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Introduction to Statistics: Political Science (Class 3) Calculating R-Squared, Dichotomous and Nominal Variables, F-tests.

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Presentation on theme: "Introduction to Statistics: Political Science (Class 3) Calculating R-Squared, Dichotomous and Nominal Variables, F-tests."— Presentation transcript:

1 Introduction to Statistics: Political Science (Class 3) Calculating R-Squared, Dichotomous and Nominal Variables, F-tests

2 R-Squared

3 R-Squared Example Measure of proportion of variance in Y explained by the IVs Coef. St.Err T P Bush FT Party Identification Constant FULL SAMPLE Coef. St.Err T P Bush FT Party Identification Constant Random Cases R 2 =.5336

4 First, we need the variance of Y Mean = 66, so: Obama FT = (-.090)(Bush FT) (Party Identification) Observed (Observed- Mean) (Observed- Mean) Variance =6190

5 Bush FTPIDPredictedObserved (Observed- Predicted) (Observed- Predicted) SSR (Sum of Squared Residuals) = Variance of Y = 6190 R 2 = ( ) 6190 =.5336

6 What is a good R 2 ? Predict feelings about Obama with: –Party ID and feelings about Bush –Education –Zodiac sign

7 Non-continuous IVs Dealing with Dichotomous and Nominal Variables

8 Democratic Peace Is sum of democracy scores the right measure? Alternative: Are the pair of countries both democracies? Indicator/dummy/dichotomous variable: –1 if both countries have democracy scores >5 –0 otherwise

9 Dichotomous IV Coef SE Coef T P Democratic Pair (1=yes) Constant R-squared = Coef SE Coef T P Democratic Pair (1=yes) Military Spending ($mil) Constant R-squared = DV: Years at peace

10 Nominal variables Speed dating survey: You have 100 points to distribute among the following attributes -- give more points to those attributes that are more important in a potential date, and fewer points to those attributes that are less important in a potential date. Attractive Fun Intelligent Sincere Ambitious Shared Interests

11 How do peoples perspective/goals affect whats important to them? What is your primary goal in participating in this event? –Seemed like a fun night out=1 –To meet new people=2 –To get a date=3 –Looking for a serious relationship=4 –To say I did it=5 Does this make sense as a linear scale?

12 Who is likely to say each of the following is important? Attractiveness? Fun? –Seemed like a fun night out=1 –To meet new people=2 –To get a date=3 –Looking for a serious relationship=4 –To say I did it=5 Does this make sense as a linear scale?

13 Effects of Nominal Variable One Variable: Seemed like a fun night out=1 To meet new people=2 To get a date=3 Looking for a serious relationship=4 To say I did it=5 Five Variables: Seemed like a fun night out (1=yes) To meet new people (1=yes) To get a date (1=yes) Looking for a serious relationship (1=yes) To say I did it (1=yes)

14 Importance of Attribute = β 0 + β 1 (Seemed Fun) + β 2 (Meet People) + β 3 (Date) + β 4 (Serious Relationship) + β 5 (Say Did) + u What would β 0 correspond to in this model?

15 Reference Group Leave one indicator out Importance of Attribute = β 0 + β 1 (Seemed Fun) + β 2 (Meet People) + β 3 (Date) + β 4 (Serious Relationship) + β 5 (Say Did) + u

16 (Remember: reference group is to say I did it) AttractivenessCoef. SE Coef.Tp Seemed Fun Meet People Date Serious Relationship Constant What if we want to know whether people who want a date and those who want a serious relationship differ in how important they think attractiveness is?

17 Easiest way: change reference category Importance of Attribute = β 0 + β 1 (Seemed Fun) + β 2 (Meet People) + β 3 (Date) + β 4 (Serious Relationship) + β 5 (Say Did) + u AttractivenessCoef.SE Coef.Tp Seemed Fun Meet People Date Say I Did Constant Do people who want a date and those who want a serious relationship differ in how important they think attractiveness is?

18 Nominal and Dichotomous IVs AttractivenessCoef. SE Coef.Tp Seemed Fun Meet People Date Say I Did Gender (1=male) Constant Estimated points allocated to attractiveness for men who attended because it seemed fun?

19 F-Tests Testing the joint significance of variables

20 F-test Way of testing joint significance of variables – i.e., whether set of variables significantly improve explanatory power When to use: –Nominal variables –Variables likely to be highly correlated, but important predictors

21 Terminology Unrestricted model – includes IVs you want to test joint significance of Restricted model – same model, excluding IVs to be tested SSR – Sum of Squared Residuals

22 Formula q = # of variables being tested n = number of cases k = number of IVs in unrestricted F = (SSR r - SSR ur )/q SSR ur /(n-(k+1)

23 Who values fun people? FunCoef. SE Coef.Tp Seemed Fun Meet People Date Say I Did Gender (1=male) Constant What if we want to know whether the reason for attending variables as a group improve the explanatory power of the model?

24 q = # of variables being tested n = number of cases k = number of IVs in unrestricted F = (SSR r - SSR ur )/q SSR ur /(n-(k+1) UNRESTRICTEDSum of SquaresdfMS Model Residual Total RESTRICTED RestrictedSum of SquaresdfMS Model Residual Total F = ( )/ /(2484-(5+1)) = 9.25

25 Statistical significance of F-test What does an F value of 9.25 mean? Similar idea to a t-test, but shape of F- distribution depends (heavily) on degrees of freedom –Numerator = number of IVs being tested –Denominator = N-(number of IVs)-1 –Here: 4 and 2478 ( )

26

27 Look up critical value in a table or use Minitab Calc Probability Distributions F Note: this will give you area under the curve up to your F-test, so use 1-p Cumulative Distribution Function F distribution with 4 DF in numerator and 2478 DF in denominator x P( X <= x )

28 Notes and Next Time Graded homework will be handed back next time and model answers will be posted online early next week New homework will be handed out next time (and due next Thursday) Next time: –Functional form in multivariate regression


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