The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

152 CHAPTER 9 Nonparametric Methods

ρ=− 16 Tnn/[ (^2 − 1 )},

where TRXRY=∑in= 1 [( ) ()]ii−^2.

As an example, let us look at the correlation between the tempera-

ture in Washington and New York over the 12 months of the year. The

reason the paired t test worked so well was because most of the variation

was due to seasonal effects that were removed through the paired differ-

ences. This variation will translate into high correlation between X i , the

temperature in New York on the 15th of month i , with Y i , the temperature

Washington, DC on the 15th of the i th month. Using the ranks, we show

in Table 9.4 how the Spearman correlation is calculated in this case.

Table 9.4
Daily Temperatures for Two Cities: Spearman Rank Correlation

Day Washington mean
temperature ( ° F)

New York mean

temperature ( ° F)

Ranked

pairs

Term

Y (rank) X (rank) [ r ( x ),

ry )]

[ r ( y i ) − r ( x i )]^2

1. January 15 31 (2) 28 (3) (3, 2) 1


2. February 15 35 (4) 33 (4) (4. 4) 0


3. March 15 40 (5) 37 (5) (5, 5) 0


4. April 15 52 (6) 45 (6) (6, 6) 0


5. May 15 70 (8) 68 (8) (8, 8) 0


6. June 15 76 (10) 74 (10) (10, 10) 0


7. July 15 93 (12) 89 (12) (12. 12) 0


8. August 15 90 (11) 85 (11) (11, 11) 0


9. September 15 74 (9) 69 (9) (9, 9) 0


10. October 15 55 (7) 51 (7) (7, 7) 0


11. November 15 32 (3) 27 (2) (2, 3) 1


12. December 15 26 (1) 24 (1) (1.1) 0
    T 2
    ρ = 1 − 6 T /
    [ n ( n 2 − 1)]

= 1 − 1 2 /

[12(143)]

= 142/143

= 0.9930