# t-tests and F-tests in regression

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1 t-tests and F-tests in regression Johan A. Elkink University College Dublin 5 April 2012 Johan A. Elkink (UCD) t and F-tests 5 April / 25

2 Outline 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

3 Outline Simple linear regression 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

4 Outline Simple linear regression Model 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

5 Linear equations Simple linear regression Model y = Intercept +Slope x Johan A. Elkink (UCD) t and F-tests 5 April / 25

6 Linear equations Simple linear regression Model Intercept y = Intercept +Slope x Johan A. Elkink (UCD) t and F-tests 5 April / 25

7 Linear equations Simple linear regression Model x y = Intercept +Slope x Johan A. Elkink (UCD) t and F-tests 5 April / 25

8 Linear equations Simple linear regression Model Slope x x y = Intercept +Slope x Johan A. Elkink (UCD) t and F-tests 5 April / 25

9 Simple linear regression Simple regression model Model y x Johan A. Elkink (UCD) t and F-tests 5 April / 25

10 Simple linear regression Simple regression model Model y x y = Intercept +Slope x Johan A. Elkink (UCD) t and F-tests 5 April / 25

11 Simple linear regression Simple regression model Model y x y = β 0 +β 1 x Johan A. Elkink (UCD) t and F-tests 5 April / 25

12 Simple linear regression Simple regression model Model y x y i = β 0 +β 1 x i +ε i Johan A. Elkink (UCD) t and F-tests 5 April / 25

13 Notation Simple linear regression Model y i x i x ε i β k ˆβ k ŷ i Value on the dependent variable for case i Value on the independent variable for case i Mean value on the independent variable for case i The error for case i: ε i = y i ŷ i True coefficient for variable k Estimated coefficient for variable k Predicted value on the dependent variable for case i Johan A. Elkink (UCD) t and F-tests 5 April / 25

14 Ordinary Least Squares Simple linear regression Model Quickly put, the regression line is chosen to minimize the RSS; it has slope ˆβ 1, intercept ˆβ 0, and goes through the point ( x,ȳ). Furthermore, the estimate for σ 2 is ˆσ 2 = RSS/(n 2) (Verzani 2005: 280). Johan A. Elkink (UCD) t and F-tests 5 April / 25

15 Outline Simple linear regression Variance and R 2 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

16 Breakdown of variance Simple linear regression Variance and R 2 Total Sum of Squares (TSS): Explained Sum of Squares (ESS): Residual Sum of Squares (RSS): N i=1 (y i ȳ) 2 N i=1 (ŷ i ȳ) 2 N i=1 (y i ŷ i ) 2 = N i=1 ε2 i TSS = ESS +RSS Johan A. Elkink (UCD) t and F-tests 5 April / 25

17 Breakdown of variance Simple linear regression Variance and R 2 Total Sum of Squares (TSS): Explained Sum of Squares (ESS): Residual Sum of Squares (RSS): N i=1 (y i ȳ) 2 N i=1 (ŷ i ȳ) 2 N i=1 (y i ŷ i ) 2 = N i=1 ε2 i TSS = ESS +RSS Sometimes the second is called regression sum of squares (RSS) and the third errors sum of squares (ESS), which might in fact be more accurate, since ε really represents errors, not residuals, in this specification. Beware the confusion! Johan A. Elkink (UCD) t and F-tests 5 April / 25

18 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? Johan A. Elkink (UCD) t and F-tests 5 April / 25

19 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? R 2 = 1 RSS N TSS = 1 i=1 (y i ŷ i ) 2 N N i=1 (y i ȳ) = i=1 (ŷ i ȳ) 2 2 N i=1 (y i ȳ) 2 Can be interpreted as the proportion of total variance explained by the model. Johan A. Elkink (UCD) t and F-tests 5 April / 25

20 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? R 2 = 1 RSS N TSS = 1 i=1 (y i ŷ i ) 2 N N i=1 (y i ȳ) = i=1 (ŷ i ȳ) 2 2 N i=1 (y i ȳ) 2 Can be interpreted as the proportion of total variance explained by the model. For this interpretation, the model must include an intercept. Generally, one should not attach too much value to having a high R 2 - it is usually more important to understand whether x affects y and by how much, rather than to understand how much of y has not yet been explained. Johan A. Elkink (UCD) t and F-tests 5 April / 25

21 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? R 2 = 1 RSS N TSS = 1 i=1 (y i ŷ i ) 2 N N i=1 (y i ȳ) = i=1 (ŷ i ȳ) 2 2 N i=1 (y i ȳ) 2 Can be interpreted as the proportion of total variance explained by the model. For this interpretation, the model must include an intercept. Generally, one should not attach too much value to having a high R 2 - it is usually more important to understand whether x affects y and by how much, rather than to understand how much of y has not yet been explained. For simple linear regression (i.e. one independent variable), R 2 is the same as the correlation coefficient, Pearson s r, squared. Johan A. Elkink (UCD) t and F-tests 5 April / 25

22 Outline Inference 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

23 Outline Inference t-test 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

24 Inference from regression Inference t-test In linear regression, the sampling distribution of the coefficient estimates form a normal distribution, which is approximated by a t distribution due to approximating σ by s. Johan A. Elkink (UCD) t and F-tests 5 April / 25

25 Inference from regression Inference t-test In linear regression, the sampling distribution of the coefficient estimates form a normal distribution, which is approximated by a t distribution due to approximating σ by s. Thus we can calculate a confidence interval for each estimated coefficient. Johan A. Elkink (UCD) t and F-tests 5 April / 25

26 Inference t-test Inference from regression In linear regression, the sampling distribution of the coefficient estimates form a normal distribution, which is approximated by a t distribution due to approximating σ by s. Thus we can calculate a confidence interval for each estimated coefficient. Or perform a hypothesis test along the lines of: H 0 :β 1 = 0 H 1 :β 1 0 Johan A. Elkink (UCD) t and F-tests 5 April / 25

27 Inference from regression Inference t-test To calculate the confidence interval, we need to calculate the standard error of the coefficient. Johan A. Elkink (UCD) t and F-tests 5 April / 25

28 Inference from regression Inference t-test To calculate the confidence interval, we need to calculate the standard error of the coefficient. Rule of thumb to get the 95% confidence interval: β 2SE < β < β +2SE Johan A. Elkink (UCD) t and F-tests 5 April / 25

29 Inference from regression Inference t-test To calculate the confidence interval, we need to calculate the standard error of the coefficient. Rule of thumb to get the 95% confidence interval: β 2SE < β < β +2SE Thus if β is positive, we are 95% certain it is different from zero when β 2SE > 0. Johan A. Elkink (UCD) t and F-tests 5 April / 25

30 Inference from regression Inference t-test To calculate the confidence interval, we need to calculate the standard error of the coefficient. Rule of thumb to get the 95% confidence interval: β 2SE < β < β +2SE Thus if β is positive, we are 95% certain it is different from zero when β 2SE > 0. (Or when the t value is greater than 2 or less than 2.) Johan A. Elkink (UCD) t and F-tests 5 April / 25

31 Outline Inference F-test 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

32 Breakdown of variance Inference F-test Total Sum of Squares (TSS): Explained Sum of Squares (ESS): Residual Sum of Squares (RSS): N i=1 (y i ȳ) 2 N i=1 (ŷ i ȳ) 2 N i=1 (y i ŷ i ) 2 = N i=1 ε2 i TSS = ESS +RSS Johan A. Elkink (UCD) t and F-tests 5 April / 25

33 Inference F-test F-test In simple linear regression, we can do an F-test: H 0 :β 1 = 0 H 1 :β 1 0 Johan A. Elkink (UCD) t and F-tests 5 April / 25

34 Inference F-test F-test In simple linear regression, we can do an F-test: H 0 :β 1 = 0 H 1 :β 1 0 F = with 1 and n 2 degrees of freedom. ESS/1 RSS/(n 2) = ESS ˆσ 2 F 1,n 2 Johan A. Elkink (UCD) t and F-tests 5 April / 25

35 Inference F-test F-test In simple linear regression, we can do an F-test: H 0 :β 1 = 0 H 1 :β 1 0 F = with 1 and n 2 degrees of freedom. ESS/1 RSS/(n 2) = ESS ˆσ 2 F 1,n 2 For multiple regression, this would generalize to: F = ESS/(k 1) RSS/(n k) F k 1,n k Johan A. Elkink (UCD) t and F-tests 5 April / 25

36 Outline Exercises 1 Simple linear regression Model Variance and R 2 2 Inference t-test F-test 3 Exercises Johan A. Elkink (UCD) t and F-tests 5 April / 25

37 Exercises Exercise Open oecd 1960.sav. Repeat for both industry (IND) and services (AGR): 1 Regress percentage in sector on income per capita. 2 Interpret the regression results. 3 Evaluate the model fit. 4 Interpret the t- and F-tests. Johan A. Elkink (UCD) t and F-tests 5 April / 25

38 Exercises Exercise Open bes class data.sav and investigate: whether left-right self-placement (lr) and explains trust in politics (trustpol) whether left-right self-placement influences attitude towards EU membership (eumember) Johan A. Elkink (UCD) t and F-tests 5 April / 25

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