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Understanding Simple Linear Correlation: Calculation, Interpretation, & Significance, Study notes of Statistics

Bryan CollegeStatistics

An in-depth explanation of simple linear correlation, including its definition, calculation steps, interpretation, and significance testing. It covers the correlation coefficient (r), its range, the difference between positive, negative, and no association, and the use of scatter plots. The document also discusses the hypothesis testing process and the importance of reporting the correlation coefficient and its significance level.

Typology: Study notes

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SIMPLE LINEAR CORRELATION
• Simple linear correlation is a measure of the degree to which two variables vary
together, or a measure of the intensity of the association between two variables.
• Correlation often is abused. You need to show that one variable actually is affecting
another variable.
• The parameter being measure is (rho) and is estimated by the statistic r, the
correlation coefficient.
• r can range from -1 to 1, and is independent of units of measurement.
• The strength of the association increases as r approaches the absolute value of 1.0
• A value of 0 indicates there is no association between the two variables tested.
• A better estimate of r usually can be obtained by calculating r on treatment means
averaged across replicates.
• Correlation does not have to be performed only between independent and dependent
variables.
• Correlation can be done on two dependent variables.
• The X and Y in the equation to determine r do not necessarily correspond between a
independent and dependent variable, respectively.
• Scatter plots are a useful means of getting a better understanding of your data.
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Positive association Negative association No association
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Download Understanding Simple Linear Correlation: Calculation, Interpretation, & Significance and more Study notes Statistics in PDF only on Docsity!

SIMPLE LINEAR CORRELATION

  • Simple linear correlation is a measure of the degree to which two variables vary together, or a measure of the intensity of the association between two variables.
  • Correlation often is abused. You need to show that one variable actually is affecting another variable.
  • The parameter being measure is  (rho) and is estimated by the statistic r, the correlation coefficient.
  • r can range from - 1 to 1, and is independent of units of measurement.
  • The strength of the association increases as r approaches the absolute value of 1.
  • A value of 0 indicates there is no association between the two variables tested.
  • A better estimate of r usually can be obtained by calculating r on treatment means averaged across replicates.
  • Correlation does not have to be performed only between independent and dependent variables.
  • Correlation can be done on two dependent variables.
  • The X and Y in the equation to determine r do not necessarily correspond between a independent and dependent variable, respectively.
  • Scatter plots are a useful means of getting a better understanding of your data. ...... ...... ........... .. …... …........ Positive association Negative association No association

The formula for r is:

( ) ( ) (SSX)(SSY)

SSCP

X X Y Y

n

X Y

XY -

2 2

r Example 1 X Y XY 41 52 2132 73 95 6935 67 72 4824 37 52 1924 58 96 5568 ∑X = 276 ∑Y = 367 ∑XY =21, ∑X^2 = 16,232 ∑Y^2 = 28,833 n = 5 Step 1. Calculate SSCP

5

SSCP = 21,383− =

Step 2. Calculate SS X

  1. 8 5

SSX 16,232 -

2 = = Step 3. Calculate SS Y

  1. 2 5

SSY 28 , 233 -

2 = =

  1. As n approaches 100, the r value to reject Ho:  = 0 becomes fairly small. Too many people abuse correlation by not reporting the r value and stating incorrectly that there is a “significant correlation”. The failure to accept Ho:= 0 says nothing about the strength of the association between the two variables measured.
  2. The correlation coefficient squared equals the coefficient of determination. Yet, you need to be careful if you decide to calculate r by taking the square root of the coefficient of determination. You may not have the correct “sign” is there is a negative association between the two variables.

Where: Zi^ '^ = 0. 5 ln ( 1 + ri ) ( 1 − ri )

Zw^ '^ =

∑[ (ni −^3 )Zi^ '^ ]

∑(ni −^3 )

χ 2 = ∑[(ni − 3 )(Z^ 'i^ − Zw^ '^ )^2 ]

df = n − 1 for χ 2 test Step 2. Look up tabular χ 2 value at the α = 0.005 level. χ^2 0.005, 2 df = 10. Step 3. Make conclusions Because the calculated χ^2 (0.388) is less than the table χ^2 value (10.6), we fail to reject the null hypothesis that the r - values from the three locations are equal.

Step 4. Calculate pooled r ( rp ) value Wheree 2. e 1 e 1 'W i W 2Z 2Z =

rp = Therefore 0. 341 e 1 e 1 2(0.356) 2(0.356) =

rp = Step 5. Determine if rp is significantly different from zero using a confidence interval. CI = ThereforeLCI 0.100andUCI 0.

rp 1. 96 = =

∑ ni

Since the CI does not include zero, we reject the hypothesis that the pooled correlation value is equal to zero.

Simple Linear Correlation Analysis

The CORR Procedure 2 Variables: (^) y x Simple Statistics Variable N Mean Std Dev Sum Minimum Maximum y (^) 5 73.40000 21.76695 367.00000 52.00000 96. x (^) 5 55.20000 15.78607 276.00000 37.00000 73. Pearson Correlation Coefficients, N = 5 Prob > |r| under H0: Rho= y x y (^) 1.00000 0.

x (^) 0.

  • The top value is the correlation value and the bottom value is the Prob>| r| to test the null hypothesis Ho: ρ=0.
  • The values on the diagonal are always 1.0.
  • The values above and below the diagonal are symmetrical.