Biomedical Statistics系所別 : 中央大學電機工程學系 (NCUEE) 指導教授 :蔡章仁 (Jang-Zern Tsai)姓名 : 凃建宇 (Jacky Tu)

Outline Two Sample Hypothesis Testing for Correlation Multiple Correlation Spearman’s Rank Correlation

Two Sample Hypothesis Testing for Correlation

Case1: Independent samples

Case2: Dependent samples

Two Sample Hypothesis Testing for Correlation with independent samples

Example:A sample of 40 couples from London is taken comparing the husband’s IQ with his wife’s. The correlation coefficient for the sample is .77. Is this significantly different from the correlation coefficient of .68 for a sample of 30 couples from Paris?

Some excel functions:

FISHER equ.

SQRT equ. square root(number)

NORMSDIST equ.

Then we can perform either one of the following tests:

Two Sample Hypothesis Testing for Correlation with dependent samplesWhat difference?two correlations have one variable in common or because the two variables are correlated at one moment in time and again at another moment in time

Example: IQ tests are given to 20 couples. The oldest son of each couple is also given the IQ test with the scores displayed in Figure 1. We would like to know whether the correlation between son and mother is the significantly different from the correlation between son and father.

use the following test statistic

S is the 3 × 3 sample correlation matrix and

Since p-value = .042 < .05 = α  we reject the null hypothesis, and conclude that the correlation between mother and son is significantly different from the correlation between father and son.

Multiple Correlation

We can also calculate the correlation between more than two variables

Definition 1: multiple correlation coefficient

multiple coefficient of determination Rz,xy^2

R^2

x,y:independent variables z:dependent variable R

Multiple Correlation(Cont.)

Definition 2adjusted multiple correlation coefficient

 k = the number of independent variables

Example

By using Excel’s Correlation data analysis tool, we can get correlation coefficients for data in Example 

We use the data above to obtain the values , rPW , rPI , and rWI 

Definition 3:partial correlation(x and z holding y constant)

semi-partial correlation(x and y is eliminated,  x and z and y and z not)

ExampleIf we want to know the relationship between GPA (grade point average) , salary and IQbut maybe IQ correlates well with both GPA and Salary.To test this need to determine the correlation between GPA and salary eliminating the

influence of IQso the partial correlation r(GS,I)

If we continue calculate r(PW,I),rP(W,I)

Then we can proof the property by:

Since the coefficients of determination is a measure of the portion of variance attributable to the variables involved, we can look at the meaning of the concepts defined above using the following Venn diagram, where the rectangular represents the total variance of the poverty variable

calculate the breakdown of the variance for poverty:

we can calculate B in a number of ways: (A + B –  A, (B + C) – C, (A + B + C) – (A+ C)

where D = 1 – (A + B + C)

Follow the property 1:

If the independent variables are mutually independent:

Spearman’s Rank Correlation

Definition : the same as correlation coefficient r has the range -1~1 but is on the rank.Example: If IQ associates with the number of hours listen to rap music per month

Pearson’s correlation = CORREL(A4:A13,B4:B13) = -0.036

Spearman’s rho = CORREL(C4:C13,D4:D13) = -0.115

Can use Excel’s functionRANK.AVG(A4,A$4:A$13,1)

shows there isn’t much of a correlation between IQ and listening to rap music, although the Spearman’s rho is closer to zero (indicating independent samples) than the Pearson’s.

If we plot the example

no ties in the ranking, there is alternative way of calculating Spearman’s rho using the following property

di = rank xi – rank yi

If we use the property above to do the example again:

the same as the CORREL(C4:C13,D4:D13) = -0.115

Example: Repeat the analysis for Example of 

One Sample Hypothesis Testing for Correlation using Spearman’s rho

Spearman’s rho is the correlation coefficient on the ranked data, namely CORREL(C5:C19,D5:D19) = -.674

A study is designed to check the relationship between smoking and longevity. A sample of 15 men 50 years and older was taken and the average number of cigarettes smoked per day and the age at death was recorded, as summarized in the table in Figure 1. Can we conclude from the sample that longevity is independent of smoking?

We now use the table in Spearman’s Rho Table to find the critical value for the two-tail test where n = 15 and α = .05. Interpolating between the values for n = 14 and 16, we get a critical value of .525. Since the absolute value of rho is larger than the critical value, we reject the null hypothesis that there is no correlation between cigarette smoking and longevity.

Since n = 15 ≥ 10, we can use a t-test instead of the table

Since |t| = 3.29 > 2.16 = tcrit = TINV(.05,13), we again conclude that there is a significant negative correlation between the number of cigarettes smoked and longevity.

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