ttests and Ftests in regression


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1 ttests and Ftests in regression Johan A. Elkink University College Dublin 5 April 2012 Johan A. Elkink (UCD) t and Ftests 5 April / 25
2 Outline 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
3 Outline Simple linear regression 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
4 Outline Simple linear regression Model 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
5 Linear equations Simple linear regression Model y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
6 Linear equations Simple linear regression Model Intercept y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
7 Linear equations Simple linear regression Model x y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
8 Linear equations Simple linear regression Model Slope x x y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
9 Simple linear regression Simple regression model Model y x Johan A. Elkink (UCD) t and Ftests 5 April / 25
10 Simple linear regression Simple regression model Model y x y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 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 Ftests 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 Ftests 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 Ftests 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 Ftests 5 April / 25
15 Outline Simple linear regression Variance and R 2 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 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 Ftests 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 Ftests 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 Ftests 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 Ftests 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 Ftests 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 Ftests 5 April / 25
22 Outline Inference 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
23 Outline Inference ttest 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
24 Inference from regression Inference ttest 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 Ftests 5 April / 25
25 Inference from regression Inference ttest 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 Ftests 5 April / 25
26 Inference ttest 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 Ftests 5 April / 25
27 Inference from regression Inference ttest To calculate the confidence interval, we need to calculate the standard error of the coefficient. Johan A. Elkink (UCD) t and Ftests 5 April / 25
28 Inference from regression Inference ttest 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 Ftests 5 April / 25
29 Inference from regression Inference ttest 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 Ftests 5 April / 25
30 Inference from regression Inference ttest 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 Ftests 5 April / 25
31 Outline Inference Ftest 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
32 Breakdown of variance Inference Ftest 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 Ftests 5 April / 25
33 Inference Ftest Ftest In simple linear regression, we can do an Ftest: H 0 :β 1 = 0 H 1 :β 1 0 Johan A. Elkink (UCD) t and Ftests 5 April / 25
34 Inference Ftest Ftest In simple linear regression, we can do an Ftest: 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 Ftests 5 April / 25
35 Inference Ftest Ftest In simple linear regression, we can do an Ftest: 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 Ftests 5 April / 25
36 Outline Exercises 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 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 Ftests. Johan A. Elkink (UCD) t and Ftests 5 April / 25
38 Exercises Exercise Open bes class data.sav and investigate: whether leftright selfplacement (lr) and explains trust in politics (trustpol) whether leftright selfplacement influences attitude towards EU membership (eumember) Johan A. Elkink (UCD) t and Ftests 5 April / 25
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