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Chapter 08correlation
 

Chapter 08correlation

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    Chapter 08correlation Chapter 08correlation Presentation Transcript

    • Chapter 8 Simple Linear Correlation
      • Up to now, the statistical methods you have learnt concern with single variable only
      • Such as
      • To estimate the average height among high school students
      • To compare the average height of high school students between city and country side
      • However, the relationship between two variables is often concerned in practice:
      • Example: For high school students,
      • Height and Age – linear relation?
      • Height and Weight – linear relation?
      • In this chapter, we are going to study:
      • two variables,
      • linear relationship between two variables
      • Two types of questions:
      • Whether there is a linear relationship?
      • -- Linear correlation
      • How to predict one variable by another variable?
      • -- Linear regression
      • Example 7.1   To explore the correlation between
      • systolic pressure and diastolic pressure (mmHg),
      • 665 girls aged from 6 to 10 years were measured.
      • Two random variables X and Y
      • Sample: 665 girls
      • The individuals in the sample should be
      • independent each other.
      • Example 7.2 To explore the correlation between
      • the heights of father and son, 20 graduate male
      • students were randomly selected from a name
      • list of graduates in a high school. The heights
      • (cm) of fathers and sons were measured
      • Table 7.1 Heights (cm) of 20 pairs of father and son
    • 12.1 Linear correlation
    • Scatter Diagram : Fig. 7.1 Scatter diagram of systolic and diastolic blood pressures (mmHg) of 665 girls of 6 to 10 years old
    •  
    •  
    • 7.2 Correlation Coefficient 7.2.1 Population correlation coefficient
      • Pearson’s product-moment linear correlation coefficient:
      • The mean of “product of the two standardized variables”
      • ---- simple correlation coefficient
      -- covariance between X and Y
    • Pearson’s product moment sample correlation coefficient r 7.2.2 Sample correlation coefficient
    • A measurement of linear relationship: 1) Whether there is a correlation If the correlation coefficient is 0 or not big enough -- no correlation 2) If correlation coefficient is big enough The direction of correlation? -- positive or negative The strength of correlation? high or not? -- Is the absolute value big enough? Complete correlation : +1 or -1,
    • Example 7.3 Calculate the correlation coefficient between the heights of father and son.
      • r is sample correlation coefficient, change from sample to sample
      • There is a population correlation coefficient, denoted by ρ
      • Question : Whether ρ =0 or not?
      • Assumption:
      • X and Y follow a bi-variable normal distribution
      7.3 Inference on Correlation Coefficient 7.3.1 Hypothesis test
      • H 0 : ρ =0, H 1 : ρ ≠0 α =0.05
      • (1) Checking a special table (Table 8 in appendix 2 )
      • H 0 is rejected
      • ---- positive correlation between the heights of father and son.
      • Question:
      • Since , very small, can we say
      • the correlation is very strong ?
      • Does a small P value mean that the correlation
      • is strong ?
    • Question : If r =0.90, can you claim the two variables are correlated each other? Table 8 Critical values for r
    • (2) t test (Assume normal distribution) H 0 : ρ =0, H 1 : ρ ≠0
      • If P -value < α , then reject H 0 , conclude that
      • the population correlation coefficient is significantly different from 0.
      •  =20-2=18,
      • The population correlation coefficient might not be 0.
    • 7.3.2 Interval estimation
      • Assumption: X and Y follow a bi-variable normal distribution.
      • Pre-knowledge:
      • (1) hyperbolic tangent and its inverse
      • Hyperbolic tangent ( 双曲正切 )
      • Inverse  of hyperbolic tangent ( 反双曲正切 )
      • approximately follows a normal distribution
      • Confidence interval of :
      • or
      • Taking a transformation of
    • Example 7.5 After getting , please find out a 95% confidence interval for population correlation coefficient .
      • It is useful to:
      • ranked data
      • As well as measurement data
      • ---- not follow a normal distribution;
      • or not sure about the distribution;
      • or not precisely measured
      • or X or Y are ordinal variables
      7.4 Rank Correlation 7.4.1 Spearman ’ s rank correlation coefficient
    • Spearman ’ s rank correlation coefficient
      • sort (x 1 ,x 2 ,…,x n ), get rank p i for x i
      • sort (y 1 ,y 2 ,…,y n ), get rank q i for y i
      • n pairs of observations, (x 1 ,y 1 ), …, (x n ,y n )
    • Example 7.6 An etiology study on liver cancer has collected data on liver-cancer-specific death rate and the relative content of aflatoxin in certain food for 10 Counties. Putting the ranks into the formula of Spearman’s correlation coefficient
    • Table 9 Critical value for Spearman’s rank correlation coefficient 7.4.2 Hypothesis test for (1) Checking a special table (Table 9) P <0.02 and it is significant
      • (2) t test
      • Same as the t test for Pearson’s correlation
      • coefficient
      • If p is small, then reject
    •