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Review of ANOVA & Inferences About The Pearson

Correlation Coefficient

Heibatollah Baghi, and Mastee Badii

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Review of ANOVA (1)

Source SS DF MS Fc Fα

-------------- ------ ------ -------- ------ ------

Between 70 2 35.0 9.13 3.88

Within 46 12 3.83

------- ------ ---- ----- ----- -------

Total 116 14

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Review of ANOVA (2)

Source SS DF MS Fc Fα

-------------- ----- ------ -------- ------ ------

Explained 70 2 35 9.13 3.88

Unexplained

46 12 3.83

------- ------ ---- ----- ----- -------

Total 116 14

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Review of ANOVA (3)

S.V. SS DF MS Fc Fα

-------------- ------ ------ ------ ----- -----

Systematic Effect 70 2 35 9.13 3.88

Random Effect 46 12 3.83

------- ---------- ----- ----- -------

Total 116 14

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Practical Significance or Effect Size in ANOVA

Statistical significance does not provide information about the effect size in ANOVA.

The index of effect size is η2 (eta-squared)

η2 = SSB / SST or η2 = 70/116 = .60

60 % of the variability in stress scores is explained by different treatments.

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Practical Significance or Effect Size in ANOVA, Continued

Source SS DF MS Fc Fα η2

--------------- ------ ----- ------- ----- ----

Between 70 2 35.0 9.13 3.88 .60

Within 46 12 3.83

------- ------ ---- ----- ----- -------

Total 116 14

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Sample Size in ANOVA

To estimate the minimum sample size needed in ANOVA, you need to do the power analysis.Given the:

α = .05, effect size = .10, and a power ( 1- beta) of .80, 30 subjects per group would be needed.

(Refer to Table 7-7, page 178).

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Inferences About The Pearson Correlation

Coefficient

Refer to Session 5GPA and SAT Example

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STUDENTS Y(GPA) X(SAT)

A 1.6 400 -0.97 -145.80 141.43

B 2.0 350 -0.57 -195.80 111.61

C 2.2 500 -0.37 -45.80 16.95

D 2.8 400 0.23 -145.80 -33.53

E 2.8 450 0.23 -95.80 -22.03

F 2.6 550 0.03 4.20 0.13

G 3.2 550 0.63 4.20 2.65

H 2.0 600 -0.57 54.20 -30.89

I 2.4 650 -0.17 104.20 -17.71

J 3.4 650 0.83 104.20 86.49

K 2.8 700 0.23 154.20 35.47

L 3.0 750 0.43 204.20 87.81

Sum 30.80 6550.0 378.33

Mean 2.57 545.80

S.D. 0.54 128.73

)X - (X)Y - (Y )X - (X )Y - (Y

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Calculation of Covariance & Correlation

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Population of visual acuity and neck size “scores”

ρ=0

Sample 1 EtcSample 2 Sample 3

r = -0.8 r = +.15 r = +.02

Rel

ativ

e F

requ

ency

r: 0

µr

The development of a sampling distribution of sample v:

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Steps in Test of Hypothesis

1. Determine the appropriate test 2. Establish the level of significance:α3. Determine whether to use a one tail or two tail test4. Calculate the test statistic5. Determine the degree of freedom6. Compare computed test statistic against a

tabled/critical valueSame as Before

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1. Determine the Appropriate Test

Check assumptions:Both independent and dependent variable (X,Y) are measured on an interval or ratio level.Pearson’s r is suitable for detecting linear relationships between two variables and not appropriate as an index of curvilinear relationships.The variables are bivariate normal (scores for variable X are normally distributed for each value of variable Y, and vice versa)Scores must be homoscedastic (for each value of X, the variability of the Y scores must be about the same)

Pearson’s r is robust with respect to the last two specially when sample size is large

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2. Establish Level of Significance

α is a predetermined value

The convention• α = .05

• α = .01

• α = .001

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3. Determine Whether to Use a One or Two Tailed Test

H0 : ρXY = 0

Ha : ρXY ≠ 0

Ha : ρXY > or < 0

Two Tailed Test if no direction is specified

One Tailed Test if direction is specified

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4. Calculating Test Statistics

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5. Determine Degrees of Freedom

For Pearson’s r df = N – 2

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6. Compare the Computed Test Statistic Against a Tabled Value

α = .05

Identify the Region (s) of Rejection.

Look up tα corresponding to degrees of freedom

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Formulate the Statistical Hypotheses. Ho : ρXY = 0 Ha : ρXY ≠ 0

α = 0.05

Collect a sample of data, n = 12

Example of Correlations Between SAT and GPA scores

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Data

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Calculation of Difference of Y and mean of Y

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Calculation of Difference of X and Mean of X

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Calculation of Product of Differences

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Covariance & Correlation

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Calculate t-statistics

2738.2

r1 S 82.1

S

r t

0.50 )73.128)(54.0(

39.34

SS

S r

2

rr

c

Y X

XYXY

n

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Identify the Region (s) of Rejection. tα = 2.228

Make Statistical Decision and Form Conclusion.

tc < tα Fail to reject Ho

p-value = 0.095 > α = 0.05 Fail to reject Ho

Or use Table B-6: rc = 0.50 < rα =.576 Fail to reject Ho

Check Significance

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Practical Significance in Pearson r

Judge the practical significance or the magnitude of r within the context of what you would expect to find, based on reason and prior studies.

The magnitude of r is expressed in terms of r2 or the coefficient of determination.

In our example, r2 is .50 2 = .25 (The proportion of variance that is shared by the two variables).

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Intuitions about Percent of Variance Explained

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Sample Size in Pearson r

To estimate the minimum sample size needed in r, you need to do the power analysis. For example, Given the:

α = .05, effect size (population r or ρ) = 0.20, and a power of .80, 197 subjects would be needed. (Refer to Table 9-1).

Note: [ρ = .10 (small), ρ=.30 (medium), ρ =.50 (large)]

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Magnitude of Correlations

ρ = .10 (small)

ρ = .30 (medium)

ρ = .50 (large)

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Factors Influencing the Pearson r

Linearity. To the extent that a bivariate distribution departs from normality, correlation will be lower.

Outliers. Discrepant data points affect the magnitude of the correlation.

Restriction of Range. Restricted variation in either Y or X will result in a lower correlation.

Unreliable Measures will results in a lower correlation.

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Take Home Lesson

How to calculate correlation and test if it is different from a constant