Dr. C. Ertuna1 Statistical Relationship (Lesson – 02E) As One Set of Data Move in One Direction...
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Transcript of Dr. C. Ertuna1 Statistical Relationship (Lesson – 02E) As One Set of Data Move in One Direction...
Dr. C. Ertuna 1
Statistical Relationship
(Lesson – 02E) As One Set of Data Move in One Direction What Do the Other Set
of Data Do?
Dr. C. Ertuna 2
Statistical Relationship
1/A B C
2 S&P500 Tracway
3 15300 100.00$
4 15438 101.50$
5 15867 102.50$
6 14984 97.10$
7 15468 97.70$
8 15608 99.50$
9 16218 103.10$
The price for TRC stock and S&P 500 index are given on the left.
Is there any relationship between those two?
To answer this question we need to measure the “Statistical Relationship” between those two.
Data: St-CE-Ch02-x1-Examples-Slide 60
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Statistical Relationship
The descriptive statistics that measures the degree of relation between 2 variables are called correlation coefficients.
Three measures for statistical relationship are:
Scale data Pearson’s r •Normal Distribution
•Linearity
Ordinal (or above data) Kendall’s Tau-b •Distribution free
•Monotonicity
Nominal (or above data) Chi-Square Test •Raw frequency > 5
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Statistical Relationship (Cont.)
• Pearson Correlation coefficient (ρ, r) measures the strength of linear relationship between two variables (X and Y) assuming normal distribution. {significance!}
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Statistical Relationship (Cont.)
• Correlation coefficient will range from -1 to +1• A correlation of 0 indicates that there is no linear
relationship between two variables• Even a high correlation could be observed just by
chance; to be sure we need to run a statistical test. • Correlation between two variables does not mean
causal relationship between them• Correlation Matrix provides pair-wise correlation
between more than two variables.
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Statistical Relationship (Cont.)
Square of Pearson’s r (r2)can be interpreted as explained variance if there is a Dependent Variable (DV) Independent Variable (IV) relationship exists.
• For example if r = 0.933 than r2 = 0.87049 that means IV explains 87.05% of the variations in the DV.
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Correlation Test Assumptions
Parametric Correlation Test
• Pearson’s r:– Interval data– Normality– Equal Variance (not needed if n > 30)
– Linearity
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Statistical Relationship (Cont.)
• Kendall's tau-b
A distribution-free (nonparametric) measure of association for ordinal (or ranked) variables that take ties into account. The sign of the coefficient indicates the direction of the relationship, and its absolute value indicates the strength, with larger absolute values indicating stronger relationships. Possible values range from -1 to 1.
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Statistical Relationship (Cont.)
• Spearman’s rho
Commonly used distribution-free (nonparametric) measure of correlation between two ordinal variables. For all of the cases, the values of each of the variables are ranked from smallest to largest, and the Pearson correlation coefficient is computed on the ranks.
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Correlation Test Assumptions
Non-Parametric Correlation Test
• Kandell’s tau-b & Spearman’s rho:– Ordinal data– Monotonicity
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2 -Test
Chi-square test is suitable for analyzing nominal and ordinal data. (Interval and ratio data should be grouped first)
Chi-square test is used for - Goodness-of-fit (1-Way classification; 1-DV, 1-IV)
- Test for independence (2-Way classification; 1-DV, 2+IV)
Categorical data in Rows
Ordinal data in Columns
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2 -Test Assumptions
– Categorical data– Any cell’s raw frequency > 5 – Random Sampling
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2 -Test
PHStat2 / Multiple-Sample Tests /
/ Chi-Square Test
Significance Level: to be entered
Number of Raws: to be entered
Number of Columns: to be entered
If p_value < 0,05 There is a relationship
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2 -Test
)1(*
_'
kN
statisticstestSquareChiVsCramer
Strength of the Relationship is measured by
Where N = total number of observations
k = min( #rows, #columns)
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2 -Test
• Cramer’s V has a value between 0 and 1
• Where 0 means independence or no relationship and 1 means perfect relation ship.
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Interpreting the Association
Although there is no theoretical guideline on how to interpret the value of association, here are some guidelines:
• Interpret the squared value of the association• 1.00 – 0.80 High (strong) association• 0.80 – 0.60 Moderately high association• 0.60 – 0.40 Moderate association• 0.40 – 0.20 Weak association• 0.20 – 0.00 Very weak association
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Interpreting of Cramer’s V
Although there is no theoretical guideline on how to interpret the value of association, here are some guidelines for Cramer’s V:
• 1.00 – 0.40 Worrisomely High (strong) association• 0.40 – 0.35 Very High (strong) association• 0.35 – 0.30 High (strong) association• 0.30 – 0.25 Moderately high association• 0.25 – 0.20 Moderate association• 0.20 – 0.10 Weak association• 0.10 – 0.00 Very weak association/Not acceptable
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SPSS – Nominal Association
Cathegories should be coded first:
• Data / Weight Cases /
(variable that stands for frequencies)
• Analyze / Discriptives / Crosstabs /
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Example: Statistical Relationship
1/A B C
2 S&P500 Tracway
3 15300 100.00$
4 15438 101.50$
5 15867 102.50$
6 14984 97.10$
7 15468 97.70$
8 15608 99.50$
9 16218 103.10$
The price for TRC stock and S&P 500 index are given on the left.
1. Compute the correlation between S&P500 and TRC
2. Explain the meaning of the result.
Data: St-CE-Ch02-x1-Examples-Slide 60
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Example: Statistical Relationship
• Analyze/ Correlate/ Bivariate
• Select the variables & move to the right pane
• Select Pearson, Kendall’s tau_b, Spearman.
(2-tailed* ; 1-tailed)
• Ok
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Example: Statistical Relationship
CorrelationsS&P 500 Trackway
S&P 500 Pearson Correlation1 0.933Sig. (2-tailed). 0.000N 126 126
Trackway Pearson Correlation0.933 1Sig. (2-tailed) 0.000 .N 126 126
CorrelationsS&P 500 Trackway
Kendall's tau_bS&P 500 Correlation Coefficient1.000 0.675Sig. (2-tailed). 0.000N 126 126
Trackway Correlation Coefficient0.675 1.000Sig. (2-tailed) 0.000 .N 126 126
Spearman's rhoS&P 500 Correlation Coefficient1.000 0.836Sig. (2-tailed). 0.000N 126 126
Trackway Correlation Coefficient0.836 1.000Sig. (2-tailed) 0.000 .N 126 126
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Example: Statistical Relationship (cont.)
1/A B C D E F G H
2 S&P500 Tracway Corellation Sig.
3 15300 $100.00 Pearson 0.933 0.000
4 15438 $101.50 Kendall 0.675 0.0005 15867 $102.50 Spearman 0.836 0.0006 14984 $97.10
7 15468 $97.70
8 15608 $99.50
9 16218 $103.10
Data: St-CE-Ch02-x1-Examples-Slide 60
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Example: Statistical Relationship (cont.)
The correlation between Tracway stock price and S&P 500 index is 0.93.
• Explain the meaning of the result.
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Meaning: Statistical Relationship (cont.)
Pearson correlation coefficient of 0.93 indicates that there is a
1 strong (most of the time holding) ,
2 positive (when one changes, the other one changes in the same direction) ,
3 linear
relationship between S&P 500 index and Tracway stock price (assuming both normally distributed).
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Next Lesson
(Lesson - 03A) Random Variables & Probability
Distribution