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Correlations between 2 variables and Controlling for 3 rd variables

Correlations between 2 variables and Controlling for 3 rd variables. Binary variables: an example. Computing  (lambda). Predicting Item Sex unknown: Prediction = “No” 13 correct, 6 wrong Errorratio = 6/19 Sex known: Prediction if Female: “No” if Male: “Yes”

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Correlations between 2 variables and Controlling for 3 rd variables

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  1. Correlations between 2 variables and Controlling for 3rd variables

  2. Binary variables: an example

  3. Computing  (lambda) • Predicting Item • Sex unknown: • Prediction = “No” • 13 correct, 6 wrong • Errorratio = 6/19 • Sex known: • Prediction • if Female: “No” • if Male: “Yes” • 14 (=9+5) correct, 5 wrong • Error ratio = 5/19 • Proportional reduction of error ratio =  When predicting “Item” one makes 17% less errors if “Sex” is known than if “Sex” is not known

  4. Controlling for age • Predicting Item • Sex known and Young • errors=2 • Sex known and Old • errors=1 • Error ratio = 3/19 • Proportional reduction of error ratio: Proportional reduction of error when predicting “Item” if “Age” and "Sex" are known compared to if only “Sex” is known equals 40%

  5. Marriages / Divorces / Births (per 1000 inhabitants) in the US Las Vegas!

  6. Marriages / Divorces / Births (per 1000 inhabitants) in the US : dichotomized data

  7. Lambda () (GoodMale & Kruskal, 1954) To what extent does the prediction of the rows improve when the columns are known in a 22 contingency table (and vice versa)? Marriage ratio predicted WITH knowledge on divorce ratio : High if Divorces HighLow if Divorces Low Marriage ratio predicted WITHOUT knowing divorce ratio : High (25>23)

  8. Lambda is not (always) symmetrical  predicting Marriages % = = .522  predicting Divorce % = = .542 • predicting Marriages % = .304 •  predicting Births % = .304  predicting Births % =  predicting Divorce % =

  9. Controlling for 3rd variable: Partial lambda  predicting Divorces = .542 Error ratio predicting Divorces when Marriages known = 11/48 To which extent does this proportion decrease if not only marriages but also births are known " predicting Divorces controlling for Marriages" The prediction of divorces from births AND marriages is NOT better than the prediction of divorces based on marriages only

  10. Lambda and Partial lambda  predicting divorces  predicting Divorces controlling for Births The prediction of Divorces based on knowledge of both Births AND Marriages is 26.7% better than the prediction of Divorces based on knowledge of Births only

  11. yi xi Generalization y x

  12. Generalization: relating continuous variables  : To what extent does prediction of rows improve when columns are known in a 22 contingency table (and vice versa)? Generalizaton r² : To what extent does prediction of variable y improve when one uses the knowledge of the regression line y=ax+b than if one does not use this information? Sum of squared errors predicting WITHOUT knowledge of X Sum of squared errors predicting WITH knowledge of X Improvement:

  13. Partial r² Prediction of y from z only Find az en bz such that is minimal Prediction of y knowing both x and z Find azxb1 and b2 so that is minimal To what extent does the error made when predicting y decrease if z is taken into account on-top-of x

  14. Application X = Births Y = Divorces Z = Marriages Wording the result: • When predicting the divorce rate in a state from knowing its marriage rate, the number of errors can be reduced with another .007 (less than 1%) if the state's birthrate is taken into account as well • In essence birth rate and divorce rate are unrelated when controlling for marriage rate • When holding marriage rates constant there is (almost) no relation between a state's birth ratio and its divorce ratio

  15. Relation between a continuous variable and a dichotomous variable y = 9.6 + 1.4x Regression line Dummy variable Autocratic = 0 Democratic = 1

  16. Computing r² When knowing the leadership style in a team we can predict its productivity 15% better than if we don't have that information

  17. r² depends on the difference between the group means and the within group variance Higher r² Higher r²

  18. r and the t-test Ggeneralizing a finding in a sample to the population H0 : there is no relation between a dichotomous variable x (groupmembership) and the continuous variable y H0 is unlikely if r² is high. WHAT IS "HIGH"? Statistics  if in a population there is no relation between x and y, then r for sample data is distributed as

  19. -3 -2 -1 0 1 2 3 Student t-distribution versus Normaldistribution N(0,1) t(100) .4 t(5) .3 .2 Type I wrong .1

  20. Example - Marriages / Divorces / Births (per 1000 inhabitants) in the US Correlations (N=48-1) Las Vegas! Does a correlation of .347 suffice in order to conclude that a relation between berths and divorces exists in the population? Rule of thumb!Table is better > 2 Conclusion: most likely this relation exists in the population

  21. Example – leadership style and productivity r² between leadership style and productivity equals .15(N=10 teams, r = .39) Conclusion: the observed relation is not strong enough to reject the null hypothesis which states that leadership style and productivity are unrelated However: this does NOT imply that the null hypothesis is correct!

  22. Autocratic teams Democratic teams Classical approach for t-test

  23. BE CAREFUL !! Statistically significantTheoretically relevant One can ALWAYS choose N large enough to make the t-test indicate a significant relation … example r=.04 r²=.0016 N=3000 Yields t = 2,19

  24. Tolerance towards homosexuals • I won't associate with known homosexuals if I can help it. (+) • I would not be afraid for my child to have a homosexual teacher. (-) • It is normal for magazine stands to openly display homosexual magazines. (-) • Homosexuality, as far as I'm concerned, is not sinful. (-) • The increasing acceptance of homosexuality in our society is aiding in the deterioration of morals. (+) • I would vote for a homosexual in an election for public office. (-) • A TV comedy series centered on a homosexual couple would be a nice change from most series concentrated on heterosexual couples. (-) • Many homosexuals are good role models. (-) • If I were a parent, I could accept my son or daughter being gay. (-) • Homosexual websites, chat rooms, and programs should be banned from the internet. (+) 1=Strongly agree 2=Agree 3=Disagree 4=Strongly disagree

  25. Descriptives Non-catholics Catholics

  26. Histogram

  27. Boxplot

  28. Scatterplot Scoreintolerance Y=15.24 + 6.80x Catholicism

  29. Computation of T-test OR Conclusion: REJECT null-hypothesis which states that catholics are as tolerant towards homosexuals as non-catholics are

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