Negative Correlation No Correlation Perfect Correlation
- 10. Types of Correlation Strong Correlation Weak Correlation
- 11. Sample Problem x y xy x2 y2 5 5 25 25 25 10 2 20 100 4 4 8 32 16 64 8 3 24 64 9 2 8 16 4 64 7 5 35 49 25 9 5 45 81 25 6 7 42 36 49 1 10 10 1 100 12 3 36 144 9 Σx = 64 Σy = 56 Σxy = 285 Σx2 = 520 Σy2 = 374
- 12. Computation y
- 13. Computation
- 14. Computation
- 15. 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
- 16. 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
- 17. 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.
- 18. Interpretation Coefficient Value Interpretation r 0.899 Very Strong negative relationship between no. of hospitals and MMR (the more hospitals in a city the lesser its MMR) r2 0.808 80.8% of the variation in MMR is influenced by the variation in the no. of hospitals 1-r2 0.192 19.2% of the variation in MMR is influenced by factors other than no. of hospitals
- 19. 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
- 20. 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