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Statistical Relationship

(Lesson – 02E) As One Set of Data Move in One Direction What Do the Other Set

of Data Do?

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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

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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