Analyzing Statistical Results
- 2. Our Test Statistics Chi-square Gamma Z and t Statistics Correlation coefficient and the R2 value Significance and the p-value 1-sided and 2-sided tests 2
- 3. 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 variables and testing for associations 3
- 4. c 2 Test of Independence 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 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 4
- 5. Statistical Independence Sample 1 5
- 6. The c 2 Test Statistic Sample 2 53 47 18 77 69 26 149 132 49 (45%) (40%) (15%) “Observed Value” “Expected Value” Remember: This number does not reflect the strength of an association. It only provides evidence that an association actually exists. 6
- 7. The c 2 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 0.2 0.18 0.16 0.14 0.12 0.1 0.08 Area = a = 0.10 0.06 0.04 0.02 0 0 5 7.78 10 15 20 25 30 35 40 d.f. = 4 7
- 8. The c 2 Distribution 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 0.2 0.18 0.16 0.14 0.12 0.1 0.08 Area = a = 0.10 0.06 0.04 0.02 0 0 5 7.78 10 15 20 25 30 35 40 d.f. = 4 8
- 9. The c2 Distribution 0.18 3x3 0.16 0.14 0.12 Increasing d.f. 0.1 0.08 Increasing d.f. 0.06 0.04 0.02 0 5x5 -0.02 0 10 20 30 40 d.f. = 5 d.f. = 10 d.f. = 20 Degrees of Freedom = (# of rows – 1) x (# of columns – 1) 9
- 10. So, how do we determine the strength and direction of an established association between variables? 10
- 11. Concordant Y X (x2, y3) (x1, y2) 11
- 12. Discordant Y X (x1, y2) (x2, y1) 12
- 13. 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) 13
- 14. Measure of Association C = Concordant Pairs D = Discordant Pairs lies between -1 and 1, where a value of 0 means that there is no relation between the two ordinal variables and | | = 1 represents the strongest associations. 14
- 15. Testing Independence with g H0: There is no association between variables (g value is near zero) H1: There is an association between variables g follows a normal distribution such that our z statistic becomes: z= s.e. Comparing our z statistic with a, we can determine if the variables are statistically independent Standard Error (s.e.) = sample standard deviation / √ n 15
- 16. Testing Independence with g Assuming an a of 0.10 then our benchmark is z0 = 1.285 0.045 0.04 0.035 0.03 0.025 Area = a = 0.10 0.02 0.015 0.01 0.005 0 -4 -3 -2 -1 0 1 1.2852 3 4 If z > z0 = 1.285 then reject H0 Rejection of the H0 means that the variables are not independent (see previous slide) Using g to test for independence does not rely on the degrees of freedom 16
- 17. Why do we not just use g and forget about c2 since g is a more powerful statistic? 17
- 18. c 2 and g An ordinal measure of association (i.e. g ) may equal 0 when the variables are actually statistically dependent c2 = 437, g = 0 18
- 19. Association vs. Causation Stock Market If then ? Hemline Theory = Association without Causation 19
- 20. 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) 20