Interpreting Analysis Results

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Chi-square

Gamma

Z and t Statistics

Correlation coefficient and the R2 value

Significance and the p-value

1-sided and 2-sided tests

Our Test Statistics

Variable Types• Nominal, Ordinal, and Interval

Nominal Example: Type of Anesthetic (e.g. Local, General, and Regional)

Ordinal Example: Hospital Trauma Levels (e.g. Level-1, Level-2, Level-3)

Interval Example: Drug Dosage (e.g. mg of substance)

• There are multiple kinds of tests for each type of variable depending on assumptions being made and what is being tested for (e.g. means, variances, association, etc.)

Our focus is on ordinal variablesand testing for associations

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Does not measure strength of an association, but, rather, it indicates how much evidence there is that the variables are or are not independent

The c2 statistic ranges from 0 to ∞ (always ≥ 0)

The larger the c2 value, the more confident we can be that the variables are not independence

c2 Test of Independence

Example: Assume we have two ordinal variables (Hospital Size and the Relative Cost of the Laryngoscope being used):

H0: There is no association between variables (c2 value is near zero)

H1: There is an association between variables

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Sample 1

Statistical Independence

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The c2 Test Statistic

Remember: This number does not reflect the strength of an association. It only provides evidence that an association actually exists.

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(45%) (40%) (15%)

7753

149

4769

132

18

4926

“Observed Value”

“Expected Value”

Sample 2

The c2 Distribution Okay…so what does c2 = 83 tell us?

With 4 degrees of freedom (d.f.) and an a = 0.10, our benchmark (critical value) for dependence is 7.78

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30 35 40

d.f. = 4

7.78

Area = a = 0.10

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The c2 Distribution

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H0: There is no association between variables (c2 value is near zero)

H1: There is an association between variables

We reject H0 if our c2 value is > our “benchmark” value of 7.78

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30 35 40

d.f. = 4

7.78

Area = a = 0.10

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 10 20 30 40

d.f. = 5 d.f. = 10 d.f. = 20

Increasing d.f.

Degrees of Freedom = (# of rows – 1) x (# of columns – 1)

Increasing d.f.

3 x 3

5 x 5

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The c2 Distribution

So, how do we determine the strength and direction of an established association

between variables?

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Concordant

(x1, y2)

(x2, y3)

X

Y

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Discordant

(x1, y2)

(x2, y1)

X

Y

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The Gamma Value (g) Appropriate for ordinal-by-ordinal data

Generally more powerful test statistic than the c2 test statistic (i.e. we are able to detect significant differences with greater ease)

Because ordinal variables can be ordered, we can talk about the direction of association between them (i.e. positive or negative)

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C = Concordant Pairs

D = Discordant Pairs

lies between -1 and 1, where a value of 0means that there is no relation between the two ordinal variables and | | = 1 represents the strongest associations.

Measure of Association

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g follows a normal distribution such that our z statistic becomes:

Comparing our z statistic with a, we can determine if the variables are statistically independent

Testing Independence with g

z =s.e.

Standard Error (s.e.) = sample standard deviation / √ n

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H0: There is no association between variables (g value is near zero)

H1: There is an association between variables

Testing Independence with g

Using g to test for independence does not rely on the degrees of freedom

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-4 -3 -2 -1 0 1 2 3 4

Assuming an a of 0.10 then our benchmark is z0 = 1.285

If z > z0 = 1.285 then reject H0

Rejection of the H0 means that the variables are not independent (see previous slide)

1.285

Area = a = 0.10

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Why do we not just use g and forget about c2 since g is a more

powerful statistic?

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c2 and g An ordinal measure of association (i.e. g ) may equal 0

when the variables are actually statistically dependent

c2 = 437, g = 0

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Association vs. Causation

If then ?

Stock Market

Hemline Theory = Association without Causation

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Association vs. Causation Causation can often be obscure or counterintuitive so

how do we tell the difference from Association? Performing controlled comparisons

Increasing the variable resolutions (i.e. the number

of data points)

Sequence Analysis (time becomes a variable)

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