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 - PowerPoint PPT Presentation
Biomedical Statistics: (NCUEE) :(Jang-Zern Tsai): (Jacky Tu)
OutlineTwo Sample Hypothesis Testing for CorrelationMultiple CorrelationSpearmans Rank Correlation
Two Sample Hypothesis Testing for Correlation
Case1: Independent samples
Case2: Dependent samplesTwo Sample Hypothesis Testing for Correlation with independent samples
Example:A sample of 40 couples from London is taken comparing the husbands IQ with his wifes. 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:
SQRT equ. square root(number)
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 statisticSis the3 3 sample correlation matrix and
Since p-value = .042 < .05 = we reject the null hypothesis, and conclude that the correlation between mother and son is significantly different from the correlation between father and son.
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 2adjustedmultiple correlation coefficient
k= the number of independent variablesExample
By using Excels 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(xandz holding y constant)
semi-partial correlation(xand y is eliminated, xandzand y andz 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)
whereD= 1 (A + B + C)
Follow the property 1:
If the independent variables are mutually independent:
Spearmans 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
Pearsonscorrelation= CORREL(A4:A13,B4:B13) = -0.036
Spearmansrho = CORREL(C4:C13,D4:D13) = -0.115
Can use Excels functionRANK.AVG(A4,A$4:A$13,1) shows there isnt much of acorrelationbetween IQ and listening to rap music, although theSpearmansrho is closer to zero (indicating independent samples) than the Pearsons.
If we plot the example
no ties in the ranking, there is alternative way of calculating Spearmans rho using the following property
di= rank xi rank yiIf 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 ofOne Sample Hypothesis Testing for Correlationusing Spearmans rho
Spearmans rho is the correlation coefficient on the ranked data, namely CORREL(C5:C19,D5:D19) = -.674A 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 inSpearmans Rho Tableto find the critical value for the two-tail test wheren= 15 and= .05. Interpolating between the values forn= 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.
Sincen= 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.Thank you for your listensing