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Data Interpretation and Statistical Methods

Analyze data using Spearman’s rank correlation coefficient (rho)

  • It measures the relationship between two logically related variables. Like the conventional correlation coefficient (r), Spearman rho can have any value between -1 to +1. A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear. In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. Monotonic relationship can be nonlinear. In a linear relationship, the variables move in the same direction at a constant rate.

CLICK HERE to refer to the diagrams of Linear, Nonlinear and monotonic relationships.

In Statistics, Correlation is largely a measure of an association between variables. In logically correlated data, the change in the magnitude of 1 variable is related to a corresponding change in the magnitude of another variable, either in the same (positive correlation: High-High, Low-Low) or in the opposite (negative correlation: Low-High or High-Low) direction. Very often, the term correlation is used in the context of a linear relationship between 2 continuous variables and expressed as Pearson product-moment correlation. For monotonically (nonlinear) distributed continuous ordinal data(indicate the order or rank of things) or for data with relevant outliers,  Spearman rank correlation can be used as a measure. Spearman rho refers to the ranked values rather than the original measurements.

Refer: IARS journal for more information

Determine how these two variables are correlated using Spearman’s rank correlation coefficient?

Variable A
(arranged in an ascending order-smallest to largest)
Rank (R1)
(assigned after arranging in order)
Variable BRank (R2)Difference in rank
(R1-R2)=d
200106.21+981
110095.12+749
190083.14+416
240074.63+416
310062.55+11
390051.57-24
420041.86-24
530030.410-749
590021.38-636
650010.59-864
Total d² = 320

Spearman rank (rho) = 1-( \frac{6 \sum d^2 }{n^3-n })

where n = number of paired observations

rho = 1-( \frac{6 \cdot 320 }{1000 - 10 })

= 1-( \frac{1920 }{990 }) = 0.939

There are (n) degrees freedom (10)

Significance Level

Compare your result to the critical values for Spearman's Rank correlation (ignoring + / -). If rho is greater than or equal to the critical value, then there is a significant correlation and the null hypothesis can be rejected.
Number of pairs of measurement
(n)
p = 0.05
(95%)
(+ or -)
p = 0.01
(99%)
(+ or -)
51
60.8861.000
70.7860.929
80.7380.881
90.6830.833
100.6480.818
110.6230.794
120.5910.780
130.5660.745
140.5450.716
150.5250.689
160.5070.666
170.4900.645
180.4760.625
190.4620.608
200.4500.591
250.4000.526
300.3640.478
350.3360.442
400.3140.413

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