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Measures of Correlation PDF

This document discusses correlation and regression analysis. It defines correlation as measuring the strength and direction of association between two quantitative variables. The Pearson product-moment correlation coefficient is used to analyze the correlation between continuous variables. Regression predicts the value of a dependent variable based on the value of an independent variable using the regression equation y= a + bx. The document provides an example of using correlation and regression to analyze the relationship between weight and hemoglobin A1c level from medical records. It finds an inverse high correlation and predicts hemoglobin A1c levels based on weight values.

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Nathaniel Pulido
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249 views14 pages

Measures of Correlation PDF

This document discusses correlation and regression analysis. It defines correlation as measuring the strength and direction of association between two quantitative variables. The Pearson product-moment correlation coefficient is used to analyze the correlation between continuous variables. Regression predicts the value of a dependent variable based on the value of an independent variable using the regression equation y= a + bx. The document provides an example of using correlation and regression to analyze the relationship between weight and hemoglobin A1c level from medical records. It finds an inverse high correlation and predicts hemoglobin A1c levels based on weight values.

Uploaded by

Nathaniel Pulido
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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S.M.A.

- Pearson – Product
Moment Correlation
Measures Coefficient
- Spearman Rank – Order
Correlation Coefficient
Used to analyze a collection of paired sample data
and determine whether these appears to be a
relationship between the two quantitative variables.
One variable (y) is treated as the response (dependent
or outcome) variable

The other variable (x) is the explanatory (independent


or predictor) variable
- Measures the strength of
association
Linear - The sign describes the direction
while the absolute value of the
Correlation correlation coefficient describes
Coefficient the magnitude of the relationship

- Denoted by rho (𝜌) or r


−1 ≤ ρ/r ≤ 1
CORRELATION INTERPRETATION X Y
COEFFICIENT
+ 0.91 - +1.00 VERY HIGH
+ 0.71 - +0.90 HIGH
+
+ 0.41 - +0.70 MODERATE
+ 0.21 - +0.40 LOW
-
+ 0.00 - +0.20 VERY LOW
Pearson
Product
Moment Karl Pearson

Coefficient Correlation of continuous variables


of
Correlation
n xy    x  y 
rxy
n x    x n y    y 
2 2 2 2
Weight HbA1C
A researcher abstracted level (%)
hemoglobin A1C and 269 7.3
weight from consenting 120 12.1
participants’ medical 224 7.5
records. 223 9.4
211 5.9
x y x² y² xy 𝑛 𝑥𝑦 − 𝑥 𝑦
269 7.3
𝑟𝑥𝑦 =
72361 53.29 1963.7 𝑛 𝑥2 − 𝑥 2 𝑛 𝑦2 − 𝑦 2

120 12.1 14400 146.41 1452 5 8436.8 − 1047 42.2


𝑟𝑥𝑦 =
224 7.5 50176 56.25 1680 5 231187 − 1047 2 5 379.12 − 42.2 2

223 9.4 49729 88.36 2096.2 42184 − 44183.4


𝑟𝑥𝑦 =
211 5.9 44521 34.81 1244.9 1155935 − 1096209 1895.6 − 1780.84

∑x= ∑y= ∑x²= ∑y²= ∑xy= −1999.4


379.12 8436.8 𝑟𝑥𝑦 =
1047 42.2 231187 59726 114.76
***Therefore, there is an inverse −1999.4
high relationship between the 𝑟𝑥𝑦 =
6854155.76
weight and hemoglobin A1C level
of the participants. −1999.4
𝑟𝑥𝑦 =
*** As their weight increases , their 2618.04
hemoglobin A1C level decreases 𝑟𝑥𝑦 = −0.76
and vice versa.
 Regression is the next step after correlation.
 Correlation analysis is concerned with strength of
the relationship while regression analysis predicts
the value of a dependent variable (y) based on the
value of at least one independent variable (x).
 Regression is a statistical model. A mathematical
formula or assumption that describes a real world
situation.
𝑦= a + bx
where: a= y-intercept
b= slope
𝑦 = predicted value of the dependent variable

𝑛 𝑥𝑦 − 𝑥 𝑦
b=
𝑛 𝑥2 − 𝑥 2
a= 𝑦 - b𝑥
Weight HbA1C
A researcher abstracted level (%)
hemoglobin A1C and 269 7.3
weight from consenting 120 12.1
participants’ medical 224 7.5
records. 223 9.4
211 5.9
x y x² y² xy 𝑛 𝑥𝑦 − 𝑥 𝑦
b= 𝑎 = 𝑦 − 𝑏𝑥
269 7.3 𝑛 𝑥2 − 𝑥 2
72361 53.29 1963.7
𝑦 42.2
120 12.1 𝑦= = = 8.44
14400 146.41 1452 5 8436.8 − 1047 42.2 𝑛 5
b=
224 7.5 5 231187 − 1047 2 𝑥 1047
50176 56.25 1680 𝑥= = = 209.4
𝑛 5
223 9.4 49729 88.36 2096.2 −1999.4
b= 𝑎 = 𝑦 − 𝑏𝑥
59726
211 5.9 44521 34.81 1244.9 𝑎 = 8.44 − (−0.03)(209.4)

∑x= ∑y= ∑x²= ∑y²= ∑xy= b=-0.03 𝑎 = 14.72


1047 42.2 231187 379.12 8436.8

𝑦= 𝑎 + 𝑏𝑥 a. Weight = 130 b. x= 255

𝑦 = 14.72 − 0.03𝑥 𝑦 = 14.72 − 0.03𝑥


𝑦 = 14.72 + (−0.03)𝑥 𝑦 = 14.72 − 0.03(130) 𝑦 = 14.72 − 0.03(255)
𝑦 = 10.82 𝑦 = 7.07
𝑦 = 14.72 − 0.03𝑥
Alcausin, G. Garcia, E., & Manikis, M. (1989).
Makati: Salesiana Publishers, Inc.
Goodman, M. (2018). Biostatistics for Clinical
and Public Health Research. New York:
Routledge.

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