Regression and Correlation Analysis Violeta Bartolome
Senior Associate Scientiest PBGB-CRIL
[email protected]
Correlation Analysis • A measure of association between two numerical variables. • Example (positive correlation) o As
soil fertility increases, rice grain yield also increases
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Example
For seven randomly nitrogen content in the soil and the grain yield were recorded.
Nitrogen Content (%)
Grain Yield (kg/ha)
0.12
1652
0.14
2056
0.15
2598
0.16
2734
0.19
3238
0.22
4824
0.23
4858
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Grain YieldCancel of Rice at differntAnd levels Download Print of Soil Nitrogen Content 6000 ) 5000 a h / g k ( d l e i Y 3000 n i a r G
2000 1000 0.1
0.15
0.2
0.25
Nitrogen Content (%)
How “strong” is the linear relationship? IRRI-PBGB-CRIL
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Measuring the Relationship In order to print this document from Scribd, you'll first need to download it. Cancel
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Pearson’s Sample Correlation Coefficient, r measures the direction and the strength of the linear association between two numerical paired variables.
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Direction of Association Download And Print
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Positive Co Correlation
Negative Co Correlation
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Strength of Linear Association In order to print this document from Scribd, you'll first need to download it. Cancel
r
value
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Interpretation
1
perfect positive linear relationship
0
no linear relationship
-1
perfect negative linear relationship IRRI-PBGB-CRIL
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Strength of Linear Association Cancel
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Perfect Linear Positive Correlation
No Linear Correlation
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Other Strengths of Association Cancel
r
value
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Interpretation
.
s rong assoc a on
0.5
moderate association
0.25
weak association IRRI-PBGB-CRIL
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Other Strengths of Association Cancel
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Strong Positive Linear Correlation
Moderate Negative near or re a on
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Formula
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= n = number of paired items x i = input variable x = x-bar = mean of ’s x ’s s x = standard deviation of x ’s ’s
y i = output variable y = y-bar = mean of ’s y ’s s y = standard deviation of y ’s ’s
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Correlation Coefficient (r) Cancel
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r=0 does not necessarily mean no relationshi . Relationshi ma be nonlinear.
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Correlation Coefficient Cancel
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Correlation Coefficient (r) Cancel
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A significant r does not necessarily mean a strong linear relationship
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Correlation Coefficient Cancel
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50 0
r = .25** n = 234
45 0
40 0
When no. of o serva ons s large, a low r-value may still be significant.
35 0 t o l p / d 30 0 l e i Y
25 0
20 0
15 0
10 0 0
5
10
15
20
Tiller/plant
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Correlation Coefficient (r) Cancel
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To be able to conclude that 2 variables have a strong linear relationship, r should be both high and si nificant
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Correlation Coefficient Cancel
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6
r = .90** n = 60
5
4 ) a h / t ( 3 d l e i Y
2
1
0 20
30
40
50
60
70
80
90
100
110
No. of spikelet/panicle
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Test of significance for r Download And Print
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Degrees of Freedom
Probability, p
0.05
0.01
0.001
1
0.997
1.000
1.000
2
0.950
0.990
0.999
3
0.878
0.959
0.991
4
0.811
0.917
0.974
5
0.755
0.875
0.951
6
0.707
0.834
0.925
.
.
.
8
0.632
0.765
0.872
9
0.602
0.735
0.847
10
0.576
0.708
0.823
11
0.553
0.684
0.801
12
0.532
0.661
0.780
13
0.514
0.641
0.760
14
0.497
0.623
0.742
15
0.482
0.606
0.725
16
0.468
0.590
0.708
17
0.456
0.575
0.693
18
0.444
0.561
0.679
19
0.433
0.549
0.665
20
0.423
0.457
0.652
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r is significant if the absolute value is reater that the tabular value.
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CORRELATION CORRELATION ANALYSIS ANALYSIS
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PEARSON CORRELATION CORRELATION ANALYSIS Nitrogen.Content Grain.Yield Nitrogen.Content Nitrogen.Content Coef 1 0.99 P-value 1 1e-04 . . P-value 1e-04 1
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R
r
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i n An l
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Scientific Question Cancel
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What is the growth rate of a rice plant? Growth rate can be defined as the change in height er unit of time.
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Data Collection Cancel
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DAS
Height (cm)
0
0
10
12
30
55
60
80
90
110
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Statistical Questions
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120 )100 m c ( t 80 h g i 60 e H t 40 n a l P 20
0 0
20
40
60
Days Days after Se eding
80
100
• What What is the the re rela lati tion onsh ship ip between age and height? Linear • How do I describe ibe or or quanti y t e re ations ip? Regression • Is the the associ ociation ion significant? Statistical Test
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Linear Regression Cancel
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• A general method for estimating or describing association between a continuous outcome variable (dependent) and one or multiple predictors in one equation. o o
One predictor: Simple linear regression Multiple predictors: Multiple linear regression IRRI-PBGB-CRIL
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Statistical Model Download And Print Cancel
Data = Model Fit + Residual i =
Y
ˆ
i
i
ˆ = β + β X Y i i 0 1 Inte Interrcept Slope
X
Y i = µ + α i + ε i IRRI-PBGB-CRIL
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Least Squares Estimates Cancel
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ˆ + ε Y i = Y i i
ˆ = β + β X Y 0 1 i i
To estimate the intercept and slope, minimize residual sum of squares (RSS (RSS ) RSS = ∑ε i
2
∂ RSS ∂β 0
=
∂ RSS ∂β 1 ==>
=
∑ (Y i − Y ˆi ) 2 =∑ (Y i − β 0 − β 1 X i ) 2
∑ (Y i − β 0 − β 1 X i ) ∂β 0 ˆ β 0
==>
=
=
2
2∑ (Y i − β 0 − β 1 X i ) = 0
=−
ˆ X Y − β 1
∑ (Y i − Y + β 1 X − β 1 X i ) 2
ˆ = β 1
∂β 1
=−
2∑ ( X i − X )( X )(Y Y i − Y + β 1 X − β 1 X i ) = 0
X )(Y Y i − Y ) Y ) ∑ ( X i − X )( X ) 2 ∑ ( X i − X )
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We don’t have to do the estimation by hand. R/CropStat or other statistical packages can do the work for us. 26
Print document In order to print this document from Scribd, you'll first need to download it. ANALYSIS ANAL YSIS
LINEAR REGRESSION REG RESSION Dependent Variable: Height
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Analysis of Variance SV Df Sum Square DAS 1 8201.389781 Residuals 3 257.810219
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Mean Square 8201.389781 85.93674
F value 95.435198
Pr (>F) 0.002279
Parameter Estimates Parameter Estimate
Std. Error
t value
Pr (> |t|)
(Intercept) DAS
6.311259 6.311259 0.125227
0.778356 9.769094
0.493109 0.002279
Model Summary quare Adj. R Squared
. 0.959364
4.912409 1.223358
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Example: Growth Rate Data Cancel
Parameter Estimates Parameter Estimate Estimate (Intercept) 4.912409 DAS 1.223358
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Std. Error 6.311259 6.311259 0.125227
t value 0.778356 9.769094
Pr (> |t|) 0.493109 0.002279
140 120 Height =4.9+ 1.223DAS 1.223DAS r = 0.98
) 100 m c ( t h 80 g i e H 60 t n a l P 40
Intercept: The height at age 0 is 4.9 cm. Slope: The height increase per day after seed seedin ing g is 1.22 1.223 3 cm. cm.
20 0 0
20
40
60
80
100
Days after Seeding
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Prediction
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140
Given the regression line, it can be predicted that the hei ht at 40 da s after seeding will be 53.8 cm.
120 Height =4.9+ 1.223DAS r = 0.98
) 100 m c ( t h g i e H 60 t n a l P 40
20 0 0
20
40
60
80
100
Days Days after Se eding
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Example: Growth Rate Data Cancel
Analysis of Variance SV Df Sum Square DAS 1 8201.389781 Residuals 3 257.810219
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Mean Square 8201.389781 85.93674
F value 95.435198
Pr (>F) 0.002279
Model Summary Adj. R Squared
. 0.959364
Sums of Squares
∑ (Y i − Y ) =∑ (Y i − Y ˆi + Y ˆi − Y ) =∑ (Y ˆi − Y ) 2
SST Degrees of freedom n-1 R
2
=
ˆ − Y ) (Y ∑ i = SST ∑ (Y i − Y ) 2
SSM
2
2
2
+
∑ (Y i − Y ˆi )
SSM
SSE
1
n-2
2
RRI-PB GB-CRIL 30 X. R 2 is Ithe fraction of variation in Y explained by
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Linear Regression vs. ANOVA Cancel
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ANOVA Dependent: Continuous Independent: Categorical
Linear regression Dependent: Continuous Independent: Continuous
Linear models ANOVA and regression are the same thing!!!
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Misuse of Regression and Correlation Cancel Analysis Download And Print
• Performing regression and correlation on spurious data could could give significant results. But this this is not a valid indication of a linear relationship.
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Misuse of Regression and Correlation Cancel Analysis Download And Print
• Extrapolation of results o scope of data is extended. Example If the relationship of yield IR8 and stemborer
o
If the relationship between grain yield and protein content from varietal trials is assumed to be applicable to other types of experiments such as fertilizer trials
functional relationship is assumed to hold beyond the range of X values tested IRRI-PBGB-CRIL
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Misuse of Regression and Correlation Cancel Analysis Download And Print 11000
10000 y = 23.751x + 4307.2 r = 0.987**
9000
There is no evidence if a near re a ons p s o s above N = 180 kg/ha
) a h / g 8000 k ( d l e i Y 7000 n i a r G
6000
5000
4000 0
30
60
90
120
150
180
210
240
N-rate (kg/ha)
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Coefficient of Determination (R 2) Cancel
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Percentage of the total variation that is explained by the linear function.
For example, with an R 2 value of 0.64, the implication is 64% [(0.64)(100) = 64] of the variation in the variable Y can be explained by the linear function of the variable X.
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Problems with R 2 •
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R 2
tends to increase as additional variables are included to a regression equation, regardless of their true importance in determining the values of the dependent variable 2
Ra2
•
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=
1−
2 a
n −1 n − ( p + 1)
(1 − R 2 )
where n
=
no . of observatio ns
p
=
no . of independen t var iables
Gives no information on the appropriateness of the model IRRI-PBGB-CRIL
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Problems with R 2
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Curvilinear data fitted by a straight line with high R2
Segregated data fitted by a straight line with high R2
For detecting these kinds of departures from the regression model there is no substitute to plotting the data IRRI-PBGB-CRIL
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Thank you!
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